PREFACE THE principal object in writing this book has been the same as that which has governed the author in writing other mathematical text-books ; viz., to bring out the fundamental utilities which underlie and grow out of the principles pre- sented. Not only is the fundamental source of new power in Trigonometry frequently emphasized, but each new process is taken up, not arbitrarily, but for the sake of the economy or new power which it gives. Among other special features of the book, the following may be mentioned : Under each case in the solution of triangles two groups of examples are given ; one with the degree divided sexa- gesimally, and the other with the degree divided decimally. The inclusion of the examples in terms of the decimally divided degree meets the new requirements of Harvard, Yale, and Princeton. A chapter is given on logarithms and their properties. Practical examples are included in this chapter which are not only interesting in themselves, but which afford a review of and a correlation with other branches of mathematics. When use is made of the line equivalents of the trigono- metric ratios, it is specially shown that such treatment is merely a convenient substitute for the ratio treatment, and the method of this substitution is shown and its processes carefully safeguarded. A chapter is given in which the applications of trigo- nometry are reduced to a system. 3 219350 4\ i: ; f C- : - { : * : :\ . * : PREFACE The subject-matter of the text-book is enlivened and made more vital and human by a chapter on the history of trigo- nometry. Attention is also called to the method in which logarithmic work is arranged. This form of tabulation is used, for instance, in the designing room in the United States Navy Department and by engineers in general. Among the ad- vantages of this method of arranging logarithmic work are the following : (1) It abbreviates the work by omitting the equality marks. (2) It includes within itself the actual numbers whose logarithms are being used. (3) It facilitates the correction of mistakes by including and presenting in order all the steps of a logarithmic reduc- tion. (4) The arrangement of the work is such that after the pupil has acquired facility in logarithmic computation, some of the steps in the tabulation may be omitted without chang- ing the general form of tabulation. The author wishes to express his especial indebtedness to Mr. Howard Smith of the Hill School, Pottstown, Pa., to whom most of the examples are due, and who has made important suggestions concerning other parts of the work. The writer is also under obligation to his colleague, Mr. J. H. Keener, to whom the examples in the General Review Exer- cise are mainly due. Professor William Betz of the East Rochester High School, Rochester, N.Y., Dr. Henry A. Con- verse of the Polytechnic Institute, Baltimore, Md., and Professor William H. Metzler of Syracuse University have also aided the writer by important corrections and suggestions. FLETCHER DUKELL. LAWRENCEVILLE, N.J., January 10, 1910. TABLE OF CONTENTS CHAPTER I PAGE LOGARITHMS 7 CHAPTER II DEFINITIONS. TRIGONOMETRIC FUNCTIONS . . . . . .24 CHAPTER III RIGHT TRIANGLES 52 CHAPTER IV GONIOMETRY 73 CHAPTER V GONIOMETRY (Continued} 93 CHAPTER VI OBLIQUE TRIANGLES 107 CHAPTER VII PRACTICAL APPLICATIONS 131 CHAPTER VIII CIRCULAR MEASURE. GRAPHS OF TRIGONOMETRIC FUNCTIONS . . 142 CHAPTER IX INVERSE TRIGONOMETRIC FUNCTIONS 152 CHAPTER X COMPUTATION OF TABLES. TRIGONOMETRIC SERIES .... 157 CHAPTER XI HISTORY OF TRIGONOMETRY . 162 5 6 TABLE OF CONTENTS CHAPTER XII PACK INTRODUCTION TO SPHERICAL TRIGONOMETRY 185 CHAPTER XIII THE RIGHT SPHERICAL TRIANGLE 191 CHAPTER XIV OBLIQUE SPHERICAL TRIANGLES 203 CHAPTER XV SOME APPLICATIONS OF SPHERICAL TRIGONOMETRY .... 230 PLANE TRIGONOMETRY CHAPTER I LOGARITHMS 1. The logarithm of a number is the exponent of that power of another number, taken as the base, which equals the given number. Thus, 1000 = 10 3 , hence log 1000 = 3, 10 being taken as the base; again, if 8 be taken as the base, 4 = 8% hence Iog4 = f; also, if 5 be taken as the base, log 125 = 3, log -^ = 2, etc. The base used is sometimes stated in the context as above ; but, when desirable, it is indicated by writing it as a small subscript to the word log. Thus the above expressions might be written, lo glo 1000 = 3 ; Io g8 4 = i ; Iog 5 125 = 3 ; Iog 5 & = - 2 ; etc. In general, by the definition of a logarithm, number = (base) logarithm , or N= B l ; hence log B N= I 2. Uses or Utility of Logarithms. One of the principal uses of . logarithms is to simplify numerical work. For instance, by logarithms the numerical work of multiplying two numbers is converted into the simpler work of adding the logarithms of these numbers. To illustrate this principle we may take the simple case of multiplying two numbers which are exact powers of 10, as 1000 and 100. Thus TRIGONOMETRY 1000 = 10 3 100 = 10 2 hence 1000 x 100 = 10 5 = 100,000, the multiplication being performed by the addition of exponents. Similarly, if 384 = 10 2 - 58 * 53 * and 25 = 10 1 - 39794+ , 384 may be multiplied by 25 by adding the exponents of 10 2 - 584334 - and 10 i.39794 +j thus obtaining lO 3 - 9822 ^, and then getting from a table of loga- rithms the value of 10 3 - 98227+ , viz. 9600. In like manner, by the use of logarithms, the process of dividing one number by another is converted into the simpler process of subtracting one exponent, or log, from another ; the process of involution is converted into the simpler process of multiplication; and the extraction of a root into the simpler process of division. The saving of labor effected by the use of logarithms can be increased by committing to memory the logs of certain much used numbers as of 2, 3, 9, TT, I/TT, , V 2, J/3, etc. TT Also by use of the slide rule, the practical use of logarithms is reduced to sliding one rod along another and reading off the number corresponding to the terminal position of one end of a rod. If the teacher can find time, it will be a use- ful exercise to teach the class the use of the slide rule in con- nection with the study of this chapter. 3. Systems of Logarithms. Any positive number except unity may be made the base of a system of logarithms. Two principal systems are in use : 1. The Common (or Decimal) or Briggsian System, in which the base is 10. This system is used almost exclusively when logarithms are employed to facilitate numerical compu- tations. LOGARITHMS 9 r 2. The system termed Natural or Napierian, in which the base is 2.7182818 + . This system is generally used in alge- braic processes, as in demonstrating the properties of algebraic expressions, etc. EXERCISE I 1. Give the value of each of the following : Iog 3 9, Iog 3 27, Iog 4 64, log 4T V, logg |, logj^r, log 10T V, logio-01, Iog 10 .001. 2. Also of Iog 2 32, Iog 2 gV, !og 2 yis > }o ^ 8, Iog 8 16. 3. Simplify Iog 2 4 -f Iog 3 9 + Iog 10 .1 Iog 3 1. 4. Write out the value of each power of 2 up to 2 20 (thus 2 1 = 2, 2 2 = 4, 2 3 = 8, etc.) in the form of a table. 5. By means of this table multiply 32 by 8, converting the multi- plication into an addition of exponents. 6. In like manner convert each of the following multiplications into an addition : 32 x 16 ; 64 x 32 ; 1024 x 16 ; 512 x 64. 7. Also convert each of the following divisions into a subtraction : 1024-16; 512-64; 32768-8-1024. 8. Also convert each of the following involutions into a multiplica- tion : (32) 3 ; (64) 2 ; (32) 4 . 9. Also convert each of the following root extractions into a divi- sion: ->/64; -S/1024; A/4096. 10. Let the pupil make up two examples like those in Ex. 6 ; in Ex. 8 ; in Ex. 9. 11. Let the pupil construct actable of powers of 3 and make up similar examples concerning it. COMMON SYSTEM 4. Characteristic and Mantissa. If a given number, as 384, be not an exact power of the base, its logarithm, as 2.58433 + , consists of two parts, the whole number called the characteristic, and the decimal part called the mantissa. To obtain a rule for determining the characteristic of a given number (the base being 10), we have, 10,000 = 10 4 , hence log 10,000 = 4 ; 1000 = 10 3 , hence log 1000 = 3 ; 100 = 10 2 , hence log 100 = 2 ; 10 = 10 1 , hence log 10 = 1. 10 TRIGONOMETRY f, Hence any number between 1000 and 10,000 has a loga- rithm between 3 and 4 ; that is, the log consists of 3 and a fraction. But every integral number between 1000 and 10,000 contains four digits. Hence every integral number containing/bwr figures has 3 for a characteristic. Similarly every number between 100 and 1000, and there- fore containing three figures to the left of the decimal point, has 2 for a characteristic ; every number between 10 and 100 (that is, every number containing two integral figures) has 1 for a characteristic ; and every number between 1 and 10 (that is, every number containing one integral figure) has for a characteristic. Hence, the characteristic of an integral or mixed number is one less than the number of figures to the left of the decimal point. 5. Characteristic of a Decimal Fraction. 1 = 10. ' .-. log 1 = ; .1 = ^ = 10- .-.log .!=-!; m ===^- '<* <** !<* -01 = -3, eta. Hence the logarithm of any number between .1 and 1 (as of .4 for 'instance) will lie between - 1 and and hence will consist of -- 1 plus a positive fraction ; also the logarithm of every number between .01 and .1 (as of .0372 for instance) will be between 2 and 1, and hence will consist of plus a positive fraction ; and so on. Hence, the characteristic of a decimal fraction is negative, and is numerically one more than the number of zeros between the decimal point and the first significant figure. There are two ways in common use for writing the char- acteristic of a decimal fraction. Thus, (1) log .0384=2.58433, the minus sign being placed over the char- acteristic 2, to show that it alone is negative, the mantissa being positive. LOGARITHMS 11 Or (2) 10 is added to and subtracted from the log, giving ^ log .0384 = 8.58433 - 10. In practice the following rule is used for determining the characteristic of the logarithm of a decimal fraction : Take one more than the number of zeros between the decimal point and the first significant figure., subtract it from 10, and annex 10 after the mantissa. EXERCISE 2 Give the characteristic of : 1. 452. 6. .08267. 11. 7. 2. 16730. 7. 1.0042. 12. 6267.3. 3. 767.5. 8. 7.92631. 13. .000227. 4. 64.56. 9. .007. 14. 100.58. 5. 9.22678. 10. .0000625. 15. 23.7621. 16. How many figures to the left of the decimal point (or how many zeros immediately to the right) are there in a number the characteristic of whose logarithm is 3? 2? 5? 1? 0? 4? 8-10? 7-10? 9-10? 17. Can you make up a rule for fixing the decimal point in the number which corresponds to a given logarithm? 6. Mantissas of numbers are computed by methods, usually algebraic, which lie outside the scope of this book. After being computed the mantissas are arranged in tables, from which they are taken when needed. In this connection it is important to note that The position of the decimal point in a number affects only the characteristic, .not the mantissa, of the logarithm of the member. Thus, if log 6754 = 3.82956 C.T'U log 67.54 = log ~ = log - = log 10 1 - 82956 = 1.82956. In general log 6754 = 3.82956 log 675.4 = 2.82956 log 67.54 = 1.82956 log 6.754 = 0.82956 log 0.6754 = 9.82956 - 10 log 0.06754 = 8.82956 - 10, etc. 12 TRIGONOMETRY 7. Direct Use of a Table of Logarithms ; that is given a number, to find its logarithm. For methods in detail see Introduction to Logarithmic Tables (Arts. 2-5 and 17). EXERCISE 3 Using five-place tables find the logarithm of each of the following numbers : 1. 7627. 10. .00672. 19. 17.6287. 2. 6720. 11. .000007. 20. 42. 3. 82. 12. 400000. 21. .000001. 4. 7862. 13. 14.6235. 22. .0186789. 5. 75. 14. .00226725. f23. 32679. 6. 157. 15. 87. 24. 3267.9. 7. 36278. 16. .76. 125. 326.79. 8. 67.222. 17. .000125. (26. 32.679. 9. 3.3427. 18. 100.25. V27. 3.2679. 28. Commit to memory the mantissa for each of the following : 2, 3, 5, TT. Then write at sight the log of each of the following, 200, 3000, 50, 100 TT, 20, .002, 30, .0005, --, .3, .2, 10 *, 20,000. 100 Using four-place tables, find the logarithm of each of the fol- lowing : 29. 12.67. 36. 24.68. 43. .000036775. 30. 762.8. 37. .11116. 44. .0026382. 31. 42.68. 38. 11.685. 45. 28966. 32. 1.2267. 39. .0012678. 46. 19.572. 33. .0263. 40. 965.3. 47. .8625. 34. .0012678: 41. 1.4676. 48. .0100267. 35. 1.0026. 42. 1.7628. 49. 2.225. 50. Work Ex. 28 for four-place tables. 8. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it (called its antilogarithm). See Introduction to the Logarithmic Tables (Arts. 6 and 17). LOGARITHMS 13 EXERCISE 4 Using five-place tables, find the antilogarithm of each of the follow- ing: ' 1. 1.41863. ; ,,- 4. 7.68416. 7. 6.59068. 2. 2.19756. 5. 9.22321-10. 8. 5.74706-10. 3. 0.98349. 6. 6.42857-10. 9. 8.00400. 10. Find log of 2.34578. 15. Find antilog of 3.21678. 11. Find antilog of 2.34578. 16. Find antilog of 6.00371. 12. Find log of 1.03678. 17. Find log of 6.00371. 13. Find antilog of 1.03678. 18. Find antilog of 4.98672. 14. Find log of 3.21678. 19. Find log of 4.98672. Find the number corresponding to each of the following logarithms, using four-place tables. 20. 1.4082. 23. 9.1546-10. 26. 8.0283-10. 29. 2.6575. 21. 2.7332. 24. 2.0326. 27. 7.1170-10. 30. 4.3490-10. 22. 3.2335. 25. 1.0135. 28. 5.0019-10. 31. 2.8177. 32. Find antilog of 2.3041. 35. Find antilog of 0.4975. 33. Find log of 2.3041. 36. Find antilog of 1.6924. 34. Find log of 0.4975. 37. Find log of 1.6924. COMPUTATIONS BY USE OF LOGARITHMS 9. Properties of Logarithms used in Numerical Computations. It is shown in algebra that a* . tf = a x+y ' y and also that (a x ) p = a p *. Using these properties of exponents, it can be shown that 1. log (mn) = log m 4- log n. 3. logm p P log m. fm\ PI log m 2. log f J = logm-logn. 4. logVm= For 771 = 10*. .-. logm = a?. n = 10 y . .*. logn ?/. .-. mn = 10* +y or log mn = x + y = logm + log n. (1) Also = : = 10*-", or log ^ = x - y = log m - log n. (2) 14 TRIGONOMETRY Also m p = (W X ) P = W px . .-. log m p =px=p - log m, (3) and Vm = W p . .-. log Vm = = -. (4) P P Hence : I. To multiply numbers : Add their logarithms and find the antilogarithm of the sum. This will be the product of the numbers. II. To divide one number by another : Subtract the logarithm of the divisor from the logarithm of the dividend and obtain the antilogarithm of the difference. This will be the quotient. III. To raise a number to a required power : Midtiply the logarithm of the number by the index of the required power and find the antilogarithm of the product. IV. To extract the required root of a number : Divide the logarithm of the number by the index of the required root and find the antilogarithm of the quotient. Ex. 1. Multiply 561.75 by .03286 by the use of loga- rithms. % log (561.75 x .03286)= log 561.75 + log .03286 log 561.75 = 2.74954 log .03^86 = 8.51667-10 antilog 1.26621 =18.4591, Product. The following, however, is the arrangement of work used by many practical computers. It has the advantage of show- ing all the steps in a complex logarithmic computation. (See p. 12, etc.) 561.75 log 2.74954 .03286 log 8.51667 - 10 Answer = 18.4591 log 1.26621 Observe that "561.75 log 2.74954" reads "561.75, its log is 2.74954," etc. Ex. 2. Compute the amount of $ 1 at 5 per cent com- pound interest for 20 years. LOGARITHMS 15 The amount of $ 1 at 5% for 20 years = (1.05) 20 . 1.05 log 0.02119 ; 20 log 0.42380 Amount = 2.65338 log 0.42380. If the student will compute the value of (1.05) 20 by con- tinued multiplication, and compare the labor in such a pro- cess with that involved in the above process, he will have a good illustration of the usefulness of logarithms. Ex. 3. Extract approximately the cube root of 532.768. 532.768 log 2.72653 1 log 0.90884. Root = 8.1066 log 0.90884. 10. Cologarithm. In operations involving division, instead of subtracting the logarithm of the divisor, it is usual to add its cologarithm. The cologarithm of a number is obtained by subtracting the logarithm of the number from 10 10. Hence adding the cologarithm of the divisor gives the same result as subtracting its logarithm. The use of the cologa- rithm saves figures, and gives a more orderly and compact statement of the work. ;he cologarithm of a number is obtained directly from a e of logarithms by the following rule : Subtract each figure of the logarithm of the given number from 9 except the last significant figure, which subtract from 10. Ex. 1. Find the colog of 37.16. log 37.16 = 1.57008. Hence, colog 37.16 = 8.42992 - 10. Ex. 2. Divide 52678 by 37.16 by the use of the cologa- rithm of the divisor. 52678 log 4.72163 37.16 log 1.57008 colog 8.42992 - 10. Quotient = 1417.58 log 3.15155. 11. In the extraction of the root of a decimal number it is best to add to and subtract from the logarithm of the decimal 16 TRIGONOMETRY number such a multiple of 10 that the last term of the quotient shall be 10. Ex. Extract the seventh root of .0854329. .0854329 log 8.93162 - 10 60 -60 7)68.93162 - 70 Root = .703667 log 9.84737 - 10 12. Computations involving Negative Numbers. In com- puting, by the use of logarithms, the value of expressions containing one or more negative factors, first, determine the sign of the result ; second, determine the magnitude of the result by treating all the factors as if they were positive and using logarithms. Ex. Compute ~ . / y) The result must be negative, since a negative number divided by a positive number gives a negative quotient. The magnitude of the result is determined by computing 876 EXERCISE 5 the value of 795 Compute-by mean's of five-place logarithms the value of each of the following : 1. 85 x 627. 5. 45 x 27.68 x .0967 x 4.2678. 2. 26.27 x 52.67. , 6. (2.67) 3 . 3. 8.25x25675. ? 27.8675 1768 18.678' ' 211.6' 8. (.5278) 7 . 9. -\/156.78. Also, if you can, extract the cube root of 156.78 with- out the use of logarithms. About how much more work in this process than in the logarithmic process ? Which process is more likely to be accurate, the long or the short one ? 10. -\/.86785. Also extract the square root of the square root of LOGARITHMS 17 .86785. About how much longer is this process than the logarithmic work ? 11. \/- 76.526. 12. -\/-. 00021. 13. -fy - .00062367 x 7.867. Find the compound interest on : 14. $ 15375 for 20 years at 6%. Make the computation without the use of logs. What fraction of the work is avoided by the use of logs ? 15. $ 323.50 for 12 years at 8%. 16. In 1623 the Dutch bought Manhattan Island from the Indians for $ 24. What would this sum amount to at the present time, if it had been placed on interest at 6%, the interest to be compounded annually ? 17. By aid of the logs committed to memory in Ex. 28, page 12, 200 100 TT 300 x 500 compute each 01 the following : ^=^; ; - oTo oo " TT 18. Also obtain the colog of 43560 (the number of square feet in an acre) and use it to find the area in acres of a field 200 ft. x 300 ft. ; one 300 ft. x 500 ft. ; one 1000 ft. x 2000 ft. Using four-place logarithms, compute the value of the following: 19. 1.2634 x 26.42. 20. .001467 x 96.8 x 47.37. J22.93 \I6^T E 556.85 x .00016277 x 4.6. .0016666 (12.67) 3 . ' .00042635' (3.176) 7 . 26. V42.67 x .10126 x 9.2. 27. A/.0000073. 28. Work Exs. 17 and 18 by the four-place tables. 29. Why are four-place logarithmic tables sufficiently accurate for the work of a carpenter or land surveyor ? Find the compound interest on : 30. $ 359.67 for 8 years at 6%. 31. $ 100 for 37 years at 4 % . 32. $4962.75 for 16 years at 5%. Try to compute this without the use of logs. About how much longer is the process without logs ? Which process is more likely to be accurate ? 13. Complex Computations. By the use of the properties of logarithms demonstrated in Art. 9, the value of a complex numerical expression may be computed. 18 TRIGONOMETRY V215 a ~ ' eo - by the use of logarithms. 67 x 52 J 2 = * log 6T^52 = *<* 215 + C log 67 + colog 62 >' Before looking up the logarithm of any number in the table, it is im- portant to make a scheme or outline of the work, leaving blank the places which are to be filled in by logs taken from the table. Thus the preliminary outline for Ex. 1 would be as follows : 215 log 67 log colog 52 log colog 2) Answer = log After looking up and inserting the logarithms and completing the computation, the work will appear as follows : 215 log 2.33244 67 log 1.82607 colog 8.17393 - 10 52 log 1.71600 colog 8.28400 - 10 2)18.79037 - 20 Answer = .248422 log 9.39519 - 10 One advantage of the above method of tabulating logarithmic work is that without essential change in the form of the tabulating, the work may be presented in the above complete form, or in a more condensed form (at the option of the teacher), as by omitting the logs of 67 and 52 and giving only their respective cologs in the tabulation. V2L8 . A/.03678 , Ex. 2. Compute - - by the use 01 logarithms. .28756 21.8 log 1.33846 -J log 0.66923 .03678 log 8.56561 - 10 1 log 9.52187 - 10 ' .28756 log 9.45873 - 10 colog 0.54127 Answer = 5.39975 log 0.73237 14. Exponential Equations. An exponential equation is one in which the unknown quantity occurs in the exponent of some term or factor, as a x l}. An equation of this kind can often be solved by the use of logarithms. Ex. Find the value of x in the equation .3* = 2. LOGARITHMS 19 Taking the logarithm of each member of the equation, a; log .3 = log 2. lo 2 0.30103 0.30103 9.47712- 10 (X522 * = EXERCISE 6 Using five-place tables, compute the value of the following : (Do not fail to make an outline of the work in each example before looking up any logarithms.) V2L82 x V.0071725 . /.59 x 2209 .926Z8 ^ 47 x .3481 2. Y ~ W ' 4 " V(.19678) 2 - (.072567) 2 . -V/.00231 X V76l9~ . /267.S5 x 7 x .000925 x 468.765 D. (21.67) 2 x .00096725 x^/567.256 7. Using the logarithms committed to memory in Ex. 28, Exer- cise 3, compute each of the following: '300 x 500. j 37 8. If there are 39.37 inches in a meter, convert the following into feet: 500 meters; 7294 meters; 300 meters (height of Eiffel Tower). What logs used in the first of these computations could be retained and used in the other computations ? Solve for x : 9. 6* = 67. 11. 2.8* = .1967. 10. 14 2 * + 3 = 2167. 12. .85* = . 01978. * If the teacher prefers, the remainder of the work for this example may be arranged as follows : log x + log (log .3) = log (log 2). /. log x = I log 2 - 1 log .3. 2 log 0.30103 1 log 9.47861 - 10. .3 log 9. 47712 10 (or ,.52288) 1 log (-) 9.71840 10 colog 0.28160. x = - .5757+ log 9.76021 - 10. 20 TRIGONOMETRY 13. Find the side of a square whose area is equal to that of a parallelogram whose base is 22.678 and whose altitude is 17.375. 14. Find the side of a square whose area is equal to that of a circle whose radius is 13.56. 15. Calculate the value of K in the equation, /f = s(s - a)(s - b)(s- c), wheni9 _ < lA_^ and a = 17.6, 6=21.675, c = 26.427. 16. Calculate the value of 6 in the equation, b = Va 2 c 2 , when a = .17623 and c= .12673. (Use 6= V(a + c)(a c), etc.) 17. Find the volume of a sphere whose radius is 14.7, if V= f irR 3 and TT = 3.1416. la Given t = 8, a = 32.17, find s, if s = J- at 2 . 19. Given s = a + b + c and a = .1732, b = .14326, c = .2242, find hj if h = - Vs(s d)(s b)(s c). c 20. Given # = 14.16 and TT = - 2 f , find 5, if 5 = 4 7r# 2 . 21. Given * = -^ and D = 23.8, find F, when V = i TrZ) 3 . 22. In how many years will $1 at compound interest at 5 % amount to $25? Using four-place tables, compute the value of the following : 23. 24. */ 529 1(5.78 |i O O'T \67 X 51.8 25 A/ 6.78 / .3756 x .265 26. V(125) (67) 2 . \ .227 x . 27. 28. 29. 1678 47.326 / 55400 X 8 .10021 V 123456 x .007 -^.216 7 ^ /21.67 1 .16765 _ \32.77 V 1.76364 UN '1^673 (26.72) 2 1 . (36.27)^ X .01267 Solve for x : 30. 2* =-19. 32. 19.38 3 * = 81672. 31, 4 2jc -3 ,= n+i, 33. .17' = .4782, LOGARITHMS 21 34. Find the side of a square whose area is equal to that of a rectangle whose base is 17.628 and whose altitude is 8.263. 35. Find the volume of a sphere whose radius is 1.1124, using 36. Given t = 12 and g = 32.17, find s, if s = 37. Work Exs. 16-19 above by the use of four-place tables 38. Work Exs. 7 and 8 above by the use of four-place tables. GENERAL PROPERTIES OP SYSTEMS OF LOGARITHMS 15. The logarithm of unity in any system of logarithms is zero. For, if a be the base, 1 = a. .-. log, 1 = 0. 16. The logarithm of the base in any system of loga- rithms is unity. For a = a 1 . .-. log,, a = 1. 17. The logarithm of zero in any system whose base is greater than unity is negative infinity ; that is, as the number approaches 0, the logarithm approaches negative infinity. or, since >1, = = ^ = or 00 . .'. log = - oo. ut in any system whose base is less than unity, the logarithm of zero is positive infinity. For, since a < 1, = a 00 . /. log a = oo. 18. Logarithm of a Product, Quotient, Power, and Root in any system. If a be taken as the base, and m and n be any two numbers, it can be shown in a manner similar to that used in Art. 9 that 1. log a mn = log a m + log a n. wi/ 2. log a = log m log a n. [Let the pupil supply the 3. proof. See Art. 9 ; use -j.tog.rn. J fa 22 TRIGONOMETRY 19. Changing the Base of a System of Logarithms. Given the logarithm of a given number, r, to a base a, to find the logarithm of r to another base k, we use the following formula: } For, let logj. r = x. Then #" =r ........ (1) by definition of a logarithm. Take the logarithm of each member of (1) to base a, then x log a k = log a r. Hence, x = 1 a ? log a & lo r or It follows as a special case that if r = a, log* a = -, or log* a-log a k = 1. Ex. Find the logarithm of .7 to the base 5. By the formula just proved, 9.84510-10 Iog 10 5 0.69897 EXERCISE 7 In working the first twelve examples in the following exercise use four-place tables in solving the even-numbered examples, and five-place tables in solving the odd-numbered examples. Find the value of : 1. Iog 5 60. 5. log^VB. 9. Iog 2 .7261. 2. Iog 6 9.3. 6. logsolS. 10. log^ -08275. 3. Iog 3 . 7 26.2. 7. logj.8 .17362. 11. log L2 .9267. 4. Iog 4 .93. 8. log. 8 .2631. 12. Iog 7 V3.1416. LOGARITHMS 23 Find without the use of tables : 13. Iog 3 27. 15. lo'g 9 J T . 17. logo .125. 14. Iog 2 32. 16. logj_8. 18. Iog 2 .0625. 19. Find the base of the system of logarithms in which the log of 16 = 4. 20. If the log of 27 = f , find the base. 21. If i = the log of 5, find the base. 22. Given the log of 5^ = f , find the base. 23. If the log of 64 = 1.2, find the base. 24. In how many years will a sum of money double itself at 4 % compound interest? at 6 % ? 25. If $1520 amounts to $10,701.46 in 40 years at compound inter- est, what is the rate per cent ? 26. Who invented logarithms, and when (see p. 169)? Find out all you can about this man and the way in which he invented logarithms. 27. What nation first divided the circle into 360 degrees, and one degree into 60 minutes ? CHAPTER II DEFINITIONS. TRIGONOMETRIC FUNCTIONS 20. Source of New Power. Illustrations. A spring of water is situated at the point A and a house at B. It is desired to find the length of a pipe needed to connect B with A, A and B being separated by a swamp. How can the length of the pipe be determined without going through the swamp? 32.0 yd. FIG. 1. 510 yd. FIG. 2. FIG. 3. If the swamp is situated as in Fig. 1, so that a point C can be taken where CA and CB form a right angle, then CA and CB can be measured and the length of AB computed by the methods of plane geometry. Let the pupil compute AB of Fig. 1. But if the swamp is situated as in Fig. 2, the above method of computing AB cannot be followed. However, if we take a convenient point C in Fig. 2 and measure the lines A C, CB, and the Z (7, the distance AB can be computed provided we have a table giving the ratios of the sides of all possible right triangles. Thus from this table we form the triangle given (on enlarged scale) in Fig. 3. Then by the properties of similar triangles we have the proportion 10 : 5.2 = 420 yd. : AD. 24 TRIGONOMETRIC FUNCTIONS 25 From this proportion AD is obtained ; afterward AB may be computed from the right triangle ADB by geometry. Hence the source of new power in trigonometry is a set of tables giving the ratio of each pair of sides in all possible right triangles. By the aid of such tables it will be found that we are able to find the unknown parts of many tri- angles which cannot be solved by ordinary geometry. Thus it will be found that if one side AB (Fig. 4) and any two angles (as A and B) of a triangle be known, the other sides (AC and CR] may be com- puted. By this method, for instance, the FlG - 4 - distance from the earth to the moon is computed. (For other illustrations of the new power given by trigonometry see Chapter VII.) 21. Trigonometry, as first considered, is that branch of mathematics which determines the remaining parts of a t triangle from certain given parts. PThus it will be found that if any three parts of a triangle are given, provided one of them is a side, the remaining parts maybe determined. Later the word trigonometry comes to have a more ex- tended meaning so as to cover the theory of the functions of angles in general wherever these angles may be found. Hence it comes to include much of the theory of wave motion and therefore of particular cases of wave motion, as of sound, light, and electricity. It also becomes largely algebraic in nature. Plane Trigonometry treats of plane triangles. See if you can find the derivation of the word trigonometry. 22. Trigonometric Functions of an Acute Angle. The fun- damental tools or instruments used in trigonometry are the functions of an angle now to be described and defined. 26 TRIGONOMETRY From any point B in one side of an acute angle BAC let fall a perpendicular BC to the other side, forming the right triangle ABC. FIG. 5. 6 FIG. 6. T) ri Then the ratio - is termed the sine of the angle A. Similarly, AC AC AB cosine A = , cotangent A = -^^, cosecant A = - tangent A = , secant J..= - , versed sine A = l- , coversed sine A \ or, in general, in a right triangle : The sine o/ an acwte angle is the ratio of the opposite Mg to the hypotenuse. The cosine is the ratio of the adjacent leg to the hypotenuse. The tangent is the ratio of the opposite leg to the adjacent kg. The cotangent is the ratio of the adjacent leg to the opposite kg. The secant is the ratio of the hypotenuse to the adjacent leg. The cosecant is the ratio of the hypotenuse to the opposite kg. The versed sine is 1 minus the cosine. The coversed sine is I minus the sine. These eight ratios are called the trigonometric ratios, or the trigonometric functions. The versed sine and the coversed sine are used so little in TRIGONOMETRIC FUNCTIONS 27 elementary work that we confine our attention mainly to the other six functions. Hence when we speak of the " six functions " we mean the first six trigonometric functions as given above. The abbreviations sin, cos, tan, cot, sec, esc, vers, covers, are ordinarily used for the eight functions. The cosine, cotangent, cosecant, and coversed sine are termed the co-functions of the sine, tangent, secant, and versed sine respectively. In the above triangle (Fig. 6), denoting the side AB by c, AC by &, and BC\>y a, we have sm^i = - aj/j^r sec^4 = - c 6 b rf * ** c cos A = - esc A = - c a tan A = - vers A = 1 b , c - covers^ = l-- ^ioailarly i a -p, c esc B = c b tan B- vers J5 = 1 a c cotJ5 = - covers B = l- b b c Or using abbreviations, sin of either acute Z = 1 PP* cot of either acute Z = 1 hyp. -L opp. cos of either acute Z = 1 sec of either acute Z = 2Ei- hyp.' J-- I Vi n tan of either acute Z = EC, esc of either acute Z = 2^- adj. -Lopp. 28 TRIGONOMETRY The method of indicating a power of a trigonometric function is shown by the following example: for " the square of the sine of the angle A" that is, for " (sin^.) 2 ," we write "sin 2 ^.." How then would "the cube of cos^L" be written? "The nth power of tanJ.?" In this book unless the contrary is stated, in the right triangle ABC, the letter C is supposed to be placed at the vertex of the right angle. 23. Utility of the Trigonometrical Ratios. It will be found that the numerical value of the above trigonometrical ratios for every angle from to 90 may be computed and arranged in tables whence they may be taken and used when needed. These numerical values are used by what is vir- tually the geometrical principle of similar triangles in solving triangles. Later, however, they become units and elements which can be variously grouped and used in many kinds of algebraic processes. 24. The value of a trigonometric function of an anorle depends only on the size of the ; then of AB. 23. On Fig. 3, what is the value of sin A' ? 24. On Fig. 6, if AB = 125, Z = 27, and sin 27 =.454, compute AC. 25. Can you suggest some practical problem similar to that given in Art. 20, which could be solved by trigonometry and not by geom- etry ? What is the source of new power in trigonometry which enables us to do this ? 26. If by the methods of trigonometry we are able to solve any triangle in which one side and any two angles are given, suggest some practical problem which could be solved by this means (and not by geometry). In a rt. A. given : 27. a = Vp* 4- g 2 , b = V2pg, find sin A and cos A. 28. a = 2 mn, c = m 2 + n 2 , determine sin A, sec A, and tan A. 29. b = 2pq, c=p* + q 2 , find tan A, sin A, esc A. 30. a Vm 2 + mn, b = Vwn + n 2 , find all the functions of B. 31. If a = 2 Vmn and c = m -f n, find all the functions of B. 32. If a = 60 and c = 61, find sec A, tan B, cot B, sin A. 33. If b = 2.64 and c = 2.65, find the functions of B. 34. If a = 2 b, find the functions of A. 35. If b | c, find the functions of A. 36. If a + b = | c, find the functions of B. 37. If a b = ^ c, find the functions of A. 38. Find the functions of B, if a = 4 d and 6 = 3 d By use of squared paper construct a rt. A, given : 39. c = 4 and tan A = %. 40. b = 3 and sin^=|.. 41. Find b if cos A = .36 and c = 4.5. 42. On Fig. 8, sin A = what ? cos B = what ? Does sin ^4 = cos B ? In like manner, show that cos A = sin B } tan ^4 = cot B, cot ^4 = tan B, sec u4 = esc JB, esc ^L = sec B. 43. Show the same on Fig. 6. TRIGONOMETRIC FUNCTIONS 31 44. In Fig. 6, since c is the hypotenuse, it is evident that it is greater than either leg. Hence sin A, or -, is always less than 1. c What other function of A is always less than 1? Which functions of A are always greater than 1 ? Which may be either greater or less than 1 ? 45. Which of the six functions are always proper fractions? improper fractions? may be either proper or improper fractions? Verify this on Fig. 8. 46. If A is any acute angle, is it correct to say that sec^l is always greater than sin A ? Why ? 47. The values of which of the six functions of A (on Fig. 6) have c for a denominator ? a ? b ? 48. How many of the above examples can you work at sight (i.e. for how many can you give results without the use of pencil and paper)? 26. Functions of the Complement of an Angle. From F. 6 (page 26). sin A = - ; also cos B = - . c c sin A = cos B, sin A = cos (90 - A\ since B = 90 - A. Let the pupil show in like manner that cos A = sin B = sin (90 - A), tan A = cot B = cot (90 - A) 9 and sec A = esc B = esc t (90 - A). Hence, in general, Any trigonometric function of an angle is equal to the co- function of the complement of the angle. By the use of this property, Any trigonometric function of an angle between 45 and 90 can be reduced to the function of an angle between and 45. Thus, sin 88 10' = cos 1 50'. 32 TRIGONOMETRY 5. csc 21 24' 30". 6. sec8'416'. 7. sin 89 59'. 8. cos 1 18'. EXERCISE 9 Express each of the following trigonometric functions as a function of the complementary angle : 1. sin 60. 2. cos 15. 3. tan 65 24'. 4. cot 55 36'. 9. Given tan 60 = V3, find cot 30. 10. Given sin 30 = |, find cos 60. 11. Given cos A = -, find sin (90 A). 12. Given sin A = p, find cos (90 A). 13. How many of the examples in this exercise can you work at sight ? RELATIONS OF TRIGONOMETRIC FUNCTIONS OF AN ANGLE 27. Three pairs of reciprocals exist among the trigoB- metric functions of an acute angle, viz. sin and csc cos and sec tan and cot For a b FIG. 9. sin A x csc A = 1 . cos A x sec A = 1. tan A x cot A = 1 . 28. Four equations connect the trigonometric functions of an acute angle in important ways. For, from Fig. 9, a* + b* = a 2 a EXERCISE II Prove each of the following identities : (In the solution of identities, the first of the two methods given above is to be preferred, since its use helps fix in mind the fundamental equations and formulas given in Arts. 28 and 29.) 1. cos A tan A = sin A. 5. sin A = cos A tan A. 2. sin A sec A = tan A. e. sin A 1 cos A 3. cos A esc A = cot A. 1 + sin A cos A 4. cos A = sin A cot A. cos A 1 sin A , 8. sin 2 A cos 2 A = 2 sin 2 A \. 9. (1 - sin 2 A) tan 2 .4 = sin 2 A. 10. (tan A + cot ^4) -sin A cos A = l, 11. (1 sin 2 ^4) esc 2 yl = cot 2 A. 12. (sin yl + cos^4) 2 = 1 + 2 sin ^4 cos A. 13. (sin J. + cos ^4) 2 + (sin J. cos yl) 2 = 2. 14. (esc 2 ,4 1) sin 2 A = cos 2 A 15. cos A sin 1 + cot 2 A 17. tan ^4 + cot A sec A esc ^4. 18. tan^ + cot^= sec ^ + C8c2 A sec^L x esc A 19. sin 4 A cos 4 ^4 = sin 2 - A cos 2 -4. o sin ^4 cos -4 1 cot A 1 tan A Ml + cos A 22. TRIGONOMETRIC FUNCTIONS 39 1 + tan A __ 1 tan A l-|-.cot.l cot ^4 1 1 23. cot A 4- tan A = sin A cos ^4. 24. tan 2 A sin 2 .1 = tan 2 A sin 2 .4 . 25. esc 4 .1-2 esc 2 A == cot 4 .1-1. 26. sec 4 A (I sin 4 .1) = 2 tan 2 .4+ 27 . _J^A_ cosA tan A 4- cot J. 28. ~ cot2 ^ = sin 2 ^l - cos 2 .!. 1 4- cot- A cot -4 cos A cot .1 cos .1 cot A cos .1 cot A 4- cos J 30. 1- cot 4 .1 = 2 csc 2 .! -csc 4 .!. 31. Vl sin 2 A tan A = sin A. 32. sin 6 A 4- cos 6 .1 = 1 3 sin 2 A cos 2 A. . cos 3 A sin 3 A = (cos A sin A)(\. 4- sin .4 cos A). [ Reduce tan 6 x sec 4 x to the form (tan 8 x 4- tan 6 x) sec 2 x. Transform : 35. tan 8 x into (tan 6 x tan 4 x 4- tan 2 x 1) sec 2 a? 4- 1 . 36. sec 10 y into sec 2 ?/ (1 4- 4 tan 2 ?/ 4- 6 tan 4 y 4- 4 tan 6 ?/ 4- tan 8 y). GOSX 37. VI 4- sin x into Vl sin x 38. ; into sec 2 x sec x tan x. 1 + Sill X 1 4- sin x , o 39. - into sec* x 4- sec x tan #. cos 2 a; 40. See if you can make up or discover any other', trigonometrical identities for yourself. 41. How many of the above examples can you work at sight? TRIGONOMETRIC FUNCTIONS OF PARTICULAR ANGLES 32. Functions of 45. The trigonometric functions of 30, 45, and 60 are used so frequently that it is of service to determine their values and commit these values to 40 TRIGONOMETRY memory. It is helpful to notice that we determine these values in each case by the use of a right angle, the hypote- nuse of which is taken as 1. Let ABC (Fig. 11) be an isos- celes right triangle, the hypotenuse of which, AB, is 1. Then, by geome- try, each leg is lV2_(for Z B = 45, .-.AC^= BC; but AC* + BC* = I 2 , FIG. 11. ' 2BC = I 2 , etc.). C By the definitions of the trigonometric functions, sin 45 = cos 45 = cot 45 = 1V2 V2 - 1V2 V9 9 sec 45 =l-_ = -^ 2 V2 V2 33. Functions of 30 and 60. Let ABD (Fig. 12) be an equilateral triangle in which the length of one side is 1. Let AC be .BD. Then, by geometry 6.0 "" and Z BAG =30. Also .4 (7 bisects BD, hence AC = FIG. 12. Then in the right triangle ABC 9 sin 30 = 1. cos 30 = 1V3. TRIGONOMETRIC FUNCTIONS 41 tan 30 = ^ = - = = 1V3. V3 3 cot 30 = sec 30 = 1V3 V3 esc 30 = T = 2. Let the pupil write out in like manner the functions of 60 (that is, of Z ABC in the A ABC). Of the results obtained in Arts. 32 and 33 those which are most used may be conveniently arranged i/i a table thus: 30 45 60 sin i IV2 iV3 cos iVs iVs i tan ^Vs 1 V3 34. Functions of 0. Let ABC (Fig. 13) be a right triangle in which the hypotenuse AB = 1 and the angle BAG is small and is diminished and made to approach as a limit. Then if AB remains fixed in length, BC approaches zero and AC approaches 1. At the limit, FIG. 13. sin = - = 0. cos = = 1. tan == ^ = 0. sec = j = 1. esc = TT = GO. versO= 1-1 = 0, cot = -= = oo. covers = 1 - = 1 42 TRIGONOMETRY 35. Functions of 90. Let ABC (Fig. 14) be a right triangle in which BAG is nearly a right angle and approaches 90 as a limit. AB remains fixed in length; hence BC approaches 1 as a limit and AC approaches 0. -C Fio. 14. At the limit. sin 90 = = 1. cos 90 = y = 0. tan 90 = TT = oo . cot 9a = ~ = 0. sec 90 = - = oo . esc 90 = 1 = / vers 90 - 1 - - 1. covers 90 = 1 - 1 = 0. The results obtained in Arts. 34 and 35 may be conveniently arranged in a table thus : 36. Representation of the Trigonometric Functions of an Acute Angle by Lines. If a quadrant of a circle OAB be drawn with center and radius OB equal to 1, the sine of any angle AOP' is M'P' = M'P' 1 = M'T. OP' Similarly the sine of and sine of Z AOP" = M"P". In other words the sine of any angle AOP in a quadrant whose radius is 1 is represented by the perpendicular let fall from P upon the radius OA. M" M' M' A FIG. 15. TRIGONOMETRIC FUNCTIONS 43 Hence it is easy to see. that, since MP is the sine of Z A OP, if AOP becomes very small and = 0, MP = 0, and at the limit sin = 0. Also if /.AOP" increases and = 90, sin /_ AOP" or M"P" = OB or 1. Hence at the limit sin 90 = 1. Similarly cos Z AOP f = OM' OM' = OM! Hence also OP' 1 cos Z AOP = OM, cos Z AOP" = OJf." In other words the cosine of any angle A OP in a quadrant whose radius is 1 Dented by the part of OA intercepted between and uno foot of the line representing the sine. Hence cos = A or 1, and as Z A OP changes from to 90, the cosine changes from 1 to 0. Similarly, (Fig. 16), AT AT AOT = cot^AOT= OA OT OA 1 OT BR 'OB OR OB BE = OT. = BR. o FIG. 16. OR N The various lines which represent the trigonometric functions of an acute angle AOP may be combined in a single figure (Fig. 17). Let the pupil find the lines on the figure which represent vers Z A OP and covers ZJ.OP. 37. Tables of Trigonometric Functions of Angles from to 90 called Natural Functions. By methods which will be explained later (see Art. 116) the values of the trigonometric O M - FIG. 17. 44 TRIGONOMETRY functions for angles of every degree and minute from to 90 may be calculated. These values are arranged in tables called Tables of Natural Trigonometric Functions. EXERCISE 12 By the use of squared paper, construct the following angles, making use of their natural functions : 1. 30. (Use sin 30 = 1 ) 2. 45. 3. 60. 4. If tan 61 37' = 1.85, construct the angle 61 37' on squared paper. By use of the table of natural tangents, construct : 5. 42 30'. 6. 56 37'. 7. 47.24. 8. 72.37. By use of the table of natural sines, construct : 9. 61 23'. 10. 47 15'. 11. 52.35. 12. 63.84. Find the numerical value of : 13. 2 sin 30 + cos 60 + sin 90. 14. b tan 30 + c cot 60 + tan 0. 15. 4 tan + 4 sin 2 45 -f 2 cos 45. 16. tan 30 cos 90 - 4 sin 60 + cos 2 0. ^ tan 30 cot 30 - 2 sin 45 tan 45 - 6 cos 60 cot 45 + sin 90. 18. sec 60 cos 60 - tan 30 cot 60 + tan 60 cot 30 - 20 sin 30. 19. Show that (sin 60 sin 45) (cos 30 -f- cos 45) = i. * If P = 0, Q = 30, ^ = 45, = 60, T=90, find the value of each of the following expressions : 20. sin Q + cos R 1. 21. tan 2 P + tan 2 Q-f-tan 2 ^. 22. cos P cos Q cos R + sin R sin S sin T. 23. sec P + 2 sin Q + 2 cos 2 R + tan 2 S + cosec T. 24. Does twice the tangent of 45 = the tan of 90 ? Why ? 25. Does sin 30 -f sin 45 = sin 75? 26. Does cot 30 + cot 45 = cot 75 ? 27. Draw a diagram showing the trigonometric functions as lines when Z A OP is less than 45. 28. Also when /-AOP is greater than 45. 29. Also when AOP equals 45. TRIGONOMETRIC FUNCTIONS 45 30. Given that x is greater than 45 and less than 90, 'show on a diagram similar to Fig. 17 that tan x ifc greater than cot x. 31. Given that x is less than 45, show that sec x is less than esc x. 32. Show that cos x is always less than cot x. 33. Show that sin x < tan x < sec x. 34. Show that cot x < esc x. 35. If a flagstaff is at a distance of 150 ft. and the angle of elevation (see Art. 88) of the top of the flagstaff is 30, find the height of the flagstaff. 36. Find its height if the angle of elevation of the top (at the same distance) is 45. Is 60. 37. Make up two examples similar to Ex. 35. 38. The Washington Monument is 555 ft. high. At a certain place the angle of elevation of its top is 30. Find the distance of the monument from this place. 39. At a certain spot 165 ft. from the top of a particular part of *-a Falls the angle of depression (see Art. 88) of the bottom of Is is 45. What is the perpendicular extent of the falls ? 40. How many of the examples in this exercise can you work at sight? 38. Many trigonometric equations involving only acute angles may now be solved. Ex. 1. Find the value of x which satisfies the equation sin x = ^. Since sin 30 = -*-, in the given equation x = 30, Ans. Ex. 2. Solve sinx = cosx. Dividing each member by cos x, tan x = 1. .-. a = 45, Ans. Ex. 3. Solve tan x 1 = 2 sin a; 2 cos x. .. Substituting for tan x, ^HL? _ 1 = 2 sin x - 2 cos x. COSX Hence, sin x cos x = 2 sin x cos x 2 cos 2 x. Factoring, (sin x cos a;)(l 2 cos x) = 0. Hence, sin x cos x = 0. .-. tano; = l, x = 45. Also 1 2 cos a; = 0. .-. cos x = %, x = 60. Hence, x = 45, 60, Ans, 46 TRIGONOMETRY Ex. 4. Given sin x = cos 4 x, find x. By Art. 26 we may substitute for sin x its equal, cos (90 x}. Then cos (90 x) = cos 4 x. ^ = 90. x = 18, EXERCISE 13 Solve each of the following equations : 3. cot x'= 3 tan x. 4. cot 2 x = 1 o 5. Vl sin 2 x 1 + sin x. 2 sin ?/ -h esc ?/ = 3. 13. 2 sin x V3 + 4 cos x = 5. 14. sec x = 2 tan x. 15. 4 sin 2 x tan 2 x = cot 2 x. 16. cot a? -K2 tan a? = 7. tan x 4- cot x = 2. 8. sec x = V2 tan x. 9. cos 2 x sin 2 x = sin x. - 11. 3 cot 2 x 4- cot x = 4. Solve : 23. sin x = cos 5 x. 24. tan y cot 8 ?/. 25. cos \x sin x. 17. 3 cos#4- tanx = 1 4-3 sino?. 18. tan# = 2cot# 1. 19. esc ?/ = 2 cot y. 20. 2 sin x + cos x = 2. 21. 2 sec x cosx = 1. 22. sin 2 x 4- sin a; = |. 26. sec (45 4- #) = csc a;. 27. sin ?/ = cos ny. 28. sin 3 x = cos 2 x. 29. If a church steeple is at a distance of 80 ft., and the steeple is 80 ft. high, find the angle of elevation of the top of the steeple. 30. If the height of the steeple is 80.5 ft. and the distance of the base is 100 ft., see if you can find the angle of elevation of the top of the steeple by use of the table of natural tangents (pp. 91-96 of the tables). 31. Make up an example similar to Ex. 29. 32. Make up an example similar to Ex. 30. 33. In a right triangle given c = 62, a 31, find A. 34. Given c = 150, a = 75, find B. 35. Given c = 120, b = 60 V3, find A. 36. How many of the examples in this exercise can you work at A TRIGONOMETRIC FUNCTIONS 47 39. Tables of Logarithms of the Trigonometric Functions from to 90. In performing numerical work involving trigonometric functions, it is usually more expeditious to proceed by the use of logarithms. Hence the logarithms of the natural trigonometric functions have been obtained once for all and arranged in tables called Tables of Logarithmic Trigonometric Functions. The use of these tables is ex- plained in the Introduction to the Tables (Arts. 7-11). EXERCISE 14 By the use of five-place tables, find : 1. log sin 26 18'. 9. log sin 4 6' 55". 2. log cos 12 16'. 10. log cos 17 17' 30". 3. log tan 36 18'. 11. log cot 37 28' 50". 4. log cot 76 18'. 12. log sin 78 59' 30". 5. log tan 55 16'. 13. log tan 86 46' 5". 6. log tan 15 18'. 14. log tan 4 44' 50". 7. log cos 86 52'. 15. log cos 45 48 '48". 8. log tan 36. 16. log cot 60 52' 6". 17. We have proved (see Art. 33) that sin 30 = .5. Obtain log .5 and thus show that the value of log sin 30 as given in the table is correct. 18. Similarly verify the value of log sin 45, and of log tan 60, as given in the table. 19. In the rt. A ABC, a = b tan A. (Why ?) If A = 18 16' and b = 18.63, find a. 20. In the rt. A ABC, b = c cos A. (Why ?) Find b if c = 18.675 and A = 36 36' 36". By the use of four-place * tables, find : 21. log sin 15.3. 24. log tan 78.8. 22. log cos 47.5. 25. log sin 27.35. 23. log cot 33.7. 26. log cos 26.36. * When the term "four-place tables " is used in connection with angles, the four-place logarithmic tables for the decimally divided degree are meant. See Arts. 18-19 of the tables. 48 TRIGONOMETRY 27. log tan 63.78. 29. log cos 40.16. 28- log cot 12.65. 30. log cot 29.23. 31. In the rt. A BA C, b = a cot A. (Why ?) If .4 = 18.67 and a = .2167 feet, find 6. 32. In the rt. A ABC, a = c sin A (Why?) If c = 17.65 and A = 59.72, find a. Also find 6, if b = c cos A EXERCISE 15 Using five-place tables, find A, given : 1. log sin A = 9.59632 - 10. 7. log cos A = 9.53390 - 10. 2. log tan A = 9.73777 - 10. 8. log tan A = 1.06575. 3. log cos A = 9.90951 - 10. 9. log sin A = 9.95788 - 10. 4. log cot A = 10.07029 - 10. 10. log cot A = 1.02921. 5. log sin A = 9.96159 - 10. 11. log sin A = 8.84501 - 10. 6. log tan A = 0.44540. 12. log cos A = 8.84501 - 10. By use of four-place tables, find A, given : 13. log sin A = 9.6495 - 10. 20. log cos 'A = 9.8409 - 10. 14. log cos A = 9.8063 - 10. 21. log tan A = 0.2575. 15. log tan A = 9.7384 - 10. 22. log cot A = 2.0248. 16. log cot A = 0.4755. 23. log tan A = 1.5718. 17. log cot A = 9.8248 - 10. 24. log sin A = 9.9596 - 10. 18. log tan A = 0.4422. 25. log cos A = 9.3129 - 10. 19. log cos A =9. 6351 -10. 26. log cot A = 0.5881. EXERCISE 16 By use of five-place tables find : 1. log sin 56' 18". 5. log cot 1 18' 36". 2. log tan 1 16' 37". 6. log cos 89 7' 19". 3. log cos 88 13' 26". 7. log sin 1 6' 12". 4. log tan 88 54' 50". 8. log cot 88 16' 32". Find the angle A if : 9. log tan ^1=7.88154 -10. 13. log tan A = 3.05992. 10. log cos A = 8.28910 - 10. 14. log cot A = 2.88206. 11. log sin ^4 = 8.09600 -10. 15. log sin A = 6.88800 - 10. 12. log cot A = 7.90390 - 10. 16. log cos A = 7.63702 - 10. TRIGONOMETRIC FUNCTIONS 49 For "angle whose log sin is" we may write "Z log sin," or "antilog sin," hence tind : 17. Z log sin 9.82627 -10. 18. Z log tan 10.90261 - 10. 19^ Z log cos 9. 06000 -10. 23. In the A ABC, a = c sin 18' 48." 20. Z log cot 8.09599 -10. 21. Z log cos 8.09599 -10. 22. Z log tan = 2.77651. Find a if c = 18.6 and A = 26 Find the value of the following : 528.7 x cos 83 16' 24" x tan 2 75 18' 24" 24 25. 672 cot 2 18 32' 54" x sin 69 - cos 2 15 16' 34" 265 x tan 65 18' x cos 2 14 28 f 12" 19 cot 2 11 16' 24" x sin 75 15' 45" x .7* By use of four-place tables, find : 26. log cos 88.76. 27. log sin 0.762. 28. log cot 89.267. 29. log tan 1.067. Find angle A if : 34. log cot A = 8.1067 - 10. 35. log tan A = 8.2574 - 10. 36. log cos A = 8.1360 10. 37. log sin A = 8.0440 - 10. 38. log tan A = 2.1080. 39. log cot A = 2.0532. 40. log sin A = 7.9100 10. 41. log cos A = 7.9932 - 10. 49. In the rt. A ABC, a = c sin A. and A = 1.267. 50. In the rt. A ABC, b = acot A. 30. log tan 88.763. 31. log cot 0.765. 32. log sin 1.267. 33. log cos 89.467. Find: 42. log cot 88.676. 43. log tan 88.676. 44. Z log cot 8.1078 -10. 45. Z log tan 8.0295 -10. 46. Z log cos 8.0959 - 10. 47. Z log sin 8.0371 - 10. 48. log tan 88.68. (Why ?) Find a if c = 126.27, (Why ?) Find b if a = 0.4267, and A = 2.166. 632.7 x cos 78.16 x tan 2 71.62 51. Find the value of 52. Find the value of 426.8 x sin 13.25 x cot 2 12.47 x .8 326 x tan 38.25 x cos 2 88.627 43 x cot 0.826 x sin 2 2.467 ' 50 TRIGONOMETRY EXERCISE 17. REVIEW 1. In the right A ABC, given tan A = T \ and a = 16, find b, c, and the other functions of A. 2. If cos'^L = -?-, find the value of 17 cos A cot A 3. Show that cos 60 cos 30 + sin 60 sin 30 = cos 30. 4 . Show that cot 45 + CQt 90 =1. 1 - cot 45 cot 90 (Work Exs. 5-12 without the use of tables.) 5. Which is greater, sin 49 or cos 49 ? 6. If sin A = f , is A greater or less than 45 ? 7. If tan A = 2, is A greater or less than 60 ? 8. Which is the greater, tan 37 or cot 37 ? 9. If A = 60, show that sin 1 A = /1 10. If A = 60, show that cot%A = 13. Prove 1 + cot A = sec ^ + csc A 1 cot A sec ^4 csc A 14. Prove X + ta " 2 1 + cot COS COS 15. Prove * + cos ^ = (csc J. + cot A) 2 . I cos A 16. If x = 30, show that tan 2x = 1 tan 2 x 17. If a; = 30, show that sin 3 x = 3 sin x 4 sin 3 #. 18. If x = 30, show that cos 3 x = 4 cos 3 x 3 cos x. Solve the following trigonometric equations : 19. tan x + 3 cot x 4. 20. 2 sec 2 z-tan 2 x = 5. 21. 3csc 2 a;-2cota; = 4. TRIGONOMETRIC FUNCTIONS 51 If P = 0, Q = 30, R = 45, S= 60, T = 90, find the value of : 22. cos 2 Q + cos 2 S + cos 2 T -f 2 cos Q cos $ cos T 7 . 23. sec (2(1 + tan 72) sin 3 T(cos R + sin # cos Q). 24. 1 + tan 2 2 8 + 3(cos P sin 2 fl - sin S). A tan l\i 25. If 25 sin A = 7, find cot ^4 and esc A 26. If p cot (9 = Vr 2 i> 2 , find sin 0. 27. If i denotes the angle of incidence of a ray of light falling on a piece of glass, and r the angle of refraction, then sin i = f sin r. Find r when i' = 27 17'. 28. If at a distance of 300 ft. the angle of elevation of the top of one of the big trees of California is 45, how tall is the tree? 29. If at a distance of 300 ft. the angle of elevation of the top of a tree were 42, see if you can find out how tall the tree would be. (Why are we able to determine this height by trigonometry and not by geometry ?) 30. Who first, and at what date, defined the sine of an angle as the ratio between two lines (see p. 165) ? Give the different substitutes for this idea of the sine that had been used before this time. Why is the ratio definition of the sine superior to each of these ? 31. Explain the origin and literal meaning of the word sine (see p. 166). 32. Who first invented each of the other trigonometric ratios, and at what time (see pp. 162, 164) ? 33. Give some of the various names used for these ratios, with the names of the inventors of these names. 34. What nation first used the trigonometrical identity sin 2 A + cos 2 A = l (see p. 172) ? tan x = ^^ ? cosx 35. Give an account of the computation of trigonometric tables (see pp. 168-170). CHAPTER III RIGHT TRIANGLES 40. Two Cases arise in the trigonometrical solution of right triangles. CASE I. Given one side and an acute angle. CASE II. Given two sides. In each of these cases it will be observed that three parts are really given, since the right angle is known. CASE I 41. The solution of Case I is effected as follows : Subtract the given angle from 90. This ivill give the un- known angle. The unknown sides may then be found by means of the following : 1. Either leg = (sine of ^opposite) x hypotenuse. 2. Either leg = (cosine o/Z adjacent) x hypotenuse. 3. Either leg = (tangent of/, opposite] x other leg. 4. Hypotenuse = (secant of either acute Z) x (leg adjacent to thatZ.}. Also (either leg) = (cot of Z adjacent) x (other leg); hyp. = (esc of either acute Z) x (leg opposite that Z). Proof By def ., sin A = - c A b C FIG. 18, Also Also cos B = -. c tan A = -- b sec B = - a 52 , a = c sin A. a = c cos B. a = b tan A. . c = a sec jB. RIGHT TRIANGLES 53 Similarly it may be proved that : b = c sin jB, b = c cos A, b = a tan B, and c = b sec A. Ex. 1. Given A = 55 43' 29", c = 415.18, find the remain- ing parts of the right triangle. We first draw a diagram (Fig. 19) of the triangle to be solved, and on this diagram write the known magnitudes (415.18 for c, and 55 43' 29" for A). We also indicate the parts to be computed (a, b, B) by annex- ing the = mark to each of these.* During the numerical computation, as soon as the result for any part is ascertained, this result should be entered on the diagram after the proper = mark. Z B = 90 - 55 43' 29" = 34 16' 31 ". a = 415.18 sin 55 43' 29". (Art. 41, 1) .-. log a = log 415.18 + log sin 55 43'- 29". 415.18 log 2.61824 55 43' 29" log sin 9.91716 - 10 a = 343.085 log 2.53540 Also b = 415,18 cos 55 43' 29". (Art. 41, 2) .-. log b = log 415.18 -f log cos 55 43' 29". 415.18 log 2.61824 55 43' 29" log cos 9.75064 - 10 = 233.821 log 2.36888- (As a check use a = b tan A.) FIG. 19. Ex. 2. A b FIG Given a= .0723, ^ = 31 47' 7", find the remain- ing parts of the right triangle. Z A = 90 - 31 47' 7" = 58 12' 53". b= .0723 tan 31 47' 7" .0723 log 8.85914 - 10 31 47' 7" log tan 9.79216 - 10 b = .448022 log 8.65130 - 10 c=. 0723 sec 31 47' 7" .0723 cos 31 47" 7' 5723 log 8.85914 - 10 31 47' 7" log cos 9.92943 - 10 colog cos 0.07057 c= .0850567 "Tog 8.92971 - 10 (As a check use b= c cos A.) = C . 20. 54 TRIGONOMETRY Ex. 3. By use of four-place tables solve the right triangle in which & = 21.635, .A = 47.23. Z B = 90 - 47.23 = 42.77. Also a = 21.635 tan 47.23. (Art. 41, 3) B .'. log a = log 21.635 + log tan 47.23. 21.635 log 1.3352 47.23 log tan 0.0339 a =23.394 log 1.3691 By Art. 41, 4, c = 21.635 sec 47.23 = cos 47.23 .-. log c = log 21.635 H-colog cos 47.23 ' 21j635 'C 21.635 log 1.3352 FIG. 21. 47.23 colog cos 0.1681 c = 31.864 log 1.5033 (As a check use a = c cos J5.) 42. First Estimates. Graphical Solutions. In the solutions of triangles fully one half the mistakes commonly made, and those the most important ones, are eliminated by making a rough mental forecast of the results before proceeding with the exact numerical work. Thus in solving Ex. 1 of Art. 41, the pupil should first of all observe that, the hypotenuse being 415.18, each of the legs will be less than 415.18 ; and also that, since angle B is less than angle A, side b must be less than side a. If then as a result of his exact numerical calcula- tion, the pupil finds a leg greater than 415.18, or a less than 6, he knows at once that a mistake has been made. Similarly it is useful, by means of the rule and protractor, to make a drawing according to scale of the triangle to be solved, and from the figure to determine as accurately as possible the dimensions of the unknown parts by measuring them according to scale. Such results should be accurate enough to aid in eliminating any large errors in the numeri- cal work. (Indeed, if the work be neatly done, the results obtained from the diagram will be accurate enough for many practical purposes.) RIGHT TRIANGLES 55 43. Exact checks of the numerical accuracy of the work of solving right triangles are obtained by calculating some side or angle of the triangle by a formula different from those already used in the computation, and observing whether the results thus obtained accord with those obtained in the first solution. Thus, to check the accuracy of the solution given for Ex. 1, Art. 41, determine whether tan A = --, that is, compute the value of the frac- b tion 343>Q85 and also obtain from the table the value of tan 55 43' 29" 233.821 and observe whether these two values accord. EXERCISE 18 State at- sight the formula value of x (or of x and y) in each of the following triangles : Thus in Ex. 1, (1), x = 208 sin 40. 1. (3) (1) (2) 3. Make up an example similar to Ex. 2. By use of five-place tables solve each of the following triangles, given : (In working each example outline all the work carefully before looking up any logs see Ex. 1, p. 18.) 4. .4 = 28, 6 = 12. 6. .4 = 46 18', 6 = 48.527. 5. -4 = 78,' c = 26.736. 7. .4 = 28 17', c = 24.16- 56 TRIGONOMETRY 8. B = 54 43' c = 1123. 10. A = 38 16' 24", c = 3.6289. 9. B = 37 19', 6 = 293.8. 11. B = 72 16' 42", a = 22.684. 12. Given c = . 52684, B = 63 18' 48"; find a. 13. Given A = 37 25' 20", c = .356 ; find b. Find the remaining parts in each of the following right triangles, given : 14. ^ = 63 28' 40", a = 256.43. 15. c = 13.867, A = 87 16' 30". 16. A = 51 9' 6", c = .19678. 17. a = 126.78, A = 26 18' 36". 18. Given ^4 = 5 16' 32", b = .96156; find c. 19. Given A = 37 14' 15", b = 217 ; find a. 20. If the top of the Statue of Liberty in New York harbor is 301 ft. above the water surface, and a boat in the harbor finds the angle of elevation of the top of the statue to be 12, how far is the boat from the statue ? 21. If a certain point on the brink of the Grand Canon of the Colo- rado is known to be a horizontal distance of 3 miles from the Colorado River and the angle of depression of the river is 17, how deep is the canon at that place and how far from the observer is the river in a straight line? 22. Which of the examples in Exercise 22 are you able to solve by Case I ? Solve one of these. 23. Make up a similar practical problem for yourself and solve it, as for instance one concerning the Bunker Hill monument (221 ft. high). Solve the following right triangles, by use of four-place tables, hav- ing given : 24. .4 = 32.6, 6 = 18. 28. .4=*37.67, c = 126.7. 25. .4 = 56, c = 2.678. 29. =.76.25, a = .926. 26. 5 = 38.2, c = .7685. 30. .4 = 21.32, a = 16.256. 27. 5 = 82.5, a = 12.56. 31. 5=66.27, b = .0087. 32. Given c = .6243, 5 = 51.25; find a. 33. Given A = 77.26, c = .5163; find b. ^34. Given 5 = 39.29, 6 = 41.67; find a. RIGHT TRIANGLES 57 Find the remaining parts in each of the following right triangles, given : 35. c = 13.13, A = 88.17. 36. 5 = 42.16, a = .5252. 37. Given A = 5.26, 6 = 128.6; find c. 38. Given B = 87.267, c = 22.67 ; find a. 39. Given A = 4.276, a = 26.32 ; find 6. 40. Work Exs. 20-23 by four-place tables. Solve without the use of tables, having given : 41. .4 = 30, b = 7. 45. .4 = 60, a = 2000. 42. .4 = 45, c = 12. 46. J5 = 30, c = 1200. 43. 5 = 60, 6 = 25. 47. ^4 = 45, 6 = 200. 44. 5 = 30, a = 1000. 48. .4 = 30, c = 20d. 49. Solve Exs. 6 and 7 of this exercise without the use of logarithms (i.e. by the use of the Tables of Natural Sines, etc., pp. 91-96). 50. How many of Exs. 41-48 can you solve at sight without draw- ing a figure ? 51. On the figure if AADB and DOB are right A, find BD, BC, and DC at sight. 52. On Fig. 52, p. 93, if OP = 1, what is the value of OQ ? of PQ? of QN? of ON? CASE II TWO SIDES GIVEN 44. The Solution of Case II is effected as follows: Find one of the angles of the given triangle by using that one of the following trigonometric ratios which contains the two given sides : 1. sine of either acute = -7 hyp. 2. cosine of either acute ^ = -r 3. tangent of either acute ^- = -. -TT-. i aa]. 58 TRIGONOMETRY Find the remaining parts of the triangle by Case I (but if the hypotenuse and a leg are given, the other leg may be found by one of the formulas, a = V(c + b)(c b), b = V( c .+ a)(c- a)). Ex. 1. Given a = 317, c = 438, find the remaining parts of the right triangle ABC. sin A = (Art. 44, 1) B Hence log sin A = log 317 + colog 438 317 log 2.50106 438 log 2.64147 colog 7.35853 - 10 A = 46 21' 55" log sin 9.85959 - 10 B = 90 - 46 21' 55" = 43 38' 5". b = 438 cos 46 21' 55". (Art. 41, 2) 438 log 2.64147 46 21' 55" log cos 9.83888 - 10 FIG. 22. b = 302.24 log 2.48035 (As a check use tan A = } b J Ex. 2. By use of four-place tables, solve the right triangle in which a = 3.104, = 2.965. A 3.104 2.965 ~2.965 c- 3 ' 104 c / (Art. 41) ^/ cos B log 0.4920 colog 9.5279 - 10 A 3.104 43.69 2.965 G FIG. 23. log 0.4920 colog cos 0.1408 = 46.31 log tan 0.0199 = 90 - 46.31 = 43.69. c= 4.293 log 0.6328 45. Sources of Power in Trigonometrical Solution of Tri- angles. There is danger that the pupil form mechanical habits of solving triangles without realizing the nature or RIGHT TRIANGLES 59 meaning of what he is doing. He should constantly realize that he is able to do what he is doing because some one be- fore him has computed the legs of every possible right tri- angle whose hypotenuse is 1, and the other parts when each leg is 1, and arranged the results in tables (natural sines, etc.,) and that he uses these results (and therefore uses the work done in computing them) by the geometrical principle . of similar triangles. Also that some one else has made the pupil's work easier by looking up the logarithms of all the numbers in the natural tables and arranging them in other tables, and that the pupil is using this work also. 46. Special Case. Given the hypotenuse and a leg nearly equal, the angle between them will be very small. If this angle be found directly from the parts given, it will be found in terms of the cosine. Since the cosine of a small angle changes slowly as the angle varies, such a solution will not be accurate in the last figures. A more accurate solution is obtained by first calculating the third side by the use of the formula a = ^/(c + b)(c b) and finding the angle men- tioned in terms of the sine. Ex. Given c= 412, b = 410, solve the triangle. By the formula, a = V(412 + 410) (412 - 410) = V822 x 2. .-. log a = -I- (log 822 + log 2). 822 log 2.91487 2 log 0.30103 2)3.21590 A1 4 40.546 a = 40.546 log 1.60795 Also sm A : : 40.546 log 1.60795 412 colog 7.38510 - 10 A = 5 38' 52" log sin 8.99305 - 10 B = 90 - 5 38' 52" = 84 21' 8". 60 TRIGONOMETRY EXERCISE 19 Using five-place tables, solve in full the following right triangles, given : (In working each example outline all the work carefully before looking up any logs see Ex. 1, p. 18.) 1. c=18.4, a = 10.7. 5. c = . 89672, a =.68425. 2. c = 37.266, a = 20.46. 6. 6 = 14.222, c = 21.678. 3. a = 26.725, c = 39.626. 7. a = .0628, b = .0487. 4. a = 5, 6 = 6. 8. a = .1777, c = . 25643. 9. Given a = 4 yd., 6 = 9 ft., find A. 10. Given a = 8.701 yd., b = 21.645 yd., find Z A. 11. Given b = .26725, c = .39626, find Z B. 12. Solve in full if a = 6, 6 = 6. 13. Find A if a = .02678, 6 = .05537. 14. Solve in full if c = 117.32, a = 112.67. SUGGESTION. First use 6 = Vc 2 a 2 = V(c + a) (c a). 15. Solve in full if b = 358, c = 362. 16. Solve in full if a = 26.63, c = 27.99. 17. If the Mt. Washington railway at a certain place rises 3596 ft. for 3 mi. of the length of the track, what angle on the average does the track make with the horizon ? 18. The carpenter's rule for constructing J of a right angle is to con- struct a right triangle whose legs are 5 and 12 inches and take the greater acute angle in the triangle. How far is this from being correct ? 19. Which of the examples in Exercise 22 are you able to solve by the methods of Case II ? Solve two of these. 20. Make up a similar practical problem for yourself and solve it. Solve by use of four-place tables, having given : 21. c = 23.7, a = 15.7. 25. 6 = 6.7, c = 9.7. 22. c = .562, 6 = .3962. 26. 6 = .12675, a = .14296. 23. a = 33.29, 6 = 27.28. 27. c = 132.96, 6 = 100.82. 24. a = 5, 6 = 8. 28. a = .07282, c = .11111. 29. a = 2367, 6 = 1827.6. RIGHT TRIANGLES 61 30. Given a = 11, c = 16, find A. 31. Given a = 27.82, b = 33.67, find B. 32. Given c = 156.7, b = 148.2, solve in full. First use a = Vc 2 - 6 2 = Vc + 6)(c-6). 33. Given c = 862, a = 854, solve in full. 34. Given a = 98.6, b = 63.4, find A. 35. Given c = .4367, b = .1967, find 5. 36. Work Exs. 17-20 by the four-place tables. Without the use of tables solve in full each of the following right triangles, given : 37. a = 13, 6 = 13. 41. c = 6, a=3V3. 38. c = 18, a = 9. 42. c = V2, 6 = 1. 39. c = 200, 6 = 100. 43. c = 100, a = 50V3. 40. a=V3, 6 = 1. 44. a + c=18, 6 = 6V3. 45. Solve Exs. 3 and 4 of this Exercise without the use of logarithms. 46. How many of Exs. 37-43 are you able to solve at sight without drawing a figure ? 47. Isosceles Triangles. If certain parts of an isosceles triangle be given, the unknown parts may often be deter- mined by dividing the isosceles triangle into two equal right triangles by means of a perpendicular drawn from the vertex to the base, and by solving one of the right triangles thus formed. Ex. 1. If the vertex angle of an isosceles triangle is 42 30' and a leg is 47.6, find the base. Draw the altitude OD. Then /.A OD=2l 15'. Hence, in the right A AOD, we have a side and an acute angle given, to find the base AD (Case I). Hence AD = 47.6 sin 21 15'. 47.6 log 1.67761 21 15' log sin 9.55923 -10 AD = 17.252 log 1.23684 AB = 2 AD = 34.501 FIG. 25. B 62 TRIGONOMETRY Ex. 2. By use of four-place tables, solve the isosceles A triangle whose base is 12.25 and vertex angle 28.22. Draw the altitude AD. Then Z BAD = 1(28.22) = 14.11. Z B = 90 - 14.11 = 75.89. AB = 6.125 sec 75.89 = 6 ' 125 6.125 FIG. 26. cos 75.89 6.125 log 0.7872 75.89 colog cos 0.6130 AB = 25.129 log 1.4002 48. A regular polygon may be divided into equal right tri- angles by lines drawn from the center to the vertices and by the apothems to the sides. Hence if certain parts of a reg- ular polygon are given, the remaining parts may often be determined by divid- ing the polygon into right triangles and solving one of these triangles. It is to be observed that one of the right triangles, as A CD of Fig. 27, has the radius of the circle circumscribed about the polygon for its hypotenuse AC, and the radius of the inscribed circle, 360 CD. for a leg. Hence, Z AC A '= - , where n denotes the n number of sides of the polygon, and Z A CD of the right 180 triangle = . n EXERCISE 20 Using five-place tables, solve each of the following isosceles triangles, given : 1. Base = 120, base Z = 60. 2. Leg = 216, vertex Z= 110. 3. Base Z = 56 18', leg = 8.7265. 4. Base Z. = 38 17' 50", altitude = 31.42. RIGHT TRIANGLES 63 5. Base Z = 55 18' 24", altitude = 762.89. 6. Base = 8.2364, altitude = 7-8. 7. Vertex Z = 113 17', base = .12692. 8. Altitude = 4835, base =9248. 9. One side of a regular pentagon is 12. Find the apothem, radius. perimeter, and area of the pentagon. 10. One side of a regular decagon is 1. Find the apothem, radius, perimeter, and area of the decagon. 11. The radius of a circle is 16 feet. Find the side, apothem, and area of a regular inscribed dodecagon. 12. Find the same magnitudes for a regular dodecagon which is circumscribed about a circle whose radius is 17.. 13. The diagonal of a regular pentagon is 14 ; find the side, apothem, perimeter, and area of the pentagon. 14. The apothem of a regular heptagon is 0.69786 ; find the perimeter and area of the heptagon. If m denotes the base, li the altitude, I the leg, C the vertex angle, and D the base angle of an isosceles triangle, find : 15. 7i, m, and (7, in terms of D and I. 16. D, I, and (7, in terms of m and h. 17. D, C, and m, in terms of h and I. 18. C y , h, and /, in terms of D and m. 19. D, h, and /, in terms of C and m. 20. Solve the isosceles triangle in which a leg = 2. 62731 and the altitude = 1.76683. 21. If a chord 22.67 ft. in length subtends an arc 127 23', what is the radius of the circle ? 22. If the radius of a circle is 105.27 ft., what is the length of a chord which subtends an arc of 54 13' ? 23. The side of a regular polygon of fourteen sides inscribed in a circle is 21.6 ft. ; find the side of a regular twenty-sided polygon in- scribed in the same circle. 24. The radius of a circle is R, show that each side of a regular inscribed polygon of n sides is 2 R sin I- -J, and that each side of a /180\ regular circumscribed polygon is 2 E tan ( - - ) 64 TRIGONOMETRY \/ 25. Each side of a regular polygon of n sides is m; show that the radius of the circumscribed circle is equal to esc ( - ) , and the radius of the inscribed circle is equal to cot f ] 2 \ n J 26. If the chord of an arc of 36 is 24, find the chord of an arc of 12 in the same circle. 27. If the chord of an arc of 48 is 36, find the chord of an arc of 66 in the same circle. Using four-place tables, solve the isosceles triangle in which : 28. Leg = 36.72, base Z = 32.6. 29. Base = 1600, base Z = 67.4. 30. Vertex Z = 117.72, altitude = 17.83. 31. Base = .7368, altitude = .4864. 32. Altitude = 112.67, leg = 128.7. 33. Leg = 67.87, base Z = 32.73. 34. Altitude = .11683, base Z = 76.18. 35. Base = 31.26, altitude = 21.73. 36. Vertex Z = 151.7, leg = .4363. 37. One side of a regular octagon is 14. Find the apothem and area of the octagon. 38. The apothem of a regular pentagon is 19.7. Find the perimeter of the pentagon. 39. A regular decagon is inscribed in a circle whose radius is 1.76. Find the side and apothem of the decagon. 40. Find the magnitude of the various parts of a regular heptagon circumscribed about a circle whose radius is 21. 41. The diagonal connecting two alternate vertices of a regular dodecagon is 18. Find the side, apothem, and area of the dodecagon. 42. If a chord of 37.82 ft. subtends an arc of 118.3, find the radius of the circle. 43. If the radius of a circle is 100, what is the length of a chord which subtends an arc of 67.7 ? RIGHT TRIANGLES 65 Without the use of the tables, solve the following : 44. The base of an isosceles triangle is 50, and the vertex angle is 120. Find the base angle and altitude. 45. The leg of an isosceles triangle is 100, and the altitude is 50. Find the base angle and base. 46. The altitude of an isosceles triangle is 10, and the base angle is 60. Find a leg and the base. 47. The leg of an isosceles triangle is 6V2, and the base is 12. Find the base angle, vertex angle, and altitude. 48. The radius of a circle is 2. Find the number of degrees in an arc which subtends a chord whose length is 2V3. 49. The diagonal of a square is 10. Find the side of the square. 50. How many of Exs. 44-49 can you work at sight ? AREAS 49. General Method of computing Area of a Right Triangle. If 1} denote the base, a the altitude, and K the area of a right triangle, by geometry K = ^db. .'. log K = log a + log b + colog 2. Ex. 1. Given J. = 3719', 6=308, find the area of the right triangle. To find log a and then the area we proceed as follows : B a = 308 tan 37 19'. (Art. 41) 308 log 2.48855 37 19' log tan 9.88210 - 10 a log 2.37065 / K = 308 log 2.48855 2 colog 9.69897 - 10 .fiT = 36155 log 4.55817 FIG. 28. Ex. 2. Find the area of a right triangle in which the hypotenuse is 417 and the base 356. a = Vtf^b 2 = V(417) 2 - (356) 2 = V(417 + 356)(417-356) = V773 x 61. 66 TRIGONOMETRY K= ab. ..*. log K= log a + log 6 -f colog 2. 773 log 2.88818 J log 1.44409 61 log 1.78533 J log 0.89267 356 log 2.55145 ^ SDO o- 2 colog 9.69897 - 10 FIG. 29. K= 38652.7 log 4.58718 Ex. 3. By use of four-place tables find the area of the right triangle in which A = 37.32 and 6=308 (see Fig. 28). log K = log a -f log 308 + colog 2. To find log a, a = 308 tan 37.32. 308 log 2.4886 37.32 log tan 9.8821 a log 2.3707 308 log 2.4886 2 colog 9.6990 - 10 K= 36167 log 4.5583 50. Formulas for Area of a Right Triangle. The area of a right triangle may often be obtained more readily by the use of a formula involving only the particular parts of the triangle given. Denoting the area of a right triangle by K, let the pupil show that When the two legs are given, K= |- db. When an acute angle and the hypotenuse are given, K = \ c* sin A cos A (or = J c 2 sin B cos B). When the hypotenuse and a leg are given, c-a) (or = When an acute angle and a leg are given, lT = ltt 2 tan.B (or = \ b 2 tan A) , or K = a 2 cotA (or = \ cot B) . By geometry, what is the method or formula for computing the area of an isosceles triangle? of a regular polygon? The formulas given above for computing the area of a right triangle are sometimes useful in com- puting the area of an isosceles triangle, or of a regular polygon. RIGHT TRIANGLES 67 EXERCISE 21 Using five-place tables, compute the area of the right triangle in which : 1. A = 28 18', 6 = 216. 5. 5 = 63 18', c = 124.72. 2. .5 = 72, a = 196. 6. a = 192.7, b = 212.97. 3. .4 = 21 16' 30", c = 31.967. 7. a = 0.73216, c=.9125. 4. c = 46.72, 6 = 32.54. 8. c = 927.8 ft, b = 759.8 ft 9. Given a = 2.5 and K= 4.27, find 6, c, and A 10. Given K= 7.256 and ^L = 26 18', find a, b, and c. 11. Given K = 55.686 and c = 15.67, find a, 6, and A. Compute the area of the isosceles triangle in which : 12. Base = 12.67, leg = 9.267. 13. Base = .67892, altitude = .26217. 14. Base angle = 68 18', leg = .2892. 15. Vertex angle = 105 17', altitude = 13.67. 16. Vertex angle = 113 18', leg 25.6. 17. Given area = 16.72 and base = 6.37, find altitude, leg, and base angle. 18. Given area = .9273 and base angle = 27 18', find leg, base, and altitude. 19. Given area = 22.76 and vertex angle = 117 55', find leg, base, and altitude. 20. Find the area of the regular pentagon whose perimeter is 3.35. 21. Find the area of the regular dodecagon whose apothem is 1.7267. 22. Find the area of a regular heptagon inscribed in a circle whose radius is 0.7516. 23. Given a regular octagon whose apothem is 2.27 ; find the differ- ence between its area and that of the inscribed circle. 24. Given n = 9 and K = 30, find r, c, and R. 25. Given n 11 and K = 35, find the perimeter. 26. Given n = 5 and K = 37, find p and R. 27. If n denotes the number of sides, R the radius, and C the cen- tral angle of any regular polygon, prove that K=nR 2 sin ^ C cos ^ (7. 68 TRIGONOMETRY Using four-place tables, find the area of each of the following right triangles, given: 28. A = 34.6, a = 67.8. 32. b = 8.42, c = 11.26. 29. B = 84, a = 100. 33. B = 39.24, c = 23.68. 30. A = 18.62, b = 72.36. 34. c = 5000, a = 3000. 31. a = .16376, b = .19762. 35. A = 47, a = .0087. Solve the following right triangles, given: 36. 6 = 6.37, K= 26.38. 37. K =1200, .4 = 63.18. 38. K = . 4962, c = . 1635. Find the area of each of the following isosceles triangles, given : 39. Base = .7262, leg = .5263. 40. Altitude = 12.36, leg = 17.27. 41. Altitude = 86.27, base = 111.63. 42. Base angle = 42.67, leg = 17.43. 43. Vertex angle = 100.24, altitude = 8.217. 44. Vertex angle = 78.32, leg = .6526. In an isosceles triangle : 45. Given area = 192.67 and base = 43.64, find altitude, leg, and base angle. 46. Given area = 0.7362 and base angle = 37.43, find leg, base, and altitude. 47. Given area= 1367.8 and vertex angle = 113.28, find base, leg, and altitude. 48. Given area = .1025, and leg = .4916, find the base, altitude, and angle at the base. 49. Find the area of a regular decagon whose perimeter is 27.63. 50. Find the area of a regular pentagon whose apothem is .4782. 51. Find the area of a regular heptagon inscribed in a circle whose radius is 116.2. 52. Given the side of a regular octagon as 5.33, find the difference between the area of the octagon and that of the circumscribed circle. RIGHT TRIANGLES 69 In a regular polygon : 53. Given n = 1 and K = 14, find c, r, and R. 54. Given n = 11 and K = 1000, find r, c, and R. 55. Given ?i = 9 and K = 47, find 7*, c, and 7?. 56. Given n = 14 and K= 800, find the perimeter. Without the use of the tables, find the area of each of the following right triangles, given: 57. a = 100 and A = 60. 61. a = 80 and c = 160. 58. b = 600 and c = 1200. 62. b = 40 and c = 40 V2. 59. a = 26.3 and 6 = 21.2. 63. c = 4000 and ^4 = 30. 60. B = 60 and a = 90. 64. A = 45, 6 = 120. Also of each of the following isosceles triangles, given : 65. Vertex Z = 120, leg = 100. 67. Leg = 40, altitude = 20. 66. Base Z = 30, base = 200. 68. Vertex Z = 90, leg = 400. EXERCISE 22. APPLICATIONS Solve, using either set of tables : 1. The angle of elevation (see Art. 88) of the top of a cliff, measured from a point 225 ft. from the base, is 60. How high is the cliff ? 2. At a point 170 ft. from a tower, and on a level with its base, the angle of elevation of the top of the tower is found to be 70 18' [70.3]. What is the height of the tower ? 3. The angle of elevation of the sun is 65 30' [65.5] and the length of a tree's shadow, on a level plane, is 52 ft. Find the height of the tree. 4. If the Eiffel Tower is 984 ft. high, what will be the angle of elevation of its top, when viewed at a distance of a mile ? 5. The length of a kite string is 700 ft., and the angle of eleva- tion of the kite is 44 36' [44.6]. Find the height of the kite suppos- ing the kite string to be straight. 6. One of the equal sides of an isosceles triangle is 62.8 ft., and one of the equal angles is 52 18' 36" [52.31]. Find the base, altitude, and area of the triangle. 7. What is the elevation of the sun, if a tree 82.6 ft. high casts a shadow 105.8 ft. long on a horizontal plane? 70 TRIGONOMETRY 8. A ladder, 25 ft. long, leans against a house and reaches to a point 21.6 ft. from the ground. Find the angle between the ladder and the house, and the distance the foot of the ladder is from the house. Why are we able to solve an example like this by trigonometry when- we are not able to do so by geometry ? 9. The Washington Monument is 555 ft. high. How far apart are two observers 555 who from points due west of the monument observe its angles of elevation to be 25 and 48 17' [48.28] respectively? 10. If the Grand Canon of the Colorado is 5000 ft. deep, what will be the angle of depression of the river flowing through it when viewed from the brink of the canon at a horizontal distance of 3 mi. ? 11. If a hillside has a slope of 7, a dam 10 ft. high will force the water how far back up the hillside? 12. A tower 125 ft. high stands on the bank of a river. The angle subtended by the tower at the edge of the opposite bank is 23 31' [23.52]. Find the width of the river. 13. What is the height of a hill if its angle of elevation taken at the foot of the hill is 40 18' [40.3] and if this angle taken 150 yd. from the foot of the hill and on a level with the foot is 28 42' [28.7] ? 14. From the summit of a hill, there are observed two consecutive milestones on a straight horizontal road running from the base of the hill. The angles of depression (see Art. 88) are found to be 12 and 7 respectively. Find the height of the hill. 15. A valley is crossed by a horizontal bridge, whose length is I. The sides of the valley make angles ra and n with the plane of the horizon. Show that the height of the bridge above the bottom of the valley is cotm + cotn 16. Upon a hill overlooking the sea stands a tower 70 ft. high. From a ship the angle of elevation of the base and top of the tower are respectively 15 4' [15.07] and 1540' [15.67]. What is the height of the hill and the horizontal distance of the ship from the tower ? 17. Given : Z. AKF= Z ARK= Z RTF= 90. Z KAR = 60 and AR = 12. Without the use of the tables find the length of all the other lines in the figure. A 12 R RIGHT TRIANGLES 71 18. A boy standing m feet behind and opposite the middle of a football goal, sees that the angle of elevation of the nearer crossbar is A, and the angle of elevation of the crossbar at the other end of the field is C. Prove that the length of the field is m (tan A cot C 1). 19. A railroad embankment is 7 ft. high. If the top of the embank- ment is 8 ft. wide and the sides slope at an angle of 43, what will be the width of the base ? 20. If the Metropolitan Life Insurance building of New York City is 700 ft. high, how far from the building is an observer when the angle of elevation of the top of the building is 7 36' [7.6] ? 21. A man standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is 60 ; when he retires 50 m. from the edge of the river, the angle of elevation is 30. Without the use of the tables find the height of the tree and the width of the river. s 22. Given: A T P=6m.; Z7T=Z ^=60; Z SRN = 45; and RNTP a square. Without the use of the tables find the lengths of KR, PR, RS, ST, SF, and TF. 23. A tower and a monument stand on the same horizontal plane. The height of the tower is 35.6 m. and the angles of depression of the top and base of the monument, as observed from the top of the tower, are respectively 5 16' 48" [5.28] and 8 18' 30" [8.3]. How high is the monument ? 24. A flagstaff stands on the roof of a building. From a point A on the ground the angles of elevation of the foot and the top of the flagstaff are 37 and 46, respectively. From a point B, 250 ft. farther off and in line with A and the base of the building immediately below the flagstaff, the angle of elevation of the top of the flagstaff is 27 30' [27.5]. Find the length of the flagstaff. 25. From the top of a lighthouse, 150 ft. above the sea level, the angle of depression of a buoy situated between the lighthouse and the shore was 62 14' [62.23] and that of a point on the shore in a straight line with the buoy was 12 10' [12.17]. Find the distance, in feet, of the buoy from the shore. 26. The base of a rectangle is 50.62 and its diagonal is 71.6. Find the altitude of the rectangle and the angle which the diagonal makes with the base. 72 TRIGONOMETRY 27. Given : 0.4 = 1, Express AB, OB, BC, OC in terms of trigonometric functions of x and y. 28. The Singer building of New York City is 612 ft. high. Make up some problem concerning this which can be solved by trigonometry. 29. The diagonals of a rhombus are 42.28 and 30.58. Find the sides and angles. 30. Make up (or collect) as many different examples as you can showing the practical uses of the solution of right triangles by trigo- nometry, each example being distinct from the rest either in principle or in the field of its application. 31. Who first, and at what date, taught the trigonometric solution of triangles in the same general way as is done at present ? CHAPTER IV GONIOMETRY TRIGONOMETRIC FUNCTIONS OF ANGLES IN GENERAL 51. Angles greater than 90. In solving oblique triangles, angles greater than 90 may occur. Hence it is important to learn what the trigonometric functions of an obtuse angle are. Similarly the radius of a rotating wheel, as in a dynamo, generates angles greater than 360 and by successive rota- tions generates angles unlimited in size. In astronomy, the heavenly bodies, by successive rotations about an axis, and by revolutions in an orbit, also generate angles unlimited in size. Hence a general method is needed of determining the trigonometric functions of angles unlimited in size. 52. The Four Quadrants. Definitions. Let AC (Fig. 30) be the horizontal diameter of a circle ABCD, and BD the diameter perpendicular to AC. Then AOB, BOC, COD, and DOA are termed the first, second, third, and fourth quadrants of the circle. On Fig. 31 the four parts into which a plane is divided by the lines XX' and TY 1 are also termed quadrants and are numbered in the same order as the quadrants of a circle. In treating of the properties of angles in general, it is convenient, wherever possible, to let the angles start at the same place, as OA (that is, to have the vertex and a side in common). Let the rotating radius start in the position OA and rotate toward the position OB (in the direction contrary to that in which the hands of a clock move, or counter-clockwise). 73 74 TRIGONOMETRY The AAOP 19 AOP 2 , AOP 8 , AOP 4 are called angles in the first, second, third, and fourth quadrants respectively. The initial line of an angle is the rotating radius, which generates the angle, in its first position, as AO. The terminal line of an angle is the rotating radius in its final position, as OP 2 for z AOP%. By continuing the rotation of OA, angles greater than 360 will be generated. If two angles differ by 360, or by any exact multiple of 360, they will have the same terminal line. Coterminal angles are angles which have the same termi- nal line, as 37, 397, and 757. In general an angle is said to be of or in that quadrant in which its terminal line lies. 53. Negative Angles. In algebra it is shown that negative quantity is quantity exactly opposite in some respect, as, for instance, in direction, from other quantity taken as positive. Hence if the rotating radius move from the position OA (Fig. 30) toward the position OD (that is, in the same direction with the hands of a clock, or clockwise), a nega- tive angle, as the acute Z AOP 4 , will be generated. If the radius continue to rotate in this direction, a whole series of negative angles will be formed similarly. 54. Rectangular Coordinates. In order to define the trigonometric functions of angles greater than 90, and of nega- tive angles, two straight lines, XX' and YY' (Fig. 31), inter- secting at the point and per- pendicular to each other, are taken and called axes. The signs of other lines used are de- FiG.si. termined by their position with GONIOMETRY 75 reference to these axes Lines drawn from YY' to the right (and || XX') are taken as + ; lines drawn from YY' to the left (and II XX') are taken as - . Lines drawn from XX' above (and II YY') are taken as + ; lines drawn from XX' below (and II YY') are taken as -. The origin is the point in which the axes intersect, as the point on Fig. 31. The ordinate of a point is the distance of the point above or below the axis XX'. The abscissa of a point is the distance of the point to the right or left of the YY' axis. Thus, the ordinate of P l is Mfi ; the abscissa of P l is OM^ Coordinates is the general term for abscissa and ordinate of a point.. The coordinates of a point may be written to- gether in parenthesis with abscissa first and a comma be- tween. Thus if OMt = a, and Mfi = 6, the coordinates of P l are (a, 6). The distance of a point is the line drawn from the origin to the point, thus on Fig. 31 the distance of P 1 is OPi- The distance of a point is independent of sign. 55. Definitions of Trigonometric Functions of Any Angle. Y Y Y Y ^\ f^XT \ / Tx - N S\ \J\f\ N, M 3 : /' \M< X M 2 X V X \ >\ 1 / 3 ^4 FIG. 32. FIG. 33. FIG. 34. FIG. 35. If we regard an angle as formed by an initial line and a line drawn from the origin to a point whose abscissa and ordinate are considered, then sme of an angle = ratio of ordinate to distance; cosine of an angle = ratio of abscissa to distance; 76 TRIGONOMETRY tangent of an angle = ratio of ordinate to abscissa cotangent of an angle = ratio of abscissa to ordinate; secant of an angle = ratio of distance to abscissa; cosecant of an angle ratio of distance to ordinate. Thus in Figs. 32, 33, 34, 35, sin z XOP, issa; ^^ sin Z XOP* = , sin Z XOP, = , sin Z ZOP 4 = Let the pupil point out in like manner the other trigo- nometric functions of the angles XOP^ XOP 2 , XOP 3 , XOP*. 56. Trigonometric Functions represented by Lines. If a circle (Fig. 36) be drawn with as a center and a radius OA, equal to 1, and with Mf^ M 2 P 2 , M 3 P 3 , M 4 P 4 , perpendicular to XX', ] A X A x Similarly, sin Z AOP 2 = M 2 P 2 ; sin Z AOP 8 = M 3 P 3 ; and sin ^ J.OP 4 = M 4 P 4 . , Or, in the circle as described, the' sine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the horizontal diameter. GONIOMETRY 77 Similarly if (in Fig. 37) Nf l9 N 2 P 2 , N 3 Pv N 4 P 4 are perpen- dicular to YY', i cos Z AOP, = & = Ml-NiPi ; cos cos Or, in the circle as described, the cosine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the vertical diameter. Similarly (in Fig. 38), if TT' is tangent to the circle at A, tan Z.AOP* = AT 2 ; tan Z AOP 3 = AT Z ; tan Z AOP 4 = 'AT*. Or in the circle as described, the tangent of an angle is repre- sented by a line drawn touching the initial end of the arc inter- cepted by the angle, and terminated by the radius to the other end of the arc, produced. R t R, FIG. 38. FIG. 39. Similarly (in Fig. 39), if R^ is tangent to the circle at the point B, cot Z^OP! = tan z.BOR l = ^ = ^ = BR l \ OB cot ^AOP 2 = BR* ; cot z.AOP 3 = BR S ; cot Z.AOP = BR ; or in the circle as described the cotangent of an angle is repre- 78 TRIGONOMETRY sented by a line which is the tangent of the complement of the given angle- On Fig. 38 the secants of the four angles used are readily shown to be represented by 07\, OT. 2 , OT 3 , OT 4 ; or, in general, the secant of an angle is represented by a line drawn from the center through the terminal end of the arc intercepted by the angle, and terminated by the tangent. Similarly on Fig. 39 the cosecants of the four angles used are repre- sented by ORu OR 2 ) OR 3) OR 4 ; or, in general, the cosecant of an angle is represented by a line which is the secant of the complement of the angle. It will be convenient to draw a figure for an angle in each quadrant showing the lines which represent the functions of that angle. R B The lines which represent the various trigonometric func- tions of an angle are not the same as the trigonometric functions which they represent, but they have many of the game properties as the functions or ratios. It is often GONIOMETRY 79 easier to perceive these properties by the use of the lines, than by the use of the ratios which the lines represent. In deriving the properties of the trigonometric functions of angles greater than 90 we shall derive them from the lines representing the functions ; but in such cases we give some specimen proofs showing how these properties may be derived from the ratio definitions (of Art. 55), and in other cases leave it as an exercise for the pupil to derive the proofs from the ratios if the teacher considers it desirable. 57. Signs of the Trigonometric Functions in the Different Quadrants. Of the lines representing the sines of angles in the different quadrants, viz. M^P^ M<>P^ M 3 P S , Mf (Fig. 36), the first two are above the horizontal axis, and are therefore plus in sign; the last two are below, and therefore minus. Hence the signs of the sines of angles in the four quadrants are respectively + , + , , . The students may obtain the same results from Figs. 32-35 by using the general definitions of trigonometric functions given in Art. 55. Similarly in Fig. 37 the cosine lines N V P^ -ZV" 2 P 2 ? N 4 P 4 are +, -, -, +, respectively; and in Fig. 38 the tangent lines AT,, AT,, AT 3 , AT are + , -, + , -, respectively. Since the sine of a quantity and of its reciprocal must be the same, the sign of the cotangent in the various quadrants must be the same as that of the tangent ; that of the secant, the same as the cosine ; that of the cosecant, the same as the sine. Or, proceeding geometrically, on Fig. 39, the cotangent lines BR 2 , BE,, BE A are +, -, +, -. The secant is considered as plus when it is drawn in the same direction from the center as the terminal radius (thus OT 2 , Fig. 38, is opposite in direction from OP 2 and is therefore negative). Hence the secant lines OT ly OT 2) OT 3 OT 4 have the signs +, , , +, respec- 80 TRIGONOMETRY lively. Similarly the cosecant lines (Fig. 39) OR l} OR 2 , OR 3 , OR+ have the signs -f> +> j - The results thus obtained may be arranged in a table as follows : I II Ill IV sine and cosecant + + - - cosine and secant + - - + tangent and cotangent + - -+- - EXERCISE 23 In which quadrant is each of the following angles ? 1. 123. 1 6. 415. 1 11. 1111. 2. 155. % 7. - 18. U 12. - 222. 3. 215. 3 8. -125.? 13. -1826. 4. 285. If 9. 612. ) 14. 2625. 5. 338. /j 10. -500. 15. -1500. 16. Find the signs of the functions of the angles in Exs. 1, 3, and 5. Give two positive and two negative angles each of which is co- terminal with : 17. 25. 18. -30. 19. 100. 20. -100. Find the smallest possible angle coterminal with : 21. 425. 22. 780. 23. -300. 24. 875. 25. -1760. 26. 1493. In which quadrant does an angle lie : 27. If its sin is positive and cos negative ? 28. If its tan is positive and sin negative ? 29. If its cot is negative and cos negative ? 30. If its esc is negative and cot positive ? 31. If its cos is positive and tan negative ? 32. If its sec is negative and tan negative ? 33. A railroad embankment is 9 ft. high and 43 ft. wide at the base. If each of its sides makes an angle of 27 15' [27.25] with the horizon- tal, how wide is the top of the embankment ? GONIOMETRY 81 FIG. 44. 34. If a railroad embankment is 7 ft. high and 28 ft. 9 in. wide at the top, and one side has a slope of 23 30' [23.5] and the other a slope of 32 45' [32.75], how wide is the base ? 35. Make up a similar example for yourself. 58. Functions of 0, 90, 180, 270, 360. In Arts. 34 and 35 it is shown that sin = and sin 90 = 1. Similar results are readily perceived for other quadrants by the use of a figure showing the sines as lines in the different quadrants. Thus in Fig. 44 in the first quadrant the sine increases from to 1 ; in the second quadrant it decreases from 1 to ; in the third it decreases from to - 1 ; in the fourth quadrant it increases from -1 to 0. Hence the sines of 0, 90, 180, 270, 360, in order, are 0, 1, 0, - 1, 0. Similarly in the first quadrant (Fig. 45) the cosine decreases from 1 to ; in the second quadrant it decreases from to 1 ; in the third quadrant it increases from -- 1 to ; in the fourth quadrant it increases from to 1. Hence the cosines of 0, 90, 180, 270, 360, in order, are 1,0, -1,0,1. Similarly from Fig. 38, or from the formula tan#= sm x . it is clear cos a; that the tangent in the different quadrants changes from to ; from oo to ; from to oo ; from oo to 0. Hence the tangents of 0, 90, 180, 270, 360, in Order, are 0, oc, 0, oo, 0. The changes in the value of the cotangent, the secant, and the cosecant, and the values of these functions for the above-mentioned angles may be obtained from geometrical figures in like manner, but these values are obtained more readily from the reciprocal formulas cot = ; sec = ; csc = ^- tan cos sm Thus, sec 180 = cos 180 - 1 82 TRIGONOMETRY Obtaining the values of the required functions thus and arranging all the results obtained in a table, we have 90 180 270 360 sin 1 -1 cos 1 -1 1 tan CO 00 cot 00 00 00 sec 1 00 -1 00 1 CSC 00 1 00 -1 OD In the above table co is to be taken as + or -- according to the side from which it is approached (see Art. 57). EXERCISE 24 Find the numerical value of : 1. 5 sin 90 -f 7 cos 180 + 8 sin 30. 2. m sin + p cos 90 -f c cot 360. 3. b cos 90 - c tan 180 + b cot 270. 4. ( a 2 _ C 2) cos 180 o + 4 ac sin 90 o f 5. 2 tan sin 90 - 4 sec sin 270 -f 5 esc 90 cos cot 270. 6. a cos 180 sec 360- b tan 180 sin 270- a sin 90 sec + b sin 90 cos 270. 7. m sin 270 esc 90 + n cos 180 esc 270 cot 270 - m sec 180. 8. 6 m esc 90 cos 2 - 17 n sec 2 cot 2 270 + 3 m sin 270 sec 360. 9. Show that 4 cos 2 45 sec + 6 tan 2 30 sin 270 + 12 cot 2 45 cos 180 -4 tan 2 45 esc 270 = -8. 59. Trigonometric Functions of Angles greater than 360. It is evident that the trigonometric functions of angles from 360 to 720 are the same in order as those from to 360. Similarly for every succeeding 360, the functions repeat themselves. Hence to find the functions of an angle greater than 360, Divide the- angle by 360 and find the required trigono- metric function of the remainder. GONTOMETRY 83 Ex. Sin 766 = sin (2 x 360 + 46) = sin 46. 60. Formulas for the Acute Angle extended to any Angle. The equations and formulas proved in Arts. 27-29 concern- ing the function of an acute angle are true for the functions of any angle. Thus, on each of the Figs. 40-43, MP 2 + OM 2 = OP\ That is, sin 2 x + cos 2 x = 1. Also in each quadrant the A OMP, OAT, OBR are simi- lar. .'. AT: OA = HP : OM, or tanx: 1 = sinx: cosx, sinx or tan x = - . cosx Let the pupil prove in like manner, 1 1 sin x = - , cos x = esc x sec x Or these results may be proved directly from the ratio definitions of the trigonometric functions of any angle. For if angle XOP of Figs. 32-35 be denoted by x, in any quadrant abs. P + ord. P = dist. P , /abs. P V , /ord. P V = -i ' P) (dist. PJ Hence, sin 2 x + cos 2 x = 1. Let the pupil prove in a similar manner that tan 2 x -f- 1 = sec 2 x, and cot 2 x -f- 1 = esc 2 x. ord. P Also ta , . abs. P abs. P cos x cos x dist. P Also ord'-Pydist. P =1 abs. P dist. P =1 ord. P abs. P =1 . ' dist. P ord. P ~ ' dist. P abs. P ' abs. P ord. P~ or sin x x csc x = 1, cos x x sec x = 1, tan x x cotx= 1. 84 TRIGONOMETRY 61. One function of an angle being given, the other functions may be found in a manner similar to that used in Art. 30. Owing to the fact that for angles less than 360, two angles correspond to any given function, two sets of answers are found in each example. Ex. 1 . Given cos x = -|, find the other functions of x. By the table of signs (Art. 57) a negative cosine occurs in both the second and third quadrants. 2d quadrant. sin x = Vl (f ) 2 = Vl |-f = V^V = t> ci ti nf*. = f , etc. COS X 3d quadrant. COS X sin x = Vl (|) 2 = tan x = - = I = }, etc. COS X 4 <> Ex. 2. Given tan x = 2, find the remaining functions of x. The positive tangent occurs (see Art. 57) in both the first and third quadrants. 1st quadrant, sec 2 x = 1 -j- tan 2 x = 1 -f- 4 = 5, sec x = V5, cos x = = = - V5, etc. sec x V5 ^ 3d quadrant, sec 2 x = 1 + 4, sec x = V5, cos x = = V5, etc. -V5 5 In case solutions are sought by the geometrical method, the follow- ing figures may be used in Exs. 1 and 2 respectively. p -4 PI FIG. 46. F FIG. 47. 85 EXERCISE 25 1. Find the numerical value of sin 390 ; also of cos 390, tan 390, and sec 390. 2. Find the numerical value of cos 780 ; also of tan 780, sin 780, and cot 780. 3. Find the values of sin, cos, tan, and cot of the following angles : 4. I860 . 6. -675. 8. -1740. 5. -330. 7. 750. 9. 2205. 10. Given cos x = -| , find the other functions of x. 11. Given tan x = -y, find the other functions of x. 12. Given sin x = -&, find the other functions of x. 13. Given cot x = 2 and sin x negative, find the other functions of x. 14. Given sec x = m and tan x negative, find the other functions of x. 15. Given tan x = 3, find the other functions of x when x is an angle in the fourth quadrant. 16. Given sec x = 6, find the other functions of x if tan x is posi- tive. 17. Verify geometrically the results obtained in Exs. 10-16. 18. Given cot y = f V5 and cos y negative, find sin y and esc y. 19. Given tan x = ^ V3 and cos x positive, find the other func- tions of x. % 20. If 6 is in the second quadrant and if cosec = - 1 /, find the value P cot 4- sec tan 6 -|- cos 21. Find the value of CQS ^ + cot ^, if is in the fourth quadrant esc H- sec arid tan = ^-. 62. Trigonometric Functions of 90 + o? in terms of func- tions of x. The trigonometric functions of 90 + xmay be reduced to functions of x by use of the following formulas : sin (90 + as) = cos x. cot (90 -f x) = - tan x. cos (90 + x) = - sin x. sec (90 + x) = - esc x. tan (90 + ac) = cot x. esc (90 + a?) = sec x. 86 TRIGONOMETRY For, let Z AOP (Fig 48 a) be any angle x in the first quadrant. Let POQ P be a right angle. Let OP = OQ = 1. Then /. A .*. sin (sides _L) = A MOP. (%p. cmd acute Z = ) FIG. 48 a. = 0#= -PM= -sins. tan (90 + X ) = sin ; 9 Q n o + *j = -^L = - cot x. cos90 + j -smx -smx Let the pupil supply the proofs for cot (90 + x), sec (90 + x)> and esc (90 4- x). The same results may readily be obtained for angles end- ing in the second, third, and fourth quadrants by use of the following diagrams. Q FIG. *48 6. FIG. 48 c. FIG. 48 d. Ex. 1. Find the value of sin 300. sin 300 = sin (90 + 210) = cos 210 = - sin 120 = - cos 30 = JV3. Ex. 2. Reduce tan 923 to a function of an angle less than 90. tan 923 = tan (720 + 203) = tan 203 (Art. 59) = - cot 113 = tan 23. Ex. 3. Simplify cos (630 + 4). cos (630 + A)'= cos (270 + A) = - sin (180 4- A) = - cos (90 + A) = sin A. GONIOMETRY 87 . EXERCISE 26 Find the numerical value of : 1. sin 210. 4. cot 150. 7. tan 210. 2. cos 300. 5. sec 1215. 8. sin 330. 3. tan 120. 6. sec 900. 9. cos 240. 10. cos 225 +3 sin 330 -tan 225. 11. cot 840-3 tan 420+2 sec 480. Express each of the following trigonometric ratios in terms of a ratio of some positive angle not greater than 45 : 12. sin 142. 18. cos 110. 24. sin (280 16'). 13. tan 163. 19. sin 567: ' 25. cot (2100 17 f ). 14. cos 310. 20. cot 1415. 26. esc 1325. 15. sec 185. 21. esc 1200. 27. cos 82. 16. cot 265. 22. cos 117. 28. tan 1060. 17. tan 315. 23. tan 428. 29. tan 840. 30. Prove sin 330 cos 390 = cos 570 sin 510. 31. Prove tan 45 sec 1080 cos 570 sin 510 - sin 330 tan 225 cos 390 D = 0. 32. Find the value of 6 sec 2 1080 tan 2 135 sin 1890 + 8 cot 45 cos 1140 + esc 630 tan 225 cos 720 sin 1830. Simplify the following expressions : 33. 5 sin (90 + a;) 6 cos (180 + x). 34. a sin (90 + a?) + b cos (270 + x) - c tan (180 + a?). 35. p sin (180 + x) cos (180 + x). 36. (a + b) sin (270 + x)-(a-b) cos (270 + a?). 63. Trigonometric Functions of a Negative Angle. The trigonometric functions of a negative angle may be converted into functions of a positive angle by use of the following formulas : sin ( x) = sin x. cot ( x) = cot x. cos ( x) = cos w. sec ( x) = sec x. tan ( - x) = - tan x. esc () = esc x. 88 TRIGONOMETRY For let /- A OP (Fig. 49) be a positive angle, x, and AOQ an equal negative angle. Let OP = OQ = 1. Then the right triangles 0J/P and OMQ are equal. Hence, sin ( x) = MQ = MP sin x cos ( x) = OM = cos x sin ( x) sin x tan ( x) cosx FIG. 49. cos ( x) = tan x. Let the 'pupil supply the proofs for cot ( x), sec ( x), and esc ( x). The same results are readily obtained for angles in the other quadrants by the use of appropriate diagrams. Ex. 1. Find the numerical value of cos ( 225). cos (- 225) = cos 225, = - sin 135 (Art. 62) = cos 45 = -J- V2, Ans. Ex. 2. Simplify cot (180 - A). cot (180 - A) = - tan (90 - A), = cot ( A) = cot A, Ans. 64. Reduction Tables and General Rules. Some of the reductions made by the methods of the preceding articles are usecl so frequently that it is convenient to collect the results obtained by them, and arrange them in tables for future reference. Thus sin (90 x) cos x. cos (90 - x) = sinx. tan (90 - x) = cot x. cot (90 - x) = tan x. sin (180 x) = sinx. cos (180 - x) = cos a:. tan (180 -x) = -tan a! cot (180- x) = -cot a; sec (180 -x) = -seccc esc (180 x) = esc x sec (90 - x) = cscx. esc (90 x) = sec x. Let the pupil form similar tables for the functions of 270 - x, 360 - x, 180 + x' 9 270 + x. GONIOMETRY 89 Or the following general rule may be used : Each function of 18Q a; or 360 x is equal in absolute value to the like-named function of x; but each function of 90 x or 270 x is equal in absolute value to the co-named function of x.* For example, sin (180 + x) and sin x by the above rule are equal in absolute value. But it must also be remembered that they are opposite in sign. For if, for instance, x is acute, 180 + x is an angle in the third quadrant and therefore sin (180 + x) is negative. But x mean- time 'would be an angle in the first quadrant, hence sin x would be positive. Hence, in general, sin (180 + x) = - sin x. Let the pupil show in like manner that, by the above rule, sin (360 x) = sin x ; also that sin (270 x) = cos x. In applying the above general rule to any particular example it will be found that the algebraic sign of the result is the same as the sign of the original function. Thus, sin 330 = sin (360 - 30) = - sin 30, the short way of deter- mining the sign of sin 30 being to note that sin 330 is negative since 330 is in the fourth quadrant and that sin 30 must have the same sign as sin 330. If geometrical proofs for the above reduction formulas are desired, such proofs may be obtained by following the methods of Art. 62. But in such proofs, when constructing an angle like 180 -|- x, or 270 -f x on the diagram, it is an advantage to construct the 180, or 270 first, beginning with the initial line, and then to annex the angle x to the 180, or 270, after it has been constructed. Thus, to prove that tan (270 + x)= cot x when x is an angle in the second quadrant (i.e. an obtuse angle) we first take (Fig. 50) the positive angle AOB' (270) and annex to it Z.B'OP (=x or AAOP). Then * At this point it is often advantageous to have the class study the solution of Case I of oblique-angled triangles (Arts. 74, 79). This shows the pupil an important application of the preceding principle and introduces variety into the course of study. 90 TRIGONOMETRY x)=ZAOT (as indicated by the long bent arrow), and tan (27Q + x) = AT. Also cot x (or cot AOP) = BR. But ZB'OT = ZAOR (construction) Subtracting 90 from each of these angles we have ZAOT = ZBOP. . AAOT = ABOP. (leg and acute Z =) .-. AT = .B.R, in absolute magnitude. (horn, sides of= A) .-. tan (270 + x) and cot a; are equal in absolute magnitude. But AT and BR are opposite in sign. .-. tan (270 + x) = cot x. Similarly, to prove sin 270 x = cos x when x is an angle in the second quadrant (Fig. 51) we take Z AOB' (270) and from it deduct Z B' OP' (=ZAOP or x). Hence, sin (270 - x) = MP 1 , while cos x = JVP. Since A OMP' = A O^P. MP' and ^P are equal in absolute magnitude. They are also opposite in sign. .. sin (270 x) = cos x. EXERCISE 27 Find the numerical value of : 7. sec (-240). a tan (-150). 9. sin (-135). 1. sin (-225). 4. cot (-210). 2. tan (-300). 5. tan (-600). 3. cos (- 120). 6. sin (- 900). Keduce the functions of the following negative angles to the functions of positive angles not greater than 45 : 10. -119. 13. -15. 16. -900. 11. -81. 14. -253. 17. -216 43'. 12. -195. 15. -1000. 18. -307.24. 19. Show that sin 420 cos 390 = 1 - cos (- 300) sin (- 330). 20. That 3 tan (- 60) cot (- 210) + 9 sin (- 240) cos (- 150) = f . By the general rule stated in Art. 64 reduce each of the following to a function of x : 21. cos (180 + x). 22. sin (270 + #). 23. cos (270 - x). 24. tan(180 25. sec (180 - aj). 26. esc (270 + a?). GONIOMETRY 91 Simplify the following expressions : 27. 5 sin (90 x) + 8 cos (180- x ). 28. a sin (270 - x) - b cos (270 x) + c tan (180 - x). 29. m cos (180 + A)+p cot (180 A) + q tan (270 + ^4). 30. sin (270 + a;) cos (270 - x) sin (180 - a;). 31. sin (aj - 90) + cot (x - 90) + tan (a; - 180). 65. General Solutions of Trigonometric Equations. If there be no limit to the size of an angle, an indefinite num- ber of angles will satisfy every trigonometric equation (see Art. 38). Ex. 1. Solve sin x = \. There are two angles less than 360 whose sine is |-, viz. : 30 and 150. If 360, or any multiple of 360, be added to, or subtracted from, each of these angles, the sine is unchanged. Hence, in the above example, x = 30 n (360), 150 n (360), where n = or any positive integer. Ex. 2. Solve tana:- V5- _ " 60 n(360), 120 n [240 n(360), 300 w(360). Ex. 3. Solve sin 2 x=cos 2 x. 1 COS 2 X = COS 2 X. cos x = i V2. = {45 w(360), 315 n(360), ~ |l35 (360), 225 (360). Or more briefly, x = n (180) 45. Ans. The pupil should observe that the values of a; in a trigonometric equation differ in an important respect from the values of x in an algebraic equation. Thus, in an algebraic equation the values of x are the roots of the equation and the number of values which x has equals the degree of the given equation. Whereas, for instance in Ex. 3 above, the roots are the values of cos a?, while the values of x are inferred from the values of cos x and may be unlimited in number no matter what the degree of the original trigonometric equation. 92 TRIGONOMETRY EXERCISE 28 Solve, the following trigonometrical equations, for values of x or 0. 1. sin x = J. 10. 2 V3 cot - f esc 2 0=1. 2. cos 2 x = f. 11. tan -f sec 2 = 3. 3. tan 2 a = l. 12. cos 2 + cot 2 = 3 sin 2 0. 4. tan x = i cot x. 13. 1 cot cos -f- sin = 1. 5. sin x + esc x = f . 14. sec 2 esc 2 + 2 esc 2 = 8. 6. tan 2 x sec x = 1. 15. 2 V3 tan = 3 sec 2 6. 7. 2cos 2 a;-3sina;=0. 16. 4 sec 2 - 7 tan 2 = 3. 8. tan x + cot a = 2. 17. cot + 2 tan = f sec 0. 9. cot x + esc 2 a; = 3. 18. sin + V3 cos = 2.. 19. A ship starting from a certain point sailed at the average rate of 9.25 mi. per hour on a course 22 15' [22.25] north of east. At the end of 7 hr. 45 min., how far east of her starting point would she be ? How far north ? 20. If a railroad embankment is 11 ft. high, 76 ft. wide at the base, and 49 ft. wide at the top, and its two sides have the same slope, find the angle at which each side slopes. 21. In an oblique triangle ABC, A = 127 36' [127.6], AB = 472 ft., AC =374: ft. By dividing the triangle into right triangles and solving, find BC. 22. Pisa spring of water, Q is a house, and R is a barn. If QR = 217 ft., Z PQR = 63 40' [63.67], Z PRQ = 58 15' [58.25], find the distance of the spring from the house and also from the barn, by solving right triangles only. CHAPTER V GONIOMETRY (Continued) 66. Formulas for sin (ae + 1/) and cos(a? + ?/). In Fig. 52 let AOQ be an angle x, and QOP an angle y, the sum of x and ?/ being less than a right angle. Let OP = 1. Draw PM J_ OJ., P$_L OQ, QR^PM. Then ZPPQ=Zx (sides -L), PQ = sin ?/, 0$= cos ?/. sin (x 4- ?/) = PJf = $JV"4- PP. In rt. A OQN, QN= sin x 0$ (Art. 41) = sin x cos y. In rt. A RPQ, PP = cos xPQ = cos x sin T/. H^nce, sin (a? 4- */) ?= sin a? cos y 4- cos as sin ?/. Also on Fig. 52, cos(x + y)= OM= ON-RQ. In rt. A OQN, ON= cos x OQ = cos x cos y. In rt. APPQ, PQ=sinxPQ=sinxsin2/. Hence, cos (ac + y) = cos x cos y sin a? sin y. If x and ?/ be acute angles whose sum is an obtuse angle, the above proofs will hold good without any change except that it E^ \ is necessary to notice that in the statement cos (x + y) = OM= ON- RQ, OM is a neg- ative line and is obtained by subtracting M o N the positive line RQ from the smaller FIG. 53. positive line ON. See Fig. 53. If either x or y is obtuse, the above formulas may be proved as follows : 94 TRIGONOMETRY Taking x and y as still acute, sin (90 + x + y) = cos (x + y) (Art. 62) = cos x cos y sin x sin y. But cos B= sin (90 + x),- sin z = cos (90 + z). (Art. 62) /. sin (90 + x 4- y) = sin (90 + x) cos y + cos (90 + x) sin ?/. Replacing 90 4- x by #', sin (#' + y) = sin x' cos y + cos x sin ?/, where #' is an obtuse angle. In like manner the formula can be extended to the case where y is an obtuse angle. The formula for cos (x + y) may also be extended in like manner. By successive additions of 90 to x and y, these angles may thus be made any angles however large. In like manner the formulas may be shown to be true when x and y are diminished by any integral multiple of 90. Hence, the above formulas are true when x and y are any angles. Ex. Taking the functions of 30, 45, 60 as known, find sin 75. sin 75 = sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30 , Am. 67. Formulas for sin (w y) and cos (oc y}. In Fig. 54 let AOQ be a positive acute angle x, and POQ a smaller angle y, subtracted from x. Let OP=l- draw PM-LOA, PQA. OQ, QN OA, PE QN. Then Z.RQP=^x. (sides -L) Also PQ = s'my, OQ = cosy. sin (x-y) = PM= QN- RQ. In rt. A OQN, QN= sin x OQ = sin x cos y. GONIOMETRY 95 In rt. A RQP, RQ = cos x PQ = cos x sin y. Hence, sin (x y) = sin w cos y cos oc sin y. Also on Fig. 54, cos (x-y) = OM= ON+ RP. In rt. A OQN, ON= cos x OQ = cos x cos y. In rt. A RQP, RP = sin x PQ = sin x sin y. Hence, cos (a? y) = cos a? cos y + sin ae sin t/. By the same method as that used in Art. 66 these formulas can be proved true when x and y are any angles. Ex. Obtain the numerical value of cos 15. cos 15 = cos (45 - 30), = cos 45 cos 30 + sin 45 sin 30 Ans. 68. Formulas for tan (x + y) and tan (x y). By Art. 66, sin (x + y) sin x cos y + cos x sin y = s-- cos (# 4- 2/) cos x cos y sin a: sin y Divide both numerator and denominator of the last fraction by cos x cos y. sin x cds y cos x sin ^/ m, v COS X COS 7/ COS X COS ty Then, tan (x + y) = cos>a^cos y sin ^ sin cos cos i cos x cos tan tan 1 - tan x tan y Similarly, let the pupil show that tan x - tan y tan (oc-y} = , 1 + tan x tan y . COt '05 COt ?/ T 1 and cot (x ; ru L 1 cos 2 A Ex. 1. Prove that - - = tan A. 2 sin A cos A cos A 2 sin 2 A _ sin A 2 sin A cos J. cos ^4 sin A _ sin ^t cos A cos J. Or if the teacher prefers, the proof may be put in the following form : 1 cos 2 A _~L (1 2 sin 2 A) _ 2 sin 2 A _ sin A _ . * sin 2 A 2 sin A cos ^4 2 sin A cos ^1 cos A Ex. 2. Prove sin (A + 5) sin (A - 5) = sin 2 A - sin 2 5. (sin A cos 5 + cos A sin 5) (sin ^4 cos B cos J. sinB) = sin 2 A sin 2 B. sin 2 .4 cos 2 B cos 2 ^4 sin 2 B = sin 2 ^1 sin 2 B. sin 2 ^1 (1 sin 2 B) (1 sin 2 A) sin 2 5 = sin 2 A sin 2 B. sm*A sin 2 ^4 sin 2 B sin 2 5 -f sin 2 A sin 2 5 = sin 2 A sin 2 B. sin 2 .4 - sin 2 B = sin 2 .4 - sin 2 B. 73. Functions of the Angles of a Triangle. If the sum of three angles is 180, the functions of the angles have important relations. Ex. If A + B + C = 1 80, prove that sin A + sin B + sin C = 4 cos 4 A cos 4 B cos i C. GONIOMETRY 103 Hence sin %(A + B) = sin (90 - i 0) = cos (7. sin A 4- sin 5 + sin O= sin .4 + sin 5 + sin [180 (A + B}~\ = sin A 4- sin B + sin (.4 4- B) = 2 sin i (A 4- 5) cos 1 (A - B) 4- 2 sin |- (.4 4- 5) cos | (.44- 5) (Arts. 69, 71) = 4 cos (7 cos i ^4 cos | - EXERCISE 33 Prove the following identities : j. cos 4- sin __ sin 2 + 1 cos sin 9 cos 2 B 2. 2 cos (45 + J- .4) cos (45 - -*- .4) = cos A. 3. cos (^L 4- B) cos (.4 B)= cos 2 5 sin 2 A 4. tan (45 + x) - tan (45 a;) = 2 tan 2 a;. 5. ( Vl + sin x Vl sin x) 2 = 4 sin 2 ^ x. 6 cos (a; + y) + cos (x y) _ cos (a; y) cos (a; cos a; cos y sin a; sin ?/ ? tan (45 + 4- A) + tan (45 - ^t) = CC ^ tan (45 4- -J- ^1) -7 tan (45 - a sin A cos 9. cos ^- sin ^ cos ^4 4- sin A 10. tan 1 + cos 4- cos 2 cot 1 _ 1 sin 2 cot + 1 ~ cos 2 11. x 12. ^f = cos x. 1 4- tan 2 1 a? ' If ^1 + B 4- C - 180, prove that 13. cos A 4- cos B + cos (7=1 4-4sini.4sin-i-.Bsm| C. 14. tan A 4- tan J3 4- tan C= tan .4 tan 5 tan C. 15. cos (^4 + B + 0) = cos 2 (7. 104 TRIGONOMETRY EXERCISE 34. REVIEW 1. Given cos = f and is in the third quadrant, find esc 6, cot 6, sin 0, tan (180 0), sin ( 0). 2. Given tan ^ x = 2 (and a? acute), find sin x. 3. Given sin 2 x = ^ VB, find cot ^ x. 4. Given cos |- x J, find sin 2 x and tan 2 x. 5. Given cot 30= V3, find cos 15, esc 15, and tan 15. 6. Given sin A = f and A acute, cos JB = i and 5 acute, find (a) Bin(<4-.B); (6) cos (A+B) ; (c) cos (-4-5); (ef) sin 25; (e) cos 2 B; (/) tan 2 5; (y) cot 2 .4; (fc) tan (J. - ) ; (i) cot (-4 + .B) ; 0') cos i ^- 7. Given cot 6 = 2 and is the second quadrant, find (a) sec 0; (6) tan (180 - 0) ; (c) cot (180 + 0) ; (d) cos (- 0). 8. Find sin, cos, tan, cot, of : (a) * - ; (6) (*- 0) ; () a - ; (d) ( " + x) ' where * = Prove the following : 1 cos 2 x sin x + sin 2 # 9. tan x = 12. - = tan x. sm 2x 1 + cos x + cos 2 # 10. tan^". 13. . sin A cos 11 2 sin ^1 sin 2 A _ 1 cos A 14 sin 21 + sin 5 _ tan ~ ' ' cos 21 + cos 5 " 15 cos 9 + cos 5 + cos sin 9 + sin 5 + sin . , o ^ tan a5 4- cot x + 1 2 + sin 2x 16. cos 2 a; tan 2 a; + sm 2 x cot of = 1. 19. - tan x 4- cot x 1 2 sin 2 a; 17 cos 75 + cos 15 ^ t sin 75 - sin 15 cos 2 a; -1 18 sin^ + sin5 ==coH( ^_ jB)> a> sin (s + y) = cot x + cot y< cos B cos J. sin (x y) cot ?/ cot x 22. COS A = cos x cos 24. cos 5 x + cos 3 # = 2 cos 4 x cos #. 25 sin 2 a; 1 2 tanx tan 2 x 27. GONIOMETRY 105 26. sin (45 -f x) -f sin (45 a?) = V2 cos x. l-cot 2 (?-oA 28. _ _~ - Z = - 1 _ C0 t 2 ( - + x } 1 + cot 2 ( ^ - V 4 / V 4 1 4- cos x -f cos 2 a _ sin a; + sin 2 # cos x sin a; 30. cos 12 a- + cos 6 x -+- cos 4 a; + cos 2 a = 4 cos 5 x cos 4 x cos 3 #. / , ^ . x\ 1 + sin x 31. tan[4o-f- =\ ^ 2y ^ 1 sm a; 32. (sin a; cos y cos # sin y) 2 -+- (cos cos y + sin a; sin y) 2 = 1. 33. cos 2 1 a; (tan 1 x I) 2 = 1 sin x. 34. Find the value of CSC<9 when cot (9 = - and (9 is in quad- sec 6 + sm 2 rant II. 35. Find the value of tan e + cos e w h en sin = - 1 and (9 is in the cot -f sec 5 3d quadrant. 36. Simplify cos 300 - cot ^~ + 60") + cot 150 - tan f - |Y 37. Simplify sin 660 + tan (^f - 60 0> ) + cot 330 + cos (- 30). V 2 / 38. Simplify : (a - b) sin - - (a + 6) tan 225 + (a 2 + b 2 ) cot ^ - a cos f 2 2 \ 39. If tan 2 = - 2 /, find tan and sin 0, being in the 3d quadrant. p sin (A + #) _ tan A + tan j? _ cot B + cot J. sin (A B) tan ^1 tan B cot B cot A 41. If J. is an angle in the second quadrant and sin A = f , find the value of sin 2 A + cos 2 A If .4 + B + (7= 180, prove : 42. sin A + sin B sin (7=4 sin 1 J. sin 1 5 cos |- C. 43. cot i-J. -f cot i J5 +cot i C = cot i J. cot \ B cot 1 C. 44. sin 2 J. + sin 2 J5 + sin 2 (7 = 4 sin A sin 5 sin (7. 45. cos 2 ^. -f cos 2 5 -h cos 2 (7 = (4 cos A cos B cos (7+1). 46. tan A cot .B = sec A esc 5 esc C. 106 TRIGONOMETRY In a right triangle, C being the right angle, prove 47. sin 2 -B =^^> 49. 2 2c 2 a+c 48. cos sns- so. cos 2 -.4 = 2 1 A & + c 2 2 ) c 2 2c Using sin x cos x = \ sin 2 a>, sin 2 x = 1 ~ cos2a; , cos 2 x = 1 + C ( s2a; , transform : 51. sin 2 ic cos 2 a; into ^(1 cos 4 a?). 52. sin 4 a? cos 2 a; into T ^(l cos 4 x) i sin 2 2 a; cos 2 x. 53. sin 4 a; cos 4 x into an expression in terms of the cosines of even multiples of x. 54. sin 8 a? into an expression of the same general kind as in Ex. 53. 55. What nation first used the formula for sin 1 A ? 56. What man discovered the formula for sin 2 A ? 57. Who first published the formulas for sin (A B) and cos (A B)j and at what date ? CHAPTER VI OBLIQUE TRIANGLES TRIGONOMETRIC PROPERTIES OF OBLIQUE TRIANGLES 74. Law of Sines in a triangle. In any triangle the sides are to each other as the sines of the angles opposite. In Fig. 55 the angles A and B are both acute. In Fig. 56 the angle A is acute, and angle ABC obtuse. Let (7D, denoted by p, be the altitude in each triangle. In Fig. 55, in the rt. A J. CT>, p = 6 sin J. ; (Art. 41) in the rt. A CBD, p = a sin B ; (Art. 41) . * . -6 sin A = a sin B: (Ax. 1) In Fig. 56, in the rt. AACD, p = I sin A ; in the rt. A BCD, p = a sin (180 - Z ABC) = a sin /.A BC. (Art. 64) Hence in A ABC in both figures, 1} sin A = a sin B, or a : 1} = sin A : sin B. In like manner, b : c = sin B : sin (7, and a : c = sin J. : sin (7. Or, collecting results, a sin A sn 107 sn 7" 108 TRIGONOMETRY 75. Law of Tangents in a triangle. In any triangle the sum of any two sides is to their difference as the tangent of half the sum of the angles opposite the given sides is to the tangent of half the difference of these angles. In a triangle ABC (Figs. 55 and 56), a : 1} = sin A : sin B. (Art. 74) By composition and division, a 4- b _ sin A 4- sin B a I} sin A sin B 2 sin \ ( A +.B) cos \ (A - B} (Art. 71) Or, In like manner, and a I} l^~c c + a tan^Qg It is also helpful to have a geometric proof of the Law of Tangents. This may be obtained as follows : In a given triangle ABC (CB >AC), produce A C to D, making CD= CB or a. On CB mark off CE = AC or b. Draw the straight line DB. Also EB = CB-CE-a-b. /-DCB, being an exterior angle of A ACE, = x + x = 2x. Also Z.DCB, being an exterior angle of A ACE, = A + 5 (of A Also, FIG. 57. Also A ADF and EFB are similar (two A equal). OBLIQUE TRIANGLES 109 .-. Z AFD = Z EFB. .-. AF1. DB. In &AFDandEFB, DF : FB = a + b : a b. In A AFD and AFB, tanz:tanZ J Z^LB = : AF AF By Ax. 1, a + b : a b = tan x : tan = tan %(A + -B) : tan 76. Law of Cosines in a triangle. In the triangle AB6, Fig. 55, by geometry, a 2 = 6 2 + c 2 - 2 c x AD. But in the rt. A ACD, AD = b cos A. If ^4. is an obtuse angle, Fig. 58, by geometry, a 2 = & 2 + c 2 + 2 c x .AD. But in the rt. A J.CD, J.D = & cos Z CAD = &cos (180 - A) = - b cos A. Hence in either case, 2 6c cos J. = 6 2 4- c 2 - a\ or 'A* -v 'B In like manner it may be proved that FIG - 58 ' COSjB= 77. Formulas derived from the Cosine Formula. The for- mula for cos A in Art. 76 has a numerator which is primarily a sum and difference, hence logarithms cannot be used in computing numerical values from it. In order to put this formula in such a shape that its value can be computed by the aid of logarithms, it is necessary to transform the numerator of the fraction into a product. This is done 110 TRIGONOMETRY by the use of the formula for the cosine, or of that for the sine of a half angle (Art. 70). Thus: 2 be 2 be 2 be _(b + c-\-a)(b+c a) 2 be Let 2s = a-f-6 + c; then, subtracting 2 a from each member, 2s 2a = b + c a. Hence, -8 cos^ A = Z "$ ~ 2 ffi ) , or cos In like manner, Also from Art. 70, 2 sin 2 1 ^1 = 1 - cos ^ = 1 - 2 6c 2 ftc = a_ (b - c) 2 = (a + b - c)(a-b + c) 2 be 2 be = (2 g - 2 c) (2 g - 2 6) = 4(8 - b} (s - c) 2 be 2 be Hence, sin A '- In like manner, Dividing the formula for sin ^ A by that for cos Similarly, c) -, Rs a} (s 1 OBLIQUE TRIANGLES 111 EXERCISE 35 1. Prove that the diameter of a circle circumscribed about a triangle is equal to any side of the triangle divided by the sine of the angle opposite that side. 2. By means of the property of sines, prove that the bisector of an angle of a triangle divides the opposite side into segments which are proportional to the sides forming the given angle. 3. In any triangle ABC, prove that a = b cos C + c cos B. State this property in words. Write the two similar formulas for b and c. What does the above formula become when C = 90 ? 4. Prove that the radius of an inscribed circle of a triangle is equal to sm % A S1P ^ B where c is one side of the triangle and A and B cos^-C are the angles adjacent to c, and C is the angle opposite c. 5. Prove sin A = Vs(s - a)(s - b)(s - c) if s = a + b + c . uc 6. Prove cos ^ = be 7. Find the form to which the formula gLJ = tan I ( A + B ) a-b tan $(AB) reduces, and describe the nature of the triangle, when (I) C = 90, (II) A-B = 90, and B=C. 8. What does a 2 = b 2 + c 2 2 be cos A become when (I) A = 90, (II) A = 0, (III) A = 180>? What does the triangle become in each of these cases ? 9. What does - = ^-^ become when A is a right angle ? When b sin B B is a right angle ? SOLUTION OF OBLIQUE TRIANGLES 78. Cases in the Solution of Oblique Triangles. Four cases occur in the solution of oblique triangles according as the parts given are I. One side and two angles. II. Two sides and the included angle. 'III. Three sides. IV. Two sides and an angle opposite one of them. 112 TRIGONOMETRY CASE I. ONE SIDE AND Two ANGLES GIVEN 79. To solve Case I use the law of sines (Art. 74), thus : Subtract the sum of the two given angles from 180 ; this will give the third angle. The unknown sides may then be found by the following proportion : unknown side : known side = sine of angle opposite the unknown side : sine of angle opposite the known side. In solving oblique triangles by the use of logarithms it is of special importance to make an outline or skeleton of the work before looking up any logarithms, and then to do all the work connected with the use of the tables together. Ex. 1. Given A = 67 21', B = 57 48', b = 367. Solve the oblique triangle ABC. SOLUTION C = 180 - (67 21' + 57 48') = 54 51 f . Then by the law of sines (Art. 74), (Check) a 367 sin 67 21' sin 57 48' sin 54 51' 367 sin 57 48' sin 67 21' sin 54 21' Before looking up any logarithms in the tables the pupil should outline the work as follows: 367 log 67 21' log sin 57 48' colog sin . . . . 367 log 54 51' log sin . . . . 57 48' colog sin . . . . clog . . . . 67 21' log sin 54 51 'colog sin . . . . a = log . . . . c = log .... a = log ... OBLIQU3 TRIANGLES 113 The pupil can then look up all the logarithms at once and fill in the above tabulated form. (Any logarithm occurring more than once on being taken from the tables should be entered uniformly wherever it belongs.) Proceeding thus, he should obtain 367 log 2.56467 67 21' log sin 9.96541 - 10 57 48' colog sin 0.07253 a = 400.227 log 2.60231 367 log 2.56467 54 51' log sin 9.91257 - 10 57 48' colog sin 0.07253 c = 354.625 log 2.54947 (Check) c log 2.54947 67 21' log sin 9.96541 -10 54 51' colog sin 0.08743 a log 2.60231 Ex. 2. Solve the triangle B= 83.11, and 6= 7641. ABC, given A = 18.29, 7641 FIG. 60. C = 180 - (18.29 + 83.11) = 78.6. Then by the law of sines (Art. 74), a sin 18.29 7641 sin 83.11 7641 log 3.8832 18.29 log sin 9.4967 - 10 83 11' colog sin 0.0032 a = 2416.11 log 3.3831 sin 78.6 7641 sin 83.11 7641 log 3.8832 78.6 log sin 9.9913 - 10 83.11 colog sin 0.0032 c = 7546 log 3.8777 (Check) sin 18.29 c sin 78.6 c log 3.8777 18.29 log sin 9.4967 - 10 78.6 colog sin 0.0087 a log 3.3831 114 TRIGONOMETRY The accuracy of the work in Exs. 1 and 2 might also have been checked by use of the formula a 2 = 6 2 + c 2 2 be cos A, or ,. -, A + s(s a) of cos \A = \~ be In general in solving oblique triangles the accuracy of the work in any one case can be checked by applying to the results obtained one of the rules or formulas of the other cases. EXERCISE 36 Find the remaining parts of the triangle, given : 1. a = 12.632, .4 = 65 35', 5 = 73 18'. 2. a = 300, B = 10 18', C= 35 22'. 3. b = 1000, B = 49 18', C = 72 50'. 4. c = 1640.22, (7= 18 25', B = 52 16'. 5. A= 66 18' 36", B = 43 43' 48", c = .87654. 6. C= 100 18' 42", B = 50 40' 16", c = 114.682. 7. C= 22 18' 24", B = 58 12' 24", a = 1.26984. 8. A= 68 15' 20", B = 43 18' 36", a = 1.8263. 9. = 57 23' 12", ^1 = 54 21' 18", c= .20814. 10. Given a = 5. 267, ^1 = 30, 5 = 45, solve without using the tables. 11. Given c = 1000, ^4 = 60, JB = 45, find a and b without using tables. 12. In a parallelogram given a diagonal d, and the angles m and n which this diagonal makes with the sides, find the sides. Find the sides when d = 14.632, and m = 38 18', and n = 12 32'. Using four-place tables, find the unknown parts, having given : 13. a = 14.26, A = 52.16, B = 71.11. 14. c = 200, C = 18.16, B = 80.52. 15. b = .7125, A =116.18, C = 38.25. 16. a = 63.28, B =-- 63.28, C= 36.82. 17. 6 = 4000, B = 17.28, (7 = 82.26. 18. c = 8, -4 = 79.26, 5 - 99.99. 19. a = 19.28, B = 42.8, C = 19.53. OBLIQUE TRIANGLES 115 20. c = .2265, B = 71.28, A = 52.85. 21. b = 176.8, C = 9.82, B = 68.22. 22. a == 4812, B = 75.6, O = 48.71. 23. 5 = 14.267, C = 110.6, A = 41.63. 24. c = 712.8, B = 44.18, A = 79.22. Without the use of tables, solve, having given : 25. a = 100, = 60, ^L = 60. 27. a = 500, A = 75, 5 = 60. 26. ^ = 120, B = 30, c = 200. 28. 6 = 200, A = 105, c = 45. Solve Exs. 29-31 by either set of tables. 29. A ship S can be seen from two points M and N on the shore. The distance MN is 700 ft., and the angles SMN and SNM are 57 42' [57.7] and 75 18' [75.3] respectively. Find the distance of the ship from M. 30. A balloon is directly over a straight road, and between two points on the road from which it is observed. The distance between the two points is 2652 yd., and the angles of elevation of the balloon as seen from the two points are 58 50' [58.83] and 47 24' [47.4] respectively. Find the distance of the balloon from each of the given points, and also the height of the balloon from the ground. 31. Which examples in Exercise 41 can be worked by Case I ? Work such of these examples as the teacher may direct. 32. Make up some practical problem which can be solved by the method of Case I and solve it. " CASE II. Two SIDES AND THE INCLUDED ANGLE GIVEN 80. To solve Case II we have the following method by the use of the law of tangents (Art. 75) : Subtract the given angle from 180; divide the remainder by 2. The result will be half the sum of the unknown angles. One half of their difference may then be found by the follow- ing proportion: tan \ the difference of the unknown angles : tan \ their sum = difference of the two given sides : their sum. 116 TRIGONOMETRY Then ^ sum of unknown ^ -h J their difference = greater unknown Z. \ sum of unknown ^ \ their difference = smaller unknown Z. The third side is found by Case L Ex. 1. Given a =4527, fc = 3465, C = 66 6' 28", solve the triangle.* a + b = 7792. a - 6 = 1062. .4 + 5 - 180 - 66 6' 28 = 113 53' 32". i- (A + B) = 56 56' 46". By the law of tangents (Art. 75), tan %(A - B) : tan | (A -f B) = a - b : a + 6, that is, tan 1 (A - B) : tan 56 56' 46" = 1062 : 7992. tan 1 (A B} = 1062 tan 56 56 ' 46 " 7992 1062 log 3.02612 56 56' 46" log tan 0.18659 7992 log 3.91266 - 10 colog 6.09734 - 10 (A - B) = 11 32' 28" log tan 9.31005 - 10 A = 6S 29' 14" B = 45 24' 18" The side c may now be found by Case I. c sin 66 6' 28" Thus we have 3465 sin 45 24' 18" * If only the third side, c, is required, and the numbers representing the other sides, a and &, are small, the solution may often be readily effected by the formula of Art. 76 without the use of logs. Thus given a = 5, 6 = 6, G = 60, find c. c = Va 2 + b* - 2 ab cos C = \/25 + 36 - 60 x \ = V31 = 5.5775. OBLIQUE TRIANGLES 117 3465 log 3.53970 66 6' 28" log sin 9.96109 - 10 45 24' 18" log sin 9.85254 - 10 colog sin 0.14746 c = 4448.9 log 3.64825 (What checks can you suggest for the work ?) Ex.2. Given c= 30.15, a = 18.159, =54.22, solve the triangle. c + a = 48.309. (>- a = 11.991. <7 + .4 = 180 -54.22 = 125.78. By Art. 75, tan (<7 - A) : tan -j- (C + ^4) = c - a : c + a ; that is, tan (C - .4) : tan 62.89 = 11.991 : 48.309. tan ? rc ^n = 11 - 991 tan 62>89 . 48.309 11.991 log 1.0789 62.89 log tan 0.2908 48.309 log 1.6840 colog 8.3160 - 10 $(C-A) = 25.87 log tan 9.6857 - 10 | (C + A) =62.89 $(C-A) = 25.87 (7=88.76 ^4 = 37.02 The side b may now be found by Case I. b = sin 54.22 18.591 ~ sin 37.02 18.159 log 1.2591 54.22 log sin 9.9092 - 10 37.02 log sin 9.7797 - 10 colog sin 0.2203 b = 24.467 log 1.3886 (What checks can you suggest for the work ?) 118 TRIGONOMETRY EXERCISE 37 Using five-place tables, solve the following triangles, having given: 1. a = 27.7, b = 18.6, C = 68. 2. b = 400, c = 250, A = 68 18'. 3. A = 30 12' 20", b = .24135, c = .35627. 4. B = 63 35' 30", a = .062788, c = .077325. 5. A = 123 16' 30", b =3.1625, c = 3.1536. 6. A = 52 6', 6 = 420, c = 200. 7. (7 = 60, 6 = 9, a = 7. Find c only. SUGGESTION. c= Va* -f 6 2 2 a6 cos (7. 8. c = 26.369, b = 17.268, ^ = 32 18' 30". 9. B = 168 18' 39", c = 186.27, a = 132.91. Using four-place tables, solve the following triangles, having given : 10. a = 200, b = 260, C = 51.82. 11. b = 1.763, c = 1.112, A = 28.16. 12. a = .782, c = .412, jB = 112.18. 13. b = 11.65, a = 8.26, (7 = 12.12. 14. a = 1720, c = 642, B = 78.63. 15. b = 9, c = 6, ^4 = 60. Find a only. SUGGESTION. a = V& 2 + c 2 2 6c cos A 16. c = V7, ft = VlT, ^ = 1688. Find C, JB, and a. 17. b = 79.23, a = 100.6, C = 68.25. 18. a = 1200, 6 = 2100, C = 43.18. 19. a = 12, c = 15, B = 45. Find b without the use of tables. Solve the following, using either set of tables: 20. Two trees M and P are on opposite sides of a pond. The dis- tance of M from a point K is 159.6 ft., the distance of P from K is 216.8 ft,, and the angle MKP is 75 18' [75.3]. Find the distance between the trees. OBLIQUE TRIANGLES 119 21. The length of a lake subtends at a certain point an angle of 120, and the distances of this point from the two extremities of the lake are 2 and 3 miles respectively. Find the length of the lake. 22. The point is acted on by a force OA of 12 pounds and a force OB of 17 pounds, and the angle between the lines of direction of the two forces is 120 43' [120.72]. What will be the resultant force and what angle will it make with each of the original forces ? (Use the principle of the parallelogram of forces.) 23. Two trains leave the same station at the same time on straight tracks intersecting at an angle of 21 12' [21.2]. If the trains travel at the rate of 40 and 50 miles an hour respectively, how far apart will they be in 10 minutes ? 24. The sides of a parallelogram are 172.43 and 101.31 and the angle included by them is 61 16' [61.27]. Find the two diagonals. 25. In Exercise 41 which examples can be worked by the methods of Case II ? Work such of these as the teacher may direct. 26. Make up some practical problem which can be solved by the method of Case II and solve it. CASE III. THREE SIDES GIVEN 81. The Solution of Case III is effected by the use of the formulas proved in Art. 77. In case it is desired to find only one of the angles of a given triangle it will be best to use that one of the formulas of Art. 77 which will give t-he required angle most accurately. The cosine formula may be stated in general language thus: The cosine of one half of any angle of a triangle is equal to the square root of one half the sum of the three sides multiplied by one-half the sum minus the side opposite, divided by the product of the other two sides. Thus ab 120 TRIGONOMETRY Ex. 1. If in the triangle ABC, a= 123, b = 113, c= 103, find the angle A. , = 1(123 + 113 + 103) = 169.5. s - a = 169.5 - 123 = 46.5. /169 . " \ 5x46.5 113xl03 169.5 log 2.22917 46.5 log 1.66745 113 colog 7.94692-10 103 colog 7.98716-10 2)19.83070-20 i A = 34 37' 22" log cos 9.91535-10 .-. Z A = 69 14' 44". In case the half angle (^ ^4.) to be computed is small, it is best not to use the formula for cos \ A. Why ? In case the half angle to be computed is close to 90, it is best not to use the formula for sin 1 A. Why ? In case it is desired to find all three angles of a triangle, it is best to use the tangent formula of Art. 77. For it will be found that by that method it is necessary to employ the logarithms of but four different numbers, whereas by either of the other formulas it is necessary to use the logarithms of seven different numbers. It is a further advantage to trans- form the tangent formula thus : tan i A = s(s a) Let s a . Then t an l A = , t an \ B = , t an l C = sa sb sc To test the accuracy of the work add the angles obtained. Their sum should differ very slightly from 180. OBLIQUE TRIANGLES 121 s = 169.5. s a = 46.5. s c = 66.5. Ex. 2. If in the triangle ABC, a = 123, 6 = 113, c=103, find the three angles of the triangle. 46.5 log 1.66745 56.5 log 1.75205 66.5 log 1.82282 169.5 colog 7.77083-10 2)3.01315 rlog 1.50658 r log 1.50658 56.5 colog 8.24795 -10 1 5=29 36' 25" log tan 9.75453-10 . _ J46.5 x 56.5 x 66.5 : ^ 169.5 r log 1.50658 46.5 colog 8.33255-10 i^=3437'22"logtan9.83913-10 r log 1.50658 66.5 colog 8.17718-10 i (7=25 46' 15" log tan 9.68376-10 Hence A= 69 14' 44" B= 59 12' 50" C= 51 32' 30" 180 0' 4" (check) The fact that the sum of the angles of the triangle as computed differs from 180 by four seconds is due to the fact that the logarithms used are only approximately correct in the last figure. When five-place tables are used, as in the above solution, the sum of the angles obtained should not differ from 180 by more than six or seven seconds. Ex. 3. Find the three angles of the triangle in which a= 26.16, 6 = 29.15, c=32.24. s = 43.775 8-6 = 14.625 s - a = 17.615 s - c = 11.535 . r = r .615x 14.625x11.535 43.775 r log 0.9159 17.615 colog 8.7541-10 | .4=25.07 log tan 9.6700-10 r log 0.9159 14.625 colog 8.8349-10 15=29.39 log tan 9.7508-10 17.615 log 1.2459 14.625 log 1.1651 11.535 log 1.1620 43.775 colog 8.3587-10 2)1.8317 r log 0.9J59 r log 0.9159 11.535 colog 8.9280-10 C=35.54 log tan 9.8539-10 ^=50.14 5-58.78 0=71.08 180 (check) 122 TRIGONOMETRY EXERCISE 38 By use of five-place tables solve each of the following triangles, hav- ing given: a = 100, 1. 2. 3. 5. a = .117, \ b = .261, | c - .217. [ a = 122.6, J b = 169.4, 7. 8. 13. c = 95.2. a"= 79.38, b = 48.16, c=50. j b = 125, ( c = 140. f a = 1.57, b = 1.7, c = 1.266. a = 17.03, b = 12.585, c = 11.085. fa=113, b = 147, 14. 9. 10. 11. 12. a= V14, 6 = V19, c = V33. fa = 4.1409, 6 = 4.9935, [ c = 1.8181. (a = 2.6, | 6 =5.7, [ c = 7.8. fa = 17.51, | 6 = 12.575, I c = 23.645. Find the largest angle. 15. The sides of a triangle are 10, 17, and 25. Find the smallest angle in the triangle. 16. The sides of a triangle are 3, 4, and 5.5. Find the sine of the smallest angle. 17. The sides of a triangle are 1.1, 1.3, 1.6. Find the cosine of the largest angle. 18. The sides of a triangle are 18, 21, and 25 ft. Find the length of the perpendicular from the vertex of the largest angle to the opposite side. 19. By use of four-place tables solve Exs. 1-18. 20. The distances between three towns, P, Q, R, are as follows : PQ= 51, QJ?=65, P72=20. If R is due east from P, what is the direction of each place from every other place? If R is N.E. from P, what would each of these directions be ? 21. What angle is subtended by an island 2 miles long as viewed from a point 3 miles distant from one end of the island and 4 miles from the other end ? 22. Make up two practical problems which can be solved by the method of Case III and solve them. OBLIQUE TRIANGLES 123 CASE IV. GIVEN Two SIDES AND AN ANGLE OPPOSITE ONE OF THEM 82. The Solution of Case IV, like that of Case I, is effected by the use of the law of sines (Art. 74). But it has been shown in geometry that when two sides and an angle oppo- site one of them are given, sev- eral special cases arise in the con- struction of the triangle. Thus in the triangle ABC (Fig. 64) let the given parts be the angle A and the sides a and b. Then under the following conditions the following triangles may be constructed : I. If given Z A is obtuse and 1. side opp. A > side adj. . ... . . one A. 2. side opp. A < side adj no A. II. If given Z. A is right (same results as in I). III. If given Z A is acute and 1. side opp. > side adj one A. 2. side opp. = side adj. . . . one isosceles A. 3. side opp. (side adj.) x (sin given Z.) . > two A. (2) side opp. = (side adj.) x (sin given Z.) . . one right A. (3) .side opp. < (side adj.) x (sin given Z) . . . . fno A. In practice, the cases of no solution and of one right tri- angle or one isosceles triangle as the solution do not often occur. Hence we usually need merely a method of discrimi- nating between the cases where one oblique triangle or two 124 TRIGONOMETRY oblique triangles form the solution. We may state this test in the form of question and answer thus : Q. In general, when are there two solutions in Case IV f Ans. When the side opposite the given angle is less than the other given side. Q. In this case, how may the two triangles be con- structed f Ans. Take the vertex between the two given sides as a center, and describe an arc, using the smaller side as radius. ' It is usual so to letter the figure that the vertex of the given angle comes at the left end of the unknown base. Thus given ZC = 38, b= 152, c = 103, we have . Fig. 65. . FlG 65 Hence, in solving examples in Case IV, Observe whether the side opposite the given angle is less than the other given side; if it is, there are, in general, two solutions, which construct by taking the vertex between the given sides as a center and describing an arc with the smaller side as radius. In either case find the 'unknown angle opposite the known side by the use of the following proportion : sine of unknown Z opp. known side : sine of known Z = side opp. unknown Z : side opp. known Z. In case there are two solutions, use in one triangle the angle obtained from the table, and in the other triangle the supplement of this angle. Find the third angle and third side by Case I. Ex. 1. Given a = 84, fc = 48.5, ^ = 21 31', solve the tri- angle. OBLIQUE TRIANGLES 125 Since the side opposite the given angle, 84, is greater than the other given side, 48.5, there is but one solution. sin B 48.5 .'. sin B = sin 21 31' 84 48.5 sin 21 31' 84 48.5 log 1.68574 21 31 'log sin 9.56440 -10 84 log 1.92428 colog 8.07572 - 10 B = 12 13' 33" log sin 9.32586 - 10. = 146 15' 27". By Case I we find c = 127.211. Ex. 2. a = 22, b = 34, A = 30 20', solve the triangle. Since the side a opposite the given angle A is less than the other given side (A being acute, and 22 > 34 sin 30 20') there are two solu- tions to the given triangle. In this case it is well to draw the smaller triangle separately as well as the general figure. C= C'= FIG. 67. By the law of sines (Art*. 74), sin .5 _34 sin 30 20' ~ 22* 34 log 1.53148 30 20' log sin 9.70332 - 10 22 log 1.34242 colog 8.65758 - 10 FIG. 67 a. sin B = 34 sin 30 20' 22 B = 51 18' 27" log sin 9.89238 - 10 ,'. on Fig. 67a, '=180-51 18' 27" -128 41 '33"-. To complete the solution of A ACS, Z ACE = 180 -(Z.A + ZABC) = 180 -81 38' 27" = 98 21' 33". Hence by Case I we find c = 43.098. To complete the solution of A AC'B' (Fig. 67a). = 180 - 159 1' 33" = 20 58' 27". Then by Case I we find c' = 15.5926. (What checks can be used in the case of each of the two triangles ?) 126 TRIGONOMETRY Ex. 3. Given a = 22, b = 34, A = 30.33, solve the triangle. Since the side a opposite the given angle A is less than the other given side (A being acute and 22> 34 sin 30.33), there are two solutions. In this case it is well to draw the smaller triangle separately as well as the general figure. C' = FIG. 68. By the law of sines (Art. 74), sin B = 34 sin 30.33 " 22' 34 log 1.5315 30.33 log sin 9.7033 - 10 22 log 1.3424 colog 8.6576 - 10 B = 51.32 log sin 9.8924 - 10 sm f = 34 sin 30.33 C 22 To complete the solution of AACB, ^ACB = 180 - (30.33 + 51.32) = 98.35. Hence by Case I, obtain c = 43.1. = 180 - 51.32 = 128.68. To complete the solution of AAC'B' (Fig. 68a), we have 0" = 180 - (30.33 + 128.68) = 20.99. Hence, by Case I, find c' 15.6. EXERCISE 39 State the number of solutions for each of the following and con- struct a figure for each example, lettering it according to the method specified in Art. 82 : 5. (7=80, 6 = 16, c = 15.5. 6. 5 = 54, a = 23, 6 = 36. 7. (7 = 30, a = 18, c = 9. 1. .4 = 30, 6 = 50, a = 60. 2. .6 = 30, a = 100, 6 = 70. 3. (7 = 45, a = 60, c = 60. 4. A = 60, 6 = 12, a = 10. 8. 5 = 50, a = 50, 6 = 37. 9. J. = 75.16, c = 18, a = 17.6. Using five-place tables, solve the following triangles, having given : 10. A = 38 18', 6 = 120.6, a = 138.7. 11. .4 = 61 18' ; c = 23.7, a = 21.25. OBLIQUE TRIANGLES 12? 12. (7 = 104 13' 48", 6 = 115.72, c = 165.28. 13. B = 22 22', a = .6728, 6 = .81434. 14. ^1 = 47 19', a = 100, c=120. 15. B = 15 30' 12", a = 1200, 6 = 590. 16. C = 78 18' 18", a =.26725, c = . 37926. 17. = 26 18' 36", a = 28.604, 6 = 12.678. 18. A = 131 18' 24", a = .8888, c = .4128. 19. (7 = 31 31' 15", 6 = 11.111, c = 8.267. Using four-place tables, solve the following triangles, having given : 20. B = 32.37, 6 = 126.6, a = 138.7. 21. xL = 57.366, c = 22.7, a = 20.672. 22. = 105.273, 6 = 306.72, c = 241.8. 23. (7 = 26.223, a = 66.35, c = 82.59. 24. 5 = 14.3, a = 20.17, 6 = 17.8. 25. .4 = 22.37, c = 300, a = 200. 26. 5 = 63.31, c = 7.67, 6 = 9.54. 27. (7 = 49.31, 6 = .17634, c = . 15678. 28. In a parallelogram, one side is 167, one diagonal is 295.6, and the angle included by the diagonals is 24 18' [24.3]. Find the other side and other diagonal, and also the angles of the parallelogram. 29. If the angle between two forces is 154 20' [154.33], one of the forces is 960 pounds, and the resultant of the two forces is 440.46 pounds, find the other force. AREA OP AN OBLIQUE TRIANGLE 83. I. Given two sides and the included angle, to find the area of a triangle, use the rule : The area of a triangle equals one half the product of any two sides multiplied by the sine of the angle included by these sides. For let the given sides be a and c. 128 TRIGONOMETRY In Fig. 69a, let /.B be acute ; in Fig. 696, let Z. ABC be obtuse. c c Let p be the perpendicular from C to A B or AB produced. In each figure, the area of A ABC = \c x p. In Fig. 69a, in the rt. ACBD, p = a sin B. (Art. 41) In Fig. 696 in the rt. A CBD, p = asm (180-Z^C) = a sin ABC. (Art 64) Hence, in each figure, if we denote area of A ABC by K, K=^ac sinB. In case the given parts are a, b, C 9 or 6, c, A, let the pupil state what the formula becomes. Let the pupil also state these formulas in general language. Ex. 1. J. = 66 4' 19", 6 =21.66, c= 36.94, find the area of the triangle ABC. By the formula K = \ be sin A, K= 1(21.66 x 36.94 x sin 66 4' 19"). .-. log K= log 21.66 + log 36.94 + log sin 66 4' 19" + colog 2. 21.66 log 1.33566 36.94 log 1.56750 66 4' 19" log sin 9.96097 - 10 2 colog 9.69897 - 10 Area= 365.682 log 2.56310 21.66 FIG. 70. Ex. 2. Given A = 66.07, b = 21.66, c = 36.94, find the area of the triangle ABC. OBLIQUE TRIANGLES 129 By the above rule, K= i (21.66 x 36.94 x sin 66.07). .-. log K= log 21.66 + log 36.94 + log sin 66.07 + colog 2. 21.66 log 1.3357 36.94 log 1.5675 66.07 log sin 9.9610 - 10 2 colog 9.6990 10 Area = 365.75 log 2.5632 84. IT. Given two angles and a side, find the third angle as usual. Let the given side be a, then a. second side c may be determined as follows : c : a = sin C : sin A. _ a sin C _ a sin C _ _ a sin C ' sin A ~ sin [180 -(B + C)] " sin (B + C) Substituting this result in the formula for K in Art. 83, ft* sin B sin C Hence the area may be found by substituting directly in this last formula. 85. III. Given three sides. In this case we know from plane geometry that - a)(s b)(s - c). 86. IV. In case two sides and an angle opposite one of them are given, to find the area it is necessary to find the log sin oi the angle included between the two given sides by the method of Case IY (Art. 82), and then proceed as in Art. 83, In some cases two answers may occur (see Art. 82). EXERCISE 40 Using either five-place or four-place tables, find the area of the following triangles, having given: 1. a = 16.7, 6 = 21.6, C = 36 18' 24" [36.61]. 2. a = .86, B = 52 18' [52.3], C = 66 42' [66.7]. 130 TRIGONOMETRY 3. a = 18, 6 = 14, c = 24. 4. 6 = 200, c = 150, A = 72 18' 30" [72.31]. 5. b = 600, A = 18 26' [18.43], C= 31 44' [31.73]. 6. b = 14.7, a = 18.6, A = 74 18' [74.3]. 7. a = .8167, & = .68256, c = .72623. 8. a = 100, c = 125, B = 170 16' [1 70.27]. 9. 6 = 62.8, c = 47.2, ^L = 60. 10. Given ^L = 29 32' 16" [29.54], 6 = 500, and a=300, find the difference in area between the two triangles which contain these parts. 11. In a parallelogram, given two adjacent sides, c and d, and the included angle A, obtain a formula for the area of the parallelogram in terms of the given parts. 12. Prove that the area of any quadrilateral is equal to one half the product of its diagonals and the sine of their included angle. 13. Two sides of a parallelogram are 30 and 40 respectively, and their included angle is 60. Find the area of the parallelogram without the use of tables. 14. The diagonals of a quadrilateral are 17.6 and 20.5, intersecting at an angle of 36 18' [36.3]. Find the area of the quadrilateral. CHAPTER VII PRACTICAL APPLICATIONS 87. Instruments for Measuring Angles. In order to deter- mine unknown heights or distances it is important to have an instrument for measuring angles either in the horizontal or in the vertical plane. Horizontal angles can be measured by the Surveyor's Compass. Both horizontal and vertical angles can be measured by the Transit Instrument. 88. An angle of elevation is the angle between a line drawn from the eye of the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is above the horizontal plane. Thus, on Fig. 71, ACB is the angle of elevation of A as viewed from C. An angle of depression is the angle between a line drawn from the eye of the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is below the horizontal plane. Thus, on Fig. 71, DAC is the angle of depression of C as viewed from A. 89. I. To determine the Height of an Accessible Object above a Horizontal Plane. In Fig. 71 let AB be the object whose altitude is sought, and EF the P. horizontal plane, and C the point of observation. In the right triangle ABC, what line shall we measure ? What angle ? How then can AB be computed ? y IG . u, 131 132 TRIGONOMETRY D p> F 90. II. To find the Distance on a Horizontal Plane to an In- accessible Object whose Height is Known. In Fig. 71, let AB be the inaccessible object whose height is known ; let EF be the horizontal plane and C the position of the observer. In the right triangle ABC, what side is known? What angle can be measured ? How then can BC be computed ? 91. III. To determine the Height of an Inaccessible Object above a Horizontal Plane. Let AB, Fig. 72, be the altitude which is to be meas- ured, and EF the horizontal plane. Place the transit in- strument at D and measure * IG - 72 ' the angle of elevation ADB. Measure the distance DC toward B, and measure the angle ACB. By solving the triangle ACD the line AC is found. By solving the right triangle ACB, AB is found. In case it is desired to compute AB by means of right tri- angles alone, the solution may be effected by dropping a per- pendicular CP from C to AD and solving the right triangles DCP, CPA, and CAB (let the pupil supply the exact steps in this process). Or we may proceed by the use of natural tangents thus : On Fig. 72, in A DAB, DB = AB tan Z DAB, in A CAB, CB = ABttmZ.CAB. or Subtracting, DB-CB, DC = AB (tan Z DAB - tan Z CAB). DC Hence tan Z D AB - tan Z CAB ' In case it is not possible to move directly from D toward B, we may proceed as follows: Measure Z ADB (Fig. 73). PRACTICAL APPLICATIONS 133 Measure the line DC in the horizontal plane in any con- venient direction from D. Measure BDC and DCB. Then in the triangle DCB, DB may be computed (How?). Afterward in the triangle ADB compute AB (How?). FIG. 73. 92. IV. To determine the Height of an Inaccessible Object on an Inclined Plane. Let DF (Fig. 74) be the horizontal plane, DB the inclined plane, and AB the object whose height is sought. If we measure the A ADC and ACB, and the dis- tance DC, we may then compute AC (How ?). If we then measure /_ BDF, we may compute /_ CAB (How?). Then AB may be com- FIG. 74. puted (How?). 93. V. To find the Distance of an Inaccessible Object. Let A (Fig. 75) be the position of the observer and let it be required to determine the distance from A to B. Let the pupil determine what meas- urements and computations are neces- sary in accordance with the figure. FIG. 75. 94. VI. To find the Distance between two Objects separated by an Impassable Barrier (and possibly invisible to each other). 134 TRIGONOMETRY Let it be required to find the dis- tance between A and B (Fig. 76), which are separated by a swamp or a mountain for instance. Take a sta- tion C from which both A. and B are visible. Measure the angle C and the lines CA and CB. In the triangle ABC, compute AB (How ?). 95. VII. To find the Distance between two Objects, both Inaccessible and lying in the Horizontal Plane. Let A and B (Fig. 77) be two inaccessible objects (as two islands off the shore CD). Measure the line CD and the ACD, BCD, ADC, BDC. In the triangle ACD, com- pute 'AC; in the triangle BCD, compute BC ; in the FIG. 77. triangle ABC, compute AB. 96. Range Finders. In war, both on land and sea, the use of a range finder to determine the distance of an enemy is becoming general. The essential principle of such 'an instru- ment is the finding of the distance of an inaccessible object by the solution of a triangle in which a side (called a base line) and the two angles which include the side are known (see Art. 93). On land a convenient base line is taken and measured. In naval warfare, the distance between two points on the vessel is utilized as a base line. In the range finder the triangle employed is not usually solved by numer- ical computation, but by some mechanical method, which gives the result sought much more expeditiously. 97. Coast and Geodetic Survey. The essential parts of the work of the coast and geodetic survey are as follows : PRACTICAL APPLICATIONS 135 1. The measurement of a base line AB (Fig. 78) at least 4 or 5 miles long, so accurately that the error shall not ex- ceed -^ of an inch per mile. 2. The choice of a convenient station P and the measurement of the angles PAB and PBA, and the computation of PA and PB in the triangle PAB. 3. The choice of another station Q, the measurement of the angles QBP and QPB, and hence the computation of PQ and QB. 4. Proceeding in like manner from station to station till convenient points, C and D, are reached, and the length of the line CD computed. 5. The careful measurement of CD and the comparison of its computed length with the result of the measurement. This final measurement of CD serves as a test of the accuracy of all the inter- vening work. By carrying these measurements far enough, a considerable arc of a great circle of the earth may be measured, and from this arc the radius or diameter of the earth computed. 98. Distance of the Sun and Stars. The usual method of determining the distance of the sun from the earth consists essentially in taking a line (AB, Fig. 79) nearly equal to the diameter of the earth as a base line, and observing from each end of AB the angle made by a line drawn to some convenient planet P. The distance of the planet may then be computed by Art. 93. The ratio of the dis- tance of the sun to that of the planet from the earth being Fia. 78. FIG. 79. 136 TRIGONOMETRY known by an astronomical law, the distance of the sun is readily determined. The distance of tLe sun from the earth is thus found to be approximately 93,800,000 miles. The distances of the fixed stars are found by taking the diameter of the earth's orbit as a base line, measuring the angles made by this line with lines drawn from its ends to a fixed star, and making the necessary computations. Thus the trigonometrical solution of a triangle in which a side and the two angles adjacent to it are known is seen to have very wide practical applications. 99. Application to Navigation. Trigonometry also has many applications to different departments of applied science. As an illustration of these applications we will briefly indicate its method of use in navigation. If a ship should sail from R to B on the diagram (Fig 80), crossing each meridian at the same angle, for certain purposes the A ARE (AB being the arc ~f r of a parallel of latitude) could be re- FIG. so. garded as a plane triangle and solved, when necessary, by the methods of plane trigonometry. This form of navigation is called Plane Sailing. The departure between two meridians is the arc of a par- allel of latitude comprehended between the two meridians. Thus, AB is a departure between PAP' and PBP'. Evi- dently the departure between two given meridians diminishes with the distance from the equator. The difference of longitude between two places is the angle at the pole (or the arc on the equator) included between the meridians of the two given places. Thus the difference of longitude for A and D is the angle RPS, or arc RS. In Parallel Sailing a vessel sails due east or west (i.e. on a parallel of latitude) as from A to B. The difference of PRACTICAL APPLICATIONS 137 longitude corresponding to the course sailed may be found by the formula diff. of longitude departure* sec- latitude. For on Fig. 80, n 4 diff. long. : dep. - arc RS : arc AB = OR : CA = OA : CA = ^j : 1 C-OL = sec. lat : 1. .-. diff. long. : departure = sec. lat. : 1. In Middle Latitude Sailing a ship sails between two places in a course oblique to a parallel of latitude. For short dis- tances (especially near the equator) sufficient accuracy is obtained by regarding the departure as measured on the parallel of latitude midway between the parallels of the two places, and computing the difference of longitude by the formula . = departure x sec. mid. lat. EXERCISE 41 1. In Exercise 22 point out the examples which are solved by the method of Art. 89. 2. Also those which are solved by the method of Art. 90. 3. Also those solved by principles contained or implied in Art. 91. 4. The angle of elevation of the top of a tree measured from a point 213.5 ft. from its foot is observed to be 18. Find the height, of the tree. 5. A water tower 92.5 ft. high stands on a horizontal plane. An observer finds the angle of elevation of the top of the tower to be 52. Find the distance of the observer from the base of the tower. 6. Pike's Peak when viewed from a certain point on the Colorado plain has an angle of elevation of 15 48' [15.8]. Two miles farther off the angle of elevation is 11 59' [11.98]. What is the altitude of the mountain above the Colorado plain ? If the Colorado plain is 5176 ft. above sea level, what is the altitude of Pike's Peak above sea level ? 7. From the top of a hill 350 ft. high the angle of depression of the top of a tower which is known to be 150 ft. high is 57. What is the distance from the foot of the tower to the top of the hill ? 138 TRIGONOMETRY 8. A man standing west of a tree, on the same horizontal plane, observes its angle of elevation to be 48 ; he goes north 50 yd. and finds its angle of elevation to be 41. Find the height of the tree. 9. The angle subtended by a tower on an inclined plane, is at a certain point on the plane 56 ; 200 ft. further down it is 28. The inclination of the plane is 7. Find the height of the tower. 10. From the top and bottom of a castle which is 75 ft. high the angles of depression of a ship at sea are 19 and 15 respectively. Find the distance of the ship from the bottom of the castle. 11. A monument 70 ft. high and a tower stand on the same hori- zontal plane. The angle of elevation of the top of the tower at the top of the monument is 20 40' 12" [20.67], at the base of the monument it is 53 31' 12" [53.52]. Find the height of the tower and its dis- tance from the monument. 12. The three angles of a triangle are to each other as 11 : 13 : 6 and the longest side is 11. Find the other two sides. 13. Two mountains, A and B, are respectively 12 and 16 mi. from a point O, and the angle ACB is 72 18' [72.3]. Find the distance betweeh the mountains. 14. In a parallelogram one side is 16.9 and a diagonal is 30.72, and the angle included by the diagonals is 26 36' [26.6]. Find the other side and the other diagonal, also the angles of the parallelogram. 15. A flagstaff 50 ft. in height stands on a tower. From a position near the base of the tower, and on the same horizontal plane, the angles of elevation of the top and bottom of the flagstaff are 41 36' [41.6] and 22 18' [22.3], respectively. Find the distance and height of the tower. 16. The diagonals of a parallelogram are 12.5 and 12.8 ft. respec- tively, and their included angle is 52 16' [52.27]. Find the sides of the parallelogram. 17. The sides of a triangle are 11, 13, and 16. Find the cosine of the largest angle. 18. From a point 4 mi. from one end of an island and 7 mi. from the other, the island subtends an angle of 33 33' 33" [33.56]. Find the length of the island. 19. Two buoys are 1500 yd. apart. The angles formed by lines from a boat to each buoy form angles with the line between the buoys of 77 18' [77.3] and 51 16' [51.27], respectively. Find the distance of the boat from the nearer buoy. PRACTICAL APPLICATIONS 139 20. Two straight roads cross each other at an angle of 48 24' [48.4] at the point M. Four miles from M on one road is the town of P, and 6 miles from M on the other road is the town of If. How far apart are P and If? (Two answers.) 21. The diagonals of a quadrilateral are 47.6 and 61.23 rd., respec- tively, and the angle included by the diagonals is 43 10' [43.17]. Find the area of the quadrilateral. 22. To find the distance between two trees Tand T', on opposite sides of a river, a line TK and the angles T'TIf and T'KT are measured and found to be 412 ft., 62 30' [62.5], and 57 32' [57.53], respectively. Find the distance TT. 23. Two objects which are invisible from each other on account of a hill are visible from a station whose distances from the objects are 367 yd. and 514 yd., respectively, and the angle at the station subtended by the distance between the objects is 57 36 f [57.6]. Find the distance between the objects. 24. Given a circle with radius 19.8 ft. Find the area inclosed between two parallel chords on opposite sides of the center whose lengths are 25.6 and 31.7. 25. Wishing to find the distance between two trees T and T 7 ', separated by a marsh, I take TK on the prolongation of TT' through T, 89 yd. in length, and then take KP, 165 yd. in length, at right angles to IfT. The angle T'PT is found to be 33 36' 36" [33.61]. Find the distance from T to T". 26. Two yachts start at the same time from the same point, and sail one due west at the rate of 9.75 mi. per hour, and the other due north- west at the rate of 11.5 mi. per hour. How far apart will they be at the end of 2 hr. sail ? 27. In order to find the distance from a rock R to a buoy B, dis- tances EK and KP are measured to points If and P from which both rock and buoy can be seen, the distance RK being 2500 m., and KP being 3600 m. The following angles are then measured: Z.BKR = 38 48' [38.8], Z/rP = 7554' [75.9], and ^BPIf=79 30' [79.5]. Find the distance from the rock to the buoy. 28. A ship sails due east 416 mi. in latitude 40 23'. Find the difference in longitude which she makes. 29. A ship leaves latitude 30 16' K, longitude 43 17' W., and sails N.E. 350 mi. Find the difference of latitude and departure which she makes. Hence find her new latitude and longitude. 140 TRIGONOMETRY 30. A flagstaff 30 ft. high stands on the top of a building. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are observed to be 41 and 36 respectively. Assuming the ground to be level, find the height of the building. 31. A tower stands on a hillside whose inclination to the horizon is 11 ; a line is measured straight up the hill from the base of the tower 110 ft. in length and, at the upper extremity o.f the line, the tower subtends an angle of 52. Find the height of the tower. 32. A rock 60 ft. high stands on the top of a hill whose side is inclined 21 to the horizon. An observer standing on the hillside below the rock finds the angle of elevation of the top of the rock to be 64, and a second observer, farther down the slope, and in direct line with the first observer, finds the angle of elevation of the top of the rock to be 42. Find the distance between the observers, and the distance from the first observer to the base of the rock. 33. A point at is acted on by a force which gives a velocity of 1376 ft. per second along OA, and by another force which gives a velocity of 1135 ft. per second along OB. Z AOX= 30, Z BOX = 101. What will be the magnitude and direction of the resultant velocity ? 34. Show that the projection of OA plus the projection of OB on X' OX equals the projection of the resultant of OA and OB on X'OX. 35. If, in the figure of Ex. 33, OA = 200 and the resultant = 300, find OB, the angles being unchanged. 36. A tower 190 ft. high stands on the seashore. From its top the angle of depression of two boats are 8 and 11 respectively. From the bottom of the tower the angle subtended by the distance between the boats is 101. Find the distance between the boats. 37. A man on the opposite side of a river from two trees P and Q wishes to determine the distance between the trees. H,e measures a distance A B, 287 ft. He also measures the angles PAB, QAB, PBA, and PBQ and finds them 31, 36, 51, and 42, respectively. Find the distance between the trees. 38. Two straight paths cross each other at an angle of 68. A line is drawn so as to inclose, with the two paths, an acre of ground. This line cuts one of the paths at a distance of 52 yd. from the point of PRACTICAL APPLICATIONS 141 intersection of the two paths. What angle does this line make with each path ? 39. A tower 135 ft. high stands at one corner of a triangular garden. From the top of the tower the angles of depression of the other two corners of the garden are 56 18' [56.3] and 19 36' [19.6], respectively. The side of the garden opposite the tower subtends, from the top o the tower, an angle of 66. Find the length of the sides of the garden. 40. Two towers are 144 ft. apart. The angle of elevation of one observed from the base of the other is twice, that of the first observed from the base of the second; but from a point midway between the towers, the angles of elevation of the tops of the towers are complemen- tary. Find the height of the towers. (Do not use logarithms.) 41. A railroad embankment is 9 ft. high. The length of the slope of the embankment on each side is 14 ft. Find the angle which the slope makes with the horizontal, and also find the width of the embank- ment at the base if the top is 8 ft. wide. 42. Given the triangle ABC, whose sides are AB = 87.6 yd., AO= 112.7 yd., and BC =121.6 yd. A point D is taken on the line AC produced through C, so that the angle BDC is 18 37' 48" [18.63]. Find the distance DC. 43. The area of a triangle is 3 acres and two of its sides are 92.6 and 26.72 rd. Find the angle between these sides. 44. A shooting star is observed at two places 200 mi. apart on the earth's surface ; the angle of elevation of the star at one station is 27 and at the other is 63, the star being in the same plane with the two stations and the center of the earth. Taking the radius of the earth as 3956 mi. find the height of the shooting star above the earth's surface and hence the height of the earth's atmosphere. (What is a shooting star ? What causes its light ?) 45. Show how to solve each of the cases in oblique triangles by dividing the oblique triangle into right triangles and using the methods of solving right triangles given in Chapter III. Why do we not ordinarily use this method of solving oblique triangles ? 46. Make up (or collect) all the different examples you can showing practical applications of trigonometry, each example being distinct in principle or in field of application from the other examples. CHAPTER VIII CIRCULAR MEASURE. GRAPHS OF TRIGONOMETRIC FUNCTIONS 100. Radians, or the Circular Measure of Angles. The method of measuring angles by taking a right angle as the unit, dividing the right angle into 90 degrees, dividing each degree into 60 minutes, etc., is called the sexagesimal method and originated in Babylonia (see Art. 127) in very early times. It continues to be generally used in spite of its awkwardness because of the extensive tables and large number of results stated in terms of it which have been accumulated. However, the advantages of the decimal division of any unit are so great that it is a growing custom to divide the degree of angle into tenths and hundredths instead of minutes and seconds (see many examples in this book). Also within the past century it has become customary in many kinds of work (especially algebraic or theoretic work) to use a unit of angle different from the right angle, called the radian, and to divide this unit decimally. A radian is the angle which, when its vertex is placed at the center of a circle, intercepts an arc equal to the radius of the circle. FlQ 81 Thus if the arc AC (Fig. 8) equals the radius AB, the angle ABC is a radian, or the unit angle in the so-called circular method of measuring angles. 142 t CIRCULAR MEASURE 143 Hence, to determine the number of radians in an angle whose arc and radius are given, we have the relation no. of radians in an angle = - , radius denoting the number of radians in an angle by />, the subtended arc by a, and the radius of the circle by R, p= . R Ex. 1. Find the number of radians FlG - 82> in an angle AOB whose arc is 13 and radius 5. We have, Z AOB = - 1 / = 2.6 radians, Ans. From the above relation it follows that Any two of the three quantities, number of radians in an angle, arc, and radius, being given, the other may be found. Ex. 2. An angle containing 2.4 radians subtends an arc 14 in. long. Find the radius. ^ Substituting for p and a in the formula p = , R 2.4= it^Hi. .-. R = *ilE: = 5.83+ in., Ans. R 2.4 101. I. Converting Degrees into Radians. The number of radians about a point in a plane _ circumference radius _277#_ 9 ~R~' ,. 3600 = 277, or 6.2832 radians. ^ ^ Q ^ ^.^ 180 = 77, or 3.1416 radians. 90 = |, or 1.5708 radians. 30 = |, or 0.5236 radians. 60 = ^, or 1.0472 radians. I = T^> or -01745 radians. 3 180 144 TRIGONOMETRY Hence to convert degrees into radians Multiply the given number of degrees by - - (or by .01745 + ). loU Ex. 1. How many radians in 26 17' 36"? 2617'36" = 26.293+ = (26.293+)(.01745) radians. = 0.45882+ radians, Ans. Ex. 2. Simplify sin (| + x). sinf - + x ] = sin ^ cos x + cos- sin x (Art. 66) \6 J b 6 = i cos x + i V3 sin x, Ans. (Art. 33) Where the meaning is evident from the context, it is customary to abbreviate "TT radians" into "IT." Thus also we abbreviate "sin- radians" into "sin-" and similarly for other expressions. 102. II. Converting Radians into Degrees. Since 2 TT radians =360 180 1 radian = - , TT or 1 radian = 57.29579+ = 57 IT 45" = 206265". Hence to convert radians into degrees 180 Multiply the given number of radians by - - (or 57.3-). TT Ex. Convert 2.5 radians into degrees, minutes, and seconds. 2.5 radians = 2.5 x (57.2958-) = 143.2395 = 143 14' 22", Ans. 9 Hence, if the number of degrees in an angle be denoted by A, the number of radians in it by />, etc., any two of the CIRCULAR MEASURE 145 four quantities A, p, a, R being given (provided one of them is a or R), the other two may be found by substitution of the two given quantities in the two equations a /180 = - and 103. The solution of a right triangle containing an angle less than 2 may often be conveniently effected by the use of radians. For the sine or tangent of a small angle may be taken as equivalent to the number of radians in the angle (i.e. the circular measure of the angle) without appreciable error (see Art. 115). Thus sin A = A (in radians) when A is a small angle, is an ap- proximation frequently used in Physics, and the result is accurate to within the probable degree of error in measurement. Ex. If a railroad track has a rise of 1 ft. in every 2000 ft. in its length, what angle does it make with the horizontal ? Denoting the required angle by A, sin A = = no. radians in A approximately. 2000 x 206265" = 103+ " = V 43", Ans. EXERCISE 42 1. Reduce the following angles to circular measure, expressing the results as fractions of TT : 30, 135, 60, 90, 210, 270, 225, 72, 315. 2. Express the following angles in degrees : TT 7T 7T 2 TT 4 7T 3 7T 7 TT 8 77 6' 4' 3' T' IT' ~6~' T' 15* 3. What decimal part of a radian is 1 ? 16" ? 2' 15" ? 5 14' ? 4. How many degrees (minutes and seconds) in 2 radians ? 3.2 radians ? .003 radians ? 5. A circle has a radius of 14 inches. How many radians are there in an angle at the center subtended by an arc 21 in. long ? By an arc 7 in. long ? 146 TRIGONOMETRY 6. In a circle of radius R, an arc 3 ft. 6 in. subtends an angle of 1.5 radians. Find It. 7. One angle of a triangle is 30, and the circular measure of another angle is 1.5 radians. Find the third angle in degrees. Also in radians. 8. The difference between two angles is - and their sum is 110. Find the angles in degrees ; in radians. 9. Find both in radians and degrees the complement and supple- ment of the following angles : "* JL **7r 6' 3' 4' 9' 18* 10. Write out the trigonometric ratios of the following angles : 7T 7T 7T 7T 3 7T 7 7T 7 7T 6' 3' 4' 2' T> IP T' 11. How many radians in an angle whose arc is 12 and radius 10 ? How many degrees ? 12. Show that sin (x + 1 ?r) + sin (x % TT) = sin x. Supply the two missing quantities in each of the following : 13 14 15 16 17 p a R A 2.5 .25 10 in. 12ft. 100 50 in. 1 ft. 6 in. 42 in. 130' 37 18. If a railroad track has a rise of 1 ft. in 750 ft., what angle does the track make with the horizontal ? 19. If a railroad makes an angle of 1 30' with the horizontal, what is its rise in one half mile ? 20. An irrigating ditch should have a fall of at least \ in. per rod. What angle does the bottom of the ditch make with the horizontal ? 21. If the moon is at a distance of 240,000 mi. from the earth and the radius of the moon subtends an angle of 16' as seen from the earth, what is the radius of the moon in miles ? 22. If the sun is at a distance of 92,800,000 mi. from the earth, and the diameter of the sun subtends an angle of 32.4' as viewed from the earth, what is the radius of the sun in miles ? 23. The planet Mars has a diameter of 4200 miles. When Mars is nearest the earth, its diameter subtends an angle of 24.5" as seen from CIRCULAR MEASURE 147 the earth. What is the distance of Mars from the earth at such a time? 24. Find the numerical value of 3 sin - -4 cos ^ tan - + cot ^ 25. Make up two practical problems in each of which a right triangle is solved by the use of radians as in Exs. 17-21. We shall now illustrate the use of radians, or the circular measure of angles, (1) in tracing the graphs of trigonometric functions, (2) in solving trigonometric equations. GRAPHS OF TRIGONOMETRIC FUNCTIONS 104. Graph of sin oc. To form what is called the graph of sin x use the equation y = sin x and also a pair of rectan- gular axes (see Art. 54). In the equation y = sin x, let x have convenient successive values and find the corresponding values of y. Lay off each corresponding pair of values of x and y as the abscissa and ordinate of a point. Draw a continuous curve through the terminal points thus located. It is usually convenient to make the scale of the drawing such that a unit space of the cross-section paper stands for \ or .5236 + . 6 Thus, if we desire to make a graph of y = sin x we may take the following corresponding values of x and y : x = ir, y = 0, etc. $ = IT, y = 0, etc. 148 TRIGONOMETRY Using these results, the curve AOBCDE (Fig. 83) is obtained as the graph of sin x. Such a figure shows at a glance the changes in the values of sin x as x changes in value. FIG. 83. 105. Graphs of Other Trigonometric Functions. By treat- ing the equations y = cos x, y = tan x, y = sec x 9 etc, simi- larly, the graphs of the other trigonometric functions may be constructed. It is important to observe in constructing the graph of tanz, that, as x = ^ 9 y = either + GC or - oc. For as we 2i proceed from x = and make x = o", y = + GC; but as we proceed from x= TT and make x == ^, y == oc. Hence we CIRCULAR MEASURE 149 obtain as part of the graph of tan x the curve AOB, CO'D of Fig. 84. EXERCISE 43 Graph each of the following : 1. y = sin x. 9. y = tan 1 x. 2. y = cos x. 10. y = sin x -f cos x. 3. y = tan x. 11. y == s i n x cos # 4. y = cot a?. 12. y = Vsin a;. 5. y = sec #. 13. y = sin 2 #. % 6. y = esc #. 14. y = 1 + sin x. 7. y = sin i x. 15. y = l cos a. 8. y = sin 2 x. 16. y = 106. Solutions of Trigonometric Equations. Answers not greater than 360, i.e. than 2 TT radians. Ex. 1. Find the values of x less than 2 TT radians which shall satisfy the equation sin x = J. Since sin 30 = \, and also sin 150 = |, x - or ^ radians. ^4ns. 6 6 Ex. 2. Solve 4 cos a: 3 sec # = for values of x less than 2 TT. Q 4 cos a; -- = 0. cos x 4 cos 2 x - 3 = 0. cos x = 1 V3. Hence, * = 30, 150, 210, 330, or * = I',1T' T' ^ mdiaUS ' ^ S * 107. Answers Unlimited. Ex. 1. Solve the equation cos x = \. One value of x is 60 and another value is - 60. But if 360 be added to or subtracted from the value of an angle, the value of the function is unchanged. 150 TRIGONOMETRY Hence, x = 2 mr ^ radians, where n is zero or any positive or negative integer. Ex. 2. Solve the equation sin x esc x + f = 0. Solving the equation, we obtain, sin x = 2, i. Since the sine of an angle cannot be greater than 1, no angle corre- sponds to the value 2. For , sin x = i, , (2 n + !>-, Ans. EXERCISE 44 Solve each of the following equations, expressing the answers in radians, by use of TT. 1. cot 2 6 = - 3. 12. Cot g + 1 = cos 2x. cot x 1 2. tan 2 = 3. 13. 2 sin 2 a; sin x = sin 2 x cos x. 3. cot 2 0=1. 14. cos 2 x -f- cos a? = 0. 4. sin 2 = f . 15. tan (45 + a;) + tan (45 - x) = 4. 5. cot = 2 cos 0. 16. 2 esc 2 a; V3 cot x = 5. 6. cos + sec = f. 17. sin 3 x = sin 5 x + sin a;. 7. 3 sin 2 x -h cos 2 a? = f . 18. cos 3 a; + cos a? = cos 2 #. 8. 3 cot 2 x -f tan 2 a? = 4. 19. sin 5 x sin x = cos 3 x. 9. cos x = sin 2 a?. 20. cos 3 x cos a; = sin 2 a?. 10. cos 2 x + sin a? = 4 sin 2 .T. 21. sin o cc + sin 3 x + sin # = 0. 11. sin 2 # = tan 2 x. 22. cos 5 x + cos 3 x + cos a: = 0. 108. Simultaneous Trigonometric Equations. Ex. 1. Solve x sin y = a ', - for x and x cos w = Dividing the first equation by the second, tan y = -' .\ y = /- whose tan is -, Ans. b o (For a briefer way of expressing this result see Chapter IX.) CIRCULAR MEASURE 151 From this result the value of y may be obtained. When y is known x can be obtained from either of the original equations. OTX = sin y cos y Ex. 2. Solve for x and y the equations, x cos A + y sin ^4 = a (1) x sin A - y cos A = b (2) Multiply equation (1) by cos A, then x cos 2 A + y sin A cos A=a cos ^4 (3) Multiply equation (2) by sin A, then x sin 2 A y sin ^4. cos A = b sin A (4) Add (3) and (4), using the fact that sin 2 ^4 + cos 2 JL= 1, then x = a cos A -\-b sin A, and similarly, y= a sin A b cos A. ! EXERCISE 45 Solve for x and 0, or for x and ?/ : f x cos = 86.65, fa tan = 816.95, { x sin 0=50. { x sin = 426.3. f x sin = 118.96, f x sin ?/ = 4, {a cos = 160.78. 4 ' |>cosy = 8. f sin 30 + y cos 45 = 53.28, 5 * I x cos 30 + y sin 45 = 71.58. f x sin 48 -f y cos 19 = 2634.1 , 5 * ( x cos 48 + y sin 19 = 1320.3. r sin x + sin y = 1.573, [Use Art. 71.] \ cos x + cos y = 1.207. f sin # sin y .2154, \ cos x cos y= .1231. ( x sin (0-21.5) = 771.1, \aj cos (0-32.5) = 766. f x cos J. y sin A = a, { x sin ^1 + y cos .4 = 6. V CHAPTER IX INVERSE TRIGONOMETRIC FUNCTIONS 109. Anti-sine. If y is an angle and x its sine, the relation between x and y may be expressed in either of two ways : (1) x = siny, or (2) y = sin" 1 x, which reads " y is the angle whose sine is x" or " y is the anti- sine of x r One or the other of methods (1) or (2) is used according as the angle, or its sine, has the leading place in the discus- sion. Thus if the angle, or y, is more prominent, x = sin y is used; but if the sine, x, is more prominent, y = sm~ 1 x is used. The pupil should carefully discriminate between sin" 1 ^ and the 1 power of sin x. The latter is expressed thus, (sin x)~\ Thus, = sin a; (sin a?)" 1 , and not sin' 1 x. But (sin x)~ 2 may be written sin~ 2 x. 110. Other Anti-trigonometric Functions. Similarly cos" 1 x means " the angle whose cosine is x " ; tan" 1 x means " the angle whose tangent is x." Let the pupil state the meaning of cotr l #, csc" 1 ^, vers" 1 ^. It is evident that sin (sin" 1 x) = x, since the sine of the angle whose sine is x must be x. Similarly cos (cos" 1 a;) = x, etc. Hence there is a similarity in form between a(a~ l )x = x, and sin (sin- 1 x) = x. It is because of this similarity that the system of symbols described above is used to express the anti-trigonometric functions. 152 INVERSE TRIGONOMETRIC FUNCTIONS 153 A much better symbolism for "y equals the angle whose sine is x" would seem to be "y = Zsmx," and if the pupil has difficulty in grasping the principles of this chapter, it may be well for him to use this latter method of writing inverse functions till he becomes familiar with their nature. 111. Values of Inverse Trigonometric Functions. The direct and inverse trigonometric functions have an important difference with reference to the number of values which satisfy them. Thus, if y = sin 30, y has a single value, J; but if x sin" 1 J-, x can have an indefinite number of values, viz. : 30, 150*, 390, 510, etc.; or x= 2nw+%, (2 n + ]>- (See Art. 107, Ex. 2.) o 6 For many purposes it is customary to limit the values of an inverse circular function to the smallest value that will satisfy a given expression. Thus, if = tan' 1 1, we take 0= 45. 112. Given an Anti-trigonometric Function, to find the other Related Functions. Ex. 1. Given 6 = tan" 1 f , find sin 9 ; that is, find sin (tan" 1 J-). 6 = tan" 1 1^ may be converted into the form tan = f for which a diagram may be con- structed (Fig. 86). .-. sin (tan- 1 f) = ^ Vl3 Ans. Ex. 2. Find sin 2(cos- ! J). Let x be the angle whose cosine is i. Then cos x = , sin x = Vl .-. sin. 2 a; = 2 sin x cos x = Hence, sin 2(cos~ 1 ^) = | V2, 3 FIG. 86. |V2. = |V2. 154 TRIGONOMETRY Ex. 3. If = tan l a, express the direct and inverse func- tions of 6 in terms of a. tan 6 = a, hence = tan" 1 a. 1 a = sec~ 1 Vl 4- a 2 . 1 cot = -, a sec 6 = Vl 4- a*, cos = 1 Vl 4- a s 1 FIG. 87. sin = CSC = = sin VI + a 2 a VI 4- a Ordinarily only the positive value of each radical is used. 113. Inverse Trigonometric Functions of Two Angles. Ex. 1. Find sin (sin' 1 \ 4 cos' 1 -f). Let x = sin" 1 1. .-. sin x = ^-, cos a? =^V3. Let i/ = cos" 1 f . cos y = |, .-. sin y = \y FIG. 88. 2 FIG. 89. Then sin (sin" 1 1 4- cos" 1 1) = sin (x -h y) = sin # cos y + cos x sin y = J(2+V15), Ex. 2. Prove that sin" 1 a 4- cos" 1 a = ^ ) Using the method of Ex. 1, show that sin (sin" 1 a 4- cos" 1 a) = 1 = sin f . Ex. 3. Show that tan- 1 a + tan' 1 6 - tan- 1 ^-. l-ab x = tan" 1 a. Let But .-. a = tan x, y = tan" 1 '.*. b = tan ?/. i FIG. 90. 1 FIG. 91. / N tan x 4- tan y tan (x 4- y) = ' 1 tan x tan y INVERSE TRIGONOMETRIC FUNCTIONS 155 .-. tan (tan- 1 a + tan- 1 b) = ^\ t or tan- 1 a + tan- 1 6 = tan' 1 +A. 1 a& 1 ab 114. Solution of Trigonometric Equations by Use of In- verse Trigonometric Functions. It is sometimes useful to express the answer obtained by solving a trigonometric equa- tion in terms of an inverse function. Ex. Solve 6 cos 2 x cos x = 2. Factoring, (2 cos x + 1)(3 cos x 2) = 0. .-. coscc = !, f. .-. = cos" 1 ( i), cos" 1 1, Ans. EXERCISE 46 If the pupil has any difficulty in grasping any one of the following problems, it will be well for him to translate the symbols of the problem into general language before attempting the solution. Thus Ex. 2 would read " find the cosine of the angle whose cotangent is }," and might be written in the form. "find cosZ cotf " (see Art. 110). Express the following angles first in degrees and then in radians : 1. cos-^VJ?, tan-'VS, sin- 1 !, sec^V^, csc^fVS, cot^VS* cos^i, sec- 1 2, sin-^VS, cot-^Va, tan- 1 ^ V3. Find the value of : 2. cos (cot- 1 f). 8. sin (2 tan- 1 3^). 3. tan (sin- 1 T %). 9. cos (2 sec' 1 - 1 /)- 4. sec (tan" 1 -%). 10. sin(icos -1 i). 5. sin (cot- 1 a). 11. cot(| tan- 1 - 1 /)- / ft \ 12. sin (3 sin- 1 !). 6. cot (cos- 1 - ). b' 13. sm (sm- 1 1 - cos- 1 f ). 7. tan (2 sin^i). 14. tan (tan- 1 2 + cot" 1 3). Show that : 15. tan^i + tan- 1 ^. 16. tan- 17. sin- 1 T 8 T + sin- 1 f = sin" 1 J. 18. cos- 1 f + cos- 1 f V = cos- 1 (- f|). 19. tan- 1 f + tan T \ = tan" 1 JJ. 20. cot- 1 a + cot- 1 b = cot- 1 (t6 ~ 1 . 156 TRIGONOMETRY Prove that : 21. sin (sin- 1 f + cot" 1 1) = 1. 22. (cos" 1 |f + tan -1 ^-) = sin" 1 11J-- 23. sin (2 tan- 1 a?) = w ^ g . 1 T C 24. sin" 1 # = cot" 1 . 25. cos" 1 a cos" 1 b = cos~ 1 (ab + Vl a 2 b- + a 2 6 2 ). 26. 3 cos -1 a; = cos" 1 (4 3? 3 #). 27. 3 sin - 1 x = sin- 1 (3 x - 4 or 3 ). 28. tan- 1 a -tan- 1 6 = -^^. 29. sin" 1 a + sin' 1 6 = eo8~ l (vl a 2 6 2 + a 2 6 2 ab). Express the value of each of the following in its most general form : 30. sin- 1 ! 35. cos-^VS. 31. tan- 1 V3. 36. tan" 1 ^. 32. cos-^V^. 37. cot- 1 V3. 33. cot- 1 VS. 38. sec^V^. 34. sin- 1 ^ VS. 39. sin-^-l). 40. Prove that tan (2 tan" 1 a) = 41. Prove sin (2 tan- 1 a) = 1-a 2 . a 2 42. If cos" 1 x 2 cos" 1 a?, find x. 43. Express the following angles in the inverse notation : 30, 60, 90, 45, 0; n!80, n90. Can each of these angles be expressed in more than one way in the inverse notation ? 44. Who first, and at what time, brought inverse circular functions into use in their present form (see p. 173) ? 45. : At what time did the circular method of measuring angles come into use (see p. 167) ? CHAPTER X COMPUTATION OF TABLES TRIGONOMETRIC SERIES 115. Limiting values of 05 and It is important to determine the values which - '- and anx approach when x x x = Q, x being the value of an angle expressed in circular measure (radians). Take any angle AOP (Fig. 92) less than 90 and denote it by x ; construct the angle AOP' equal to o AOP, and draw the tangents PT and P / T. These tangents will meet at I on OA produced. Draw PP' . Then OT is _L to PP' at its middle point M. By geometry, arc PP' > chord PP' ; also McPP' PM, and arc PA op> d -oi^* .'. x> sin x, and x < tan x. x 1 PT OP' sin > 1, and sin x s\ux cos x x 157 158 TRIGONOMETRY Ql Yl *y* ^11 Yl 'V 1 As x = 0, cos x = 1, hence - = 1, since - - lies between cos x and 1. AT 'A. /'sin x\ -, Hence as x == 0. limit ( ) = 1 . > x / This result may also be stated thus, as x = 0, sin x = x. A T tan x sin x /^sin xA / 1 Also - _ = (-_)[- X X COS X > X / ^COS . A sin x . -, , 1 .1 But as x = 0, - = 1, and - = - or 1. X COS X 1 Hence =lx 1, or 1. x AT -j. -, Or. as x = 0. limit ( - ) = 1. x arc A. P Since the number of radians in x= , it follows that as the angle x = 0, the number of radians in x = sin x, and also = tan x. In practical work, when x<2, sinx and tanx may be taken as = p without appreciable error. 116. Computation of the Tables of Trigonometric Func- tions. Since, as x = 0, sinx and x approach equality (Art. 115), the circular measure of a small angle is the same as the sine of that angle to a large number of decimal places. By the use of methods which are beyond the scope of this book it is found that the value of sin 1' and the circular measure of r coincide for the first fourteen decimal places. Hence in constructing tables which are to be correct for the first five decimal places, there will be no error in taking sin 1' = 1' (in radians). But, by Art. 101, I' = 3 ' 141592+ radians = .0002908882+ radians. 180 x 60 Hence sin 1' = .0002908882+. COMPUTATION OF TABLES 159 But cos r = Vl - sin 2 1' = Vl - (.0002908882+) 2 = .9999999577 + . sin 2'= 2 sin V cos r = 2 x (.0002909-)(.9999999577 + ) = .000582 + . sin 3' = sin (2' + 1') = sin 2' cos 1' + cos 2' sin r. From this the value of sin 3' may be computed. In like manner the sines of all angles less than 90 may be obtained. The cosines of these angles may be obtained similarly, or by use of the formula cos x = sin (90 x). The tangents of these angles may be computed by the use Q-I i"* /y of the formula tanx=- . To obtain the cotangents, the cosx formula cot x = tan (90 x) may be used. The above method of computing sines and cosines may be abbreviated thus : sin (x 4- y) + sin (x y) = 2 sin x cos y. (Art. 71) Let x = a + 2 b, and y=b. Then, by substitution, sin (a + 3 b) -f- sin (a + b) = 2 sin (a + 2 b) cos b. Whence sin (a -f 3 b) = 2 sin (a + 2 b) cos 6 - sin (a + 6). . . (1) In like manner, cos (a -f 3 b) = 2 cos (a + 2 &) cos 6 - cos (a + &). . (2) Let 6= r in (1) and (2). sin(a + 3 / ) = 2sin(a + 2 / )cosr-sin(a+r). . . (3) cos(a + 3') = 2 cos (a + 2') cos l'-cos (a + r). . . (4) Letting a = T, 0, 1', 2', ... in succession, we obtain from < 3 ) sin 2' = 2 sin 1' cos r. sin 3' = 2 sin 2' cos r - sin 1'. sin 4' = 2 sin 3' cos 1' - sin 2 X , etc. 160 TRIGONOMETRY Similarly from (4), cos2'=2cosl'-l. cos 3' = 2 cos 2' cos V cos Y. cos 4' = 2 cos 3' cos Y cos 2', etc. 117. Computation by the Use of Series. The computation of the numerical values of the trigonometric functions is, however, performed much more expeditiously by the use of certain trigonometric series than by the above method. The demonstration of these series lies beyond the scope of this work. The series are as follows : X 3 , X 5 X* , SmX = *- + - ^ x* 2 yf 17 x 3 , = z+- + - + + - - - 6 15 olo The student is aided in recalling these series by the fact that sin ( x) = sin x (Art. 63) ; hence sin x must equal a series composed of odd powers of x. The same is true of tan x. But since cos ( x) = cos x, cos x must equal a series composed of even powers of x. 118. Analytical Trigonometry. Theory of Functions. When trigonometry is treated in the way indicated in cer- tain preceding articles, it ceases to be merely an instrument for solving triangles and becomes the theory of quantities varying in certain periodic or rhythmic ways. Also by the use of the so-called imaginary quantities, the subject of trigonometry is still further extended. Thus, for instance, denoting V 1 by the symbol i, it is shown that (cos x + i sin x) n = cos nx + i sin nx (called De Moivre's Theorem). COMPUTATION OF TABLES 161 By the aid of this theorem and similar principles, trigo- nometry gains much additional power. This branch of the subject is termed analytical trigonometry (though it is some- times treated as a part of higher algebra). When trigonometry is extended in these various ways, it is also looked upon as a part of the larger subject, the theory of functions. EXERCISE 47 * 1. By use of De Moivre's Theorem obtain the formulas for sin 3 a/- and cos 3 x, By use of this theorem we obtain (cos x + i sin a;) 3 = cos 3 x + i sin 3 x. But (cos x + i sin x) 3 = cos 3 x -f- 3 i sin x cos 2 x -\- 3 i 2 sin 2 x cos x-\- i s cos 3 x. .'. cos 3 x -f- i sin 3 a; = cos 3 x 3 sin 2 x cos x -\- i (3 cos 2 x sin x sin 3 x). By a theorem of algebra, in an identical equation containing both real and imaginary quantities, the sum of the reals in one member is equal to the sum of the reals in the other member, and so with imagi- naries. Hence, cos 3 x = cos 3 x 3 sin 2 x cos x = 4 cos 3 x 3 cos x sin 3 x = 3 cos 2 x sin x sin 3 x = 3 sin x 4 sin 3 x. In like manner, by De Moivre's Theorem, prove : sin 4 x = 2 sin 2 x (1 2 sin 2 a;), 2 1 cos 4 x = 8 cos 4 x 8 cos 2 x + 1. f sin 5 a? = 16 sin 5 x 20 sin 3 a; -h 5 sin x, 3 I \ cos 5 a? = 16 cos 5 x 20 cos 3 x + 5 cos a?.' 4. sin 7 x 7 sin a; 56 sin 3 x + 112 sin 5 a,* 64 sin 7 x. 7*0-1) 5. cos nx = cos n x- 4 cos w - 2 x sin 2 x 2)(n - 3) - cos 71 " x sin x 1 " 4 6. sin 7ix = n cos"- 1 a; sin x - ^- ' ^ v -' cos n - 3 x sin 3 a? n (n - 1) (71 - 2) (?i -3) (n - 4) n _ 5 . 6 ~W 7. tan2x = -^ tana; 1 tan 2 x 8. Find the value of sin 225 by use of the formula for sin 5 a; in Ex. 3. CHAPTER XI HISTORY OF TRIGONOMETRY 119. Epochs in the History of Trigonometry. The begin- nings, or germs, of Trigonometry are found in the Rhind Papyrus, now preserved in the British Museum. This papy- rus, the oldest known mathematical document, was written by a scribe named Ahmes about 1400 B.C., and is a copy, so the writer states, of a more ancient work, dating, say, 3000 B.C., or several centuries before the time of Moses. In dealing with pyramids, Ahmes makes use of two of the trigonometrical ratios, viz. : that between a lateral edge of a pyramid and diagonal of the base, corresponding to the co- sine of an angle ; and another which corresponds to the trigonometrical tangent of the angle made by the lateral face of a pyramid with the plane of the base. This use of ratios is, however, too crude to be regarded as scientific trigonometry. We have the following principal epochs in the scientific development of Trigonometry : 1. Greek (at Island of Rhodes and Alexandria), 150 B.C.- 200 A.D. 2. Arab (in western Asia and in Spain), 650 A.D.-1200 A.D. 3. Hindoo, 450 A.D.-1100 A.D. 4. European, 1200 A.D.- We shall also find the three following principal stages in the development of trigonometry: I. (150 B.C.-1400 A.D.) Spherical Trigonometry studied as a part of Astronomy, with incidental use of Plane Trigonometry. 162 HISTORY OF TRIGONOMETRY 163 II. (1400 A.D.-1700 A. D.) Plane and Spherical Trigonom- etry studied as a part of Geometry. III. (1700 A.D.- ) Trigonometry as an independent science. PRINCIPAL MAKERS OF TRIGONOMETRY 120. Hipparchus. The founder of trigonometry as a science was Hipparchus, a Greek, born about 180 B.C. in Bithyiiia in the northern part of Asia Minor. Hipparchus studied at Alexandria and afterward retired to the Island of Rhodes, where he did his principal work. He was primarily an astronomer and determined, for instance, the length of the year to within six minutes. He created trigonometry as a tool or aid in his astronomical work. Hence the trigo- nometry used by him was almost exclusively spherical. 121. Ptolemy (87 A.D.-165 A.D.). The next great name- in the history of trigonometry is that of Ptolemy, also a Greek. He lived and did his work in Egypt at Alexandria. Like Hipparchus, Ptolemy was primarily an astronomer and used trigonometry merely as an aid in his astronomical investigations. He wrote a treatise on mathematical and astronomical topics, now known as the Almagest,* which was the standard authority in astronomy for 1200 years. The Almagest contains thirteen books, the first of which treats mainly of trigonometry. 122. Regiomontanus (or Johann Muller, 1436-1476 A.D.) was a German and studied at the University of Vienna. After doing important work in Germany he was called to Rome by the Pope to reform the calendar and was assas- sinated while in that city. The ephemerides calculated by * Ptolemy entitled his work fteyivrr) ^a^art/cT; tan & = - -s cot o> = - cos = a, obtaining sin = 141. Rheticus obtained the formulas sin 2 A = 2 sin A cos A 9 sin 3 A = 3 sin A 4 sin 8 A Romanus discovered the formula for sin (A + B). The formulas for sin (A B) and cos (AB) were published byPitiscus(1599). 142. Vieta (France, 1540-1603) gave the general formulas for sin nA and cos nA in terms of sin A and cos A- OTHER PROCESSES 143. Trigonometrical Series- The series for sin x and cos x in terms of powers of x and for sin" 1 x in terms of sin x were known to Sir Isaac Newton before the year 1669. Those for tan x and sec x in terms of powers of x and for tan" 1 x in terms of powers of tan x were discovered by Gregory (England, 1638-1675) in 1670. - 144. Inverse Circular Functions in their general form were introduced by John Bernouilli (1667-1748). 174 TRIGONOMETRY 145- Use of V 1 or i. John Bernoulli! first treated trigonometry as a branch of analysis. Among other alge- braic methods he introduced the use of V l y or i 9 into trigonometry and obtained real results by its use. For instance, by employing V 1 he obtained a series for tann< in term of powers of tan (f>. This use of i was followed up by Euler, who among other results obtained the formula (sin x + i cos x) n = sin nx 4- i cos nx known as De Moivre's Theorem. EXERCISE 48. GENERAL REVIEW 1. Simplify Iog 2 4 + 5 Iog 3 9 + i logic -1 - log M ViOOl. 2. Compute the value of x from the equation 5 x 3 = \/'.2784 3. Also from cos x = (.9387)*. (7.605) 8 VlO2 4. Also from tan x = ^ - (27.32)* 5. If x is an angle in the first quadrant and cos# = T 8 7 , find the value of sinx + tana: . cos x cot x 6. If x is an angle in the first quadrant arid 2 cos x = 2 sin x, find the value of tan x. 7. If tan x = -, find sin 2 cc. 6 8. If sin y = a and tan y = b, prove that (1 a 2 ) (1 + 6 2 ) = 1. 9. ABCD is a square. D is joined to _EJ, the midpoint of AB. Find the trigonometric ratios of Z ECD. 10. Determine the numerical value of sin 18 by use of the geometric method of inscribing a regular decagon in a circle. 11. If A is an angle in the first quadrant and tan A = , find the value of J"*^-9sin^ p cos A -f- g sin ^4 12. Which of the following statements are possible and which im- possible : (1) 16 sin a; =1. (2) 4 sec = 1. (3) 7 tan # = 30. GENERAL REVIEW EXERCISE 175 13. Prove that sec x + tan x = sec2 x + sec x tan x + tan x. tan a; + sec x 14. Prove that * x = 2 sm x sin a;. sin x 1 -j- cos x 15. Find the numerical value of 3 tan 3 30 sec 3 60 sin 2 90 tan 2 45 + 5 cos 90. 16. If tan 2 45 - cos 2 60 = y sin 45 cos 45 tan 60, find y. cos 2 - sec ^ tan - 17. If a; sin -cos 8 - = - J, find as, csc 4 cos e Solve each of the following right triangles, given : 18. A = 36 18' 6" [36.3], b = 217.9 ft. 19. 6 = 315.92 ft.,c = 814.23 ft. 21. B = 12 15' [12.25], c== 1001.4. 20. c = 900, b = 887. 22. ^4 = 1 20' [1.33], c = 872.56. 23. In a right triangle b = 426, J. = 38 45' [38.75]. Find a -f c and the area. 24. The hypotenuse of a right triangle is 5 ft. and one angle of the triangle is 30. Solve the triangle and find the area without the use of tables. 25. The area of a regular polygon of 11 sides is 80. Find the side, radius, and apotheni of the polygon. 26. In an isosceles triangle the leg is 21.7 and the area 32.51. Solve the triangle. 27. The legs of a right triangle are to each other as 5 : 9. Find the angles of the triangle. 28. On the steepest part of the Mt. Washington railway (Jacob's Ladder), there is a rise of 13^ inches for every 3 ft. of track. What angle does the track make with the horizontal? At this rate what would be the rise in one mile of track? Show that in a right triangle : 29. cos2^ = ^^. 30. sm3A- c 3 31. sm^-sin.B 2 +c 176 TRIGONOMETRY 32. Find the other trigonometric functions of A, when cos A = -| and A lies between 540 and 630. 33. Given sec x = f and x in the third quadrant, find the value of sin x + tan x ^os x + cot x 34. Find the trigonometric functions of 180 + x and of 270 x when tan # \. 35. For what values of x between and 360 is sin x-{- cos x positive, and for what values is it negative? 36. Find the numerical value of 3 sin 2 225 + 4 sin (- 120) tan 150 - \ cos 2 330 cot 750 + 5 sin 2 180. 37. For each of the following angles state which of the three princi- pal trigonometric ratios are positive : (1) 460. (2) -220. (3) -1200. (4) ^ 38. Trace the changes in sign and magnitude of sin A between and 360. esc A between and IT. cos x between ?r and 2 ?r. tan A between - 90 and - 270. 39. If A is in the third quadrant and tan A = -f^, find the value of sin 2 A 40. Express the cosine of an angle in the second quadrant in terms of (a) each of the other trigonometric functions of the given angle, (b) the cosine of the complement of the angle. 41. If sin A = -}-| and sin B = f , and A and B are both acute, find the numerical value of tan (A + B)\ also of tan (A B). 42. If x is an angle in the second quadrant and sin x = f , find the value of sin 2 x + cos 2 x. 2 Q 5 B 43. Express 2 cos cos as a sum or difference. o o 44. If sin \ x = 1, find the numerical value of cos x. Also of tan x. Prove that: 45. sin 2 (A + B) sin 2 (A B) = sin 2 A sin 2 B. 46. = cot$ a; . 47. sin 50 + sin 10 = sin 70. cos 3 x cos 4 a? GENERAL REVIEW EXERCISE 177 48. sin 2 15 + cos 2 15 = 1. 49. cos 55 + sin 25 = sin 85. 50. cos A + cos 2 A -f- cos 3 A 51 ' l + tan 2 (45-x) = 0\_ cos f--f 52. smi Solve each of the following oblique triangles, given : 53. A = 30 18' 12" [39.3], b = 3294, c = 2846. 54. .4 = 76 24' 36" [76.41], B = 48 42' [48.7], c = 1012. 55. a = 850, b = 760, c = 590. 56. B = 46 18' [46.3], b = 213.76, a = 192.72. 57. b = 927, ^4 = 79, B = 21 17' 12" [21.29]. 58. a = V3, 6 = V2, c = V5. 59. ^ = 51 30' [51.5], a = 294.6, 6 = 301.7. 60. a = 926.8, 6 = 842.5, C= 46 27' [46.45]. 61. Solve the triangle in which K= 20.602, a = 214.2, and b = 315.8. 62. The diagonals of a parallelogram are 347 and 264 ft., and the area of the parallelogram is 40.437 sq. ft. Find the sides and angles of the parallelogram. 63. The diagonals of a quadrilateral are 34 and 56, and they inter- sect at an angle of 67. Find the area of the quadrilateral. Solve the following equations for answers not greater than 360 or less than : 64. sec x -f tan x = V3. 67. 2 sin x sin 3 x sin 2 2 x = 0. 65. sec 2 x + cot 2 x = */. 68. sin 2 + sin = cos 2 6 + cos 0. 66. sin 2 x V3 cos x. 69. sin 2 y + V3cos 2 y = 1. 70. sin(60 -z)-sin(60+x) = iV3. 71. Give the answers to Exs. 64-70, in the unlimited form. 178 TRIGONOMETRY 72. If 2 cos 2 x 1 cos x + 3 = 0, show that there is only one value for cos a;. 73. Find the least possible positive value of which will satisfy the equation 2 V3 cos 2 6 = sin 6. 74. Solve sin x + sin 2 x + sin 3 x = 1 4- cos x + cos 2 a?. 75. If sin 3 a; + sin 2 a? = sin x, find tan #. 76. Find the length of an arc intercepted by an angle of 2.2 radians at the center of a circle whose radius is 5 ft. How many degrees in this angle ? 77. Two angles of a triangle are .5 and .4 radians. Find the third angle in radians and in degrees. 78. The sura of two angles is 2 radians, their difference is 10. How many radians are there in each of these angles ? 79. Prove cos f + x\- cos f - x\ = 2 sin x. Q K "1 Q 80. Find the numerical value of - sin 2 - + 4 cos 2 - ^ tan 2 - 81. If sin /x + ^ jsinfa ^ ) = -, find #. N / ./ 82. Simplify tan ( - x] + tan ( + A \4 ) \ 4 ) 83. An angle of 30 at the center of a circle subtends an arc AB of length - ft. Find the length of the perpendicular dropped from A on 3 84. Express each of the following angles in degrees : sin- 1 !-; COS-4V2; tan-^-l); sin^-l); coa- 85. Find tan 86. Prove that tan- 1 2 + tan- 1 ! = | 87. Find the value of x, if tan- 1 x + 2 cot" 1 a; = ~ o 88. How many degrees in sin- 1 ( |-V2) ? How many radians ? 89. Prove sin" 1 a = sec" 1 Vl-a 2 GENERAL REVIEW EXERCISE 179 90. Solve the following for x and y : sin- 1 x + sin- 1 y = 120. cos- 1 x - cos- 1 y = 60. 91. At a point 50 ft. from the base of a tower the angle of eleva- tion of the top of the tower was found by the use of a transit instru- ment to be 68 18' [68.3]. If the height of the instrument above the ground was 4.75 ft., what was the height of the tower ? 92. If the railway up Pike's Peak rises 7552 ft. in 8J mi., what angle does the railway make with the horizon on the average ? 93. Two towers are 240 and 80 ft. high, respectively. From the foot of the second the angle of elevation of the top of the first is 60. Find, without the use of tables, the angle of elevation of the second from the foot of the first. 94. An unknown force combined with one of 128 Ib. produces a resultant force of 200 Ib. The resultant makes an angle of 18 24' [18.4] with the known force. Find the magnitude of the unknown force and the angle which it makes with the known force. 95. A tree 82 ft. high stands at one corner of a garden which is in the form of an equilateral triangle. The distance from the top of the tree to the midpoint of the opposite side of the garden is 112 ft. Find a side of the garden. 96. If the earth's radius (3956 mi.) as viewed from the sun sub- tends an angle of 8.8", find the distance of the earth from the sun. 97. In a circle whose radius is 13.7, find the area of a segment whose angle is - radians. 98. In order to determine the breadth of a river, a base line of 500 yd. was measured on one shore, and at each end of the base line the angle included between the base line and a line to a rod on the other bank was measured. These angles were found to be 53 and 79 12' [79.2], respectively. What was the breadth of the river ? 99. If a barn is 40 X 80 ft., and the pitch of the roof is 45, find the length of the rafters and the area of the entire roof, the horizon- tal projection of the cornice being 1 ft. 100. If the sun's angle of elevation is 60, what angle must a stick make with the horizontal in order that its shadow on a horizontal plane may be the largest possible. 180 TRIGONOMETRY 101. If a railroad rises 1 ft. for every 1000 ft. of its length, what angle does it make with the horizontal ? 102. In surveying a circular railroad curve successive chords of 100 ft. each are laid off. Find the radius of the curve, if the angle between two successive chords is 177. 103. If the diagonal of a regular pentagon is 32.835, what is the radius of the circumscribed circle ? 104. The angle x is in the third quadrant and cos x = f ; find the value of esc x, tan x, sin ^ x, tan (180 x), and sin x. 105. Find all the values of x between and 360 which satisfy the equation sin (30 - a;) = cos (30 + x). 106. If x is an angle in the second quadrant, prove geometrically that tan (270 + x) = cot x. 107. One angle of a rhombus is 60 and the opposite diagonal is 5 inches. Without the use of tables find the sides of the rhombus and its area. 108. Give a general formula for all angles whose sine is -J-. Is J. Is -1. 109. Express cos 2x in terms of each of the functions of x. 110. Express cos A cos B as a sum. 111. If cos A = h, and tan A = k, find the equation connecting h and Jc. 112. How many radians in each interior angle of a regular hexagon ? In each exterior angle ? How many degrees in each of these angles ? 113. Prove that cos' 1 f f + .2 tan' 1 i = sin- 1 f . 114. , 3 an x cos x 115. In the isosceles right triangle ABC, D is the midpoint of AC. Prove without the use of tables that cot /. ABD : cot Z DEC =2:3. 116. If 6 lies between 180 and 270, and 3 tan = 4, find the value of 2 cot = 5 cos 4- sin 0. 117. Is it possible to have an angle whose tan is 503 ? Whose cos is | ? Whose secant is ^ ? Whose sine is 23 ? 118. Show that cos 80 + cos 40 cos 20 = 0. 119. That 2 sin f x + - sin ( x - == sin 2 x cos 2 x. GENERAL REVIEW EXERCISE 181 120. If sin (60 - x) - sin (60 + x) == i V3, find tan 2 a. 121. Express 2 sin 9 A sin ^4 in the form of a sum or difference. 122. Find the value of sin^i + Stau^jVS 2cot -1 l + sec -1 l, using values between and 90 123. If tan 2x = - 2 T 4 -, find tan x and sin x, it being given that x is an angle in the third quadrant. 124. Find by inspection one value of x when cos (10 + A] cos (10 A) + sin (10 + A) sin (10 -A) = cos x. 125. A surveyor standing on a bank of a river observes the angle subtended by a flagpole on the opposite bank to be 33 10' [33.17] and when he retires 120 ft. from the bank he finds the angle to be 18 16' [18.27]. Find the width of the river. 126. Develop cos (270 x y) in the shortest way. 127. What is the angle of elevation of the sun when the length of the shadow of a pole is V3 times the height of the pole? 128. If tan A = f and sin B = if, and A is in the third quadrant and B in the second, find sin (A -f JS), cos (A + .B), tan (A -f- -B). 129. At the Panama Canal the Gatun dam has three different slopes : the ratio of the horizontal to the vertical near the base is 16 to 1 ; in the middle of the dam this ratio is 8 to 1 ; and at the top the ratio is 4 : 1. What three different angles does the surface of the dam make with the horizontal ? 130. If A is an angle in the first quadrant, and sec 2 A esc 2 A 4 = 0, find the numerical value of cot A. 131. If is an angle in the third quadrant, and sec 2 = 2 + 2 tan 0, find sin 0. 132. Find all the values of x between and 500 which satisfy the equation tan (45 x) + cot (45 x) = 4. 133. Graph y = sin" 1 x. 134. Also, y = tan' 1 x. 135. From. the top of a' mountain 3 mi. high, the angle of depression of the horizon is 2 13' 50" [2.23]. Hence determine the diameter of the earth. 136. Can an angle exist such that 9 sin 2 x + 3 sin x = 20 ? Why ? 137. Find the numerical value of tan 2 -+- cos 2 -^ -f- sin 2 ^- 3 46- 138. Find the sines of all angles less than 2 TT whose tangents are equal to cos 135. 182 TRIGONOMETRY 139. Given cos f - + x J = a, find cot f + x\ 140. What is the most general value of x which satisfies both of the equations cot x = V3 and esc x = 2. 141. Show that 2 sin f- + A\ cos f- -f B\ = cos ( J. + B) + sin (4 - J3). 142. Find the length of a circular arc whose radius is 5 ft. and whose subtending angle is 3 units of circular measure. 143. In the triangle ABC, B is 45, and C is 120, and a is 40. Without the use of tables find the length of the perpendicular drawn from A to BC produced. 144. Prove that - T i*Si**- = tan a. 1 -f cos x -f- cos 2 x 145. When y = ^, find the numerical value of 4 sin 2 y cos 2 y + 2 tan y sec 2 y. 146. Prove the identity sin" 1 y + tan- 1 y = sin' VT 147. Is sin a; 2 cos x + 3 sin a? 6 = a possible equation ? 148. A vertical pole stands at the center of a circular mill pond and rises 100 ft. above the surface of the water. From a point on the shore the angle of elevation of the top of the pole is 20. Find the area of the pond. 149. When the planet Venus is most brilliant, its diameter subtends an angle of 40" as seen from the earth. If the diameter of the planet is 7600 mi., what is the distance of the planet from the earth at such a time? 150. Verify the statement -cot 2 - + 3sin 2 ?-2csc 2 ^-?tan 2 ^ = 36 3 3463 151. Find the value of sin x, if tan ( - + ] tan ( ^ x ) -f 2 = 0. \3 J \3 / 152. What sign has sin x cos x for the following values of x : 140% 278, -356, -1125? 153. If 1 + sin 2 x = 3 sin x cos x, find tan x. 14. If i denotes the angle of incidence of a ray of light falling on water, and r the angle of refraction, and ^-^ = 1.423, find r when < = 34.37. smr GENERAL REVIEW EXERCISE 183 a 2 -4- b 2 155. When is sin# = - possible, and when impossible? 156. Show that 157. Solve sin 2 x cos 2 x sin x -\- cos x = 0. 158. Solve x = sin" 1 1 + tan~ J 1. 159. Trace the changes in sign and magnitude of - - - as x in- cos 2 creases from to - 160. Two trains leave a railroad crossing at the same time on straight tracks, including an angle of 21 12' (21.2). If they travel at the rate of 40 and 50 mi. per hour respectively, how far apart will they be in 45 ruin. ? 161. Show that = CQt , A j- Qt ^ _ , . 11 Ia4-b , la b 2 sin A 162. In a right triangle show that \/ h\/ = = *a b v a + b y cos 2 B 163. Prove _ = esc A 164. In any triangle prove that c = a cos J5 + & cos ^4, and hence show that sin (A + B) = sin A cos 5 4- cos ^1 sin 5. 165. Determine the angles in a right triangle in which a > 6, and c a = a &. 166. Prove cos 2 (a; y) 2 cos (# y} cos # cos y = sin 2 x cos 2 y. 167. If sin x cos a; + 4 cos 2 x = 2, find the ratio of tan x to sec x. 168. If ^ + = 225, prove that . \l + cot A) \1 -f cot y 2 169. The shadow of a. tower is found to be 60 ft. larger when the sun's altitude is 30 than when it is 45. Find the height of the tower without the use of tables. 170. A workman is told to make a triangular enclosure having 50, 41, and 21 yd. as its sides. If he malgdlthe first side one yard too long, of what length must he make the other two sides in order to inclose the required area, and keep the perimeter of the triangle unchanged ? 171. If sin A is a geometric mean between sin B and cos B, prove cos 2 A = 2 sin (45 - B) cos (45 -f B). 184 TRIGONOMETRY 172. If the diameter of the earth's orbit about the sun is 186,000,000 miles, and this diameter when viewed from the nearest fixed star sub- tends an angle of 1.52", find the distance of the star from the earth. 173. In a circle whose radius is 111.3 find the area inclosed between two parallel chords, on the same side of the center whose lengths are 129.3 and 97.4. . 174. If 2 tan- 1 x = cos- 1 - - cos" 1 - find x. ' 175. If tan 2 (180 - x) - sec (180 + a?) = 5, find cos x. 176. In order to fix the distance between two islands C and D, a base line, AB, 900 ft. long, is measured on the shore. Also, Z BAG was found to be 110 50' [110.83], Z DAB, 67 51' [67.85], Z CBA, 49 51' [49.85], ZABD, 85 19' [85.32]. What was the distance between the islands? SPHERICAL TRIGONOMETRY CHAPTER XII INTRODUCTION 146. Need and Utility of Spherical Trigonometry. Illus- trations- In case two places on the earth's surface, as A. and H, have the same longitude, RS, and their latitudes, RA and RH, are known, the number of miles in the arc AH may be readily determined by geometry (regarding the earth as a sphere). Let the pupil explain how. Also if two places, as A and B, have the same latitude, the number of miles in the arc of a small circle connecting them may be computed by plane trigonometry (see Art. 99). But if the longitudes of two places, as A and (7, are dif- ferent, and also their latitudes, the number of miles in the arc of a great circle AC connecting them cannot be deter- mined either by geometry or plane trigonometry. It can be determined, however, by taking the spherical triangle APC in which the two sides and the included angle are known (let the pupil point out these known parts), and solving the triangle by methods which are now to be considered. 147. Spherical Trigonometry is that branch of mathema- tics which treats primarily of the solution of spherical tri- angles. It will be found that when any three of the six parts of a spherical triangle are given, the other three parts 185 186 SPHERICAL TRIGONOMETRY may be found. Thus a spherical triangle differs from a plane triangle in that when three angles are known the three sides may be found. Since a trihedral angle is closely related to a spherical triangle, it will be found that spherical trigonometry also determines the remaining parts of any trihedral angle when three parts are given. Since certain definitions and principles of spherical geom- etry are frequently used in spherical trigonometry, it will be useful to make, at the outset, a brief statement of the lead- ing principles of spherical geometry. REVIEW OF SPHERICAL GEOMETRY 148. Definitions. A sphere is a solid bounded by a sur- face every point of which is equally distant from a fixed point within called the center. (Every section of a sphere made by a plane is a circle.) A great circle is a circle whose plane passes through the center of the sphere. What is a small circle ? The axis of a circle of a sphere is that diameter of the sphere which is perpendicular to the plane of the circle. The poles of a circle are the extremities of its axis. A spherical triangle is that portion of the surface of a sphere which is bounded by three arcs of great circles each less than a semicircumference. If a great circle be made to pass through any two points on the sur- face of a sphere, the great circle will be divided into two arcs by the points. If these arcs are unequal, the smaller arc is less than a semi- circumference. It greatly simplifies the subject of spherical trigonom- etry to consider only triangles bounded by arcs each less than a semi- circumference unless there be some special reason for the contrary. Let the student define birectangular triangle, trirectangu- lar triangle, quadrantal triangle. INTRODUCTION 187 A polar triangle is a triangle formed by taking the ver- tices of a given spherical triangle as poles and describing arcs with a radius equal to a quadrant of a great circle of the sphere. 149. Properties of Points and Lines on a Sphere. 1. Any two great circles of a sphere intersect at points 180 apart, i.e. they bisect each other. For the plane of each of the two great circles passes through the center, hence their line of intersection passes through the center and is a diameter. 2. The pole of a great circle is at a quadrant's distance from each point on the great circle. For the polar axis makes a right angle with each radius of the great circle, and a right angle is measured by a quadrant. 3. But one great circle can be made to pass through two points less than 180 apart on the surface of a sphere. For the plane of a great circle must also pass through the center of the sphere, and three points determine a plane. 4. If a point is' at a quadrant's distance from two other points on a sphere, it is the pole of the great circle through those points. 150. Properties of a Spherical Triangle. In spherical geometry it is also proved that in any spherical triangle 1. The sum of any two sides is greater than the third side. 2. The greater side is opposite the greater angle and vice versa. 3. The sum of the three sides lies between and 360. 4. The sum of the three angles lies between 180 and 540. 151. Polar and Supplemental Properties. Of especial importance are the polar and supplemental properties of spherical triangles. 188 SPHERICAL TRIGONOMETRY opposite in the other triangle. Thus in Fig. 96, A + a' = 180, 1. In a spherical triangle and its polar each vertex is the pole of the side opposite in the other triangle. Thus if A'B'C' be constructed as the polar triangle of ABC, then reciprocally is ABC the polar of A'B'C'. 2. In a spherical triangle and. its polar each angle is the supplement of the side '4-6 =180, <7'+c=180. 152. Relation of the Parts of a Trihedral Angle to the Parts of a Spherical Triangle. The planes of the great cir- cles which make three sides of a spher- ical triangle meet at the center of the sphere and thus form a trihedral angle whose vertex is the center of the sphere. It is of much importance to observe the relation between the parts of the trihedral angle and the parts of the spherical triangle. The face angles of the trihedral angle, viz. AAOC, AOB, HOC, are measured by (i.e. are equal to, or contain the same number of degrees as) the sides or the spherical triangle, viz. AC, AB, and BC. Also the dihedral angles of the trihedral angle are equal to the corresponding angles of the spherical triangle ; thus the dihedral angle C-OA-B has the same measure as the spherical angle CAB, viz. the plane angle 2 AS made by the two straight lines TA and AS tangent to the arcs AC and AB respectively, and therefore perpendicular to the radius OA. . Hence the six parts of the trihedral angle correspond to the six parts of the spherical angle ABC. FIG. 97. INTRODUCTION 189 A property of the six parts is sometimes perceived or derived more readily from these parts as arranged in the trihedral angle, sometimes more readily from the spherical triangle. In general, it is more convenient to obtain methods of solution from the trihedral angle; on the other hand, the solution of problems relating to the trihedral angle are usually obtained more readily by use of the spherical triangle. EXERCISE 49 1. If PP 1 is the diameter of a sphere, and Q any point on the sur- face of the sphere except P and P', show that the sum of the arcs PQ and P'Q is constant. 2. What must the sides and angles of a spherical triangle be in order that the triangle may coincide with its polar ? 3. How large must the sides of a spherical triangle be in order that its polar lie wholly within the triangle ? 4. By use of cardboard construct a spherical triangle whose sides are 45, 60, and 60, the radius of the sphere being 3 in. 5. Also (by aid of cardboard and a protractor) construct a spheri- cal triangle whose sides are 40, 55, and 65. Also construct the polar of this triangle. 6. Make up and work a similar example for yourself. 7. If A, B, (7, be the angles of a spherical triangle, show that B + C> 180 - A. (Use Art. 150, 4.) 8. Also show that B -f C < 180 + A. (Draw the polar triangle and use Art. 150, 1.) 9. Hence show that the spherical excess of a spherical triangle must be less than twice the smallest angle. 10. If two angles of a triangle are 55 and 110, find the maximum value of the third angle. 11. If each of the legs of a right spherical triangle is less than 90, prove that the hy- potenuse and oblique angles are each less than 90. (SUGGESTION. Let ABC be the right spheri- cal triangle, and construct the trirectangular triangles A'B'C, AB'D, ABE.) 190 SPHERICAL TRIGONOMETRY 12. If the le'gs of a right spherical triangle are unlike in species, show that the hypotenuse is greater than 90, and that the angle oppo- site the greater leg is obtuse. (SUGGESTION. Produce the hypotenuse and one leg to form a lune.) 13. If both legs are greater than 90, show that the hypotenuse is less than 90 and that both oblique angles are obtuse. CHAPTER XIII THE RIGHT SPHERICAL TRIANGLE 153. Trigonometric Properties of the Right Spherical Triangle. On a sphere with center and unit radius, let ABC be a spherical tri- angle in which Z A CB is a right angle. Hence' plane OBC is given _L plane OAC. From B draw BD JL OA and meeting OA in D. Also in the plane OAC, from D draw Z)^ 7 JL CU and meeting OC inF. Draw 57^. Then OD is _L plane DBF (Geometry, Art. 509). Hence plane DBF is _L plane 0^4. C which passes through OD (Geometry, Art. 555). Since planes DBF and OBC are both JL plane OAC, BF is _L plane AOC (Geometry, Art. 560). Hence BF is _L both OF and DF (Geometry, Art. 505). Hence we have two right triangles as follows : FIG. 99. II cos C FIG. 100. D D ff FIG. 101. 191 192 SPHERICAL TRIGONOMETRY / From I, by Art. 41, cos c = cos a cos &. / ... (1) From II, by Art. 41, sin a = sin c sin A ..... (2) Similarly by drawing lines from A instead of from B (Fig. 99), it may be proved that sin & = sin c sin &'? . . t . . (3) Also from I, DF = cos a sin &, and from II, DF = sin c cos ^1. By Ax. 1, sin c cos A = cos a sin b. Substituting for sin b from (3), sin c cos A = cos a sin c sin J5. Whence cos A = cos a sin J5.A ... (4) In like manner, cos It = cos & sin ^l./. ... (5) From (1), cos c = cos a cos &. Substituting for cos a and cos b from (4) and (5), _ cos A cos B ~ 7} ' ~ 7 ? sin B sin ^L or cos c = cot A cot J5: . ... (6) From (2), sin a = sin c sin J_. Substituting for sin c and sin A from (3) and (5), -, sm jfc> cos b or sin = tan & cot JB.^ ... (7) Similarly, sin b = tan a cot ^/ . . . . (8) From (4), cos A = cos a sin 5. Substituting for cos a and sin ^ from (1) and (3), ,, cos c sin b cos A = --- , cos b sin c or cos A = cot c tan b// . ... (9) Similarly, cos B = cot c tan a/ . . . , (10) THE RIGHT SPHERICAL TRIANGLE 193 FIG. 102. In the above proof it has been assumed that the parts of the given triangle other than the right angle are each less than 90. But the ten formulas proved can be shown to be true in a B similar manner when the parts of the triangle are greater than 90. For instance, if the leg a be greater than 90 and b less than 90, we have the adjoining diagram (see Ex. 12, p. 190) in which, in A ODF, cos c = cos a cos b. Similarly the other nine may be proved true under the given conditions. As to the derivation of the formulas by use of the ratio definitions of the trigonometric functions, see Art. 60. 154. Napier's Rule of Circular Parts. The ten formulas proved in Art. 153 may be reduced to a single rule by the use of what are called circular parts. The circular parts of a right spherical triangle are the parts of the triangle modi- fied by omitting the right angle and taking the complement of the hypotenuse and of the angles adjacent to it. Thus use of the circular parts in symbols are co. A, co. c, co. B, a, b. co.JS FIG. 104. Any one of the five circular parts may be taken as the middle part; the two parts adjacent to the part thus taken 194 SPHERICAL TRIGONOMETRY are then called the adjacent parts ; and the remaining two. parts are called the opposite parts. Thus, if b is taken as the middle part, the adjacent parts are co. A. and a, and the opposite parts are co. c and co. B. Napier s Rule for Circular Parts is then as follows : The sine of the middle part = the product of the tangents of the adjacent parts, or of the cosines of the opposite parts. It is an aid in memorizing this rule to observe that in the leading word of each of the three parts of the rule, the first vowel in the two distinctive words is the same. Thus i is the first vowel in sme and 4fc/ddle, a in tangent and adjacent, and o in cosine and opposite. As an illustration of the application of Napier's Rule, if b be taken as the middle part, we have sin b = tan (co. A) tan a = cos (co. c) cos (co. B) = cot A tan a = sin c sin B (see (8) and (3) of Art. 153). Let the pupil obtain in like manner the eight other formulas of Art. 153, by taking each of the circular parts as the middle part in turn. 155. Application of Napier's Rule to the Solution of the Right Spherical Triangle. By use of the ten formulas of Art. 153 or by Napier's Rule, any two parts of a right tri- angle being given any other part may be found. In apply- ing Napier's Rule, if the two given and the one required parts are adjacent, take the middle one of the three parts as the mid- dle part, the other two as the adjacent parts. If the three parts are not adjacent, take the part standing alone as the middle part and the other two of the three parts as the opposite parts. Ex. 1. Solve the right spherical triangle in which a = 45 15', c = 72 30'. 4515 / TJ AT > - By Napier's Rule cos B = tan 45 15' cot 72 30 r . 45 15' log tan 0.00379 72 30' log cot 9.49872 - 10 B = 71 27' 16" log cos 9.50251 - 10 Also THE RIGHT SPHERICAL TRIANGLE cos 72 30' = cos b cos 45 15'. 195 Also cos 45 15' 72 30' log cos 9.47814 - 10 45 15' colog cos 0.15242 b = 64 42' 51" log cos 9.63056 - 10 sin 45 15' = sin A sin 72 30'. sin 45 15' .*. sin A = sin Y. 30'' 45 15' log sin 9.85137 - 10 72 30' colog sin 0.02058 A = 48 7' 44" log sin 9.87195 - 10 After solving any triangle, as a check formula use that formula involving the three required parts. Thus, in the above example use cos 71 27' 16" = sin 48 7' 44" cos 64 42' 50". Also 48 7' 44" log sin 9.87195 - 10 64 42' 51" log cos 9.63056 - 10 71 27' 16" log cos 9.50251 - 10 Ex. 2. In the right spherical triangle in which A = 64.25 and . = 48.4, find b. tyo/r' Taking co. B as the middle part we have cos 48.4 = sin 64.25 cos b, cos 48.4 hence, cos b = sin 64.25 48.4 log cos 9.8221 - 10 64.25 colog sin 0.0454 b = 42.51 log cos 9.8675 10 156. Species of Parts Found. Where quantities greater than 90 are used, in case any part given (or used) is greater, than 90, it is important to watch the signs (of the functions) carefully, since in the second quadrant the cos, tan, and cot are minus, and the sine is plus. If the cos, tan, Imd cot of a computed part is found to have a minus value, the angle ob- tained from the table is to be subtracted from 180, 196 SPHERICAL TRIGONOMETRY Ex. Find B in the right spherical triangle in which a= 150 and c 80. - + cos B = tan 150 cot 80 FIG. 107. It is convenient to write the sign of each factor above the factor as is done above. Since tan 150 is minus and co J80 is plus, cos B = product of a negative quantity by a positive quantity. Hence B is greater than 90. Let the pupil complete the solution. If the sine of an unknown part is found, since the sine of an angle and its supplement are the same and both plus, the acute angle found from the table and its supplement must both be solutions unless there are other conditions which make one or the other of the solutions impossible. Two of these conditions are as follows : In any right spherical triangle, an angle and the side oppo- site it must be of the same species } i.e. both greater than 90, or both less than 90. For since sin b = tan a cot A. and sin b is always + , tan a and cot A must be both + or both -. If both are 4- , a and A are both less than 90. If both are , a and A are both greater than 90. Hence in the above example, since B is greater than 90, AC must be greater than 90. Is A greater or less than 90? Also, in any right spherical triangle, the hypotenuse is less than or greater than 90 according as the two legs are alike or unlike in species. Let the pupil show that this follows from the formula cos c = cos a cos b. 157. Case of Two Solutions. When the parts given are a side and the angle opposite it, there are two triangles which THE RIGHT SPHERICAL TRIANGLE 197 answer the given conditions. This is readily seen from the figure. In the A ABC, let the A< given parts be the angle A and FlG 108 BC the leg opposite. Produce the unknown sides AB and AC to meet at A. Then arc ABA = arc A CA = 180. Also Z A'= Z. A and ^BCA= 90. Hence the known parts of the A A 'EC are the same as the known parts of ABC. From the method of constructing the figure it is also evident that the unknown parts of the one triangle are the supplements of the unknown parts of the other triangle. Hence also, to construct the two triangles, construct one triangle and then produce the hypotenuse and unknown leg till they meet. Ex. Solve the right spherical triangle in which 1} = 23 and ^=31. Taking a the middle part, sin a = tan 23 cot 31. 23 log tan 9.62785 - 10 31 log cot 0.22123 a = 44 56' 46" log sin 9.84808 - 10 and a' = 135 3' 14". Also cos 31 = cos 23 sin A, sn and Also cos 23 31 log cos 9.93307 - 10 23 colog cos 0.03597 A -68 37' 15" log sin 9.96904 - 10 ^" = 111 22' 45". sin 23 = sin c sin 31, and sin 31 23 log sin. 9.59188 10 31 colog sin 0.28816 c = 49 20' 44" log sin 9.88004 - 10 c= 130 39' 16". 198 SPHERICAL TRIGONOMETRY 158. Other Special Cases. Certain special cases often call for special treatment. Thus, if a and b are given, and c re- quired, and it is found that the value of c is close to either or 180, this value can be found with greater accuracy by first computing A or B, and then c. (Why is this ?) Also if a and c are given and b required, and it is found that the value of b is close to either or 180, the value of b can be found more accurately by use of the formula tan 2 J b = tan |(c - a)tan ^(c + a) (obtained from the formula for tan ^b of Art. 70, and substi- tuting for cos by from cos a = cos b cos c). Also in certain cases the data of the problem may be such that a solution of the problem is impossible. Thus, if A 27 and B= 35, by Art. 150, 4, a solution is impossible. A complete statement of the conditions which make the solution of a given triangle impossible is given in Art. 172, in those cases where A is taken as equal to a right angle. EXERCISE 50 Given parts as follows, solve the following right spherical triangles, checking results : (In working each example outline the work carefully before looking up any logs.) 1. a = 36, 6-83. 2. a = 21 15', c = 5448'. 3. ^1=64, 5 = 38. 4. c = 77 30', B = 48 18'. 5. a = 20 20' 20", B = 42 6' 40". 6. A = 54 54' 42", c = 75 15' 25". 7. A = 115 18' 36", b = 62 18' 24". 8. a = 132 6', b = 77 51'. 9. = 144 32' 24", c=120. 10. c = 99 15' 36", a = 133 31 ' 12". 11. A = 100, B = 154 37' 12". 12. B = 75 25' 12", b = 42 24'. THE RIGHT SPHERICAL TRIANGLE 199 13. A = 71, a = 37J 14. a -116 44' 12", ^=100 16' 24" (100.27). 15. a = 96 18' 24 ",B = 55 6' 15". 16. a = 56 15', b = 24 45'. 17. J[ = 69 52' 36", B = 105 18' 42". 18. b = 16 18', c = 38 25' 48". 19. c = 116 18' 30", A = 50 18' 36". 20. 5 =63 48', 6 = 41 12'. 21. .4 = 8 21', 5 = 87 15' 36". 22. ^ = 88 31 12", a = 87 43' 12". Find B only. 23. A = 89 24' 18", B = 88 31' 48". Find c only. 24. c = 160 30 '36", a =162 28' 48". 25. Why are we able to solve examples like the preceding in Spheri- cal Trigonometry and not in Spherical Geometry ? Solve by use of four-place tables, having given : 26. a = 36, b = 83. 36. A = 100, B = 154.62. 27. a = 21.25, c=54.8. 37. 5=75.42, 6=42.4. 28. ^4 = 64, 5 = 38. 38. A = 71, a = 37. 29. c = 77.5, jB = 48.3. 39. a = 116.74, A = 100.27. 30. a = 20.34, B = 42.11. 40. a = 96.31, 5 = 55.11. 31. A = 54.91, c = 75.26. 41. a = 56.25, 6 = 24.75. 32. A = 115.31, b = 62.31. 42. yl = 69.88, B = 105.31. 33. a = 132.1, 6 = 77.85. 43. b = 16.3, c = 38.43. 34. = 144.54, c = 120. 44. c = 116.31, .4 = 50.31. 35. c = 99.26, a = 133.52. 45. B = 63.8, 6 = 41.2. 46. A = 8.35, B = 87.26. 47. ^1 = 88.52, a = 87.72. Find B only. 48. ^1 = 89.405, B = 88.53. Find c only. 49. c= 160.51, a = 162.48. Prove that in a right spherical triangle, 50. sin 2 a -f- sin 2 b sin 2 c = sin 2 a sin 2 b. 51. sin 2 ^1 cos 2 c = sin 2 JL sin 2 a. 52. tan ^ (c +a) tan 2 (c a) = tan 2 ^- b. 200 SPHERICAL TRIGONOMETRY 53. sm*- = sin 2 %os 2 ^-f cos 2 ^ sin 2 ! 2i 2i 2i a a 54. tan'iB = "" -" sin (c -h a) 55. sin (c 6) = tan 2 \ A sin (c + V). 56. sin (c a) = sin b cos a tan B. 57. sin 2 A = cos 2 .B + sin 2 a sin 2 jB. 58. Prove that, for any angle A, tan J- (90 -4) = A/- ^4 and * 1 -j- sin A. hence for the right spherical triangle that tan 2 (45 - 1 A) = tan 1 (c - a) cot | (c + a). When is this formula useful in solving a right spherical triangle ? 59. Prove that for the right spherical triangle, tan 2 \ B = sin (c a)csc (c -f- a) and show when this formula is useful. 60. Treat in the same way tan 2 ^ c == cos (A -f- B) sec (A B) . Also tan 2 (45 - 1 c) = tan | (A a) cot % (A + a) . 159. Quadrantal Triangles. A quadrantal spherical tri- angle is one which has one of its sides equal to a quadrant. By the supplemental property of spherical triangles (Art. 151) the polar triangle of a quadrantal triangle is a right spherical triangle. Hence to solve a quadrantal triangle, Solve the polar triangle of the given quadrantal triangle and take the supplements of the results. 160. Isosceles Spherical Triangles. It is shown in spherical geometry that if the arc of a great circle be drawn from the vertex of an isosceles spherical triangle to the mid- point of the base, it will be perpendicular to the base, will bisect the vertex angle, and divide the isosceles spherical tri- angle into two symmetrical right spherical triangles. The solution of an isosceles spherical triangle is thus reduced to the solution of a right spherical triangle. EXERCISE 51 Given parts as follows, solve the following quadrantal triangles, checking results : THE RIGHT SPHERICAL TRIANGLE 201 I/ 1. A = 104 54' 42" [104.91], b = 144 30' 24" [144.51], c = 90. 2. a = 160, b = 105, c = 90. 3. .4 = 115 47' 24", 5 = 130 31' 12", c = 90. 4. 5 = 106 54', 6 = 100 48' 36", c=90. In the following isosceles spherical triangles, given a = b and parts as follows : 5. a = 56, c = 99, find A, B, C. 6. a = 75 5' 18", A = 35 29' 36", find C, c. 7. a = 52 30', (7= 129, find A, c. 8. c = 161 31', (7= 182 24', find A, a, b. 9. Show that the solution of a spherical polygon may be reduced to the solution of a right spherical triangle. f Solve by use of four-place tables, having given : 10. A = 104.91, b = 144.51, c = 90. 11. a = 160, b = 105, c = 90. 12. A = 115.79, 5 = 130.52, c = 90. 13. B = 106.9, 6 = 100.81, c = 90. Also in the isosceles spherical triangle in which 14. a = 56, c = 99, find A, B, C. 15. a = 75.09, A = 35.49, find C, c. 16. a = 52.5, (7= 129, find ^, c. 17. c = 161.5, <7 = 182.4, find A, a, 6. 18. In a quadrantal triangle in which c = 90, prove that tan a tan 6 + sec (7=0. 19. Compute the dihedral angles made by the faces of the five regu- lar polyhedrons. 20. Find the surface and volume of a regular dodecahedron whose edge is 10. 21. If the side of a spherical square is m, find the angle M of the square. 22. A marble cutter cuts a block of marble with a rectangular base, and four lateral edges, each making an angle of 45 with the base at its corners. What is the dihedral angle between any two adjacent lat- eral faces and also the inclination of each lateral face to the base ? 202 SPHERICAL TRIGONOMETRY 23. Each lateral face of a frustum of a square pyramid makes an angle of 81 with the base. What are the face angles at the corner of the base of the frustum, and also the dihedral angle between any two adjacent lateral faces ? 24. A monument has a rectangular base. One lateral face makes an angle of 72 30' [72.5] with the base, and one of the lateral edges bounding this face makes an angle of 54 48' [54.8] with the base. What angle does the adjacent lateral face make with the first face ? 25. Find the dihedral angle made by any two adjacent lateral faces of a regular twelve-sided pyramid, it being given that the angle at the vertex, made by two adjacent lateral edges, is equal to 20. 26. Collect, or make up, and work three examples containing con- crete applications of the solution of right, quadrantal, or isosceles spherical triangles. CHAPTER XIV OBLIQUE SPHERICAL TRIANGLES TRIGONOMETRIC PROPERTIES OP OBLIQUE SPHERICAL TRIANGLES 161. Law of Sines. In a spherical triangle the sines of the sides are to each other as the sines of the angles opposite. FIG. 110. Let ABC be a spherical triangle with CD a perpendicular drawn from C to AB. If this perpendicular fall between A and B, as in Fig. 110, in the right A ACD, by Art. 154, Also in the right A CDB, sin p = sin a sin B. .-. by Ax. 1, sin b sin A = sin a sin B, or sin a _ sin A sin b sin B In case the perpendicular CD falls outside of the triangle (Fig. Ill), the same relations are true, except that sin Z CBD is used instead of sin B. But sin Z CBD= sin(180 -B) = sin B. Hence the same result. is obtained as for Fig. 110. 203 204 SPHERICAL TRIGONOMETRY sin a sin A Similarly, and sin c sin C ' sin b = sin B sin c sin C 162. Law of Cosines in a spherical triangle. To obtain the relation between an angle and the three sides of a spheri- cal triangle in Fig. 110 of Art. 161, in right A AC I), we have %, 7 COS = COS ft COS X, and in right A BCD, cos a cosp cos(c x). Equating the values of cosp, cos a _cos 1} cos (c x) cos x -rj cos 1) cos(c-x) Hence, cos a = cos x _ cos b (cos c cos x -f sin c sin x) COS X = cos b cos c 4- cos b sin c tan x. . . . (1) But in right A ACD, , 7 , cos b tan x cos ^L = cot b tan x = sin b sin b cos A cos 6 Substituting for tan x in equation (1) above, 7 7 . sin b cos ^4 cos a = cos b cos c + cos b sin c - , cos b or cos a = cos & cos c + sin & sin c cos ^4. Similarly, cos b = cos r cos c + sin sin c cos 1, cos c = cos # cos b + sin sin & cos C. To obtain the relation of a side to the three angles of a spherical triangle. OBLIQUE SPHERICAL TRIANGLES 205 Let A'B'C' be the polar A of ABC. Then in A A'B'C', by the property proved above, cos a' = cos b' cos c' + sin &' sin c' cos A'. But a! = 180 -A, &'=180-.B, etc. cos(180 - A) = cos(180 - B) cos (180- C) + sin (180-^) sin (180- (7) cos (180 -a). Hence, - cos A = ( cos B)( cos (7) 4- sin B sin (7( cos a), or cos A = cos 1? cos C+ sin 1? sin C cos a. Let the pupil state the values of cos B, and of cos C, obtained similarly. 163. Formulas for the Half Angles. By the formulas ob- tained in Art. 162, when the three sides of a spherical triangle are given it will be possible to determine the angles. But with formulas as stated in Art. 162, it is not possible to use logarithms in the computations. To obtain formulas adapted to logarithmic computations we proceed as follows : By Art. 162, sin b sin c cos A = cos a cos b cos c. cos a cos b cos c Hence, cos A = sin b sin c Subtracting both members from unity, we have -i * -i cos a cos b cos c 1 cos A = 1 - ; - sin b sin c _ cos b cos c + sin b sin c cos a t sin b sin c Hence, by Arts. 70 and 67, (1) 2 sin 2 1 A = sin b sin c 206 SPHERICAL TRIGONOMETRY Hence, by Art. 71, ,_ sin b sin c 3 i j _si- - sn sn c Denoting the sum of the sides, a + 6 + c, by 2 s, we have and a . o -, ,, sin (s &) sin (s c) Hence. sin 2 1 A = '- , sm o sm c sini^ = 7 7 ~. - (2) sin 6 sm c 1 D A/sin (s c) sin (s a) /Q \ In like manner, sin A J3 * V- ? (o) sin c sin - - sm (* a sm ( s ~ and sin a sm Again, by adding unity to both members of (1), we have -, , cos a cos & cos c l-hcosJ. = l + - sin 6 sm c _ cos a (cos 1} cos c sin b sin c) sin 6 sin c Hence, by Art. 70, 2 1 , cos a cos (& 4- c) 2 cos 2 i J. = - sm 6 sin c Hence, by Art. 71, 2 , _ 2 sin |f?? + c + a)sin^-f?> + c a) W 7T -". : ~-. ; sm b sm c Putting a 4- & + c = 2 s, whence, 6 + c - a = 2 (s - a), we have o -, ,, sin s sin(s a) cos w ^ A = - sm o sm c Or, cos 1,1 = V- - (5) sm b sin c \ OBLIQUE SPHERICAL TRIANGLES 207 T vi 1 75 ^ /sin s sin (s 6) In like manner, cos f B = \ - -f- , sin c sin -, ~ ^ /sin s sin (s c) and cos-J<7=\- \ '. sm a sm 6 Dividing (2) by (5), we have tan 1 J. = \/ sm ( s ~ fr) sm ( s ~^) -J sin 6 sin c sin 6 sin c sin s sin (s a) = /sin (s b) sin (s c) /g\ sin s sin (s a) T vi 1 r> ^ /sin (s c) sin (s a) /m In like manner, tan I> = \- v 7X ? (9) sm s sin (s - 6) and, tanl^ 8 - -(10) sin s sin (s c) 164. Formulas for the Half-Sides. By A 1 ^- 1^2, sin ^ sin C cos a = cos A + cos ^ cos C. cos ^L + cos B cos 6^ /n >. Hence. cosa = - ..... (1) sin B sm C rrn -i cos A + cos B cos (7 Then, 1 - cos a = 1 - . sin B sm C 0-81 - (cos -5 cos (7 sin ^ sin (7) cos ^4. Ur, ^ sm ^ a = - ; ; - sin B sm C cos (B+C) + cos ^4 sin B sin (7 Hence, 2 sm 2 i a= - sin JD sin 6 Denoting the sum of the angles ^1 + B 4- (7 by 2 $, we have -rj -21 COS >^COS (/S' ^4.) Hence, sm 2 1 a = -- -A L - sm B sm 6 Or, .- sm ^ sin (7 208 SPHERICAL TRIGONOMETRY In like manner, + 1 cos S cos (SB) /Q \ = \- . : - ', .... (3) sin C sin ^1 . -, . cos $ cos (S C) //n and, smlc=\- -A =-^ ..... (4) sm A sin ^ Again, adding 1 to both members of (1), we have cos A + cos B cos C 1 + cos a = 1 + sin ^ sin (7 cos J- + cos B cos (7+ sin B sin (7 _ sin B sin ( cos J. + cos (B C) Then, 2 cos 2 i a = - .' . sin B sin (7 2 cosi^ + ^-<7 cos sin B sin (7 ,/ cos H^ + ^-ff] cos l [- Or, cos 2 A a = - - ^ : ^J- sin ^ sm (7 But A + B-C=2(S- C), and .A - B + (7= 2 Whence, cos j i a = - sm ^ sm C cos $cos (8 A) = ~cos (> g- In like manner, Jcos(S-B)cos(S-C) Or, cosi = \- . p . -V" -^. . . . (5) sm B sm (7 In like manner, ' S 7^ C . OS( f-^ ) , . (6) sm G sin ^L . /COS (A^ A) COS (/S 5) /^x and, coslc = \- : f. ^- -^. . . . (7) sm A sm J5 Dividing (4) by (5), we have, OBLIQUE SPHERICAL TRIANGLES 209 -, i %/ cos /S cos (/S (7) /lm and. tan*e = \- ^ y ^ . . .(10) cos(- J.) cos(S-B) 165. Gauss's Equations and Napier's Analogies. Since cos ( J. + B) = cos ( J. + \ B\ by Art. 66, cos |( J. + B) = cos l J. cos l J? - sin 1 J. sin 1 5. Substituting for cos ^ J-, cos 1 j5 ? sin 1 A, sin ^ ^, the values obtained in Art. 163, . Ism s sm (s a) ^ Ism s sin (s =\ ; 7 v . z \- sin 6 sin c sin a sm c _ ^/sin (s b) sin (s c) ..sin (s a) sin (s c) sin b sin c sin a sin c _ sin s sin (s c) ^/sin (s a) sin (s b) sin c sin a sin b But by Art. 71, sin s sin (s c) = 2 sin -| c cos (s -J c). Also by Art. 69, sin c = 2 sin \ c cos \ c. 2 sin l c cos c But *---c = - .-. cos l(^ + J9) - cosc Hence, cos^(^l + .B) cos^c = cos i(rc + 6) sin In like manner, sin J + 1* cos-^ = cos a b cos . ... cos 101 -J5) sin Jc=sinJ(-l-6) sin | C. ... Ill sin |(^ - J5) sin | c = sin ( - 6) cos \ C. . . . IV These four equations are called Gauss's Equations. Dividing II by I, tani(^ + ^) = COS ti a 7^ c <>t l C cos Dividing IV by III, tan(A-B)= g- cot i <7. 210 SPHERICAL TRIGONOMETRY 4 Dividing III by I, tan J (a + 6) = -f tan -| . cos Dividing IV by II, tan J (a - b) =- -* tan J c. sin -g- (A. -f- H) These equations are called Napier's Analogies. EXERCISE 52 1. In the first of Napier's Analogies, show- that tan \(A + B) and cos J (a + 6) must always have like signs. Show also that according as a + b < 180, = 180, or > 180, then A + B < 180, = 180, or > 180. 2. From the third of Napier's Analogies, show that according as A + B < 180, = 180, or > 180, then a + b < 180, = 180, or > 180. 3. State in general language the two laws of cosines (Art. 162). 4. State in general language the results obtained in Art. 163. 5. Also in Art. 164. 6. If a, 6, c are the sides of a spherical triangle, and a r , &', c' the sides of its polar triangle, prove that sin a : sin b : sin c = sin a' : sin b' : sin c'. 7. If the bisector of the angle A of the spherical triangle ABC be denoted by AD, and CD be denoted by b' and BD by c', prove sin b : sin c = sin 6' : sin c'. SOLUTION OP OBLIQUE SPHERICAL TRIANGLES 166, Cases in the Solution of Oblique Spherical Triangles. Six cases occur in the solution of oblique spherical triangles according as the parts given are I. Two sides and the included angle. II. Two angles and the included side. III. Three sides. IV. Three angles. V. Two sides and an angle opposite one of them. VI. Two angles and a side opposite one of them. OBLIQUE SPHERICAL TRIANGLES 211 CASE I. Two SIDES AND THE INCLUDED ANGLE GIVEN 167. To solve Case I first find the unknown angles by the use of the first two of Napier's Analogies ; the third side may then be found either by use of the third or fourth of Napier's Analogies, or by one of Gauss's Equations. Which of the methods of finding the third side involves the looking up of the fewest new logarithms ? Ex. 1. Given a = 68 20' 25", 6 = 52 18' 15", (7=117 12' 20", solve the spherical triangle. o= 68 20' 25" 6= 52 18' 15" a + b = 120 38' 40" $ (a + b) =60 19' 20", cot 58 36' 10". cot 58 36' 10". C= 58 36' 10", By the first two of Napier's Analogies, cos 8 1' 5" tan -J3) = cos 60 19' 20" sin 8 1' 5" sin 60 19' 20" 8 1' 5" log cos 9.99574 - 10 58 36 f 10" log cot 9.78557 - 10 60 19' 20" colog cos 0.30529 + B) = 50 40' 30" log tan 0.08660 8 1' 5" log sin 9.14453 - 10 58 36' 10" log cot 9.78557 - 10 60 19' 20" colog sin 0.06107 - B) = 5 35' 47" log tan 8.99117 Therefore, A = 56 16' 47", B = 45 4' 43". If we proceed to find c by the law of sines (Art. 161), we shall ob- tain two values for c both greater than a and we shall not know which of the two values is to be taken. Proceeding therefore, by the use of Gauss's first equation (Art. 165). cog i c = cos 60 19' 20" sin 58 36' 10" cos 50 40' 30" 212 SPHERICAL TRIGONOMETRY 60 19' 20" log cos 9.69471 - 10 58 36' 10" log sin 9.93124 - 10 50 40' 30" colog cos 0.19811 c = 48 10' 17" log cos 9.82406 - 10 c = 96 20' 34". CHECK. A log sin 9.92000 a log sin 9.96820 9.95180 B log sin 9.85009 b log sin 9.89833 C log sin 9.94909 c log sin 9.99733 9.95176 43.3, 6 = tri- 9.95176 Ex. 2. Given c 19.4, C= 74.37, solve the angle. We obtain i (a + 6) =31.35, J (a -6) =11.95, 1(7 = 37.18. By the first two of Napier's Analogies (Art. 165), = cosll -? 5 cot 37.18, cos 31.35 = sinll.95 cot sin 31.35 11.95 log cos 9.9905 37.18 log cot 0.1200 31.35 colog cos 0.0686 = 56.49 log tan 0.1791 11.95 log sin 9.3161 - 10 37.18 log cot 0.1200 31.35 colog sin 0.2839 1 (A - B) = 27.69 log tan 0.2190 - 10 Hence, A = 84.17, 5 = 28.81. By use of the first of Gauss's Equations (Art. 165), _ cos 31.35 sin 37.18 cos 56.49 31.35 log cos 9.9315 -10 37.18 log sin 0.7813 -10 56.49 colog cos 0.2580 c = 20.79 log cos 9.9708 - 10 c = 41.58. OBLIQUE SPHERICAL TRIANGLES 213 CHECK. A log sin 9.9978 B log sin 9.6829 C log sin 9.9836 a log sin 9.8362 b log sin 9.5213 c log sin 9.8219 0.1616 0.1616 0.1617 EXERCISE 53 Given parts as follows, solve the following triangles, checking results: JL/ / A*l- 1. b = 64, c = 46' 18". A = 56 24'. ^ 2. a = 73, c = 47, B = 113 42'. 3. a = 120 25' 12", b = 80 22' 48", C = 54 33' 4". 3. A = 124 42' 36", C = 76 36' 36", b = 48 49' 12". 4. ^4 = 111 39' .35", 5 = 127 41' 45", c = 62 47' 40". 5. a = 124 42', 5 = 106 54', C = 145 18. 6. C = 133 51', A = 48 48' 36", b = 68 43'. 7. A = 48 16' 48", B = 32 12' 24", c = 116 18' 36". Solve by use of four-place tables, having given 8. jB = 79, (7=51, a = 44. 9. .1 = 41, B = 27, c = 148.5. 10. A = 124.71, (7= 76.61, b - 48.82. 11. A = 111.66, 5 = 127.68, c = 62.78. 216 SPHERICAL TRIGONOMETRY 12. a = 124. 7, B = 106. 9, C= 145.3. 13. C = 133.85, A = 48.81, b = 68.72. 14. J. = 48.28, B = 32.21, c = 116.31. CASE III. THREE SIDES GIVEN 169. The Solution of Case III may be effected by use of the formulas of Art. 162, but it is more convenient to use the formulas for the half angles obtained in Art. 163. Why is this ? When it is required to compute only one of the angles of the given triangle, it is most convenient to use the formula for the cosine of the half angle. Let the pupil determine why this formula is more convenient than that for the sine or tangent of the half angle. Ex.1. Given a= 76 35' 36", 6 = 50 10' 30", c=400' 10", find A. We have s = 83 23' 8", and s - a = 6 47' 32". Hence by Art. 163, L A = /sin 83 23' 8" sin 6 47' 32" V sin 50 10' 30" sin 40 0' 10"' 83 23' 8" log sin 9.99710 - 10 6 47' 32" log sin 9.07288 - 10 50 10' 30" colog sin 0.11464 40 0' 10 "colog sin 0.19190 2)19.37652 - 20 i A = 60 48' 8" log cos 9.68826 - 10 .-.^ = 121 36' 16". In case it is required to compute all three of the angles of the triangle, it is more convenient to use the tangent formula for the half angle. Let the pupil determine why by show- ing how many different logarithms would need to be used in order to compute the three angles by use of the cosine form- ula, and how many -by use of the tangent formula. OBLIQUE SPHERICAL TRIANGLES 217 The work of computing all three of the angles may be further facilitated by use of the following transformation : sm s sm (8 a) - \/ sm ( 6> ~ a ) s * n ( s ' sin s sin' 2 (s a) \/ s sm ( s ~ a ) sm ( s ~ sin (s a) sin s T r . /sin (s a) sin (s b] sin (s c) , If we denote \- '- by r, sins tan i A = sin ( - a) Likewise tan i B = - , sin (s &) tan l C = - c) Ex. 2. Given a - 124.21, & = 54.3, c = 97.21, solve the triangle. We have s = 137.86, s - b = 83.56, s - a = 13.65, s - c = 40.65. Using the above formula for r, we have y =A/ a 13.65 sin 83.56 sin 40.65 sin 137.86 13.65 log sin 9.3729 - 10 83.56 log sin 9.9973 -10 40.65 log sin 9.8139 - 10 137.86 colog sin 0.1733 2)19.3574^20 r log 9.6787 - 10 tani ^ = r_ .'. r log 9. 7687 -10 sin 13.65 13.65 colog sin 0.6271 \A = 63.68 log tan 0.3058 218 SPHERICAL TRIGONOMETRY Hence A = 127.36. In like manner, B = 51.3, C = 72.45. Let the pupil check the work by the use of the Law of Sines. (See Ex. 1, p. 212.) EXERCISE 55 Given parts as follows, solve the following triangles, checking results : 1. a = 52, b = 37, c = 43. 2. a = 150, b =,125, c = 43. \ 3. a = 40 0' 10", b = 50 10' 30", c = 76 Stf 36". 4. a = 65 39' 46", 6 =124 7' 28", c = 159 50' 4". * 5. Given a = 70 14' 20", b = 38 46' 10", c = 49 24' 10", find A. 6. a = 72 7' 12", & = 111 30' 24", c = 44 21' 36", find B. 7. a = 59 48', & = 115 43' 12", c = 135, find <7. By use of four-place tables solve the following, having given : 8. a = 52, 6 = 37, c = 43. 9. a = 150, 6 = 125, c = 43. 10. a = 40.003, 6 = 50.175, c = 76.599. 11. a = 65.66, 6 = 124.12, c = 159.83. 12. a = 70.24, 6 = 38.75, c = 49.4, find A. 13. a = 72.12, b = 111.51, c = 44.36, find B. 14. a = 59.8, 6 = 115.72, c = 135, find C. CASE IV. THREE ANGLES GIVEN 170. The solution of Case IV is best effected by the use of the formulas of Art. 164. How else might the solution be effected, and why is this second method of solution in- ferior to the first? In case but one side is required, the computation is best performed by the use of th.e formula for the sine of the half side. Why ? Since the cosine in the second quadrant is minus, it is important in using the formulas of this case to observe the sign of each function used and to record it above the func- tion in the formula. OBLIQUE SPHERICAL TRIANGLES 219 Ex. 1. Given A = 58, B= 45, C= 123, find c. We have S = 113, S-C=- 10. >'.-' 113 log cos 9.59188 - 10 10 log cos 9.99335 - 10 45 colog sin 0.15051 58 colog sin 0.07158 2)19.80732 - 20 ic= 53 13' 48" log sin 9.90366 =10 c= 106 27 '36". In case it is required to compute all three of the sides of the triangle, it is more convenient to use the tangent for- mulas. Why ? The work of computing all three of the sides may be further facilitated by use of the following trans- formation : / cosSvos(ti-A) J~ -cos 8 cos 9 (8- A) cos (S- A) cos (S-E) cos (8- C) :-WI cos (S- A) cos (S- B) cos (S- C) If we denote cos a , r> ^^A by E, cos (S- A) cos (S-B) cos (S- C) Likewise tan \ b = -R cos (S J5), Ex. 2. Given J. = 20.17, ^=55.88, (7= 114.34, solve the triangle. AVe have S = 95.2, ^ - B = 39.32, >S - A = 75.03, - C = - 19.14. 220 SPHERICAL TRIGONOMETRY By the above formula cos 95.2 C cos 75.03 cos 39.32 cos (- 19.14) It is noted that the minus signs compensate. 95.2 log cos 8.9573 - 10 75.03 colog cos 0.5878 39.32 cologcosO.il 14 19.14 colog cos 0.0247 2)19.6812-20 R log 9.8406 - 10 tan I a = R cos 75.03, R log 9.8406 - 10 75.03 log cos 9.4122 - 10 \ a = 10.15 log tan 9.2528 - 10 Hence a = 20.3. In like manner b = 56.38, c = 66.42. Let the pupil check the work by the use of the Law of Sines. EXERCISE 56 Given : / 1. A = 142, B = 105, C= 85. Solve in full. 2. A = 97 54', B = 106 48' 36", C = 127 35' 24". Find c only/ 3. A = 48 18', B = 100, C = 100. Solve in full. " 4. A = 73 35' 24", B = 98 7' 48", C = 39 12'. Find a. * 5. A = 76 30' 36", B = 83 25' 48", C = 62 49' 12". Solve in full. 6. A = 76 29' 18", B = 98 18' 36", C = 122 T 42". Find b. 7. A = 27 30 r, B = 94.18, (7 = 83 12 '. Solve in full. 8. .4 = 105 8' 10", 5 = 129 5' 28", (7= 142 12' 42". Find c. 9. Show that it is impossible to solve the triangle whose angles are 142, 125, and 65'. By use of four-place tables, having given 10. A = 142, B = 105, C = 85, solve in full. 11. ^4 = 97.9, B= 106.81, C= 127.59, find c only. 12. A = 48.3, B = 100, C= 100, solve in full. 13. A = 73.59, B = 98.13, C = 39.2, find a. 14. A =76.51, B = 83.43, C= 62.82, solve in full. OBLIQUE SPHERICAL TRIANGLES 221 15. A = 76.49, B = 93.31, C = 122.13, find b. 16. A = 27.5, B = 94.3, C = 83.2, solve in full. 17. A = 105.14, B = 129.09, C = 142.21, find c. CASE Y. Two SIDES AND AN ANGLE OPPOSITE ONE OF THEM GIVEN 171. To solve Case V first find the unknown angle oppo- site a gwen side by use of the law of sines ; then find the third side and the third angle by use of Napier's Analogies. 172. Number of Solutions in Case V. Under certain con- ditions two solutions of an oblique spherical triangle are possible. Thus, if the given parts are a, &, A, and A is acute while a + b < 180, b > 90, and a < b, but > CD (LAB), i.e. sin a > sin b sin A 9 it may be shown that two solu- tions are possible. N Similarly if A is X. / p |' V \ acute, a + 6>180, X1 ' % &<90, and a>b, there are two solu- tions. The following table shows the number of solutions under all possible circumstances in Case V. I. When A is less than a right angle, ab and a + ~b< 180 ..... one solution a>b and a + 6 = 180 or > 180 . . no solution .a b no solution 900 SPHERICAL TRIGONOMETRY a9Q\a 180 a = 6 or > b two solutions one solution no solution II. When A is equal to a right angle, !a < b or a = b no solution a > b and a + b < 180 one solution a>bsmda + b = or > 180 . . . no solution a < b or a > b no solution a = b infinite number of solutions a < b and a + b > 180 one solution a < b and a 4- b = 180 or < 180 . . no solution a = b or a > b no solution , _ 6 > 90 III. 6 = yu When ^4. is greater than a right angle, a < by or a = b ..... . . no solution a > b and a + b = 180, or < 180 a > b and a + b > 180 ... a < by or a = b 7 a> b . . a < b and a + ~b > 180 ... a < b and a + b= 180, or < 180 one solution two solutions no solution -i , . two solutions one solution no solution a = b one solution a>b two solutions In the cases in which two solutions are indicated, there will be no solution if sin a be less than sin b sin A. It will be seen from the above investigations that if a lies between b and 180 b, there will be one solution; if a does not lie between b and 180 &, either there are two solutions or there is no solution (this does not include the cases in which a = 6, or =180 -b). The above table may be verified geometrically by use of the accompanying diagram. On the diagram, ED is the OBLIQUE SPHERICAL TRIANGLES 223 projection of a great circle drawn perpendicular to the great circle ADA'E. If Z A is acute, it is represented by /- PAD (or by the equal Z DH'P). Thus, for example, when A is acute, b < 90, a < b, we have (in general) the two A APB and APB' as solutions (B and B' taken in lower part of the diagram, arcs PB and PB' not being drawn). If A is acute, b < 90, a > b, and a + b < 180, the point B would fall between A' and jff, and there would be one solution. If A is acute, b < 90, a>b, and a + b> 180, ^ would fall at B or j5' in the upper part of the figure, and there would be no solution. The results given in the table may also be obtained from an analysis of the formulas used in the solution of spherical triangles, but this investigation lies beyond the scope of this book. Ex. 1. Given a = 55, 6=138, J. = 42, solve the tri- angle. Since, b > 90, o < 6, a -+- b > 180, there is but one solution. By the law of sines (Art. 161), sin B = sin 42 sin 138 sin 55 42 log sin 9.82551 -10 138 log sin 9.82551 -10 55 colog sin 0.08664 log sin 9.73766 - 10 The angle whose log sin is 9.73766 - 10 is either 33 8 f , or 146 52'. But, since in the given triangle the greater angle must be opposite the greater side, JB=14652'. 224 SPHERICAL TRIGONOMETRY Using the second and fourth of Napier's Analogies, we obtain (7 = 54 18' 46", c = 95 59' 12". Let the pupil check the work by the use of the Law of Sines. EXERCISE 57 Given parts as follows, solve the following spherical triangles, checking results : 1. ^ = 102, a = 55 24', 6 = 32 36'X 2. A = 114 20' 14", 6 = 56 19' 42", a = 66 20' 39"/ 3. (7 = 44 22' 10", c = 50 45' 20", 6 = 69 12' 40".^ 4. A = 52 18' 24* a = 68 26' 36", b = 78 30' 30". V 5. B = 95 48' 36", b = 100 42', a = 65 27'. Find A 6. B = 29 18' 35", b = 42, c = 117 37' 12". Find C. 7. C= 129 54', c = 136 25' 12", b = 59 48'. V Solve by use of four-place tables, having given : 8. A = 102, a = 55.4, b = 32.6. 9. A = 114.34, 6 = 56.33, a = 66.34. 10. C = 44.37, c = 50.76, 6 = 69.21. 11. A = 52.31, a = 69.44, 6 = 78.51. 12. # = 95.81, 6 = 100.7, a = 65.45 (find A only). 13. 5 = 29.31, b = 42, c = 117.62 (find only). 14. C= 129.9, c = 136.42, b = 59.8. CASE VI. Two ANGLES AND A SIDE OPPOSITE ONE OF THEM GIVEN 173. To Solve Case VI first find the unknown side opposite a given angle ; then find the third side and third angle by use of Napier's Analogies. In Case VI, the number of solutions is determined by taking the polar triangle of the given triangle and using Art. 172. Ex. Given J. = 115, ^-80, Z> = 84. Solve the tri- angle. OBLIQUE SPHERICAL TRIANGLES 225 Since, in the polar triangle, a' = 65, b' = 10, B' = 96, taking B' as the primary angle instead of A, we have B' > 90, '<90, b' > a', a' + b'< ISO . Hence there is but one solution. By the law of sines sin 84 sin 115 Smtt= sin 80 * 84 log sin 9.99761 -10 115 log sin 9.95728 -10 80 colog sin 0.00665 log sin 9.96154 - 10 There are two angles whose log sin is 9.96154 10, viz. : 66 14' 30" and 113 45' 30". Since in the given triangle the greater side must be opposite the greater angle, a = 113 45' 30". By use of Napier's Analogies, we find = 78 59' 46", c = 82 26' 10". Let the pupil check the work. EXERCISE 58 Given parts as follows, solve the following spherical triangles,' checking results : 1. ^1 = 73, 0=60, a = 40. " 2. ^=6624', 5 = 51 48', a = 40. l/ 3. B = 148 48', C = 122 24', c = 75 34' 30". * 4. A = 130 24' 36", C= 100, a = 150 36' 36". 5. C = 36 36' 58", A = 48 23' 24", c = 40 24' 36". 6. A = 92 30', a = 25 42', B = 56 30'. 7. B = 71, C= 120, c = 78. Is a solution possible ? 8. A = 133, B = 140, b = 126 (find a only). Solve by use of four-place tables, having given: 9. .4 = 73, 0=60, a = 40. 10. A = 66.4, B = 51.8, a = 40. 11. B = 148.8, (7=122.4, c = 75.575. 12. A = 130.41, C= 100, a = 150.61. 226 SPHERICAL TRIGONOMETRY 13. C= 36.62, A = 48.39, c = 40.41. 14. A = 92.5, a = 25.7, B = 56.5. 15. B = 71, C= 120, c = 78. Is a solution possible ? 16. .4 = 133, B = 140, 6 = 126 (find a only). AREA OP A SPHERICAL TRIANGLE 174. When the three angles of a spherical triangle are known, the area of the triangle may be found by the follow- ing formula proved in spherical geometry: where E = A + B + C- 180 (called the spherical excess). Ex. 1. Find in terms of R the area of the spherical triangle in which J.= 78 12' 24", ^=68 24' 32", C= 52 35' 28". We obtain E = 19 12' 24" or 69,144". . 7r.fl 2 19 12' 24" _ TT fi 2 69144" ^80" 648000" TT log 0.49715 69144 log 4.83975 648000 colog 4.18842-10 .335215 log 9.52532- 10 .-.K=. 335215 H 2 Ans. Ex. 2. Find in terms of R the area of the spherical triangle in which A = 78.21, ^=68.41, C= 52.59. We obtain E = 19.21. # 2 19.21 Hence K = 180 C TT log 0.4971 19.21 log 1.2835 180 colog 7.7447-10 .33523 log 9.5253- 10 7T = . 33523 OBLIQUE SPHERICAL TRIANGLES 227 175. When the three sides are known, E may be computed by the formula tan' 2 J E = tan \ s tan ^ (s a) tan ^ (s b) tan \ (s c), called 1'Huilier's Formula. The area may then be found by the method of Art. 174. Ex. 1. Find E in a spherical triangle in which a =144, 6=64, and c= 133. We obtain s = 170 30', s - b = 106 30 f , s - a = 26 30', s - c = 37 30'. Hence tan 2 J E = tan 85 15' tan 13 15' tan 53 15' tan 18 45'. 85 15' log tan 1.08043 13 15' log tan 9.37193 53 15' log tan 0.12683 18 45' log tan 9.53078 2)0.10997 \ E = 48 37' 5" log tan .05499 E = 194 28' 20". The proof of the above formula is as follows : From the first of Gauss's Equations (Art. 165), cos %(A + B) _ cos 1 (q + 6) sin ^- C cos ^ c But sin l (7= cos (90 - \ C). (Why ?) Therefore CO8 *(^- + - B ) _cosl(a + 6)_ cos(90-i(7)~ cosic Then, by division and composition, cos %(A + B)- cos (90 - i- C) = cos ^ (a + b) - cos^ c . cos %(A + B) + cos (90 -1(7) cos^ (a + 6) + cosi c ' Using Art. 71, and taking ^4. and B as any angles, we have _ tan i ( ^ + j g )tan l^l-^). (2) cos A 4- cos B Substituting in (2) for A and B,%(A + B) and 90 - 1 (7, respectively, and taking J. and ^5 as any angles, we have 228 SPHERICAL TRIGONOMETRY cos I (A + B}- cos (90 -1(7) cos I ( J. + B) + cos ( 90 - 1 (7 ) But J& = J. + J5+ (7-180). = tani(360-2 <7-h^L + + (7- 180) = tan 1(360 -2 = tan [90 -(2 (7- = cot 1(2 x. 1. Find the antilog of 2.42376. The mantissa .42376 does not occur in the table, and the next lower mantissa is .42374. The difference between .42376 and .42374 is .00002. If a difference of 16 in the last two figures of the mantissa makes a difference of 1 in the fourth figure of the antilog, a difference of 2 in the last figure of the mantissa will make a difference of T 2 of 1 or .125 (or -13) with respect to the fourth figure of the antilog. Hence we have antilog 2.42376 = 265,313- Ans. 374 16)2.00(.13~ 16 40 Ex.2. If log x = 7.26323-10, find x. Nearest less mantissa = .26316, whose number is 1833. Tab. diff. = 24. 7-*-24 = .29 + . Hence a; =.00183329, Ans. The first part of Exercise 4 of Durell's Plane Trigonom- etry should be worked at this point. TABLE II. LOGARITHMS AND COLOGARITHMS OP MUCH-USED NUMBERS (p. 4O) This table explains itself. TABLE III. FIVE-PLACE LOGARITHMS OP TRIGONOMETRIC FUNC- TIONS FOR EVERY MINUTE OF THE QUADRANT (pp. 41-86) 7. Description of Table III. This table gives the loga- rithms of the sine, cosine, tangent, and cotangent of each *-*> minute of angle from up to 90. 10 TRIGONOMETRY Where 10 is a part of the characteristic of the log function it is omitted for the sake of economy of space. This omission occurs at the end of the log function of each angle except for log tangents from 45 to 90, and log cotangents from to 45. For angles between and 45, the required functions are printed at the top of the columns, the number of degrees at the top of the page, and the number of minutes in the hand column. For angles between 45 and 90, the required functional printed at the bottom of the columns, the number of degrees at the bottom of the page, and the number of minutes in the right-hand column. Thus, log sin 26 37' = 9.65130 - 10 (p. 68). log tan 67 48' = 0.38924 (p. 64). log sin 58 16' = 9.92968 - 10 (p. 73). log cot 12 23' = 0.65845 (p. 54). Let the pupil determine why each column of the table has the name of a trigonometric function at the top and the name of the corresponding co-function at the bottom of the column. Let him also determine why 10 is to be annexed at the end of some log trigonometric functions as taken from the tables, and not at the end of others. DIRECT USE OF TABLE III 8. Given the degrees, minutes, and seconds of an angle, to find a logarithmic trigonometric function of the angle. After finding the log function for the given number of degrees and minutes, the log function for the given number of degrees, minutes, and seconds is found by interpolation. Ex. 1. Find the log sin 37 42' 53". The log sin 37 42' is 9.78642, and the difference between this and log sin 37 43' is 16- Since an increase of 1' in the angle makes an increase of 16 in the INTRODUCTION TO TABLES 11 last two places of the log sin, an increase of 53" or |~| of 1' will make an increase of || o f 16 in the log of the function. Hence we have log sin 37 42'= 9.78642 - 10 Diff. for 53" = ff of 16= 14 log sin 37 42' 53" = 9.78656 - 10 x. 2. Find the log sin 53 27' 18". log sin 53 27' = 9.90490 - 10 Diff. for 18" = M of 9 = 3 log sin 53 27' 18" = 9.90493 - 10 Ex.3. Find log cos 23 48' 12". Since the cosine of an angle decreases as the angle increases, the log of 23 49' is less than the log cos 23 48'. Hence the correction for 12" must be subtracted from the log cos 23 48'. Thus log cos 23 48' = 9.96140 - 10 Diff. for 12" = 1| of 5 = 1 log cos 23 48' 12" = 9.96139 -10 Ex. 4. Find log cot 57 18' 43". log cot 57 18' = 9.80753 - 10 Diff. for 43" = 28 x = 20 log cot 57 18' 43" = 9.80733 - 10 Hence, in general, Obtain from the table the log function for the given number of degrees and minutes; Also obtain from the table the log function for the angle, 1 minute greater; find the difference between these two log func- tions; multiply this difference by - ; this will give the correction for seconds; Add the correction for seconds in case of sine and tangent (direct functions') ; Subtract the correction in case of cosine and cotangent (com- plementary functions'). 12 TRIGONOMETRY 9. Log Secants. To find the log secant of an angle, use the formula sec x - .*. log sec x = + colog cos x. cosx Thus log sec 39 28' 23" - colog cos 39 28' 23". But log cos 39 28' 23" = 9.88757 - 10. colog cos 39 28' 23" or log sec 39 28' 23" = 0.11243. 10. Log Functions of Angles greater than 90. ByfllB methods of Chapter IV, a trigonometric function of aiy angle greater than 90 can be reduced to a trigonometric function of an angle less than 90. Thus, since sin A = sin (180 - A), sin 113 27' = sin 66 33'. .-. log sin 113 27' = log sin 66 33' = 9.96256 - 10. Also cos A = - cos (180 - A). Hence, log cos A = log cos (180 - A)(ri), the small n being annexed to show that the function whose log is being used is a negative quantity. Thus log cos 142 18' = log cos 37 42' (n) = 9.78642 - 10 (n). At this point work the first part of Exercise 14 of Durell's Plane Trigonometry. INVERSE USE OF TABLE III 11. Given the logarithm of a function to find the correspond- ing acute angle (or find antilog sin, antilog cos, etc. or /.log ' sin, /.log cos, etc.) Obtain from the table, if possible, the number of degrees and minutes corresponding to the given logarithmic function. Ex. If log tan A = 9.92535 - 10, find the angle A. By consulting the table, tangent column, we find that ^4 = 40 6'. Or antilog tan 9.92535 - 10 = 40 6'. If the given logarithmic function does not occur in the table : INTRODUCTION TO TABLES 13 Obtain from the table the next less logarithm of the same func- tion, noting the corresponding number of degrees and minutes; subtract this logarithm from the given logarithm; Divide the difference so obtained by the tabular difference for V and multiply by 60"; the result will be the correction, in seconds, to be added in case of sine and tangent, and sub- fw^ in case of cosine and cotangent, to the angle already n 1 Ex. 1. Find antilog sin 9.78538 - 10. Z log sin 9.78538 - 10 = 37 35' + 9.78527 - 10 11 Since a difference of 16 in the log makes a difference of 1' (or of 60") in the angle, a difference of 11 in the log makes a difference of -J-^- of 60", or 41", in the angle. .-. antilog sin 9.78538-10 = 37 35' 41", Ans. Ex. 2. Find antilog cos 9.96623 - 10. antilog cos 9.96623 - 10 = 22 19' - 9.96619 - 10 - of 60" = 48" o antilog cos 9.96623 - 10 = 22 18' 12", Ans. Ex. 3. Find antilog cot 0.57603. antilog cot 0.57603 = 14 52' - 0.57601 of 60" = 2" 51 antilog cot 0.57603 = 14 51' 58", Ans. Ex. 4. Find antilog cos 9.60172 - 10. antilog cos 9.60172 - 10 = 66 27'- 9.60157 - 10 of 60" = 31", antilog cos 9.60172-10 = 66 26' 29", Ans. 14 TRIGONOMETRY At this point work the first part of Exercise 15 of Durell's Trigonometry. TABLE IV. AUXILIARY FIVE PLACE TABLE FOR SMALL ANGLES (pp. 87-89) 12. The Auxiliary Table of Logarithms of Sine and gent for Small Angles is needed because when an an smaller than 2, the logarithms of the sine and tangent vary so rapidly that ordinary methods of interpolation are not sufficiently accurate. (The same is true for the cosine, cotangent, and tangent when the angle is between 88 and 90, but there are other indirect methods of meeting such cases.) Table IV is based on Art. 115 of Plane Trigonometry, where it is shown that the sine (or tangent) of a small angle is approximately the same in value as the number of radians in the angle. Hence, for example, to find sine 1 21' 37", we divide the number of seconds in 1 21' 37" by the num- ber of seconds in a radian, viz. 206,265. This process is facilitated by Table IV. The column headed " in this table gives the number of seconds in each angle containing an exact number of minutes, and hence is an aid in converting any given angle into seconds. In the column headed S' is given the log of 206,265 (viz. 5.31443), modified by a slight correction owing to the change in the slight differences between the sine of a small angle and the radian measure of that angle. Similarly the column headed T' gives log of 206/265 in use of the tangent. (The columns headed S and T give the cologs corresponding to the S' and T' columns.) The column headed log sin gives the log sin or final answer for each even minute, these num- bers being needed also in guiding the work in the inverse use of the table. Hence INTRODUCTION TO TABLES 15 13. To find the log sin or tangent of an angle less than 2. Find the number of seconds in the given angle and find the log of this number in Table I ; Add to this log the corresponding log in column S or T ac- cording as the log sin or log tan is desired. T. Find log sin 1 26' 13". ' 1 26' 13" = 5173" log 5173 = 3.71374 S (or colog 206265) = 4.68553 - 10 .-. log 1 26' 13" = 8.39927 - 10, Ans. 14. To find the angle corresponding to a given log sine or log tangent (less than 8.54282 - 10). Look up in the L. Sin column the number nearest in size to the given log; and set down the number on the same row with this in column S' or T', according as the given function is a sine or tangent; Add the given log function to the number set down from the table; Find the antilog of the result; this will be the number of seconds in the required angle. Ex. Find antilog tan 8.39307. In L. Sin column, the nearest number is 8.39310. Corresponding to this is T = 5.31434 Given tan = 8.39307 antilog 13.70741 = 5098" = 1 24' 58", Ans. The reason for the above process is seen from the fact that , 5098" rin of required ^=206265^' .-. 206265 x (sin of required Z) = 5098". .-. log 206265 + 8.39307 = log 5098", 16 TRIGONOMETRY 15. Other Uses of the Auxiliary Table IV. The log cosine of an angle between 88 and 90 changes so rapidly as to make direct interpolation inaccurate. In such cases use the formula cos A = sin ^ (990 _ A ^ Thus, for example, log cos &8 47' = log sin 1 13', and the value of log sin 1 13' can be obtained by Art. 14. The log cot A, when A is between 88 and 90, may tained similarly. Also, if A is an angle between 88 and 90, the log tan A changes so rapidly that interpolation is inaccurate. In this case use tan A = 7. cot A log tan A = colog cot A = colog tan (90 A). Thus, for example, log tan 88 47' = colog tan 1 13', etc. At this point work the first part of Exercise 16 of Durell's Trigonometry. TABLE V. FOUR-PLACE TABLE OF THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT FOR EVERY TEN MINUTES OF THE QUADRANT (pp. 91-96) 16. Method of using Table V. By natural trigonometric functions are meant the actual numerical (not logarithmic) values of these functions. Thus ^ is the natural sine of 30. Interpolation for this table is made in the same general way as for Table V^/' \ Ex. Find natural sine 27 48'. N. Sine 27 40' = 0.4643 ft of 26 = 21 K Sine 27 48' = 0.4664, Ans. TABLE VI. FOUR-PLACE TABLE OF LOGARITHMS OF NUMBERS 1-2OOO (pp. 97-1O1) 17. Method of using Table VI. In using the four-place log of a number, when the first signifi- cant figure of the number is 1, use pp. 100-101 ; otherwise use pp. 98-99. INTRODUCTION TO TABLES 17 In finding the antilog of a four-place log, if the given log is less than .3010, use pp. 100-101; otherwise use pp. 98-99. At this point work the latter part of Exercises 3 and 4 of Durell's Plane Trigonometry. TABLE VII. POUR-PLACE LOGARITHMIC TABLE OF THE TRIGONO- 3TRIC FUNCTIONS FOR ANGLES OF THE QUADRANT EXPRESSED DECIMALLY DIVIDED DEGREES (pp. 1O3-113) Method of using Table VII. The explanation of the methods of using Table III given in Arts. 811 of this Intro- duction apply in general to the use of Table VII. Hence we need only illustrate by examples the application of these methods to the table in hand. Ex. 1. Find log sin 48.34. log sin 48.4 = 9.8738 - 10 log sin 48.3 = 9.8731 - 10 log sin 48.3 = 9.8731 - 10 T % of 7 = _ 3 7 log sin 48.34 = 9.8734 - 10, Ans. Ex. 2. Find the antilog tan 0.2165. Z log tan 0.2165 = 58.7+ 2161 Z log tan 0.2165 = 58.72, Ans. At this point work the latter part of Exercises 14 and 15 of Durell's Trigonometry. 19. Four-place Log Functions of Angles near or 90. As is explained in Art. 12 of this Introduction, when an angle is less than 2, the logarithms of the sine and tangent vary so rapidly that ordinary methods of interpolation are not sufficiently accurate. To get an accurate log function in this case we use the result obtained in Art. 106 of Plane Trigonometry, viz : sine or tangent of a very small Z x -,. - Z x in degrees = no. radians in Z x, or = - -- 18 TRIGONOMETRY .-. log sin (or tan) of small Zx = log x + colog 57.296 = log x+ 8.2419 -10. 1 57.296 Also when x is small cot x = - = : -- tan x x in degrees .-. log cot small Z x= 1.7581 + colog x. Interpolation ' also is not accurate for log cos, log cot, of angles between 88 and 90. When A. is an angle between 88 and 90 proceec follows : cos A = sin (90 -A). /. log cos A = log sin (90 - A) = 8.2419 - 10 + log (90 -A}. cot A = tan (90 -A). .-. log cot J. = log tan (90 -J_) = 8.2419 -10 + log (90 -J.)- tan A = 3L ... log tan A = 1.7581 -log (90- A). cot A Ex. 1. Find sin 0.876. log 0.876 = 9.9425 - 10 colog 57.296 = 8.2419 - 10 .-. log sin 0.876 = 8.1844 - 10, Ans. Ex. 2. Find Z log sin 7.9592 - 10. 17.9592 - 20 8.2419 - 10 antilog 9.7173 - 10 = 0.522- .-. Z log sin 7.9592 - 10 = 0.522-, Ans. At this point work the latter part of Exercise 16 of Durell's Trigonometry. TABLE VIII. TABLE FOR CONVERTING Mi JUTES AND SECONDS INTO THE DECIMAL PART OF A DEGREE (p. 114) 20. The method of using Table VIII is evident from the form of the table, but it should be remembered that in each INTRODUCTION TO TABLES 19 decimal equivalent ending in a significant figure the last figure is supposed to repeat indefinitely. Hence, for example, we have 36 46' = 36.766 0+ = 36.77 Also 35 43' = 35.716 20" = .006 .-. 35 43' 20" = 35.722 = 35.72,^s. TABLE IX. TABLE FOB CONVERTING THE DECIMAL PARTS OP A DEGREE INTO MINUTES AND SECONDS (p. 114) 21. The method of using Table IX is also evident from the table itself. TABLE I COMMON LOGARITHMS OF NUMBERS PAET I )GARiTHMS (WITH CHARACTERISTICS) OF NUMBERS 1-100 N. Log. N. Log. H. Log. N. Log. M Infinity 30 31 32 33 1.47 712 60 61 62 63 1.77 815 90 91 92 93 1.95 424 0.00 000 0.30 103 0.47 712 1.49 136 1.50 515 1.51 851 1.78 533 1.79 239 1.79 934 1.95 904 1.96 379 1.96 848 0.60 206 0.69-897 0.77 815 34 35 36 1.53. 148 l! f 5"5630 64 65 66 1.80 618 1.81 291 1.81 954 94 95 96 1.97 313 1.97 772 1.98 227 h 9 10 Ls 13 0.84 510 0.90 309 0.95 424 37 38 39 40 41 42 43 1.56 820 1.57 978 1.59 106 67 68 69 70 71 72 73 1.82 607 1.83 251 1.83 885 97 98 99 100 1.98 677 1.99 123 1.99 564 1.00 000 1.60 206 1.84 510 2.00 000 1.04 139 1.07 918 1.11 394 1.61 278 1.62 325 1.63 347 1.85 126 1.85 733 1.86332 .14 15 16 1.14 613 1.17 609 1.20 412 44 45 46 1.64 345 1.65 321 1.66 276 74 75 76 1.86 923 1.87 506 1.88 081 17 18 19 20 21 22 23 1.23 045 1.25 527 1.27 875 47 48 49 50 51 52 53 1.67 210 1.68 124 1.69 020 77 78 79 80 81 82 83 1.88 649 1.89 209 1.89 763 1.30 103 1.69 897 1.90 309 1.32 222 1.34 242 1.36 173 1.70 757 1.71 600 1.72 428 1.90 849 1.91 381 1.91 908 24 25 26 1.38 021 1.39 794 1.41 497 54 55 56 1.73 239 1.74 036 1.74 819 84 85 86 1.92 428 1.92 942 1.93 450 27 28 29 30 1.43 136 1.44 716 1.46 240 57 58 59 60 1.75 587 1.76 343 1.77 085 87 88 89 90 1.93 952 1.94 448 1.94 939 1.47 712 1.77 815 1.95 424 [21] PART II MANTISSAS OF NUMBERS 1-10,000 N. 100 01 02 03 04 05 06 07 08 09 110 11 12 13 14 15 16 17 18 19 120- 21 22 23 24 25 26 27 28 29 130 31 32 33 34 35 36 37 38 39 140 41 42 43 44 45 46 47 48 49 150 | O 00 000 1 043 087 JL 130 -L 173 5 217 260 7 303 8 346 389 1 432 860 01 284 703 02 119 531 938 03 342 743 475 903 326 745^ 160 572' 979 383 782 518 945 368 787 202 612 *019 423 822 561 988 410 828 243. 653 *060 463 862 604 *030 452 870 284 694 *100 503 902 647 *072 494 912 325 735 *141 543 941 689 *115 536 953 366 776 *181 583 981 732 *157 .J78 995 407 816 *222 623 *021 775 *199 620 *036 449 857 *262 663 *060 817 1 *242 1 662 *078 490 898 *3^^J 8c^ *269 652 *032 408 781 *151 518 882 04 139 179 218 258 297 336 376 415 454 532 922 05 308 690 06 070 446 819 07 188 555 571 961 <346 729 108 483 856 225 591 610 999 385 767 145 521 893 262 628 650 *038 423 805 183 558 930 298 664. 689 *077 461 843 221 595 967 335 700 727 *115. 500 881 258 633 *004 372 737 766 *154 538. 918 296 670 *041 408 773 805 *192 576 956 333 707 *078 445 809 844 *231 614 994 371 744 *115 482 846 918 954 990 *027 *063 *099 *135 *171 *207 *243 08 279 636 991 09 342 691 10 037 380 721 11 059 314 672 *026 377 726 072 415 755 093 350 707 *061 412 760 106 . 449 789 126 386 743 *096 447 795 140 483 823 160 422 778 *132 482 830 175 517 857 193 458 814 *167 517 864 209 551 890 227 493 849 *202 552 899 243 585 924 261 529 884 *237 587 934 278 619 958 294 ~62~8~ 565 920 *272 621 968 312 653 992 327 600 955 *307 656 *003 346 687 *025 361 394 428 461 494 528 561 594 661 694 727 12 057 385 710 13 033 354 672 988 14 301 760 090 418 743 066 386 704 *019 333 793 123 450 775 098 418 735 *051 364 826 156 483 808 130 450 767 *082 395 860 189 516 840 162 481 799 *114 426 893 222 548 872 194 513 830 *145, 457 926 254 581 905 226 545 862 *176 489 959 287 613 937 258 577 893 *208 520 992 320 646 969 230 609 925 *239 551 *024 352 678 *001 322 640 956 *270 582 613 644 675 706 737 768 799 829 860 891 922 15 229 534 836 16 137' 435 732 17 026 319 953 259 564 866 167 465 761 056 348 983 290 594 897 197 495 791 085 377 *014 320 625 927 227 52-4 820 114 406 *045 351' 655 957 256 554 850 143 435 *076 381 685 987 286 584 879 173 464 *106 412 715 *017 316 613 909 202 493 *137 442 746 *047 346 643 938 231 522 *168 473 776 *077 376 673 967 260 551 *198 503 806 *107 406 1 702 997 1 289 580 1 609 638 667 696 725 754 782 811 840 869 | N. 1 2 3 4 5 6 7 8 9 [22] .2-7 [23] N. O 1 2 3 4 5 6 7 8 9 200 30 103 125 146 168 190 211 233 255 276 298 1 01 320 341 363 384 406 428 449 471 492 514 02 535 557 578 600 621 643 664 685 707 728 03 750 771 792 814 835 856 878 29 920 942 04 963 984 *006 *027 *048 *069 *091 *112 *133 *154 05 31 175 197 218 239 260 281 302 323 345 366 06 387 . 408 429 450 471 492 513 534 555 576 07 597 618 639 660 681 702 723 744 765 785 08 806 827 848 869 890 911 931 952 973 994 09 32 015 035 056 077 098 118 139 160 181 201 210 222 243 263 284 305 325 346 366 387 403 11 428 449 469 490 510 531 552 572 593 61M 12 634 654 675 695 715 736 756 777 797 13 838 858 879 899 919 940 960 980 *001 H 14 33 041 062 082 102 122 143 163 183 203 22^ 15 244 264 284 304 325 345 365 385 405 425 ; 16 445 465 486 506 526" 546 566 586 606 626 17 646 666 686 706 726 746 766 786 806 826 18 846 866 885 905 925 945 965 985 *005 *025 19 34 044 064 084 104 124 143 163 183 203 223 220 242 262 282 301 321 341 361 380 400 420 21 439 459 479 498 518 537 557 577 596 616 22 635 655 674 694 713 733 753 772 792 811 23 830 850 869 '889 908 928 947 967 986 *005 24 35 025 044 064 083 102 1.22 141 160 180 199 25 218 238 257 276 295 315 334 353 372 392 26 411 430 449 468 488 507 526 545 564 583 27 603 622 641 660 679 698 717 736 755 774 28 793 813 832 851 870 889 908 927 946 965 29 984 4 ^=003 *021 *040 *059 *078 *097 *116 *135 *154 230 36 173 192 211 229 248 267 286 305 324 342 31 361 380 399 418- 436 455 474 493 511 530 32. 549 568^ 586 605 ' 624 642 661 680 698 717 33 736 754 773 791 810 829 847 866 884 903 34 922 940 959 977 996 *014 *033 *051 *070 *088 35 37 107 125 144 162 181 199 218 236 254 273 36 291 310 328 346 365 383 401 420" 438 457 37 475 493 511 530 548 566 585 603 n 621 639 38 658 676 694 712 731 749 767 785 803 822 1 39 840 858 876 894 912 931 949 967 985 *003 1 240 38 021 039 057 075 093 112 130 148 166 184 41 202 220 238 256 274 292 310 ~328~~ 346 ~364 1 42 382 399 417 435 453 471 489 507 525 543 43 561 578 596 614 632 650 668 686 703 721 1 44 739 757 775 792 810 828 846 863 881 899 I 45 917 934 952 970 987 *005 *023 *041 *058 *07-6 I 46 39 094 111 129 146 164 182 199 217 235 252* 1 47 270 287 305 322 340 358 375 393 410 428 1 48 445 463 480 498 515 533 550 568 585 602.1 49 620 637 655 672 690 707 724 742 759 777 1 250 794 811 829 846 863 881 898 915 933 950 1 N. O 1 2 3 4 5 6 7 8 9 [24] N. O 1 2 3 4 5 6 7 8 9 250 51 52 53 39 794 811 829 846 863 881 898 915 933 950 967 40 140 312 985 157 329 *002 175 346 *019 192 364 *037 209 381 *054 226 398 *071 243 415 *088 261 432" *106 278 _449 *123 295 466 54 55 56 483 654 824 500 671 841 518 688 858 535 705 875 552 722 892 569 739 909 586 756 926 603 773 943 620 790 960 637 807 976 57 58 59 260 61 62 63 993 41 162 330 *010 179 347 *027 196 363 *044 212 380 *061 229 397 *078 246 414 *095 263 430 *111 280 447 *128 296 464 *145 313 481 497 514 531 547 564 581 597 614 631 647 664 830 996 681 847 *012 697 863 *029 714 880 *045 731 896 *062 747 913 *078 764 929 *095 780 946 *111 797 963 *127 814 979 *144 64 65 66 42 160 325 488 177 341 504 193 357 521 210 374 537 226 390 553 243 406 570 259 423 586 275 439 602 292 455 619 308 472 635 67 68 69 270 71 72 73 651 813 v 975 667 830 991 64 846 *008 700 862 *024 716 878 *040 732 894 *056 749 911 *072 765 927 *088 781 943 *104 797 ' 959 *120 43 136 152 169 185 201 217 233 249 265 281 297 457 616 sis 473 632 329 489 648 345 505 664 361 521 680 377 537 696 393 553 712 409 569 727 425 584 743 441 600 759 74 75 76 775 933 44 091 791 949 107 807 965 122 823 981 138 838 996 154 854 *012 170 870 *028 185 886 *044 201 902 *059 217 917 *075 232 77 78 79 280 81 82 83 248 404 560 264 420 576 279 436 592 295 451 607 311 467 623 326 483 638 342 498 654 358 514 669 373 529 685 389 545 700 716 731 747 762 778 793 809 824 840 855 871 45 025 179 886 040 194 902 056 209 917 071 225 932 086 240 948 102 255 963 117 271 979 133 286 994 148 301 *010 163 317 84 85 86 332 484 637 347 500 652 362 515 667 378 530 682 393 545 697 408 561 712 423 576 728 439 591 743 454 606 758 469 621 773 87 88 89 290 91 92 93 788 939 46 090 803 954 105 818 969 120 834 984 135 849 *000 150 864 *015 165 879 *030 180 894 *045 195 909 *060 210 924 *075 225 240 255 270 285 300 315 330 345 359 374 389 538 687 404 553 702 419 568 716 434 583 731 449 598 746 464 613 761 479 627 776 494 642 790 509 -657-- 805 523 -672 820 94 95 96 '835 982 47 129 850 997 144 864 *012 159 879 *026 173 894 *041 188 909 *056 202 923 *070 217 938 *085 232 953 *100 246 967 *114 261 97 98 99 300 276 422 567 290 436 582 305 451 596 319 465 611 334 480 625 349. 494 640 363 509 654 378 524 669 392 538 683 407 553 698 712 727 741 756 770 784 799 813 828 842 N. 1 2 3 4 5 6 7 8 9 [25] N. O 1 2 3 4 5 6 7 8 9 300 47 712 727 741 756 ^M 770 ^v 784 799 813 828 842 01 857 871 885 900 914 929 943 958 972 986 02 48 001 015 029 044 058 073 087 101 116 130 03 144 159 173 187 202 216 230 244 259 273 04 287 302 316 330 344 359 373 387 401 416 05 430 444 458 473 487 501 515 530 544 558 06 572 586 601 615 629 643 657 671 686 700 07 714 728 742 756 770 785 799 813 827 841 08 855 869 883 897 911 926 940 954 968 982 09 996 *010 *024 *038 *052 *066 *080 *094 *108 *122 310 49 136 150 164 178 192 206 220 234 248 262 11 276 290 304 318 332 346 360 374 388 402 12 415 429 443 457 471 485 499 513 527 541 13 554 568 582 596 610 624 638 651 665 679 14 693 707 721 734 748 762 776 790 803 817 15 831 845 859 872 886 900 914 927 941 955 16 969 982 996 *010 *024 *037 *051 *065 *079 *092 17 50 106 120 133 147 161 174 188 202 215 229 18 243 256 270 284 297 311 325 338 352 365 19 1 379 393 406 420 433 447 461 474 488 501 320 515 529 542 556 569 583 596 610 623 637 21 651 664 678 691 705 718 732 745 759 772 22 786 799 813 826 840 853 866 880 893 907 23 920 934 947 961 974 987 *001 *014 *028 *041 24 51 055 068 081 095 108 121 135 148 162 175 25 188 202 215 228 242 255 268 282 295 308 26 322 335 348 362 375 388 402 415 428 441 27 455 468 481 495 508 521 534 548 561 574 28 587 601 614 627 640 654 667 680 693 706 29 720 733 746 759 772 786 799 812 825 838 330 851 865 878 891 904 917 930 943 957 970 31 983 996 *009 *022 *035 *048 *061 *075 *088 *101 32 52 114 127 140 153 166 179 192 205 218 231 33 244 257 270 284 297 310 323 336 349 362 34 375 388 4QJr 43,4 427 440 453 466 479 492 35 504 517 53b 543 556 569 582 595 608 621 36 634 647 660 673 686 699 711 724 737 750 37 763 776 789 802 815 827 840 853 866 879 38 892 905 917 930 943 956 969 982 994 *007 39 53 020 033 046 058 071 084 097 110 122 135 340 148 161 173 186 199 212 224 237 250 263 41 275 288 301 314 326 339 352 364 377 390 42 403 415 428 441 453 466 479 491 504 517 43 529 542 555 567 580 593 605 618 631 643 44 656 668 681 694 706 719 732 744 757 769 45 782 794 807 820 832 845 857 870 882 895 46 908 920 933 945 958 970 983 995 *008 *020 47 54 033 045 058- 070 083 095 108 120 133 145 48 158 170 183 195 208 220 233 245 258 270 49 283 295 307 320 332 345 357 370 382 294 350 407 419 432 444 456 469 481 494 506 518 N. O 1 2 3 4 5 6 7 8 9 [26] N. O 1 2 3 4 5 6 7 8 9 350 54 407 419 432 444 456 469 481 494 506 518 , 51 531 543 555 568 580 593 605 617 630 642 52 654 667 679 691 704 716 728 741 753 765 53 777 790 802 814 827 839 851 864 876 888 54 900 913 925 937 949 962 974 986 998 *011 55 55 023 035 047 060 072 084 096 108 121 133 56 145 157 169 182 194 206 218 230 242 255 57 267 279 291 303 315 328 340 352 364 376 58 388 400 413 425 437 449 461 473 485 427 59 509 522 534 546 558 570 582 594 606 618 360 630 642 654 666 678 691 703 715 727 739 61 751 763 775 787 799 811 823, 835 847 859 62 871 883 895 907 919 931 943 955 967 979 63 991 *003 *015 *027 *038 *050 *062 *074 *086 *098 64 56 110 122 134 146 158 170 182 194 205 217 65 229 241 253 265 277 289 301 312 324 336 66 348 360 372 384 396 407 419 431 443 455 67 467 478 490 502 514 526 538 549 561 573 68 585 597 608 620 632' 644 656 667 679 691 69 703 714 726 738 750 761 773 785 797 808 370 820 832 844 855 867 879 891 902 914 926 71 937 949 961 972 984 996 *008 *019 *031 *043 72 57 054 066 078 089 101 113 124 136 148 159 73 171 183 194 206 217 229 241 252 264 276^ 74 287 299 310 322 334 345 357 368 380 392 75 403 415 426 438 449 461 473 484 496 507 76 519 530 542 553 565 576 588 600 611 .623 77 634 646 657 669 680 692 703 715 726 738 78 749 761 772 784 795 807 818 830 841 852 79 864 875 887 898 910 921 933 944 955 967 380 978 990 *001 *013 *024 *035 *047 *058 *070 *081 81 58 092 104 115 127 138 149 161 172 184 195 82 206 218 229 240 252 263 274 286 297 309 83 320 331 343 354 365 377 388 399 410 422 84 433 444 456 467 478 490 501 512 524 535 85 546 557 569 580 591 602 614 625 636 647 86 659 670 681 692 704 715 726 737 749 760 87 771 782 794 805 816 827 838 850 861 872 ' 88 883 894 906 917 928 939 950 961 973 984 89 995 *006 *017 *028 *040 *051 *062 *073 *084 *095 390 59 106 118 129 140 151 162 173 184 195 207 91 218 229 240 251 262 273 284 295 306 318 92 329 340 351 362 373 384 395 406 417 428 93 439 450 461 472 483 494 506 517 528 539 94 550 561 572 583 594 605 616 627 638 649 95 660 671 682 693 704 715 726 737 748 759 96 770 780 791 802 813 824 835 846 857 868 97 879 890 901 912 923 934 945 956 966 977 98 988 999 *010 *021 *032 *043 *054 *065 *076 *086 99 60 097 108 119 130 141 152 163 173 184 195 400 206 217 228 239 249 260 271 282 293 304 If. O 1 2 3 4 *^*^*mmmm 5 6 7 8 9 [27] O 5 400 60 206 217 228 239 249 260 271 282 293 304 01 314 325 336 347 358 369 379 390 401 412 02 423 433 444 455 466 477 487 498 509 520 03 531 541 552 563 574 584 595 606 617 627 04 638 649 660 670 681 692 703 713 724 735 05 746 756 767 778 788 799 810 821 831 842 06 853 863 874 885 895 906 917 927 938 949 07 959 970 981 991 *002 *013 *023 *034 *045 *055 08 61 066 077 087 098 109 119 130 140 151 162 09 172 183 194 204 215 225 236 247 257 268 410 278 289 300 310 321 331 342 352 363 374 11 384 395 405 416 426 437 448 458 469 479 12 490 500 511 521 532 542 553 563 ' 574 584 13 595 606 616 627 637 648 658 669 679 690 14 700 711 721 731 742 752 763 773 784 794 15 805 815 826 836 847 857 868 878 888 899 16 909 920 930 941 951 962 972 982 993 *003 17 62 014 024 034 045 055 066 076 086 097 107 18 " 118 128 138 149 159 170 180 190 201 211 19 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 387 397 408 418 21 428 439 449 459 469 480 490 500 511 521 22 531 542 552 562 572 583 593 603 613 624 23 634 644 655 665 675 685 696 706 716 726 24 737 747 757 767 778 788 798 808 818 829 25 839 849 859 870 880 890 900 910 921 931 26 941 951 961 972 982 992 *002 *012 *022 *033 27 63 043 053 063 073 083 094 104 114 124 134 28 144 155 165 175 185 195 205 215 225 236 29 246 256 266 27S 286 296 306 317 327 337 430 347 357 367 377 387 397 407 417 428 438 31 448 458 468 478 488 498 508 518 528 538 32 548 558 568 579 589 599 609 619 629 639 33 649 659 669 679 689 699 709 719 729 739 34 749 759 769 779 789 799 809 819 829 839 35 849 859 869 879 889 899 909 919 929 939 36 949 959 969 979 988 998 *008 *018 *028 *038 37 64 048 058 068 078 088 098 108 118 128 137 38 147 157 167 177 187 197 207 217 227 237 39 246 256 266 276 286 296 306 316 326 335 440 345 355 365 375 385 395 404 414 424 434 41 444 454 464 473 483 493 503 513 523 532 42 542 552 562 572 582 591 601 611 621 631 43 640 650 660 670 680 689 699 709 719 729 44 738 748 758 768 777 787 797 807 816 826 45 836 846 856 865 8V5 885 895 904 914 924 46 933 943 953 963 972 982 992 *OC2 *011 *021 47 65 031 040 050 060 070 079 089 099 108 118 48 128 137 147 157 167 176 186 196 205 215 49 225 234 244 254 263 273 283 292 302 312 450 321 331 341 350 360 369 379 389 398 408 mmmmi^mm N. ^M^^^^B O 1 2 3 t^mtmi^^m 4 5 6 7 8 9 [28] X. O 1 2 3 4 5 6 7 8 9 450 65 321 331- 341 350 360 369 379 389 398 408 51 418 427 437 447- 456 466 475 485 495 504 52 514 523 533 543 552 562 571 581 591 600 53 610 619 629 639 648 658 667 677 686 696 54 706 715 725 734 744 753 763 772 782 792 55 801 811 820 830 839 849 858 868 877 887 56 896 906 916 925 935 944 954 963 973 982 57 992 *001 *011 *020 *030 *039 *049 *058 *068 *077 58 66 087 096 106 115' 124 134 143 153 162 172 59 181 191 200 210 219 229 238 247 257 266 460 276 285 295 304 314 323 332 342 351 361 61 370 380 389 398 408 417 427 436 445 455 62 464 474 483 492 502 511 521 530 539 549 63 558 567 577 586 596 605 614 624 633 642 64 652 661 671 680 689 699 708 717 727 736 65 745 755 764 773 783 792 801 811 820 829 66 839 848 857 867 876 885 894 904 913 922 67 932 941 950 960 969 978 987 997 *006 *015 68 67 025 034 043 052 062 071 080 089 099 108 69 117 127 136 145 154 164 173 182 191 201 470 210 219 228 237 247 256 265 274 284 293 71 302 311 321 330 339 348 357 367 376 385 72 394 403 413 422 431 440 449 459 468 477 73 486 495 504 514 523 532 541 550 560 569 74 578 587 596 605 614 624 633 642 651 660 75 669 679 688 697 706 715 724 733 742 752 76 761 770 779 738 797 806 815 825 834 843 77 852 861 870 879 888 897 906 916 925 934 78 943 952 961 970 979 988 997 *006 *015 *024 79 68 034 043 052 061 070 079 088 097 106 115 480 124 133 142 151 160 169 178 187 196 205 81 215 224 233 242 251 260 269 278 287 296 82 305 314 323 332 341 350 359 368 377 386 83 395 404 413 422 431 440 449 458 467 476 84 485 494 502 511 520 529 538 547 556 565 85 574 583 592 601 610 619 628 637 646 655 86 664 673 681 690 699 708 717 726 735 744 87 753 762 771 780 789 797 806 815 824 833 88 842 851 860 869 878 886 895 904 913 922 89, .931 940 949 958 966 975 984 993 *002 *011 490 69 020 028 037 046 055 064 073 082 090 099 91 108 117 126 135 144 152 161 170 179 188 92 197 205 214 223 232 241 249 258 267 276 93 285 294 302 311 320 329 338 346 355 364 94 373 381 390 399 408 417 425 434 443 452 95 461 469 478 487 496 504 513 522 531 539 96 548 557 566 574 583 592 601 609 618 627 97 636 644 653 662 671 679 688 697 705 714 98 723 732 740 749 758 767 775 784 793 801 99 810 819 827 836 845 854 862 871 880 888 500 897 906 914 923 932 940 949 958 966 975 U. O 1 2 3 4 5 6 7 8 9 [29] R. O 1 2 3 4 5 6 7 8 9 500 69 897 906 914 923 932 940 949 958 966 975 01 984 992 *001 *010 *018 *027 *036 *044 *053 *062 02 70 070 079 088 096 105 114 122 131 140 148 03 157 165 174 183 191 200 209 217 226 234 04 243 252 260 269 278 286 295 303 312 321 05 329 338 346 355 364 372 381 389 398 406 06 415 424 432 441 449 458 467 475 484 492 07 501 509 518 526 535 544 552 561 569 578 08 586 595 603 612 621 629 638 646 655 663 09 672 680 689 697 706 714 723 731 740 749 510 757 766 774 783 791 800 808 817 825 834 11 842 851 859 868 876 885 893 902 910 919 12 927 935 944 952 961 969 978 986 995 *003 13 71 012 020 029 037 046 054 063 071 079 088 14 096 105 113 122 130 139 147 155 164 172 15 181 189 198 206 214 223 231 240 248 257 16 265 273 282 290 299 307 315 324 332 341 17 349 357 366 374 383 391 399 408 416 425 18 433 441 450 458 466 475 483 492 500 508 19 517 525 533 542 550 559 567 575 584 592 520 600 609 617 625 634 642 650 659 667 675 21 684 692 700 709 717 725 734 742 750 759 22 767 775 784 792 800 809 817 825 834 842 23 850 858 867 875 883 892 900 908 917 925 24 933 941 950 958 966 975 983 991 999 *008 25 72 016 024 032 041 049 057 066 074 082 090 26 099 107 115 123 132 1^0 148 156 165 173 27 181 189 198 206 214 222 230 239 247 255 28 263 272 280 288 296 304 313 321 329 337 29 346 354 362 370 378 387 395 403 411 419 530 428 436 444 452 460 469 477 485 493 501 31 509 518 526 534 542 550 558 567 575 583 32 591 599 607 616 624 632 640 648 656 665. 33 673 681 689 697 705 713 722 730 738 746 34 754 762 770 779 787 795 803 811 819 827 35 835 843 852 860 868 876 884 892 900 908 36 916 925 933 941 949 957 965 973 981 989 37 997 *006 *014 *022 *030 *038 *046 *054 *0,62 *070 38 73 Q78 086 094 102 111 119 127 135 143 151 39 159 167 175 183 191 199 207 215 223 231 540 239 247 255 263 272 280 288 296 304 312 41 320 328 336 344 352 360 368 376 384 392 42 400 408 416 424 432 440 448 456 464 472 43 480 488 496 504 512 520 528 536 544 552 44 560 568 576 484 592 600 608 616 624 632 45 640 648 656 664 672 679 687 695 703 711 46 719 727 735 743 751 759 767 775 783 791 47 799 807 815 823 830 838 846 854 862 870 48 878 886 894 902 910 918 926 933 941 949 49 957 965 973 981 989 997 *005 *013 *020 *028 550 74 036 044 052 060 068 076 084 092 099 107 N. O 1 2 3 4 5 6 7 8 9 [30] n. 1 2 3 4 5 6 7 8 9 550 74 036 044 052 060 068 076 084 092 099 107 51 115 123 131 139 147 155 162 170 178 186 52 194 202 210 218 225 233 241 249 257 265 53 273 280 288 296 304 312 320 327 335 343, 54 351 359 367 374 382 390 398 406 414 421 55 429 437 445' 453 461 468, 476 484 492 500 56 507 515 523 531 539 547 554 562 570 578 57 586 593 601 609 617 624 632 640 648 656 58 663 671 679 687 695 702 710 718 726 733 59 741 749 757 764 772 780 788 796 803 811 560 819 827 834 842 850 858 865 873 881 889 61 896 904 912 920 927 935 943 950 958 - 966 62 974 981 989 997 *005 *012 *020 *028 *035 *043 63 75 051 059 066 074 082 089 097 105 113 120 64 128 136 143 151 159 166 174 182 189 197 65 205 213 220 228 236 243 251 259 266 274 66 282 289 297 305 312 320 328 335 343 351 67 358 '366 374 381 389 397 404 412 420 427 68 435 442 450 458 465 473 481 488 496 504 69 511 519 526 534 542 549 557 565 572 580 570 587 595 603 610 618 626 633 641 648 656 71 664 671 679 686 694 702 709 717 724 732 72 740 747 755 762 770 778 785 793 800 808 73 815 823 831 838 846 853 861 868 876 884 74 891 899 906 914 921 929 937 944 952 959 75 967 974 982 989 997 *005 *012 *020 *027 *035 76 76 042 050 057 065 072 080 087 095 103 110 77 118 125 133 140 148 155 163 170 178 185 78 193 200 208 215 223 230 238 245 253 260 79 268 275 283 290 298 305 313 320 328 335 580 343 350 358 365 373 380 388 395 403 410 81 418 425 433 440 448 455 462 470 477 485 82 492 500 507 515 522 530 537 545 552, 559 83 567 574 582 589 597 604 612 619 626 634 84 641 649 656 664 671 678 686 693 701 708 85 716 723 730- 738 745 753 760 768 775 782 86 790 797 805 812 819 827 834 842 849 856 87 864 871 879 886 893 901 908 916 923 930 88 938 945 953 960 967 975 982 989 997 *004 89 77 012 019 026 034 041 048 056 063 070 078 590 085 093 100 107 115 122 129 137 144 151 91 159 166 173 181 188 195 203 210 217 225 92 232 240 247 254 262 269 276 283 291 298 93 305 313 320 327 335 342 349 357 364 371 94 379 386 393 401 408 415 422 430 437 444 95 452 459 466 474 481 488 495 503 510 517 96 525 532 539 546 554 561 568 576 583 590 97 597 605 612 619 627 634 641 648 656 663 98 670 677 685 692 699 706 714 721 728 735 99 743 750 757 764 772 779 786 793 801 808 600 815 822 830 837 844 851 859 866 873 880 X. O 1 2 3 4 5 6 7 8 9 [31] N. o 1 2 3 4 5 6 7 8 9 600 77 815 822 830 837 844 851 859 866 873 880 01 887 895 902 909 916 924, 931 938 945 952 02 960 967 974 981 988 996 *003 *010 *017 *025 03 78 032 039 046 053 061 068 075 082 089 90 097 04 104 111 118 125 132 140 147 154 161 168 05 176 183 190 197 204 211 219 226 233 240 06 247 254 262 269 276 283 290 297 305 312 07 319 326 333 340 347 355 362 369 376 383 08 390 398 405 412 419 426 433 440 447 455 09 462 469 476 483 490 497 504 512 519 526 610 533 540 547 554 561 569 576 583 590 597 11 604 611 618 625 633 640 647 654 661 668 12 675 682 689 696 704 711 718 725 732 739 13 746 753 760 767 774 781 789 796 803 810 14 817 824 831 838 845 852 859 866 873 880 15 888 895 902 909 916 923 930 937 944 951 16 958 965 972 979 986 993 *000 *007 *014 *021 17 79 029 036 043 050 057 064 071 078 085 092 18 099 106 113 120 127 134 141 148 155 162 19 169 176 18 190 197 204 211 218 225 232 620 239 246 25$ 260 267 274 281 288 295 302 21 309 316 323 330 337 344 351 358 365 372 22 379 386 393 400 407 414 421 428 435 442 23 449 456 463 470 477 484 491 498 505 511 24 518 525 532 539 546 553 560 567 574 581 25 588 595 602 609 616 623 630 637 644 650 26 657 664 671 678 685 ' 692 699 706 713 720 27 727 734 741 748 754 761 768 775 782 789 28 796 803 810 817 824 831 837 844 851 858 29 865 872 879 886 893 900 906 913 920 927 630 934 941 948 955 962 969 975 982 989 996. 31 80 003 010 017 024 030 037 044 051 058 065 32 * 072 079 085 092 099 106 113 120 127 134 33 140 147 154 161 168 175 182 188 195 202 34 209 216 223 229 236 243 250 257 264 271 35 277 284 291 298 305 312 318 325 332 339 36 346 353 359 366 373 380 387 393 400 407 37 414 421 428 434 441 448 455 462 468 475 38 482 489 496 502 509 516 523 530 536 543 39 550 557 564 570 577 584 591 598 604 611 640 618 625 632 638 645 652 659 665 672 679 41 686 693 699 706 713 720 726 733 740 747 42 754 760 767 774 781 787 794 801 808 814 43 821 828 835 841 848 5 862 868 875 882 44 889 895 902 909 916 922 929 936 943 949 45 956 963 969 976 983 990 996 *003 *010 *017 46 81 023 030 037 043 050 057 064 070 077 084 47 090 097 104 111 117 124 131 137 144 151 48 158 164 171 178 184 191 198 204 211 218 49 224 231 238 245 251 258 265 271 278 285 650 291 298 305 311 318 325 331 338 345 351 N. O 1 2 3 4 5 6 7 8 9 [32] N. 1 2 3 4 5 6 7 8 9 MM*^ 650 ^BMM^M^MBM 81 291 298 305 311 318 325 331 338 345 351 51 358 365 371 378 385 391 398 405 411 418 52 425 431 438 445 451 458 465 471 478 485 53 491 498 505 511 518 525 531 538 544 551 54 558 564 571 578 584 591 598 604 611 617 55 624 631 637 644 651 657 664 671 677 684 56 690 697 704 710 717 723 730 737 743 750 57 757 763 770 776 783 790 796 803 809 816 58 823 829 836 842 849 856 862 869 875 882 59 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 981 987 994 *000 *007 *014 61 82 020 027 033 040 046 053 060 066 073 079 62 086 092 099 105 112 119 125 132 138 145 63 151 158 164 171 178 184 191 197 204 210 64 217 223 230 236 243 249 256 263 269 276 65 282 289 295 302 308 315 321 328 334 341 66 347 354 360 367 373 380 387 393 400 406 67 413 419 426 432 439 445 452 458 465 471 68 478 484 491 497 504 510 517 523 530 536 69 543 549 556 562 569 575 582 588 595 601 670 607 614 620 627 633 640 646 653 659 666 71 672 679 685 692 698 705 711 718 724 730 72 737 743 750 756 763 769 776 782 789 795 73 802 808 814 821 827 834 840 847 853 860 74 866 872 879 885 892 898 905 911 918 924 75 930 937 943 950 956 963 969 975 982 988 76 995 *001 *008 *014 *020 *027 *033 *040 *046 *052 77 83 059 065 072 078 085 091 097 104 110 117 78 123 129 136 142 149 155 161 168 174 181 79 187 193 200 206 213 219 225 232 238 245 680 251 257 264 270 276 283 289 296 302 308 81 315 321 327 334 340 347 353 359 366 372 82 378 385 391 398 404 410 417 423 429 436 83 442 448 455 461 467 474 480 487 493 499 84 506 512 518 525 531 537 544 550 556 563 85 569 575 582 588 594 601 607 613 620 626 86 632 639 645 651 658 664 670 677 683 689 87 696 702 708 715 721 727 734 740 746 753 88 759 765 771 778 784 790 797 803 809 816 89 822 828 835 841 847 853 860 866 872 a79 -^90 885 891 897 904 910 916 923 929 935 942 91 948 954 960 967 973 979 985 992 998 *004 92 84 Oil 017 023 029 036 042 048 055 061 067 93 073 080 086 092 098 105 111 117 123 130 94 136 142 148 155 161 167 173 180 186 192 95 198 205 211 217 223 230 236 242 248 255 96 261 267 273 280 286 292 298 305 311 317 97 323 330 336 342 348 354 361 367 373 379 98 386 392 398 404 410 417 423 429 435 442 99 448 454 460 466 473 479 485 491 497 504 700 510 516 522 528 535 541 547 553 559 566 X. O 1 2 3 4 5 6 7 8 9 [33] N. O 1 2 3 4 5 6 7 8 9 700 84 510 516 522 528 535 541 547 553 559 566 01 572 578 584 590 597 603 609 615 621 628 02 634 640 646 652 658 665 671 677 683 689 03 696 702 708 714 720 726 733 739 745 751 04 757 763 770 776 782 788 794 800 807 813 05 819 825 831 837 844 850 856 862 868 874 06 880 887 893 899 905 911 917 924 930 936 07 942 948 954 960 967 973 979 985 991 997 08 85 003 009 016 022 028 034 040 046 052 058 09 065 071 077 083 089 095 101 107 114 120 710 126 132 138 144 150 156 163 169 175 181 11 187 193 199 205 211 217 224 230 236 242 12 248 254 260 266 272 278 285 291 297 303 13 309 315 321 327 333 339 845 352 358 364 14 370 376 382 388 394 400 406 412 418 425 15 431 437 443 449 455 461 467 473 479 485 16 491 497 503 509 516 522 528 534 540 546 17 552 558 564 570 576 582 588 594 600 606 18 612 618 625 631 637 643 649 655 661 667 19 673 679 685 691 697 703 709 715 721 727 720 733 739 745 751 757 763 769 775 781 788 21 794 800 806 812 818 824 830 836 842 848 22 854 860 866 872 878 884 890 896 902 908 23 914 920 926 932 938 944 950 956 962 968 24 974 980 986 992 998 *004 *010 *016 *022 *028 25 86 034 040 046 052 058 064 070 076 082 088 26 094 100 106 112 118 124 130 136 141 147 27 153 159 165 171 177 183 189 195 201 207 28 213 219 225 231 237 243 249 255 261 267 29 273 279 285 291 297 303 308 314 320 326 730 332 338 344 350 356 362 368 374 380 386 31 392 390 404 410 415 421 427 433 439 445 32 451 457 463 469 475 431 487 493 499 504 33 510 516 522 528 534 540 546 552 558 564 34 570 576 581 587 593 599 605 611 617 623 35 629 635 641 646 652 658 664 670 676 682 36 688 694 700 705 711 717 723 729 735 741 37 747 753 759 764 770* 776 782 788 794 800 38 806 812 817 823 829 835 841 847 853 859 39 864 870 876 882 888 894 900 906 911 917 740 923 929 935 941 947 953 958 964 970 976 41 982 988 994 999 *005 *011 *017 *023 *029 *035 42 87 040 046 052 058 064 070 075 081 087 093 43 099 105 111 116 122 128 134 140 146 151 44 157 163 169 175 181 186 192 198 204 210 45 216 221 227 233 239 245 251 256 262 268 46 274 280 286 291 297 303 309 315 320 326 47 332 338 344 349 355 361 367 373 379 384 48 390 396 402 408 413 419 425 431 437 442 49 448 454 460 466 471 477 483 489 495 500 750 506 512 518 523 529 535 541 547 552 558 N. O 1 2 3 4 5 6 7 8 9 [34] 5 750 87 506 512 518 523 529 535 541 547 552 558 51 564 570 576 581 587 593 599 604 610 616 52 622 628 633 639 645 651 656 662 668 674 53 679 685 691 697 703 708 714 720 726 731 54 737 743 749 754 760 766 772 777 783 789 55 795 800 806 812 818 823 829 835 841 846 56 852 858 864 869 875 881 887 892 898 904 57 910 915 921 927 933 938 944 950 955 961 58 967 973 978 984 990 996 *001 *007 *013 *018 59 88 024 030 036 041 047 053 058 064 070 076 760 081 087 093 098 104 110 116 121 127 133 61 138 144 150 156 161 167 173 178 184 190 62 195 201 207 213 218 224 230 235 241 247 63 252 258 264 270 275 281 287 292 298 304 64 309 315 321 326 332 338 343 349 355 360 65 866 372; 377 383 389 395 400 406 412 417 66 * 423 429 434 440 446 451 457 463 468 474 67 480 485 491 497 502 508 513 519 525 530 68 536 542 547 553 559 564 570 576 58T 587 69 593 598 604 610 615 621 627 632 638 643 770 649 655 660 666 672 677 683 689 694 700 71 705 711 717 722 728 734 739 745 750 756 72 762 767 773 779 784 790 795 801 807 812 73 818 824 829 835 840 846 852 857 863 868 74 874 880 885 891 897 902 908 913 919 926 75 930 936 941 947 953 958 964 969 975 981 76 986 992 997 *003 *009 *014 *020 *025 *031 *037 77 89 042 048 053 059 064 070 076 081 087 092 78 098 104 109 115 120 126 131 137 143 148 79 154 159 165 170 176 182 187 193 198 204 780 209 215 221 226 232 237 243 248 254 260 81 265 271 276 282 281 293 298 304 310 315 82 321 326 332 337 343 348 354 360 365 371 83 376 382 387 393 398 404 409 415 421 426 84 432 437 443- 448 454 459 46~5" ~470 476 481 85 487 492 498 504 509 515 520 526 531 537 86 542 548 553 559 56J 570 575 581 586 592 87 597 603 609 614 620 625 631 636 642 647 88 653 658 664 669 675 680 686 691 697 702 89 708 713 719 724 730 735 741 746 752 757 790 763 768 774 779 785 790 796 801 807 812 91 818 823 829 834 840 845 851 856 862 867 92 873 878 883 889 894 900 905 911 916 922 93 927 933 938 944 949 955 960 966 971 977 94 982 988 993 998 *004 *009 *015 *020 *026 *031 95 90 037 042 048 053 059 064 069 075 080 086 96 091 097 102 108 113 119 124 129 135 140 97 146 151 157 162 168 173 179 184 189 195 98 200 206 211 217 222 227 233 238 244 249 99 255 260 266 271 276 282 287 293 298 304 800 309 314 320 325 331 336 342 347 352 358 N. O 1 2 3 4 5 6 7 iMHHHHi 8 9 [35] N. O 1 2 3 4 5 6 7 8 9 800 90 309 314 320 325 331 336 342 347 352 358 01 363 369 374 380 385 390 396 401 407 412 02 417 423 428 434 439 445 450 455 461 466 03 472 477 482 488 493 499 504 509 515 520 04 526 531 536 542 547 553 558 563 569 574 05 580 585 590 596 601 607 612 617 623 628 06 634 639 644 650 655 660 666 671 677 682 07 687 693 698 703 709 714 720 725 730 736 08 741 747 752 757 763 768 773 779 784 789 09 795 800 806 811 816 822 827 832 838 843 810 849 854 859 865 870 875 881 886 891 897 11 902 907 913 918 924 929 934 940 945 950 12 956 961 966 972 977 982 988 993 998 *004 13 91 009 014 020 025 030 036 041 046 052 057 14 062 068 073 078 084 089 094 100 105 110 15 116 121 126 132 137 142 148* 153 158 164 16 169 174 180 185 190 196 201 206 212 217 17 222 228 233 238* 243 249 254 259 265 270 18 275 281 286 291 297 302 307 312 318 323 19 328 334 339 344 350 355 360 365 371 376 820 381 387 392 397 403 408 413 418 424 429 21 434 440 445 450 455 461 466 471 477 482 22 487 492 498 503 508 514 519 524 529 535 23 540 545 551 556 561 566 572 577 582 587 24 593 598 603 609 614 619 624 630 635 640 25 645 651 656 661 666 672 677 682 687 693 26 698* 703 709 714 719 724 730 735 740 745 27 751 756 761 766 772 777 782 787 793 798 28 803 808 814 819 824 829 834 840 845 850 29 855 861 866 871 876 882 887 892 897 903 830 908 913 918 924 929 934 939 944 950 955 31 960 965 971 976 981 986 991 997 *002 *007 32 92 012 018 023 028 033 038 044 049 054 059 33 065 070 075 080 085 091 096 101 106 111 34 117 122 127 132 137 143 148 153 158 163 35 169 174 179 184 189 195 200 205 210 215 36 221 226 231 236 241 247 252 257 262 267 37 273 278 283 288 293 298 304 309 314 319 38 324 330 335 340 345 350 355 361 366 371 39 376 381 387 392 397 402 407 412 418 423 840 428 433 438 443 449 454 459 464 469 474 41 480 485 490 495 500 505 511 516 521 526 42 531 536 542 547 552 557 562 567 572 578 43 583 588 593 598 603 609 614 619 624 629 44 634 639 645 650 655 660 665 670 675 681 45 686 691 696 701 706 711 716 722 727 732 46 737 742 747 752 758 763 768 773 778 783 47 788 793 799 804 809 814 819 824 829 834 48 840 845 850 855 860 865 870 875 881 886 49 891 896 901 906 911 916 921 927 932 937 850 942 947 952 957 962 967 973 978 983 988 N. 1 ~ -rl Z Z 7 8 9 [36] o 3 jzra 850 92 942 947 952 957 962 967 973 978 983 988 51 993 998 *003 *008 *013 *018 *024 *029 *034 *039 52 93 044 049 054 059 064 069 075 080 085 090 53 095 100 105 110 115 120 125 131 136 141 54 146 151 156 161 166 171 176 181 186 192 55 197 202 207 212 217 222 227 232 237 242 56 247^ 252 258 263 268 273 278 283 288 293 57 298 303 308 313 318 323 328 334 339 344 58 349 354 359 364 369 374 379 384 389 394 59 399 404 409 414 420 425 430 435 440 445 860 450 455 460 465 470 475 480 485 490 495 61 500 505 510 515 520 526 531 536 541 546 62 551 556 561 566 571 576 581 586 591 596 63 601 606 611 616 621 626 631 636 641 646 64 651 656 661 666 671 676 682 687 692 697 65 702 707 712 717 722 727 732 737 742 747 66 752 757 762 767 772 777 782 787 792 797 67 802 807 812 817 822 827 832 837 842 847 68 852 857 862 867 872 877 882 887 892 897 69 902 907 912 917 922 927 932 937 942 947 870 952 957 962 967 972 977 982 987 992 997 71 94 002 007 012 017 022 027 032 037 042 047 72 052 057 062 067 072 077 082 086 091 096 73 101 106 111 116 121 126 131 136 141 146 74 151 156 161 166 171 176 181 186 191 196 75 201 206 211 216 221 226 231 236 240 245 76 250 255 260 265 270 275 280 285 290 295 77 300 305 310 315 320 325 330 335 340 345 78 349 354 359 364 369 374 379 384 389 394 79 399 404 409 414 419 424 429 433 438 443 880 448 453 458 463 468 473 478 483 488 493 81 498 503 507 512 517 522 527 532 537 542 82 547 552 557 562 567 571 576 581 586 591 83 596 601 606 611 616 621 626 630 635 640 84 645 650 655 660 665 670 675 680 685 689 85 694 699 704 709' 714 719 724 729 734 738 86 743 748 753 758 763 768 773 778 783 787 87 792 797 802 807 812 817 822 827 832 836 88 841 846 851 856 861 866 871 876 880 885 89 890 -895 900 905 910 915 919 924 929 934 890 939 944 949 954 959 963 968 973 978 983 91 988 993 998 *002 *007 *012 *017 *022 *027 *032 92 95 036 041 046 051 056 061 066 071 075 080 93 085 090 095 100 105 109 114 119 124 129 94 134 139 143 148 153 158 163 168 173 177 95 182 187 192 197 202 207 211 216 221 226 96 231 236 240 245 250 255 260 265 270 274 97 279 284 289 294 299 303 308 313 318 323 98 328 332 337 342 347 352 357 361 366 371 99 376 381 386 390 395 400 405 410 415 419 900 424 429 434 439 444 448 453 458 463 468 X. O 1 2 3 4 5 6 7 8 9 [37] N. 1 2 3 4 5 6 7 8 9 900 95 424 429 434 439 444 448 453 458 463 468 01 472 477 482 487 492 497 501 506 511 516 02 521 525 530 535 540 545 550 554 559 564 03 569 574 578 583 588 593 598 602 607 612 04 617 622 626 631 636 641 646 650 655 660 05 665 670 674 679 684 689 694 698 703 708 06 713 718 722 727 732 737 742 746 751 756 07 761 766 770 775 780 785 789 794 799 804 08 809 813 818 823 828 832 837 842 847 852 09 856 861 866 871 875 880 885 890 895 899 910 904 909 914 918 923 928 933 938 942 947 11 952 957 961 966 971 976 980 985 990 995 12 999 *004 *009 *014 *019 *023 *028 *033 *038 *042 13 96 047 052 057 061 066 071 076 080 085 090 14 095 099 104 109 114 118 123 128 133 137 15 142 147 152 156 161 166 171 175 180 185 16 190 194 199 204 2d9 2-13 218 223 227 232 17 237 242 246 251 256 261 265 270 275 280 18 284 289 294 298 303 308 313 317 322 327 19 332 336 341 346 350 355 360 365 369 374 920 379 384 388 393 398 402 407 412 417 421 21 426 431 435 440 445 450 454 459 464 468 22 473 478 483 487 492 497 501 506 511 515 23 520 525 530 534 539 544 548 553 558 562 24 567 572 577 581 586 591 595 600 605 609 25 614 619 624 628 633 638 642 647 652 656 26 661 666 .670 675 680 685 689 694 699 703 27 708 713 717 722 727 731 736 741 745 750 28 755 759 764 769 774 778 783 788 792 797 29 802 806 811 816 820 825 830 834 839 844 930 848 853 858 862 867 872 876 881 886 890 31 895 900 904 909 914 918 923 928 932 937 32 942 946 951 956 960 965 970 974 979 984 33 988 993 997 *002 *007 *011 *0}6 *021 *025 *030 34 97 035 039 044 049 053 058 063 067 072 077 35 081 086 090 095 100 104 109 114 118 123 36 128 132 137 142 146 151 155 160 165 169. 37 174 179 183 188 192 197 202 206 211 216 38 220 225 230 234 239 243 248 253 257 262 39 267 271 276 280 285 290 294 29 304 308 940 313 317 322 327 331 336 340 345 350 354 41 359 364 368 373 377 382 387 391 396 400 42 405 410 414 419 424 428 433 437 442 447 43 451 456 460 465 470 474 479 483 488 493 44 497 502 506 511 516 520 525 529 534 539 45 543 548 552 557 562 566 571 575 580 585 46 589 594 598 603 607 612 617 621 626 630 47 635 640 644 649 653 658 663 667 672 676 48 681 685 690 695 699 704 708 713 717 722 49 727 731 736 740 745 749 754 759 763 768 950 772 777 782 786 791 795 800 804 809 813 N. NMMB^H^B 1 2 3 4 5 6 7 8 9 [38] ff. o 1 2 3 4 5 6 7 8 9 950 97 772 777 782 786 791 795 800 804 809 813 51 818 823 827 832 836 841 845 850 855 859 52 864 868 873 877 882 886 891 896 900 905 53 909 914 918 923 928 932 937 941 946 950 lA 955 959 964 968 973 978 982 987 991 996 1,5 98 000 005 009 014 019 023 028 032. 037 041 6 046 050 055 059 064 068 073 078 082 087 57 091 096 100 105 109 114 118 123 127 132 58 137 141 146 150 155 159 164 168 173 177 59 182 186 191 195 200 204 209 214 218 223 960 227 232 236 241 245 250 254 259 263 268 61 272 277 281 286 290 295 299 304 308 313 62 318 322 '327 331 336 340 345 349 354 358 63 363 367 372 376 381 385 390 394 29.9 403 64 408 412 417 421 426 430 435 439 444 448 65 453 457 462 466 471 475 480 484 489 493 66 498 502 507 511 516 520 525 529 534 538 67 543 547 552 556 561 565 570 574 579 583 68 588 592 597 601 605 610 614 619 623 628 69 632 637 641 646 650 655 659 664 668 673 970 677 682 686 691 695 700 704 709 713 717 71 722 726 731 735 740 744 749 753 758 762 72 767 771 776 780 784 789 793 798 802 807 73 811 816 820 825 829 834 838 843 847 851 74 856 860 865 869 874 878 883 887 892 896 75 900 905 909 914 918 923 927 932 936 941 76 945 949 954 958 963 967 972 976 981 985 77 989 994 998 *003 *007 *012 *016 *021 *025 *029 78 99 034 038 043 047 052 056 061 065 069 074 79 078 083 087 092 096 100 105 109 114 118 980 123 127 131 136 140 145 149 154 158 162 81 167 171 176 180 185 189 193 198 202 207 82 211 216 220 224 229 % 233 238 242 247 251 83 255 260 264 269 273 277 282 286 291 295 84 300 304 308 313 317 322 . 326 330 335 339 85 344 348 352 357 361" 366 370 374 379 383 86 388 392 396 401 405 410 414 419 423 427 87 432 436 441 445 ' 449 454 458 463 467 471 88 476 480 484 489 493 498 502 506 511 515 89 520 524 528 533 537 542 546 550 555 559 990 564 568 572 577 581 585 590 594 599 603 91 607 612 616 621 625 629 634 638 642 647 92 651 656 660 664 669 673 677 682 686 691 93 695 699 704 708 712 717 721 726 730 734 94 739 743 747 752 756 760 765 769 774 778 95 782 787 791 795 800 804 808 813 817 822 96 826 830 835 839 843 848 852 856 861 865 97 870 874 878 883 887 891 896 900 904 909 98 913 917 922 926 930 935 939 944 948 952 99 957 961 965 970 974 978 983 987 991 996 1000 00 000 004 009 013 017 022 026 030 035 039 H. O 1 2 3 4 5 6 7 8 9 [39] TABLE II LOGS AND COLOGS OF CERTAIN MUCH-USED NUMBEBS NUMBER LOGARITHM COLOGARITHM 2 0.3010300 9.6989700-10 3 V2 V3 7T 0.4771213 0.1505150 0.2385607 0.4971499 9.5228787-10 9.8494850-10 9.7614^93-10 9.5028501-10 7T 2 0.9942997 9.0057003-10 27T 0.7981799 9.2018201-10 VTT 0.2485749 9.7514251-10 57.2957795 1.7581226 8.2418774-10 206264.806 5.3144251 4.6855749-10 FIVE PLACE 2 0.30103 9.69897-10 3 0.47712 9.52288-10 V2 0.15052 9.84948-10 V3 0.23856 9.76144-10 IT 0.49715 9.50285-10 7T 2 0.99430 9.00570-10 27T 0.79818 9.20182-10 VTT 0.24857 9.75143-10 57.2957795 1.75812 8.24188-10 206264.806 5.31443 4.68557-10 FOUR PLACE 2 0.3010 9.6990-10 3 0.4771 9.5229-10 V2 0.1505 9.8495-10 V3 0.2386 9.7614-10 7T 0.4971 9.5029-10 7T 2 0.9943 9.0057-10 27T 0.7982 9.2018-10 vV 0.2486 9.7514-10 57.2956695 1.7581 8.2419-10 206264.806 5.3144 4.6858-10 [40] TABLE III FIVE-PLACE LOGARITHMS OF THE SINE, COSINE, TANGENT, AND COTANGENT FOR EACH MINUTE OF THE QUADRANT [41] / L. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 oo 6.46 373 6.76 476 6.94 085 7.06 579 oo 6.46 373 6.76 476 6.94 085 7.06 579 oo 3.53 627 3.23 524 3.05 915 2.93 421 0.00 000 0.00 000 0.00 000 0.00 000 0.00 000 60 59 58 57 56 5 6 7 8 9 7.16 270 7.24 188 7.30 882 7.36 682 7.41 797 7.16 270 7.24 188 7.30 882 7.36 82 7.41 797 2.83 730 2.75 812 2.69 118 2.63 318 2.58 203 0.00 000 0.00 000 0.00 000 0.00 000 0.00 000 55 54 53 52 51 89 10 11 12 13 14 7.46 373 7.50 512 7.54 291 7.57 767 7.60 985 7.46 373 7.50 512 7.54 291 7.57 767 7.60 986 2.53 627 2.49 488 2.45 709 2.42 233 2.39 014 0.00 000 0.00 000 0.00 000 0.00 000 0.00 000 50 49 48 47 46 15 16 17 18 19 7.63 982 7.66 784 7.69 417 7.71 900 7.74 248 7.63 982 7.66 785 7.69 418 7.71 900 7.74 248 2.36 018 2.33 215 2.30 582 2.28 100 2.25 752 0.00 000 0.00 000 9.99 999 9.99 999 9.99 999 45 44 43 42 41 20 21 22 23 24 7.76 475 7.78 594 7.80 615 7.82 545 7.84 393 7.76 476 7.78 595 7.80 615 7.82 546 7.84 394 2.23 524 2.21 405 2.19 385 2.17 454 2.15 606 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 40 39 38 37 36 25 26 27 28 29 7.86 166 7.87 870 7.89 509 7.91 088 7.92 612 7.86 167 7.87 871 7.89 510 7.91 089 7.92 613 2.13 833 2.12 129 2.10 490 2.08 911 2.07 387 9.99 999 9.99 999 9.99 999 9.99 999 9.99 998 35 34 33 32 31 30 31 32 33 34 7.94 084 7.95 508 7.96 887 7.98 223 7.99 520 7.94 086 7.95 510 7.96 889 7.98 225 7.99 522 2.05 914 2.04 490 2.03 111 2.01 775 2.00 478 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 30 29 28 27 26 35 36 37 38 39 8.00 779 8.02 002 8.03 192 8.04 350 8.05 478 8.00 781 8.02 004 8.03 194 8.04 353 8.05 481 1.99 219 1.97 996 1.96 806 1.95 647 1.94 519 9.99 998 9.99 998 9.99 997 9.99 997 9.99 997 25 24 23 22 21 40 41 42 43 44 8.06 578 8.07 650 8.08 696 8.09 718 8.10 717 8.06 581 8.07 653 8.08 700 8.09 722 8.10 720 1.93 419 1.92 347 1.91 300 1.90 278 1.89 280 9.99 997 9.99 997 9.99 997" 9.99 997 9.99 996 20 19 18 17 16 45 46 47 48 49 8.11 693 8.12 647 8.13 581 8.14 495 8.15 391 8.11 696 8.12 651 8.13 585 8.14 500 8.15 395 1.88 304 1.87 349 1.86 415 1.85 500 1.84 605 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 15 14 13 12 11 50 51 52 53 54 8.16 268 8.17 128 8.17 971 8.18 798 8.19 610 8.16 273 8.17 133 8.17 976 8.18 804 8.19 616 1.83 727 1.82 867 1.82 024 1.81 196 1.80 384 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 10 9 8 7 6 55 56 57 58 59 8.20 407 8.21 189 8.21 958 8.22 713 8.23 456 8.20 413 8.21 195 8.21 964 8.22 720 8.23 462 1.79 587 1.78 805 ' 1.78 036 1.77 280 1.76 538 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 5 4 3 2 1 60 8.24 186 8.24 192 1.75 808 9.99 993 L. Cos. L. Cotg. L. Tang. L. Sin. r [42] / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 8.24 186 8.24 903 8.25 609 8.26 304 8.26 988 8.24 192 8.24 910 8.25 616 8.26 312 8.26 996 1.75 808 1.75 090 1.74 384 1.73 688 1.73 004 9.99 993 9.99 993 9.99 993 9.99 993 9.99 992 60 59 58 57 56 5 6 7 8 9 8.27 661 8.28 324 8.28 977 8.29 621 8.30 255 8.27 669 8.28 332 8.28 986 8.29 629 8.30 263 1.72 331 1.71 668 1.71 014 1.70 371 1.69 737 9.99 992 9.99 992 9.99 992 9.99 992 9.99 991 55 54 53 52 51 10 11 12 13 14 8.30 879 8.31 495 8.32 103 8.32 702 8.33 292 8.30 888 8.31 505 8.32 112 8.32 711 8.33 302 1.69 112 1.68 495 1.67 888 1.67 289 1.66 698 9.99 991 9.99 991 9.99 990 9.99 990 9.99 990 50 49 48 47 46 15 16 17 18 19 8.33 875 8.34 450 8.35 018 8.35 578 8.36 131 8.33 886 8.34 461 8.35 029 8.35 590 8.36 143 1.66 114 1.65 539 1.64 971 1.64 410 1.63 857 9.99 990 9.99 989 9.99 989 9.99 98,9 ' 9.99 989 45 44 43 42 41 20 21 22 23 24 8.36 678 8.37 217 8.37 750 8.38 276 8.38 796 8.36 689 8.37 229 8.37 762 8.38 289 8.38 809 1.63 311 1.62 771 1.62 238 1.61 711 1.61 191 9.99 988 9.99 988 9.99 988 9.99 987 9.99 987 40 39 38 37 36 1 25 26 27 28 29 8.39 310 8.39 818 8.40 320 8.40 816 8.41 307 8.39 323 8.39 832 8.40 334 8.40 830 8.41 321 1.60 677 1.60 168 1.59 666 1.59 170 1.58 679 9.99 987 9.99 986 9.99 986 9.99 986 9.99 985 35 34 33 32 31 88 30 31 32 33 34 8.41 792 8.42 272 8.42 746 8.43 216 8.43 680 8.41 807 8.42 287 8.42 762 8.43 232 8.43 696 1.58 193 1.57 713 1.57 238 1.56 768 1.56 304 9.99 985 9.99 985 9.99 984 9.99 984 2.99 984 30 29 28 27 26 35 36 37 38 39 8.44.139 8.44 594 8.45 044 8.45 589 8.45 930 8.44 156 8.44 611 8.45 061 8.45 507 8.45 948 1.55 844 1.55 389 1.54 939 1.54 493 1.54 052 9.99 983 9.99 983 9.99 983 9.99 982 9.99 982 25 24 23 22 21 40 41 42 43 44 8.46 366 8.46 799 8.47 226 8.47 650 8.48 069 8.46 385 8.46 817 8.47 245 8.47 669 8.48 089 1.53 615 1.53 183 1.52 755 1.52 331 1.51 911 9.99 982 9.99 981 9.99 981 9.99 981 9.99 980 20 19 18 17 16 45 46 47 48 49 8.48 485 8.48 896 8.49 304 8.49 708 8.50 108 8.48 505 8.48 917 8.49 325 8.49 729 8.50 130 1.51 495 1.51 083 1.50 675 1.50 271 1.49 870 9.99 980 9.99 979 9.99 979 9.99 979 9.99 978 15 14 13 12 11 50 51 52 53 54 55 56 57 58 59 8.50 504 8.50 897 8.51 287 8.51 673 8.52 055 8.50 527 8.50 920 8.51 310 8.51 696 8.52 079 1.49 473 1.49 080 1.48 690 1.48 304 1.47 921 9.99 978 9.99 977 9.99 977 9.99 977 9.99 976 10 9 8 7 6 8.52 434 8.52 810 8.53 183 8.53 552 8.53 919 8.52 459 8.52 835 8.53 208 8.53 578 8.53 945 1.47 541 1.47 165 1.46 792 1.46 422 1.46 055 9.99 976 9.99 975 9.99 975 9.99 974 9.99 974 5 4 3 2 1 60 8.54 282 8.54 308 1.45 692 9.99 974 L. Cos. L. Cotg. L. Tang. L. Sin. / [48] i L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 8.54 282 8.54 642 8.54 999 8.55 354 8.55 705 8.54 308 8.54 669 8.55 027 8.55 382 8.55 734 1.45 692 1.45 331 1.44 973 1.44 618 1.44 266 9.99 974 9.99 973 9.99 973 9.99 972 9.99 972 60 59 58 57 56 87 5 6 7 8 9 8.56 054 8.56 400 8.56 743 8.57 084 8.57421 8.56 083 8.56 429 8.56 773 8.57 114 8.57 452 1.43 917 1.43 571 1.43 227 1.42 886 1.42 548 9.99 971 9.99 971 9.99 970 9.99 970 9.99 969 55 54 53 52 51 10 11 12 13 14 8.57 757 8.58 089 8.58 419 8.58 747 8.59 072 8.57 788 8.58 121 8.58 451 8.58 779 8.59 105 1.42 212 1.41 879 1.41 549 1.41 221 1.40 895 9.99 969 9.99 968 9.99 968 9.99 967 9.99 967 50 49 48 47 46 15 16 17 18 19 8.59 395 8.59 715 8.60 033 8.6Q 349 8.60 662 8.59 428 8.59 749 8.60 068 8.60 384 8.60 698 1.40 572 1.40 251 1.39 932 1.39 616 1.39 302 9.99 967 9.99 966 9.99 966 9.99 965 9.99 964 45 44 43 42 41 20 21 22 23 24 8.60 973 8.61 282 8.61 589 8.61 894 8.62 196 8.61 009 8.61 319 8.61 626 8.61 931 8.62 234 1.38 991 1.38 681 1.38 374 1.38 069 1.37 766 9.99 964 9.99 963 9.99 963 9.99 962 9.99 962 40 39 38 37 36 25 26 27 28 29 8.62 497 8.62 795 8.63 091 8.63 385 8.63 678 8.62 535 8.62 834 8.63 131 8.63 426 8.63 718 1.37465 1.37 166 1.36 869 1.36 574 1.36 282 9.99 961 9.99 961 9.99 960 9.99"960 9.99 959 35 34 33 32 31 a u 30 31 32 33 34 8.63 968 8.64 256 8.64 543 8.64 827 8.65 110 8.64 009 8.64 298 8.64 585 8.64 870 8.65 154 1.35 991 1.35 702 1.35 415 1.35 130 1.34 846 9.99 959 9.99 958 9.99 958 9.99 957 9.99 956 30 29 28 27 26 35 36 37 38 39 8.65 391 8.65 670 8.65 947 8.66 223 8.66 497 865435 8.65 715 8.65 993 8.66 269 8.66 543 1.34 565 1.34 285 1.34 007 1.33 731 1.33 457 9.99 956 9.99 955 9.99 955 9.99 954 9.99 954 25 24 23 22 21 40 41 42 43 44 8.66 769 8.67 039 8.67 308 8.67 575 8.67 841 8.66 816 8.67 087 8.67 356 8.67 624 8.67 890 1.33 184 1.32 913 1.32 644 1.32 376 1.32 110 9.99 953 9.99 952 9.99 952 9.99 951 9.99 951 20 19 18 17 16 45 46 47 48 49 8.68 104 8.68 367 8.68 627 8.68 886 ' 8.69 144 8.68 154 8.68 417 8.68 678 8.68 938 8.69 196 1.31 846 1.31 583 1.31 322 1.31 062 1.30 804 9.99 950 9.99 949 9.99 949 9.99 948 9.99 948 15 14 13 12 11 50 51 52 53 54 8.69 400 8.69 654 8.69 907 8.70 159 8.70 409 8.69 453 8.69 708 8.69 962 8.70 214 8.70 465 1.30 547 1.30 292 1.30 038 1.29 786 1.29 535 9.99 947 9.99 946 9.99 946 9.99 945 9.99 944 10 9 8 7 6 55 56 57 58 59 8.70 658 8.70 903 8.71 151 8.71 395 8.71 638 8.70 714 8.70 962 8.71 208 8.71 453 8.71 697 1.29 286 1.29 038 1.28 792 1.28 547 1.28 303 9.99 944 9.99 943 9.99 942 9.99 942 9.99 941 5 4 3 2 1 60 8.71 880 8.71 940 1.28 060 9.99 940 L. Cos. L. Cotg. L. Tang. L. Sin. / [44] / L. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 8.71 880 8.72 120 8.72 359 8.72 597 8.72 834 8.71 940 8.72 181 8.72 420 8.72 659 8.72 896 1.28 060 1.27 819 1.27 580 1.27 341 1.27 104 9.99 940 9.99 940 9.99 939 9.99 938 9.99 938 60 59 58 57 56 5 6 7 8 9 8.73 069 8.73 303 8.73 535 8.73 767 8.73 997 8.73 132 8.73 366 8.73 600 8.73 832 8.74 063 1.26 868 1.26 634 1.26 400 1.26 168 1.25 937 9.99 937 9.99 936 9.99 936 9.99 935 9.99 934 55 54 53 52 51 10 11 12 13 14 8.74 226 8.74 454 8.74 680 8.74 906 8.75 130 8.74 292 8.74 521 8.74 748 8.74 974 8.75 199 1.25 708 1.25 479 1.25 252 1.25 026 1.24 801 9.99 934 9.99 933 9.99 932 9.99 932 9.99 931 50 49 48 47 46 15 16 17 18 19 8.75 353 8.75 575 8.75 795 8.76 015 8.76 234 8.75 423 8.75 645 8.75 867 8.76 087 8.76 306 1.24 577 1.24 355 1.24 133 .23 913 .23 694 9.99 930 9.99 929 9.99 929 9.99 928 9.99 927 45 44 43 42 '41 20 21 22 23 24 8.76 451 8.76 667 8.76 883 8.77 097 8.77 310 8.76 525 8.76 742 8.76 958 8.77 173 8.77 387 .23 475 .23 258 .23 042 .22 827 .22 613 9.99 926 9.99 926 9.99 925 9.99 924 9.99 923 40 39 38 37 36 3 25 26 27 28 29 8.77 522 8.77 733 8.77 943 8.78 152 8.78 360 8.77 600 8.77 811 8.78 022 8.78 232 8.78 441 1.22 400 1.22 189 1.21 978 1.21 768 1.21 559 9.99 923 9.99 922 9.99 921 9.99 920 9.99 920 35 34 33 32 31 ftfi 30 31 32 33 34 8.78 568 8.78 774 8.78 979 8.79 183 8.79 386 8.78 649 8.78 855 8.79 061 8.79 266 8.79 470 1.21 351 1.21 145 1.20 939 1.20 734 1.20 530 9.99 919 9.99 918 9.99 917 9.99 917 9.99 916 30 29 28 27 26 ou 35 36 37 38 39 8.79 588 8.79 789 8.79 990 8.80 189 8.80 388 8.79 673 8.79 875 8.80 076 8.80 277 8.80 476 1.20 327 1.20 125 1.19 924 1.19 723 1.19 524 9.99 915 9.99 914 9.99 913 9.99 913 9.99 912 25 24 23 22 21 40 41 42 43 44 8.80 585 8.80 782 8.80 978 8.81 173 8.81 367 8.80 674 8.80 872 8.81 068 8.81 264 8.81 459 1.19 326 1.19 128 1.18 932 1.18 736 1.18 541 9.99 911 9.99 910 9.99 909 9.99 909 9.99 908 20 19 18 17 16 45 46 47 48 49 8.81 560 8.81 752 8.81 944 8.82 134 8.82 324 8.81 653 8.81 846 8.82 038 8.82 230 8.82 420 1.18 347 1.18 154 1.17 962 1.17 770 1.17 580 9.99 907 9.99 906 9.99 905 9.99 904 9.99 904 15 14 13 12 11 50 51 52 53 54 8.82 513 8.82 701 8.82 888 8.83 075 8.83 261 8.82 610 8.82 799 8.82 987 8.83 175 8.83 361 1.17 390 1.17 201 1.17013 1.16 825 1.16 639 9.99 903 9.99 902 9.99 901 9.99 900 9.99 899 10 9 8 7 6 55 56 57 58 59 8.83 446 8.83 630 8.83 813 8.83 996 8.84 177 8.83 547 8.83 732 8.83 916 8.84 100 8.84 282 1.16 453 1.16 268 1.16 084 1.15 900 1.15 718 9.99 898 9.99 898 9.99 897 9.99 896 9.99 895 5 4 3 2 1 60 8.84 358 8.84 464 1.15 536 9.99 894 L. Cos. L. Cotg. L. Tang. L. Sin. / [45] t L. Sin. I. Tang. L. Cotg. L. Cos. i 2 3 4 8.84 358 8.84 539 8.84 718 8.84 897 8.85 075 8.84 464 8.84 646 8.84 826 8.85 006 8.85 185 1.15 536 1.15 354 1.15 174 1.14 994 1.14 815 9.99 894 9.99 893 9.99 892 9.99 891 9.99 891 60 59 58 57 56 5 6 7 8 9 8.85 252 8.85 429 8.85 605 8.85 780 8.85 955 8.85 363 8.85 540 8.85 717 8.85 893 8.86 069 1.14 637 1.14 460 1.14 283 1.14 107 1.13 931 9.99 890 9.99 889 9.99 888 9.99 887 9.99 886 55 54 53 52 51 10 11 12 13 14 8.86 128 8.86 301 8.86 474 8.86 645 8.86 816 8.86 243 8.86 417 8.86 591 8.86 763 8.86 935 1.13 757 1.13 583 1.13 409 1.13 237 1.13 065 9.99 885 9.99 884 9.99 883 9.99 882 9.99 881 50 49 48 47 46 15 16 17 18 19 8.86 987 8.87 156 8.87 325 8.87 494 8.87 661 8.87 106 8.87 277 8.87 447 8.87 616 8.87 785 1.12 894 1.12 723 1.12 553 1.12 384 1.12 215 9.99 880 9.99 879 9.99 879 9.99 878 9.99 877 45 44 43 42 41 20 21 22 23 24 8.87 829 8.87 995 8.88 161 8.88 326 8.88 490 8.87 953 8.88 120 8.88 287 8.88 453 8.88 618 1.12 047 1.11 880 1.11 713 1.11 547 1.11 382 9.99 876 9.99 875 9.99 874 9.99 873 9.99 872 40 39 38 37 36 85 4 25 26 27 28 29 8.88 654 8.88 817 8.88 980 8.89 142 8.89 304 8.88 783 8.88 948 8.89 111 8.89 274 8.89 437 1.11 217 1.11 052 1.10 889 1.10 726 1.10 563 9.99 871 9.99 870 9.99 869 9.99 868 9.99 867 35 34 33 32 31 30 31 32 33 34 8.89 464 8.89 625 8.89 784 8.89 943 8.90 102 8.89 598 8.89 760 8.89 920 8.90 080 8.90 240 1.10 402 1.10 240 1.10 080 1.09 920 1.09 760 9.99 866 9.99 865 9.99 864 9.99 863 9.99 862 30 29 28 27 26 35 36 37 38 39 8.90 260 8.90 417 8.90 574 8.90 730 8.90 885 8.90 399 8.90 557 8.90 715 8.90 872 8.91 029 1.09 601 1.09 443 1.09 285 1.09 128 1.08 971 9.99 861 9.99 860 9.99 859 9.99 858 9.99 857 25 24 23 22 21 40 41- 42 43 44 8.91 040 8.91 195 8.91 349 8.91 502 8.91 655 8.91 185 8.91 340 8.91 495 8.91 650 8.91 803 1.08 815 1.08 660 1.08 505 1.08 350 1.08 197 9.99 856 9.99 855 9.99 854 9.99 853 9.99 852 20 19 18 17 16 45 46 47 48 49 8.91 807 8.91 959 8.92 110 8.92 261 8.92 411 8.91 957 8.92 110 8.92 262 8.92 414 8.92 565 1.08 043 1.07 890 1.07 738 1.07 586 1.07 435 9.99 851 9.99 850 9.99 848 9.99 847 9.99 846 15 14 13 12 11 50 51 52 53 54 8.92 561 8.92 710 8.92 859 8.93 007 8.93 154 8.92 716 8.92 866 8.93 016 8.93 165 8.93 313 1.07 284 1.07 134 1.06 984 1.06 835 1.06 687 9.99 845 9.99 844 9.99 843 9.99 842 9.99 841 10 9 8 7 6 55 56 57 58 59 8.93 301 8.93 448 8.93 594 8.93 740 8.93 885 8.93 462 8.93 609 8.93 756 8.93 903 8.94 049 1.06 538 1.06 391 1.06 244 1.06 097 1.05 951 9.99 840 9.99 839 9.99 838 9.99 837 9.99 836 5 4 3 2 1 60 8.94.030 8.94 195 1.05 805 9.99 834 L. Cos. L. Cotg. L. Tang. L. Sin. / [46] / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 8.94 030 8.94 174 8.94 317 8.94 461 8.94 603 8.94 195 8.94 340 8.94 485 8.94 630 8.94 773 1.05 805 1.05 660 1.05 515 1.05 370 1.05 227 9.99 834 9.99 833 9.99 832 9.99 831 9.99 830 60 59 58 57 56 5 6 7 8 9 8.94 746 8.94 887 8.95 029 8.95 170 8.95 310 8.94 917 8.95 060 8.95 202 8.95 344 8.95 486 1.05 083 1.04 940 1.04 798 1.04 656 1.04 514 9.99 829 9.99 828 9.99 827 9.99 825 9.99 824 55 54 53 52 51 10 11 12 13 14 8.95 450 8.95 589 8.95 728 8.95 867 8.96 005 8.95 627 8.95 767 8.95 908 8.96 047 8.96 187 1.04 373 1.04 233 1.04 092 1.03 953 1.03 813 9.99 823 9.99 822 9.99 821 9.99 820 9.99 819 50 49 48 47 46 15 16 17 18 19 8.96 143 8.96 280 8.96 417 8.96 553 8.96 689 8.96 325 8.96 464 8.96 602 8.96 739 8.96 877 1.03 675 1.03- 536 1.03 398 1.03 261 1.03 123 9.99 817 9.99 816 9.99 815 9.99 814 9.99 813 45 44 43 42 41 5 20 21 22 23 24 8.96 825 8.96 960 8.97 095 8.97 229 8.97 363 8.97 013 8.97 150 8.97 285 8.97 421 8.97 556 1.02 987 1.02 850 1.02 715 1.02 579 1.02 444 9.99 812 9.99 810 9.99 809 9.99 808 9.99 807 40 39 38 37 36 25 26 27 28 29 8.97 496 8.97 629 8.97 762 8.97 894 8.98 026 8.97 691 8.97 825 8.97 959 8.98 092 8.98 225 1.02 309 1.02 175 1.02 041 1.01 908 1.01 775 9.99 806 9.99 804 9.99 803 9.99 802 9.99 801 35 34 33 32 31 84 30 31 32 33 34 8.98 157 8.98 288 8.98 419 8.98 549 8.98 679 8.98 358 8.98 490 8.98 622 8.98 753 8.98 884 1.01 642 1.01 510 1.01 378 1.01 247 1.01 116 9.99 800 9.99 798 9.99 797 9.99 796 9.99 795 30 29 28 27 26 35 36 37 38 39 8.98 808 8.98 937 8.99 066 8.99 194 8.99 322 8.99 015 8.99 145 8.99 275 8.99 405 8.99 534 1.00 985 1.00 855 1.00 725 1.00 595 1.00 466 9.99 793 9.99 792 9.99 791 9.99 790 9.99 788 25 24 23 22 21 40 41 42 43 44 8.99 450 8.99 577 8.99 704 8.99 830 8.99 956 8.99 662 8.99 791 8.99 919 9.00 046 9.00 174 1.00 338 1.00 209 1.00 081 0.99 954 0.99 826 9.99 787 9.99 786 9.99 785 9.99 783 9.99 782 20 19 18 17 16 45 46 47 48 49 9.00 082 9.00 207 9.00 332 9.00 456 9.00 581 9.00 301 9.00 427 9.00 553 9.00 679 9.00 805 0.99 699 0.99 573 0.99 447 0.99 321 0.99 195 9.99 781 9.99 780 9.99 778 9.99 777 9.99 776 15 14 13 12 11 50 51 52 53 54 9.00 704 9.00 828 9.00 951 9.01 074 9.01 196 9.00 930 9.01 055 9.01 179 9.01 303 9.01 427 0.99 070 0.98 945 0.98 821 0.98 697 0.98 573 9.99 775 9.99 773 9.99 772 9.99 771 9.99 769 10 9 8 7 6 55 56 57 58 59 9.01 318 9.01 440 9.01 561 9.01 682 9.01 803 9.01 550 9.01 673 9.01 796 9.01 918 9.02 040 0.98 450 0.98 327 0.98 204 0.98 082 0.97 960 9.99 768 9.99 767 9.99 765 9.99 764 9.99 763 5 4 3 2 1 60 9.01 923 9.02 162 0.97 838 9.99 761 L. Cos. L. Cotg. L. Tang. L. Sin. / [47] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.01 923 9.02 043 9.02 163 9.02 283 9.02 402 9.02 162 9.02 283 9.02 404 9.02 525 9.02 645 0.97 838 0.97 717 0.97 596 0.97 475 0.97 355 9.99 761 9.99 760 9.99 759 9.99 757 9.99 756 60 59 58 57 56 5 6 7 8 9 9.02 520 9.02 639 9.02 757 9.02 874 9.02 992 9.02 766 9.02 685 9.03 005 9.03 124 9.03 242 0.97 234 0.97 115 0.96 995 0.96 876 0.96 758 9.99 755 9.99 753 9.99 752 9.99 751 9.99 749 55 54 53 52 51 10 11 12 13 14 9.03 109 9.03 226 9.03 342 9.03 458 9.03 574 9.03 361 9.03 479 9.03 597 9.03 714 9.03 832 0.96 639 0.96 521 0.96 403 0.96 286 0.96 168 9.99 748 9.99 747 9.99 745 9.99 744 9.99 742 50 49 48 47 46 15 16 17 18 19 9.03 690 9.03 805 9.03 920 9.04 034 9.04 149 9.03 948 9.04 065 9.04 181 9.04 297 9.04 413 0.96 052 0.95 935 0.95 819 0.95 703 0.95 587 9.99 741 9.99 740 9.99 738 9.99 737 9.99 736 45 44 43 42 41 fi 20 21 22 23 24 9.04 262 9.04 376 9.04 490 9.04 603 9.04 715 9.04 528 9.04 643 9.04 758 9.04 873 9.04 987 0.95 472 0.95 357 0.95 242 0.95 127 0.95 013 9.99 734 9.99 733 9.99 731 9.99 730 9.99 728 40 39 38 37 36 25 26 27 28 29 9.04 828 9.04 940 9.05 052 9.05 164 9.05 275 9.05 101 9.05 214 9.05 328 9.05 441 9.05 553 0.94 899 0.94 786 0.94 672 0.94 559 0.94 447 9.99 727 9.99 726 9.99 724 9.99 723 9.99 721 35 34 33 32 31 83 u 30 31 32 33 34 9.05 386 9.05 497 9.05 607 9.05 717 9.05 827 9.05 666 9.05 778 9.05 890 9.06 002 9.06 113 0.94 334 0.94 222 0.94 110 0.93 998 0.93 887 9.99 720 9.99 718 9.99 717 9.99 716 9.99 714 30 29 28 27 26 35 36 37 38 39 9.05 937 9.06 046 9.06 155 9.06 264 9.06 372 9.06 224 9.06 335 9.06 445 9.06 556 9.06 666 0.93 776 0.93 665 0.93 555 0.93 444 0.93 334 9.99 713 9.99 711 9.99 710 9.99 708 9.99 707 25 24 23 22 21 40 41 42 43 44 9.06 481 9.06 589 9.06 696 9.06 804 9.06 911 9.06 775 9.06 885 9.06 994 9.07 103 9.07 211 0.93 225 0.93 115 0.93 006 0.92 897 0.92 789 9.99 705 9.99 704 9.99 702 9.99 701 9.99 699 20 19 18 17 16 45 46 47 48 49 9.07 018 9.07 124 9.07 231 9.07 337 9.07 442 9.07 320 9.07 428 9.07 536 9.07 643 9.07 751 0.92 680 0.92 572 0.92 464 0.92 357 0.92 249 9.99 698 9.99 696 9.99 695 9.99 993 9.99 692 15 14 13 12 11 50 51 52 53 54 9.07 548 9.07 653 9.07 758 9.07 863 9.07 968 9.07 858 9.07 964 9.08 071 9.08 177 9.08 283 0.92 142 0.92 036 . 0.91 929 0.91 823 0.91 717 9.99 690 9.99 689 9.99 687 9.99 686 9.99 684 10 9 8 7 6 55 56 57 58 59 9.08 072 9.08 176 9.08 280 9.08 383 9.08 486 9.08 389 9.08 495 9.08 600 9.08 705 9.08 810 0.91 611 0.91 505 0.91 400* 0.91 295 0.91 190 9.99 683 9.99 681 9.99 680 9.99 678 9.99 677 5 4 3 2 1 60 9.08 589 9.08 914 0.91 086 9.99 675 L. Cos. L. Cotg. L. Tang. L. Sin. / [48] t L. Sin. L. Tang. L. Cotg. L. Cos. 82 1 2 3 4 9.08 589 9.08 692 9.08 795 9.08 897 9.08 999 9.08 914 9.09 019 9.09 123 9.09 227 9.09 330 0.91 086 0.90 981 0.90 877 0.90 773 0.90 670 9.99 675 9.99 674 9.99 672 9.99 670 9.99 669 60 59 58 57 56 5 6 7 8 9 9.09 101 9.09 202 9.09 304 9.09 405 9.09 506 9.09 434 9.09 537 9.09 640 9.09 742 9.09 845 0.90 566 0.90 463 0.90 360 0.90 258 0.90 155 9.99 667 9.99 666 9.99 664 9.99 663 9.99 661 55 54 53 52 51 10 11 12 13 14 9.09 606 9.09 707 9.09 807 9.09 907 9.10 006 9.09 947 9.10 049 9.10 150 9.10 252 9.10 353 0.90 053 0.89 951 0.89 853 0.89 748 0.89 647 9.99 659 9.99 658 9.99 656 9.99 655 9.99 653 50 49 48 47 46 T 15 16 17 18 19 9.10 106 9.10 205 9.10 304 9.10 402 9.10 501 9.10 454 9.10 555 9.1,0 656 9.10 756 9.10 856 0.89 546 0.89 445 0.89 344 0.89 244 0.89 144 9.99 651 9.99 650 9.99 648 9.99 647 9.99 645 45 44 43 42 41 20 21 22 23 24 9.10 599 9.10 697 9.10 795 9.10 893 9.10 990 9.10 956 9.11 056 9.11 155 9.11 254 9.11 353 0.89 044 0.88 944 0.88 845 0.88 746 0.88 647 9.99 643 9.99 642 9.99 640 9.99 638 9.99 637 40 39 38 37 33 25 26 27 28 29 9.11 087 9.11 184 9.11 281 9.11 377 9.11 474 9.11 452 9.11 551 9.11 649 9.11 747 9.11 845 0.88 548 0.88 449 0.88 351 0.88 253 0.88 155 9.99 635 9.99 633 9.99 632 9.99 630 9.99 629 35 34 33 32 31 30 31 32 33 34 9.11 570 9.11 666 9.11 761 9.11 857 9.11 952 9.11 943 9.12 040 9.12 138 9.12 235 9.12 332 0.88 057 0.87 960 0.87" 862 0.87 765 0.87 668 9.99 627 9.99 625 9.99 624 9.99 622 9.99 620 30 29 28 27 26 35 36 37 38 39 9.12 047 9.12 142 9.12 236 9.12 331 9.12 425 9.12 428 9.12 525 9.12 621 9.12 717 9.12 813 0.87 572 0.87 475 0.87 379 0.87 283 0.87 187 9.99 618 9.99 617 9.99 615 9.99 613 9.99 612 25 24 23 22 21 40 41 42 43 44 9.12 519 9.12 612 9.12 706 9.12 799 9.12 892 9.12 909 9.13 004 9.13 099 '9.13 194 9.13 289 0.87 091 0.86 996 0.86 901 0.86 806 0.85 711 9.99 610 9.99 608 9.99 607 9.99 605 9.99 603 20 19 18 17 16 45 46 47 48 49 9.12 985 9.13 078 9.13 171 9.13 263 9.13 355 9.13 384 9.13 478 9.13 573 9.13 667 9.13 761 0.86 616 0.86 522 0.86 427 0.86 333 0.86 239 9.99 601 9.99 600 9.99 598 9.99 596 9.99 595 15 14 13 12 11 50 51 52 53 54 9.13 447 9.13 539 9.13 630 9.13 722 9.13 813 9.13 854 9.13 948 9.14 041 9.14 134 9.14 227 0.86 146 0.86 052 0.85 959 0.85 866 0.85 773 9.99 593 9.99 591 9.99 589 9.99 588 9.99 586 10 9 8 7 6 55 56 57 58 59 9.13 904 9.13 994 9.14 085 9.14 175 9.14 266 9.14 320 9.14 412 9.14 504 9.14 597 9.14 688 0.85 680 0.85 588 0.85 496 0.85 403 0.85 312 9.99 584 9.99 582 9.99 581 9.99 579 9.99 577 5 4 3 2 1 60 9.14 356 9.14 780 0.85 220 9.99 575 L. Cos. L. Cotg. L. Tang. L. Sin. t [49] I L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.14 356 9.14 445 9.14 535 9.14 624 9.14 714 9.14 780 9.14 872 9.14 963 9.15 054 9.15 145 0.85 220 0.85 128 0.85 037 0.84 946 0.84 855 y.yy b7b 9.99 574 9.99 572 9.99 570 9.99 568 bO 59 58 57 56 5 6 7 8 9 9.14 803 9.14 891 9-14 980 9.15 069 9.15 157 9.15 236 9.15 327 9.15 417 9.15 508 9.15 598 0.84 764 0.84, 673 0.84 583 0.84 492 0.84 402 9.99 566 9.99 565 9.99 563 9.99 561 9.99 559 55 54 53 52 51 10 11 12 13 14 9.15 245 9.15 333 9.15 421 9.15 508 9.15 596 9.15 688 9.15 777 9.15 867 9.15 956 9.16 0*6 0.84 312 0.84 223 0.84 133 0.84 044 0.83 954 9.99 557 9.99 556 9.99 554 9.99 552 9.99 550 50 49 48 47 46 15 16 17 18 19 9.15 683 9.15 770 9.15 857 9.15 944 9.16 030 9.16 135 9.16 224 9.16 312 9.16 401 9.16 489 0.83 865 83 776 0.83 688 0.83 599 0.83 511 9.99 548 9.99 546 9.99 545 9.99 543 9.99 541 45 44 43 42 41 20 21 22 23 24 9.16 116 9.16 203 9.16 289 9.16 374 9.16 460 9.16 577 9.16 665 9.16 753 9.16 841 9.16 928 0.83 423 0.83 335 0.83 247 0.83 159 0.83 072 9.99 539 9.99 537 9.99 535 9.99 533 9.99 532 40 39 38 37 36 81 00 25 26 27 28 29 9.16 545 9.16 631 9.16 716 9.16 801 9.16 886 9.17016 9.17 103 9.17 190 9.17 277 9.17 363 0.82 984 0.82 897 0.82 810 0.82 723 0.82 637 9.99 530 9.99 528 9.99 526 9.99 524 9.99 522 35 34 33 32 31 O 30 31 32 33 34 9.16 970 9.17 055 9.17 139 9.17 223 9.17 307 9.17 450 9.17 536 9.17 622 9.17 708 9.17 794 0.82 550 0.82 464 0.82 378 0.82 292 0.82 206 9.99 520 9.99 518 9.99 517 9.99 515 9.99 513 30 29 28 27 26 35 36 37 38 39 9.17 391 9.17 474 9.17 558 9.17 641 9.17 724 9.17 880 9.17 965 9.18 051 9.18 136 9.18 221 0.82 120 0.82 035 0.81 949 0.81 864 0.81 779 9.99 511 9:99 509 9.99 507 9.99 505 9.99 503 25 24 23 22 21 40 41 42 43 44 9.17 807 9.17 890 9.17 973 9.18 055 9.18 137 9.18 306 9.18 391 9.18 475 9.18 560 9.18 644 0.81 694 0.81 609 0.81 525 0.81 440 0.81 356 9.99 501 9.99 499 9.99 497 9.99 495 9.99 494 20 19 18 17 16 45 46 47 48 49 9.18 220 9.18 302 9.18 383 9.18 465 9.18 547 9.18 728 9.18 812 9.18 896 9.18 979 9.19 063 0.81 272 0.81 188 0.81 104 0.81 021 0.80 937 9.99 492 9.99 490 9.99 488 9.99 486 9.99 484 15 14 13 12 11 50 51 52 53 54 9.18 628 9.18 709 9.18 790 9.18 871 9.18 952 9.19 146 9.19 229 9.19 312 9.19 395 9.19 478 0.80 854 0.80 771 0.80 688 0.80 605 0.80 522 9.99 482 9.99 480 9.99 478 9.99 476 9.99U474 10 9 8 7 6 55 56 57 58 59 9.19 033 9.19 113 9.19 193 9.19 273 9.19 353 9.19 561 9.19 643 9.19 725 9.19 807 9.19 889 0.80 439 0.80 357 0.80 275 0.80 193 0.80 111 9.99 472 9.99 470 9.99 468 9.99 466 9.99 464 5 4 3 2 1 60 9 19 433 9.19 971 0.80 029 9.99 462 L. Cos. L. Cotg. L. Tang. L. Sin. / [50] I L. Sin. L. Tan. L. Cotg. L. Cos. 80 1 2 3 4 9.19 433 9.19 513 9.19 592 9.19 672 9.19 751 9.19 971 9.20 053 9.20 134 9.20 216 9.20 297 0.80 029 0.79 947 0.79 866 0.79 784 0.79 703 9.99 462 9.99 460 9.99 458 9.99 456 9.99 454 60 59 58 57 56 5 6 7 8 9 9.19 830 9.19 909 9.19 988 9.20 067 9.20 145 9.20 378 9.20 459 9.20 540 9.20 621 9.20 701 0.79 622 0.79 541 0.79 460 0.79 379 0.79 299 9.99 452 9.99 450 9.99 448 9.99 446 9.99 444 55 54 53 52 51 10 11 12 13 14 9.20 223 9.20 302 9.20 380 9.20 458 9.20 535 9.20 782 9.20 862* 9.20 942 9.21 022 9.21 102 0.79 218 0.79 138 0.79 058 0.78 978 Oi78 898 9.99 442 9.99 440 9.99 438 9.99 436 9.99 434 50 49 48 47 46 15 16 17 18 19 9.20 613 9.20 691 9.20 768 9.20 845 9.20 922 9.21 182 9.21 261 9.21 341 9.21 420 9.21 499 0.78 818 0.78 739 0.78 659 0.78 580 0.78 501 9.99 432 9.99 429 9.99 427 9.99 425 9.99 423 45 44 43 42 41 20 21 22 23 24 9.20 999 9.21 076 9.21 153 9.21 229 9.21 306 9.21 578 9.21 657 9.21 736 9.21 814 9.21 893 0.78 422 0.78 343 0.78 264 0.78 186 0.78 107 9.99 421 9.99 419 9.99 417 9.99 415 9.99 413 40 39 38 37 36 9 25 26 27 28 29 9.21 382 9.21 458 9.21 534 9.21 610 9.21 685 9.21 971 9.22 049 9.22 127 9:22 205 9.22 283 0.78 029 0.77 951 0.77 873 0.77 795 0.77 717 9.99 411 9.99 409 9.99 407 9.99 404 9.99 402 35 34 33 32 31 30 31 32 33 34 9.21 761 9.21 836 9.21 912 9.21 987 9.22 062 9.22 361 9.22 438 9.22 516 9.22 593 9.22 670 0.77 639 0.77 562 0.77 484 0.77 407 0.77 330 9.99 400 9.99 39a 9.99 396 9.99 394 9.99 392 30 29 28 27 26 35 36 37 38 39 9.22 137 9.22 211 9.22 286 9.22 361 9.22 435 9.22 747 9.22 824 9.22 901 9.22 977 9.23 054 0.77 253 0.77 176 0.77 099 0.77 023 0.76 946 9.99 390 9.99 388 9.99 385 9.99 383 9.99 381 25 24 23 22 21 40 41 42 43 44 9.22 509 9.22 583 9.22 657 9.22 731 9.22 805 9.23 130 9.23 206 9.23 283 9.23 359 9.23 435 0.76 870 0.76 794 0.76 717 0.76 641 0.76 565 9.99 379 9.99 377 9.99 375 9.99 372 9.99 370 20 19 18 17 16 45 46 47 48 49 9.22 878 9.22 952 9.23 025 9.23 098 9.23 171 9.23 510 9.23 586 9.23 661 9.23 737 9 23 812 0.76 490 0.76 414 0.76 339 0.76 263 0.76 188 9.99 368 9.99 366 9.99 364 9.99 362 9.99 359 15 14 13 12 11 50 51 52 53 54 9.23 244 9.23 317 9.23 390 9.23 462 9.23 535 9.23 887 9.23 962 9.24 037 9.24 112 9.24 186 0.76 113 0.76 038 0.75 963 0.75 888 0.75 814 9.99 357 9.99 355 9.99 353 9.99 351 9.99 348 10 9 8 7 6 55 56 57 58 59 9.23 607 9.23 679 9.23 752 9.23 823 9.23 895 9.24 261 9.24 335 9.24 410 9.24 484 9.24 558 0.75 739 0.75 665 0.75 590 0.75 516 0.75 442 9.99 346 9.99 344 9.99 342 9.99 340 9.99 337 5 4 3 2 1 bO 9.23 967 9.24 632 0.75 368 9.99 335 L. Cos. L. Cotg. L. Tang. L. Sin. 1 [51] 10 / L. Sin. L. Tang. L. Cotg. L. Cos. 79 i 2 3 4 9.23 967 9.24 039 9.24 110 9.24 181 9.24 253 9.24 632 9.24 706 9.24 779 9.24 853 9.24 926 0.75 368 0.75 294 0.75 221 0.75 147 0.75 074 9.99 335 9.99 333 9.99 331 9.99 328 9.99 326 60 59 58 57 56 5 6 7 8 9 9.24 324 9.24 395 9.24 466 9.24 536 9.24 607 9.25 000 9.25 073 9.25 146 9.25 219 9.25 292 0.75 000 0.74 927 0.74 854 0.74 781 0.74 708 9.99 324 9.99 322 9.99 319 9.99 317 9.99 315 55 54 53 52 51 10 11 12 13 14 9.24 677 9.24 748 9.24 818 9.24 888 9.24 958 9.25 365 9.25 437 9.25 510 9.25 582 9.25 655 0.74 635 74 563 0.74 490 0.74 418 0.74 345 9.99 313 9.99 310 9.99 308 9.99 306 9.99 304 60 49 48 47 46 15 16 17 18 19 9.25 028 9.25 098 9.25 168 9.25 237 9.25 307 9.25 727 9.25 799 9.25 871 9.25 943 9.26 015 0.74 273 0.74 201 0.74 129 0.74 057 0.73 985 9.99 301 9.99 299 9.99 297 9.99 294 9.99 292 45 44 43 42 41 20 21 22 23 24 9.25 376 9.25 445 9.25 514 9.25 583 9.25 652 9.26 086 9.26 158 9.26 229 9.26 301 9.26 372 0.73 914 0.73 842 0.73 771 0.73 699 0.73 628 9.99 290 9.99 288 9.99 285 9.99 283 9.99 281 40 39 38 37 36 25 26 27 28 29 9.25 721 9.25 790 9.25 858 9.25 927 9.25 995 -9.26 443 9.26 514 9.26 585 9.26 655 9.26 726 0.73 557 0.73 486 0.73 415 0.73 345 0.73 274 9.99 278 9.99 276 9.99 274 9.99 271 9.99 269 35 34 33 32 31 60 31 32 33 34 9.26 063 9.26 131 9.26 199 9.26 267 9.26 335 9.26 797 9.26 867 9.26 937 9.27 008 9.27 078 0.73 203 0.73 133 0.73 063 0.72 992 0.72 922 9.99 267 9.99 264 9.99 262 9.99 260 9.99 257 30 29 28 27 26 35 36 37 38 39 9.26 403 9.26 470 9.26 538 9.26 605 9.26 672 9.27 148 . 9.27 218 9.27 288 9.27 357 9.27 427 0.72 852 0.72 782 0.72 712 0.72 643 0.72 573 9.99 255 9.99 252 9.99 250 9.99 248 9.99 245 25 24 23 22 21 40 41 42 43 44 9.26 739 9.26 806 9.26 873 9.26 940 9.27 007 9.27 496 9.27 566 9.27 635 9.27 704 9.27 773 0.72 504 0.72 434 0.72 365 0.72 296 0.72 227 9.99 243 9.99 241 9.99 238 9.99 236 9.99 233 20 19 18 17 16 45 46 47 48 49 9.27 073 9.27 140 9.27 206 9.27 273 9.27 339 9.27 842 9.27 911 9.27 980 9.28 049 9.28 117 0.72 158 0.72 089 0.72 020 0.71 951 0.71 883 9.99 231 9.99 229 9.99 226 9.99 224 9.99 221 15 14 13 12 11 50 51 52 53 54 9.27405 9.27 471 9.27 537 9.27 602 9.27 668 9.28 186 9.28 254 9.28 323 9.28 391 9.28 459 0.71 814 0.71 746 0.71 677 0.71 609 0.71 541 9.99 219 9.99 217 9.99 214 9.99 212 9.99 209 10 9 8 7 6 55 56 57 58 59 9.27 734 9.27 799 9.27 864 9.27 930 9.27 995 9.28 527 9.28 595 9.28 662 9.28 730 9.28 798 0.71 473 0.71 405 0.71 338 0.71 270 0.71 202 9.99 207 9.99 204 9.99 202 9.99 200 9.99 197 5 4 3 2 1 60 9.28 060 9.28 865 0.71 135 9.99 195 L. Cos. L. Cotg. L. Tang. L. Cos. 1 [52] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 78 1 2 3 4 9.28 060 9.28 125 9.28 190 9.28 254 9.28 319 9.28 865 9.28 933 9.29 000 9.29 067 9.29 134 0.71 135 0.71 067 0.71 000 0.70 933 0.70 866 9.99 195 9.99 192 9.99 190 9.99 187 9.99 185 60 59 58 57 56 5 6 7 8 9 9.28 384 9.28 448 9.28 512 9.28 577 9.28 641 9.29 201 9.29 268 9.29 335 9.29 402 9.29 468 0.70 799 0.70 732 0.70 665 0.70 598 0.70 532 9.99 182 9.99 180 9.99 177 9.99 175 9.99 172 55 54 53 52 51 10 11 12 13 14 9.28 705 9.28 769 9.28 833 9.28 896 9.28 960 9.29 535 9.29 601 9.29 668 9.29 734 9.29 800 0.70 465 0.70 399 0.70 332 0.70 266 0.70 200 9.99 170 9.99 167 9.99 165 9.99 162 9.99 160 50 49 48 47 46 15 16 17 18 19 9.29 024 9.29 087 9.29 150 9.29 214 9.29 277 9.29 866 9.29 932 9.29 998 9.30 064 9.30 130 0.70 134 0.70 068 0.70 002 0.69 936 0.69 870 9.99 157 9.99 155 9.99 152 9.99 150 9.99 147 45 44 43 42 41 20 21 22 23 24 9.29 340 9.29 403 9.29 466 9.29 529 9.29 591 9.30 195 9.30 261 9.30 326 9.30 391 9.30 457 0.69 805 0.69 739 0.69 674 0.69 609 0.69 543 9.99 145 9.99 142 9.99 140 9.99 137 9.99 135 40 39 38 37 36 11 25 26, 27 28 29 9.29 654 9.29 716 9.29 779 9.29 841 9.29 903 9.30 522 9.30 587 9.30 652 9.30 717 9.30 782 0.69 478 0.69 413 0.69 348 0.69 283 0.69 218 9.99 132 9.99 130 9.99 127 9.99 124 9.99 122 35 34 33 32 31 30 31 32 33 34 9.29 966 9.30 028 9.30 090 9.30 151 9.30 213 9.30 846 9.30 911 9.30 975 9.31 040 9.31 104 0.69 154 0.69 089 0.69 025 0.68 960 0.68 896 9.99 119 9.99 117 9.99 114 9.99 112 9.99 109 30 29 28 27 26 35 36 37 38 39 9.30 275 9.30 336 9.30 398 9.30 459 9.30 521 9.31 168 9.31 233 9.31 297 9.31 361 ' 9.31 425 0.68 832 0.68 767 0.68 703 0.68 639 0.68 575 9.99 106 9.99 104 9.99 101 9.99 099 9.99 096 25 24 23 22 21 40 41 42 43 44 9.30 582 9.30 643 9.30 704 9.30 765 9.30 826 9.31 489 9.31 552 9.31 616 9.31 679 9.31 743 0.68 511 0.68 448 0.68 384 0.68 321 0.68 257 9.99 093 9.99 091 9.99 088 9.99 086 9.99 083 20 19 18 17 16 45 46 47 48 49 9.30 887 9.30 947 9.31 008 9.31 068 9.31 129 9.31 806 9.31 870 9.31 933 9.31 996 9.32 059 0.68 194 0.68 130 0.68 067 0.68 004 0.67 941 9.99 080 9.99 078 9.99 075 9.99 072 9.99 070 15 14 13 12 11 50 51 52 53 54 9.31 189 9.31 250 9.31 310 9.31 370 9.31 430 9.32 122 9.32 185 9.32 248 9.32 311 9.32 373 0.67 878 0.67 815 0.67 752 0.67 689 0.67 627 9.99 067 9.99 064 9.99 062 9.99 059 9.99 056 10 9 8 7 6 55 56 57 58 59 9.31 490 9.31 549 9.31 609 9.31 669 9.31 728 9.32 436 9.32 498 9.32 561 9.32 623 9.32 685 0.67 564 0.67 502 0.67 439 0.67 377 0.67 315 9.99 054 9.99 051 9.99 048 9.99 046 9.99 043 5 4 3 2 1 60 9.31 788 9.32 747 0.67 253 9 99 040 L. Cos. L. Cotg. L. Tang. L. Sin. / [53] / L. Sin. L. Tang. L. Cotg. L. Cos. 77 i 2 3 4 9.31 788 9.31 847 9.31 907 9.31 966 9.32 025 9.32 747 9.32 810 9.32 872 9.32 933 9.32 995 0.67 253 0.67 190 0.67 128 0.67 067 0.67 005 9.99 040 9.99 038 9.99 035 9.99 032 9.99 030 60 59 58 57 56 5 6 7 8 9 9.32 084 9.32 143 9.32 202 9.o2 261 9.32 319 9.33 057 9.33 119 9.33 180 9.33 242 9.33 303 0.66 943 0.66 881 0.66 820 0.66 758 0.66 697 9.99 027 9.99 024 9.99 022 9.99 019 9.99 016 55 54 53 52 51 10 11 12 13 14 9.32 378 9.32 437 9.32 495 9.32 553 9.32 612 9.33 365 9.33 426 9.33 487 9.33 548 9.33 609 0.66 635 0.66 574 0.66 513 0.66 452 0.66 391 9.99 013 9.99 Oil 9.99 008 9.99 005 9.99 002 50 49 48 47 46 15 16 17 18 19 9.32 670 9.32 728 9.32 786 9.32 844 9.32 902 9.33 670 9.33 731 9.33 792 9.33 853 9.33 913 0.66 330 0.66 269 0.66 208 0.66 147 0.66 087 9.99 000 9.98 997 9.98 994 9.98 991 9.98 989 45 44 43 42 41 20 21 22 23 24 9.32 960 9.33 018 9.33 075 9.33 133 9.33 190 9.33 974 9.34 034 9.34 095 9.34 155 9.34 215 0.66 026 0.65 966 0.65 905 0.65 845 0.65 785 9.98 986 9.98 983 9.98 980 9.98 978 9.98 975 40 39 38 37 36 12 25 26 27 28 29 9.33 248 9.33 305 9.33 362 9.33 420 9.33 477 9.34 276 9.34 336 9.34 396 9.34 456 9.34 516 0.65 724 0.65 664 0.65 604 0.65 544 0.65 484 9.98 972 9.98 969 9.98 967 9.98 964 9.98 961 35 34 33 32 31 30 31 32 33 34 9.33 534 9.33 591 9.33 647 9.33 704 9.33 761 9.34 576 9.34 635 9.34 695 9.34 755 9.34 814 0.65 424 0.65 365 0.65 305 0.65 245 0.65 186 9.98 958 9.98 955 9.98 953 9.98 950 9.98 947 30 29 28 27 26 35 36 37 38 39 9.33 818 9.33 874 9.33 931 9.33 987 9.34 043 9.34 874 9.34 933 9.34 992 9.35 051 9.35 111 0.65 126 0.65 067 0.65 008 0.64 949 0.64 889 9.98 944 9.98 941 9.98 938 9.98 936 9.98 933 25 24 23 22 21 40 41 42 43 44 9.34 100 9.34 156 9.34 212 9.34 268 9.34 324 9.35 170 9.35 229 9.35 288 9.35 347 9.35 405 0.64 830 0.64 771 0.64 712 0.64 653 0.64 595 9,98 930 9.98 927 9.98 924 9.98 921 9.98 919 20 19 18 17 16 45 46 47 48 49 9.34 380 ' 9.34 436 9.34 491 9.34 547 9.34 602 9.35 464 9.35 523 9.35 581 9.35 640 9.35 698 0.64 536 0.64 477 0.64 419 0.64 360 0.64 302 9.98 916 9.98 913 9.98 910 9.98 907 9.98 904 15 14 13 12 11 50 51 52 53 54 9.34 658 9.34 713 9.34 769 9.34 824 9.34 879 9.35 757 9.35 815 9.35 873 9.35 931 9.35 989 0.64 243 0.64 185 0.64 127 0.64 069 0.64 Oil 9.98 901 9.98 898 9.98 896 9.98 893 9.98 890 10 9 8 7 6 55 56 57 58 59 9.34 934 9.34 989 9.35 044 9.35 099 9.35 154 9.36 047 9.36 105 9.36 163 9.36 221 9.36 279 0.63 953 0.63 895 0.63 837 0.63 779 0.63 721 9.98 887 9.98 884 9.98 881 9.98 878 9.98 875 5 4 3 2 1 60 9.35 209 9.36 336 0.63 664 9.98 872 L. Cos. L. Cotg. L. Tang. L. Sin. / [54] 1Q / L. Sin. L. Tang. L. Cotg. L. Cos. 76 1 2 3 4 9.35 209 9.35 263 9.35 318 9.35 373 9.35 427 9.36 336 9.36 394 9.36 452 9.36 509 9.36 566 0.63 664 0.63 606 0.63 548 0.63 491 0.63 434 9.98 872 9.98 869 9.98 867 9.98 864 9.98 861 60 59 58 57 56 5 6 7 8 9 9.35 481 9.35 536 9.35 590 9.35 644 9.35 698 9.36 624 9.36 681 9.36 738 9.36 795 9.36 852 0.63 376 0.63 319 0.63 262 0.63 205 0.63 148 9.98 858 9.98 855 9.98 852 9.98 849 9.98 846 55 54 53 52 51 10 11 12 13 14 9.35 752 9.35 806 9.35 860 9.35 914 9.35 968 9.36 909 9.36 966 9.37 023 9.37 080 9.37 137 0.63 091 0.63 034 0.62 977 0.62 920 0.62 863 9.98 843 9.98 840 9.98 837 9.98 834 9.98 831 50 49 48 47 46 15 16 17 18 19 9.36 022 9.36 075 9.36 129 . 9.36 182 9.36 236 9.37 193 9.37 250 9.37 306 9.37 363 9.37419 0.62 807 0.62 750 0.62 694 0.62 637 0.62 581 9.98 828 9.98 825 9.98 822 9.98 819 9.98 816 45 44 43 42 41 20 21 22 23 24 9.36 289 9.36 342 9.36 395 9.36 449 9.36 502 9.37 476 9.37 532 9.37 588 9.37 644 9.37 700 0.62 524 0.62 468 0.62 412 0.62 356 0.62 300 9.98 813 9.98 810 9.98 807 9.98 804 9.98 801 40 39 38 37 36 25 26 27 28 29 9.36 555 9.36 608 9.36 660 9.36 713 9.36 766 9.37 756 9.37 812 9.37 868 9.37 924 9.37 980 0.62 244 0.62 188 0.62 132 0.62 076 0.62 020 9.98 798 9.98 795 9.98 792 9.98 789 9.98 786 35 34 33 32 31 30 31 32 33 34 9.36 819 9.36 871 9.36 924 9.36 976 9.37 028 9.38 035 9.38 091 9.38 147 9.38 202 9.38 257 0.61 965 0.61 909 0.61 853 0.61 798 0.61 743 9.98 783 9.98 780 9.98 777 9.98 774 9.98 771 30 29 28 27 26 35 36 37 38 39 9.37 081 9.37 133 9.37 185 9.37 237 9.37 289 9.38 313 9.38 368 9.38 423 9.38 479 9.38 534 0.61 687 0.61 632 0.61 577 0.61 521 0.61 466 9.98 768 9.98 765 9.98 762 9.98 759 9.98 756 25 24 23 22 21 40 41 42 43 44 9.37 341 9.37 393 9.37 445 9.37 497 9.37 549 9.38 589 9.38 644 9.38 699 9.38 754 9.38 808- 0.61 411 0.61 356 0.61 301 0.61 246 0.61 192 9.98 753 9.98 750 9.98 746 9.98 743 9.98 740 20 19 18 17 16 45 46 47 48 49 9.37 600 9.37 652 9.37 703 9.37 755 9.37 806 9.38 863 9.38 918 9.38 972 9.39 027 9.39 082 0.61 137 0.61 082 0.61 028 0.60 973 0.60 918 9.98 737 9.98 734 9.98 731 9.98 728 9.98 725 15 14 13 12 11 50 51 52 53 54 9.37 858 9.37 909 9.37 960 9.38 Oil 9.38 062 9.39 136 9.39 190 9.39 245 9.39 299 ' 9.39 353 0.60 864 0.60 810 0.60 755 0.60 701 0.60 647 9.98 722 9.98 719 9.98 715 9.98 712 9.98 709 10 9 8 7 6 55 56 57 58 59 9.38 113 9.38 164 9.38 215 9.38 266 9.38 317 9.39 407 9.39 461 9.39 515 9.39 569 9.39 623 0.60 593 0.60 539 0.60 485 0.60 431 0.60 377 9.98 706 9.98 703 9.98 700 9.98 697 9.98 694 5 4 3 2 1 60 9.38 368 9.39 677 0.60 323 9 98 690 L. Cos. L. Cotg. L. Tang. L. Sin. ; [55] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 75 C 1 2 3 4 9.38 3b8 9.38 418 9.38 469 9.38 519 9.38 570 9.39 677 ' 9.39 731 9.39 785 9.39 838 9.39 892 0.60 323 0.60 269 0.60 215 0.60 162 0.60 108 9.98 690 9.98 687 9.98 684 9.98 681 9.98 678 bu 59 58 57 56 5 6 7 8 9 9.38 620 9.38 670 9.38 721 9.38 771 9.38 821 9.39 945 9.39 999 9.40 052 9.40 106 9.40 159 0.60 055 0.60 001 0.59 948 0.59 894 0.59 841 9.98 675 9.98 671 9.98 668 9.98 665 9.98 662 55 54 53 52 51 10 11 12 13 14 9.38 871 9.38 921 9.38 971 9.39 021 9.39 071 9.40 212 9.40 266 9.40 319 9.40 372 9.40 425 0.59 788 0.59 734 0.59 681 0.59 628 0.59 575 9.98 659 9.98 656 9.98 652 9.98 649 9.98 646 50 49 48 47 46 15 16 17 18 19 9.39 121 9.39 170 9.39 220 9.39 270 9.39 319 9.40 478 9.40 531 9.40 584 9.40 636 9.40 689 0.59 522 0.59 469 0.59 416 0.59 364 0.59 311 9.98 643 9.98 640 9.98 636 9.98 633 9.98 630 45 44 43 42 41 W c *0 21 22 23 24 9.39 369 9.39 418 9.39 467 9.39 517 9.39 566 9.40 742 9.40 795 9.40 847 9.40 900 9.40 952 0.59 258 0.59 205 0.59 153 0.59 100 0.59 048 9.98 627 9.98 623 9.98 620 9.98 617 9.98 614 40 39 38 37 36 25 26 27 28 29 9.39 615 9.39 664 9.39 713 9.39 762 9.39 811 9.41 005 9.41 057 9.41 109 9.41 161 9.41 214 0.58 995 0.58 943 0.58 891 0.58 839 0.58 786 9.98 610 9.98 607 9.98 604 9.98 601 9.98 597 35 34 33 32 31 30 31 32 33 34 9.39 860 9.39 909 9.39 958 9.40 006 9.40 055 9.41 266 9.41 318 9.41 370 9.41 422 9.41 474 0.58 734 0.58 682 0.58 630 0.58 578 0.58 526 9.98 594 9.98 591 9.98 588 9.98 584 9.98 581 30 29 28 27 26 35 36 37 38 39 9.40 103 9.40 152 9.40 200 9.40 249 9.40 297 9.41 526 9.41 578 9.41 629 9.41 681 9.41 733 0.58 474 0.58 422 0.58 371 0.58 319 0.58 267 9.98 578 9.98 574 9.98 571 9.98 568 9.98 565 25 24 23 22 21 40 41 42 43 44 9.40 346 9.40 394 9.40 442 9.40 490 9.40 538 9.41 784 9.41 836 9.41 887 9.41 939 9.41 990 0.58 216 0.58 164 0.58 113 0.58 061 .0.58 010 9.98 561 9.98 558 9.98 555 9.98 551 9.98 548 20 19 18 17 16 45 46 47 48 49 9.40 586 9.40 634 9.40 682 9.40 730 9.40 778 9.42 041 9.42 093 9.42 144 9.42 195 9.42 246 0.57 959 0.57 907 0.57 856 0.57 805 0.57 754 9.98 545 9.98 541 9.98 538 9.98 535 9.98 531 15 14 13 12 11 50 51 52 53 54 9.40 825 9.40 873 9.40 921 9.40 968 9.41 016 9.42 297 9.42 348 9.42 399 9.42 450 9.42 501 0.57 703 0.57 652 0.57 601 C.57 550 0.57 499 9.98 528 9.98 525 9.98 521 9.98 518 9.98 515 10 9 8 7 6 55 56 57 58 59 9.41 063 9.41 111 9.41 158 9.41 205 9.41 252 9.42 552 9,42 603 9.42 653 ^ 9.42 704 9.42 755 0.57 448 0.57 397 0.57 347 0.57 296 0.57 245 9.98 511 9.98 508 9.98 505 9.98 501 9.98 498 5 4 3 2 1 60 9.41 300 9.42 805 0.57 195 9.98 494 L. Cos. L. Cotg. L. Tang. L. Sin. I [56] i L. Sin. L. Tang. L. Cotg. L. Cos. 74 C i 2 3 4 9.41 300 9.41 347 9.41-394 9.41 441 9.41 488 9.42 805 9.42 856 9.42 906 9.42 957 9.43 007 0.57 195 0.57 144 0.57 094 0.57 043 0.56 993 9.98 494 9.98 491 9.98 488 9.98 484 9.98 481 60 59 58 57 56 5 6 7 8 9 9.41 535 9.41 582 9.41 628 9.41 675 9.41 722 9.43 057 9.43 108 9.43 158 9.43 208 9.43 258 0.56 943 0.56 892 0.56 842 0.56 792 0.56 742 9.98 477 9.98 474 9.98 471 9.98 467 9.98 464 55 54 53 52 51 10 11 12 13 14 9.41 768 9.41 815 9.41 861 9.41 908 9.41 954 9.43 308 9.43 358 9.43 408 9.43 458 9.43 508 0.56 692 0.56 642 0.56 592 0.56 542 0.56 492 9.98 460 9.98 457 9.98 453 9.98 450 9.98 447 50 49 48 47 46 15 16 17 18 19 9.42 001 9.42 047 9.42 093 9.42 140 9.42 186 9.43 558 9.43 607 9.43 657 9.43 707 9.43 756 0.56 442 0.56 393 0.56 343 0.56 293 0.56 244 9.98 443 9.98 440 9.98 436 9.98 433 9.98 429 45 44 43 42 41 20 21 22 23 24 9.42 232 9.42 278 9.42 324 9.42 370 9.42 416 9.43 806 9.43 855 9.43 905 9.43 954 9.44 004 0.56 194 0.56 145 0.56 095 0.56 046 0.55 996 9.98 426 9.98 422 9.98 419 9.98 415 9.98 412 40 39 38 37 36 15 25 26 27 28 29 9.42 461 9.42 507 9.42 553 9.42 599 9.42 644 9.44 053 9.44 102 9.44 151 9.44 201 9.44 250 0.55 947 0.55 898 0.55 849 0.55 799 0.55 750 9.98 409 9.98 405 9.98 402 9.98 398 9.98 395 35 34 33 32 31 30 31 32 33 34 9.42 690 9.42 735 9.42 781 9.42 826 9.42 872 9.44 299 9.44 348 9.44 397 9.44 446 9.44 495 0.55 701 0.55 652 0.55 603 0.55 554 0.55 505 9.98 391 9.98 388 9.98 384 9.98 381 9.98 377 30 29 28 27 26 35 36 37 38 39 9.42 917 9.42 962 9.43 008 9.43 053 9.43 098 9.44 544 9.44 592 9.44 641 9.44 690 9.44 738 0.55 456 0.55 408 0.55 359 0.55 310 0.55 262 9.98 373 9.98 370 9.98 366 9.98 363 9.98 359 25 24 23 22 21 40 41 42 43 44 9.43 143 9.43 188 9.43 233 9.43 278 9.43 323 9:44 787 9.44 836 9.44 884 9.44 933 9.44 981 0.55 213 0.55 164 0.55 116 0.55 067 0.55 019 9.98 356 9.98 352 9.98 349 9.98 345 9.98 342 20 19 18 17 16 45 46 47 48 49 9.43 367 9.43 412 9.43 457 9.43 502 9.43 546 9.45 029 9.45 078 9.45 126 9.45 174 9.45 222 0.54 971 0.54 922 0.54 874 0.54 826 0.54 778 9.98 338 9.98 334 9.98 331 9.98 327 9.98 324 15 14 13 12 11 50 51 52 53 54 9.43 591 9.43 635 9.43 680 9.43 724 9.43 769 9.45 271 945 319 9.45 367 9:45 414 9.45 463 0.54 729 0.54 681 0.54 633 0.54 585 0.54 537 9.98 320 9.98 317 9.98 313 9.98 309 9.98 306 10 9 8 7 6 55 56 57 58 59 9.43 813 9.43 857 9.43 901 9.43 946 9.43 990 9.45 511 9.45 559 9-45 606 9.45 654 9.45 702 0.54 489 0.54 441 0.54 394 0.54 346 0.54 298 9.98 302 9.98 299 9.98 295 9.98 291 9.98 288 5 4 3 2 1 60 9.44 034 9.45 750 0.54 250 9.98 284 L. Cos. L. Cotg. L. Tang. L. Sin. / [57] / L. Sin. L. Tang. L. Cotg. L. Cos. 73 1 2 3 4 9.44 034 9.44 078 9.44 122 9.44 166 9.44 210 9.45 750 9.45 797 9.45 845 9.45 892 9.45 940 0.54 250 0.54 203 0.54 155 0.54 108 0.54 060 9.98 284 9.98 281 9.98 277 9.98 273 9.98 270 60 59 58 57 56 5 6 7 8 9 9.44 263 9.44 297 9.44 341 9.44 385 9.44 428 9.45 987 9.46 035 9.46 082 9.46 130 9.46 177 0.54 013 0.53 965 0.53 918 0.53 870 0.53 823 9.98 266 9.98 262 9.98 259 9.98 255 9.98 251 55 54 53 52 51 10 11 12 13 14 9.44 472 9.44 516 9.44 559 9.44 602 9.44 646 9.46 224 9.46 271 9.46 319 9.46 366 9.46 413 0.53 776 0.53 729 0.53 681 0.53 634 0.53 587 9.98 248 9.98 244 9.98 240 9.98 237 9.98 233 60 49 48 47 46 15 16 17 18 19 9.44 689 9.44 733 9.44 776 9.44 819 9.44 862 9.46 460 9.46 507 9.46 554 9.46 601 9.46 648 0.53 540 0.53 493 0.53 446 0.53 399 0.53 352 9.98 229 9.98 226 9.98 222 9.98 218 9.98 215 45 44 43 42 41 20 21 22 23 . 24 9.44 905 9.44 948 9.44 992 9.45 035 9.45 077 9.46 694 9.46 741 9.46 788 9.46 835 9.46 881 0.53 306 0.53 259 0.53 212 0.53 165 0.53 119 9.98 211 9.98 207 9.98 204 9.98 200 9.98 196 40 39 38 37 36 1fi c 25 26 27 28 29 9.45 120 9.45 163 9.45 206 9.45 249 9.45 292 9.46 928 9.46 975 9.47 021 9.47 068 9.47 114 0.53 072 0.53 025 0.52 979 0.52 932 0.52 886 9.98 192 9.98 189 9.98 185 9.98 181 9.98 177 35 34 33 32 31 AvF 30 31 32 33 34 9.45 334 9.45 377 9.45 419 9.45 462 9.45 504 9.47 160 9.47 207 9.47 253 9.47 299 9.47 346 0.52 840 0.52 793 0.52 747 0.52 701 0.52 654 9.98 174 9.98 170 9.98 166 9.98 162 9.98 159 30 29 28 27 26 35 36 37 38 39 9.45 547 9.45 589 9.45 632 9.45 674 9.45 716 9.47 392 9.47 438 9.47 484 9.47 530 9.47 576 0.52 608 0.52 562 0.52 516 0.52 470 0.52 424 9.98 155 9.98 151 9.98 147 9.98 144 9.98 140 25 24 23 22 21 40 41 42 43 44 9.45 758 9.45 801 9.45 843 9.45 885 9.45 927 9.47 622 9.47 668 9.47 714 9.47 760 9.47 806 0.52 378 0.52 332 0.52 286 0.52 240 0.52 194 9.98 136 9.98 132 9.98 129 9.98 125 9.98 121 20 19 18 17 16 45 46 47 48 49 9.45 969 9.46 Oil 9.46 053 9.46 095 9.46 136 9.47 852 9.47 897 9.47 943 9.47 989 9.48 035 0.52 148 0.52 103 0.52 057 0.52 Oil 0.51 965 9.98 117 9.98 113 9.98 110 9.98 106 9.98 102 15 14 13 12 11 50 51 52 53 54. 9.46 178 9.46 220 9.46 262 9.46 303 9.46 345 9.48 080 9.48 126 9.48 171 9.48 217 9.48 262 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 9.98 098 9.98 094 9.98 090 9.98 087 9.98 083 10 9 8 7 6 55 56 57 58 59 9.46 386 9.46 428 9.46 469 9.46 511 9.46 552 9.48 307 9.48 353 9.48 398 9.48 443 9.48 489 0.51 693 0.51 647 0.51 602 0.51 557 0.51 511 9.98 079 9.98 075 9.98 071 9.98 067 9.98 063 5 4 3 2 1 60 9.46 594 9.48 534 0.51 466 9.98 060 L. Cos. L. Cotg. L. Tang. L. Sin. i [68] i L. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 9.46 594 9.46 635 9.46 676 9.46 717 9.46 758 9.48 534 9.48 579 9.48 624 9.48 669 9.48 714 0.51 466 0.51 421 0.51 376 0.51 331 0.51 286 9.98 060 9.98 056 9.98 052 9.98 048 9.98 044 60 59 58 57 56 5 6 7 8 9 9.46 800 9.46 841 9.46 882 9.46 923 9.46 964 9.48 759 9.48 804 9.48 849 9.48 894 9.48 939 0.51 241 0'51 196 0.51 151 0.51 106 0.51 061 9.98 040 9.98 036 9.98 032 9.98 029 9.98 025 55 54 53 52 51 10 11 12 13 14 9.47 005 9.47 045 9.47 086 9.47 127 9.47 168 9.48 984 9.49 029 9.49 073 9.49 118 9.49 163 0.51 016 0.50 971 0.50 927 0.50 882 0.50 837 9.98 021 9.98 017 9.98 013 9.98 009 9.98 005 50 49 48 47 46 15 16 17 18 19 9.47 209 9.47 249 9.47 290 9-47 330 9.47 371 9.49 207 9.49 252 9.49 296 9.49 341 9.49 385 0.50 793 0.50 748 0.50 704 0.50 659 0.50 615 9.98 001 9.97 997 9.97 993 9.97 989 9.97 986 45 44 43 42 41 20 21 22 23 24 9.47 411 9.47 452 9.47 492 9.47 533 9.47 573 9.49 430 9.49 474 9.49 519 9.49 563 9.49 607 0.50 570 0.50 526 0.50 481 0.50 437 0.50 393 9.97 982 9.97 978 9.97 974 9.97 970 9.97 966 40 39 38 37 36 72 17 C 25 26 27 28 29 9.47 613 9.47 654 9.47 694 9.47 734 9.47 774 9.49 652 9.49 696 9.49 740 9.49 784 9.49 828 0.50 348 0.50 304 0.50 260 0.50 216 0.50 172 9.97 962 9.97 958 9.97 954 9.97 950 9.97 946 35 34 33 32 31 t 30 31 32 33 34 9.47 814 9.47 854 9.47 894 9.47 934 9.47 974 9.49 872 9.49 916 9.49 960 9.50 004 9.50 048 0.50 128 0.50 084 0.50 040 0.49 996 0.49 952 9.97 942 9.97 938 9.97 934 9.97 930 9.97 926 30 29 28 27 26 35 36 37 38 39 9.48 014 9.48 054 9.48 094 9.48 133 9.48 173 9.50 092 9.50 136 9.50 180 9.50 223 9.50 267 0.49 908 0.49 864 0.49 820 0.49 777 0.49 733 9.97 922 9.97 918 9.9? 914 9.97 910 9.97 906 25 24 23 2f2 21 40 41 42 43 44 9.48 213 9.48 252 9.48 292 9.48 332 9.48 371 9.50 311 9.50 355 9.50 398 9.50 442 9.50 485 0.49 689 0.49 645 0.49 602 0.49 558 0.49 515 9.97 902 9.97 898 9.97 894 9.97 890 9.97 886 20 19 18 17 16 45 46 47 48 49 9.48 411 9.48 450 9.48 490 9.48 529 9.48 568 9.50 529 9.50 572 9.50 616 9.50 659 9.50 703 0.49 471 0.49 428 0.49 384 0.49 341 0.49 297 9.97 882 9.97 878 9.97 874 9.97 870 9.97 866 15 14 13 12 11 50 51 52 53 54 9.48 607 9.48 647 9.48 686 9.48 725 9.48 764 9.50 746 9.50 789 9.50 833 9.50 876 9.50 919 0.49 254 0.49 211 0.49 167 0.49 124 0.49 081 9.97 861 9.97 857 9.97 853 9.97 849 9.97 845 10 9 8 7 6 5 4 3 2 1 55 56 57 58 59 9.48 803 9.48 842 9.48 881 9.48 920 9.48 959 9.50 962 9.51 005 9.51 048 9.51 092 9.51 135 0.49 038 0.48 995 0.48 952 0.48 908 0.48 865 9.97 841 9.97 837 9.97 833 9.97 829 9.97 825 60 9.48 998 9.51 178 0.48 822 997 821 L. Cos. L. Cotg. L. Tang. L.Sin. p [59] / L. Sin. L. Tang. L. Cotg. L. Cos. 71 i 2 3 4 9.48 998 9.49 037 9.49 076 9.49 115 9.49 153 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 0.48 822 0.48 779 0.48 736 0.48 694 0.48 651 9.97 821 9.97 817 9.97 812 9.97 808 9.97 804 60 59 58 57 56 5 6 7 8 9 9.49 192 9.49 231 9.49 269 9.49 308 9.49 347 9.51 392 9.51 435 9.51 478 9.51 520 9.51 563 0.48 608 0.48 565 0.48 522 0.48 480 0.48 437 9.97 800 9.97 796 9.97 792 9.97 788 9.97 784 55 54 53 52 51 10 11 12 13 14 9.49 385 9.49 424 9.49 462 9.49 500 9.49 539 9.51 606 9.51 648 9.51 691 9.51 734 9.51 776 0.48 394 0.48 352 0.48 309 0.48 266 0.48 224 9.97 779 9.97 775 9.97 771 9.97 767 9.97 763 60 49 48 47 46 15 16 17 18 19 9.49 577 9.49 615 9.49 654 9.49 692 9.49 730 9.51 819 9.51 861 9.51 903 - 9.51 946 9.51 988 0.48 181 0.48 139 0.48 097 0.48 054 0.48 012 9.97 759 9.97 754 9.97 750 9.97 746 9.97 742 45 44 43 42 41 20 21 22 23 24 9.49 768 9.49 806 9.49 844 9.49 882 9.49 920 9.52 031 9.52 073 9.52 115 9.52 157 9.52 200 0.47 969 0.47 927 0.47 885 0.47 843 0.47 800 9.97 738 9.97 734 9.97 729 9.97 725 9.97 721 40 39 38 37 36 18 25 26 27 28 29 9.49 958 9.49 996 9.50 034 9.50 072 9.50 110 9.52 242 9.52 284 9.52 326 9.52 368 9.52 410 0.47 758 0.47 716 0.47 674 0.47 632 0.47 590 9.97 717 9.97 713 9.97 708 9.97 704 9.97 700 35 34 33 32 31 30 31 32 33 34 9.eO 148 9.50 185 9.50 223 9.50 261 9.50 298 9.52 452 9.52 494 9.52 536 9.52 578 9.52 620 0.47 548 0.47 506 0.47 464 0.47 422 0.47 380 9.97 696 9.97691 9.97 687 9.97 683 9.97 679 30 29 28 27 26 35 36 37 38 39 9.50 336 9.50 374 9.50 111 9.50 449 9.50 486 9.52 661 9.52 703 9.52 745 9.52 787 9.52 829 0.47 339 0.47 297 0.47 255 0.47 213 0.47 171 9.97 674 9.97 670 9.97 666 9.97 662 9.97 657 25 24 23 22 21 40 41 42 43 44 9.50 523 9.50 561 9.50 598 9.50 635 9.50 673 9.52 870 9.52 912 9.52 953 9.52 995 9.53 037 0.47 130' 0.47 088 0.47 047 0.47 005 0.46 963 9.97 653 9.97 649 9.97 645 9.97 640 9.97 636 20 19 18 17 16 45 46 47 48 49 9.50 710 9.50 747 9.50 784 9.50 821 9.50 858 9.53 078 9.53 120 9.53 161 9.53 202 9.53 244 0.46 922 0.46 880 0.46 839 0.46 798 0.46 756 9.97 632 9.97 628 9.97 623 9.97 619 9.97 615 15 14 13 12 11 50 51 52 53 54 9.50 896 9.50 933 9.50 970 9.51 007 9.51 043 9.53 285 9.53 327 9.53 368 9.53 409 9.53 450 0.46 715 0.46 673 0.46 632 0.46 591 0.46 550 9.97 610 9.97 606 9.97 602 9.97 597 9.97 593 10 9 8 7 6 55 56 57 58 59 9.51 080 9.51 117 9.51 154 9.51 191 9.51 227 9.53 492 9.53 533 9.53 574 9.53 615 9.53 656 0.46 508 0.46 467 0.46 426 0.46 385 0.46 344 9.97 589 9.97 584 9.97 580 9.97 576 9.97 571 5 4 3 2 1 60 9.51 264 9.53 697 0.46 303 9.97 567 L. Cos. L. Cotg. L. Tang. L. Sin. / [60] 19 / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.51 264 9.51 301 9.51 338 9.51 374 9.51 411 9.53 697 9.53 738 9.53 779 9.53 820 9.53 861 0.46 303 0.46 262 0.46 221 0.46 180 0.46 139 9.97 567 9.97 563 9.97 558 9.97 554 9.97 550 60 59 58 57 56 5 6 7 8 9 9.51 447 9.51 484 9.51 520 9'51 557 9.51 593 9.53 902 9.53 943 9.53 984 9.54 025 9.54 065 0.46 098 0.46 057 0.46 016- 0.45 975 0.45 935 9.97 545 9.97 541 9.97 536 9.97 532 9.97 528 55 54 53 52 51 10 11 12 13 14 9.51 629 9.51 666 9.51 702 9.51 738 9.51 774 9.54 106 9.54 147 9.54 187 9.54 228 9.54 269 0.45 894 0.45 853 0.45 813 0.45 772 0.45 731 9.97 523 9.97 519 9.97 515 9.97 510 9.97 506 50 49 48 47 46 15 16 17 18 19 9.51 811 9.51 847 9.51 883 9.51 919 9.51 955 9.54 309 9.54 350 9.54 390 9.54 431 9.54 471 0.45 691 0.45 650 0.45 610 0.45 569 0.45 529 9.97 501 9.97 497 9.97 492 9.97 488 9.97 484 45 44 43 42 41 20 21 22 23 24 9.51 991 9.52 027 9.52 063 9.52 099 9.52 135 9.54 512 9.54 552 9.54 593 9.54 633 9.54 673 0.45 488 0.45 448 0.45 407 0.45 367 0.45 327 9.97 479 9.97 475 9.97 470 9.97 466 9.97 461 40 39 38 37 36 70 25 26 27 28 29 9.52 171 9.52 207 9.52 242 9.52 278 9.52 314 9.54 714 9.54 754 9.54 794 9.54 835 9.54 875 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 9.97 457 9.97 453 9.97 448 9.97 444 9.97 439 35 34 33 32 31 30 31 32 33 34 9.52 350 9.52 385 9.52 421 9.52 456 9.52 492 9.54 915 9.54 955 9.54 995 9.55 035 9.55 075 0.45 085 0.45 045 0.45 005 0.44 965 0.44 925 9.97 435 9.97 430 9.97 426 9.97 421 9.97 417 30 29 28 27 26 35 36 37 38 39 9.52 527 9.52 563 9.58 598 9.52 634 9.52 669 9.55 115 9.55 155 9.55 195 9.55 235 9.55 275 0.44 885 0.44 845 0.44 805 0.44 765 0.44 725 9.97 412 9.97 408 9.97 403 9.97 399 9.97 394 25 24 23 22 21 40 41 42 43 44 9.52 705 9.52 740 9.52 775 9.52 811 9.52 846 9.55 315 9.55 355 9.55 395 9.55 434 9.55 474 0.44 685 0.44 645 0.44 605 0.44 566 0.44 526 9.97 390 9.97 385 9.97 381 9.97 376 9.97 372 20 19 18 17 16 45 46 47 48 49 9.52 881 9.52 916 9.52 951 9.52 986 9.53 021 9.55 514 9.55 554 9.55 593 9.55 633 9.55 673 0.44 486 0.44 446 0.44 407 0.44 367 0.44 327 9.97 367 9.97 363 9.97 358 9.97 353 9.97 349 15 14 13 12 11 50 51 52 53 54 9.53 056 9.53 092 9.53 126 9.53 161 9.53 196 9.55 712 9.55 752 9.55 791 9.55 831 9.55 870 0.44 288 0.44 248 0.44 209 0.44 169 0.44 130 9.97 344 9.97 340 9.97 335 9.97 331 9.97 326 10 9 8 7 6 55 56 57 58 59 9.53 231 9.53 266 9.53 301 9.53 336 9.53 370 9.55 910 9.55 949 9.55 989 9.56 028 9.56 067 0.44 090 0.44 051 0.44 Oil 0.43 972 0.43 933 9.97 322 9.97 317 9.97 312 9.97 308 9.97 303 5 4 3 2 1 60 9.53 405 9.56 107 0.43 893 9.97 299 ' L. Cos. L. Cotg. L. Tang. L. Sin. ; [61] / L. Sin. L. Tang. L. Cotg. L. Cos. 69 i 2 3 4 9.53 405 9.53 440 9.53 475 9.53 509 9.53 544 9.56 107 9.56 146 9.56 185 9.56 224 9.56 264 0.43 893 0.43 854 0.43 815 0.43 776 0.43 736 9.97 299 9.97 294 9.97 289 9.97 285 9.97 280 60 59 58 57 56 5 6 7 8 9 9.53 578 9.53 613 9.53 647 9.53 682 9.53 716 9.56 303 9.56 342 9.56 381 9.56 420 9.56 459 0.43 697 0.43 658 0.43 619 0.43 580 0.43 541 9.97 276 9.97 271 9.97 266 9.97 262 9.97 257 55 54 53 52 51 10 11 12 13 14 9.53 751 9.53 785 9.53 819 9.53 854 9.53 888 9.56 498 9.56 537 9.56 576 9.56 615 9.56 654 0.43 502 0.43 463 0.43 424 0.43 385 0.43 346 9.97 252 9.97 248 9.97 243 9.97 238 9.97 234 50 49 48 47 46 15 16 17 18 19 9.53 922 9.53 957 9.53 991 9.54 025 9.54 059 9.56 693 9.56 732 9.56 771 9.56 810 9.56 849 0.43 307 0.43 268 0.43 229 0.43 190 0.43 151 9.97 229 9.97 224 9.97 220 9.97 215 9.97 210 45 44 43 42 41 20 21 22 23 24 9.54 093 9.54 127 9.54 161 9.54 195 9.54 229 9.56 887 9.56 926 9.56 965 9.57 004 9.57 042 0.43 113 0.43 074 0.43 035 0.42 996 0.42 958 9.97 206 9.97 201 9.97 196 9.97 192 9.97 187 40 39 38 37 36 20 25 26 27 28 29 9.54 263 9.54 297 9.54 331 9.54 365 9.54 399 9.57 081 9.57 120 9.57 158 9.57 197 9.57 235 0.42 919 0.42 880 0.42 842 0.42 803 0.42 765 9.97 182 9.97 178 9.97 173 9.97 168 9.97 163 35 34 33 32 31 30 31 32 33 34 9.54 433 9.54 466 9.54 500 9.54 534 9.54 567 9.57 274 9.57 312 9.57 351 9.57 389 9.57 428 0.42 726 0.42 688 0.42 649 0.42 611 0.42 572 9.97 159 9.97 154 9.97 149 9.97 145 9.97 140 30 29 28 27 26 35 36 37 38 39 9.54 601 9.54 635 9.54 668 9.54 702 9.54 735 9.57 466 9.57 504 9.57 543 9.57 581 9.57 619 0.42 534 0.42 496 6.42 457 0.42 419 0.42 381 9.97 135 9.97 130 9.97*126 9.97 121 9.97 116 25 24 23 22 21 40 41 42 43 44 9.54 769 9.54 802 9.54 836 9.54 869 9.54 903 9.57 658 9.57 696 9.57 734 9.57 772 9.57 810 0.42 342 0.42 304 0.42 266 0.42 228 0.42 190 9.97 111 9.97 107 9.97 102 9.97 097 9.97 092 20 19 18 17 16 45 46 47 48 49 9.54 936 9.54 969 9.55 003 9.55 036 9.55 069 9.57 849 9.57 887 9.57 925 9.57 963 9.58 001 0.42 151 0.42 113 0.42 075 0.42 037 0.41 999 9.97 087 9.97 083 9.97 078 9.97 073 9.97 068 15 14 13 12 11 50 51 52 53 54 9.55 102 9.55 136 9.55 169 9.55 202 9.55 235 9.58 039 9.58 077 9.58 115 9.58 153 9.58 191 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 9.97 063 9.97 059 9.97 054 9.97 049 9.97 044 10 9 8 7 6 55 56 57 58 59 9.55 268 9.55 301 9.55 334 9.55 367 9.55 400 9.58 229 9.58 267 9.58 304 9.58 342 9.58 380 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 9.97 039 9.97 035 9.97 030 9.97 025 9.97 020 5 4 3 2 1 60 9.55 433 9.58 418 - 0.41 582 9.97 015 L. Cos. L. Cotg. L. Tang. L. Sin. f [62] 9 jo t L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.55 433 9.55 466 9.55 499 9.55 532 9.55 564 9.58 418 9.58 455 9.58 493 9.58 531 9.58 569 0.41 582 0.41 545 0.41 507 0.41 469 0.41 431 9.97 015 9.97 010 9.97 005 9.97 001 9.96 996 60 59 58 57 56 5 6 7 8 9 9.55 597 9.55 630 9.55 663 9.55 695 9.55 728 9.58 606 9.58 644 9.58 681 9.58 719 9.58 757 0.41 394 0.41 356 0.41 319 0.41 281 0.41 243 9.96 991 9.96 986 9.96 981 9.96 976 9.96 971 55 54 53 52 51 68 10 11 12 13 14 9.55 761 9.55 793 9.55 826 9.55 858 9.55 891 9.58 794 9.58 832 9.58 869 9.58 907 9.58 944 0.41 206 0.41 168 0.41 131 0.41 093 0.41 056 9.96 966 9.96 962 9.96 957 9.96 952 9.96 947 50 49 48 47 46 15 16 17 18 19 9.55 923 9.55 956 9.55 988 9.56 021 9.56 053 9.58 981 9.59 019 9.59 056 9.59 094 9.59 131 0.41 019 0.40 981 0.40 944 0.40 906 0.40 869 9.96 942 9.96 937 9.96 932 9.96 927 9.96 922 45 44 43 42 41 20 21 22 23 24 9.56 085 9.56 118 9.56 150 9.56 182 9.56 215 9.59 168 9.59 205 9.59 243 9.59 280 9.59 317 0.40 832 0.40 795 0.40 757 0.40 720 0.40 683 9.96 917 9.96 912 9.96 907 9.96 903 9.96 898 40 39 38 37 36 25 26 27 28 29 9.56 247 9.56 279 9.56 311 9.56 343 9.56 375 9.59 354 9.59 391 9.59 429 9.59 466 9.59 503 0.40 646 0.40 609 0.40 571 0.40 534 0.40 497 9.96 893 9.96 888 9.96 883 9.96 878 9.96 873 35 34 33 32 31 30 31 32 33 34 9.56 408 9.56 440 9.56 472 9.56 504 9.56 536 9.59 540 9.59 577 9.59 614 9.59 651 9.59 688 0.40 460 0.40 423 0.40 386 0.40 349 0.40 312 9.96 868 9.96 863 9.96 858 9.96 853 9.96 848 30 29 28 27 26 35 36 37 38 39 9.56 568 9.56 599 9.56 631 9.56 663 9.56 695 9.59 725 9.59 762 9.59 799 9.59 835 9.59 872 0.40 275 0.40 238 0.40 201 0.40 165 0.40 128 9.96 843 9.96 838 9.96 833 9.96 828 9.96 823 25 24 23 22 21 40 41 42 43 44 9.56 727 9.56 759 9.56 790 9.56 822 9.56 854 9.59 909 9.59 946 9.59 983 9.60 019 9.60 056 0.40 091 0.40 054 0.40 017 0.39 981 0.39 944 9.96 818 9.96 813 9.96 808 9.96 803 9.96 798 20 19 18 17 16 45 46 47 48 49 9.56 886 9.56 917 9.56 949 9.56 980 9.57 012 9.60 093 9.60 130 9.60 166 9.60 203 9.60 240 0.39 907 0.39 870 0.39 834 0.39 797 0.39 760 9.96 793 9.96 788 9.96 783 9.96 778 9.96 772 15 14 13 12 11 50 51 52 53 54 9.57 044 9.57 075 9.57 107 9.57 138 9.57 169 9.60 276 9'60 313 9.60 349 9.60 386 9.60 422 0.39 724 0.39 687 0.39 651 0.39 614 0.39 578 9.96 767 9.96 762 9.96 757 9.96 752 9.96 747 10 9 8 7 6 55 56 57 58 59 9.57 201 9.57 232 9.57 264 9.57 295 9.57 326 9.60 459 9.GO 495 9.60 532 9.60 568 9.60 605 0.39 541 0.39 505 0.39 468 0.39 432 0.39 395 9.96 742 9.96 737 9.96 732 9.96 727 9.96 722 5 4 3 2 1 60 9.57 358 9.0 R41 0.39 359 9.96 717 L. Cos. L. Cotg. L. Tang. L. Sin. ; [63] / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.57 358 9.57 389 9.57 420 9.57 451 9.57 482 9.60 641 9.60 677 9.60 714 9.60 750 9.60 786 0.39 359 0.39 323 0.39 286 0.39 250 0.39 214 9.96 717 9.96 711 9.96 706 9.96 701 9.96 696 60 59 58 57 56 67 5 6 7 8 9 9.57 514 9.57 545 9.57 576 9.57 607 9.57 638 9.60 823 9.60 859 9.60 895 9.60 931 9.60 967 0.39 177 0.39 141 0.39 105 0.39 069 0.39 033 9.96 691 9.96 686 9.96 681 9.96 676 9.96 670 55 54 53 52 51 10 11 12 13 14 9.57 669 9.57 700 9.57 731 9.57 762 9.57 793 9.61 004 9.61 040 9.61 076 9.61 112 9.61 148 0.38 996 0.38 960 0.38 924 0.38 888 0.38 852 9.96 665 9.96 660 9.96 655 9.96 650 9.96 645 50 49 48 47 46 15 16 17 18 19 9.57 824 9.57 855 9.57 885 9.57 916 9.57 947 9.61 184 9.61 220 9.61 256 9.61 292 9.61 328 0.38 816 0.38 780 0.38 744 0.38 708 0.38 672 9.96 640 9.96 634 9.96 629 9.96 624 9.96 619 45 44 43 42 41 20 21 22 23 24 9.57 978 9.58 008 9.58 039 9.58 070 9.58 101 9.61 364 9.61 400 9.61 436 9.61 472 9.61 508 0.38 636 0.38 600 0.38 564 0.38 528 0.38 492 9.96 614 9.96 608 9.96 603 9.96 598 9.96 593 40 39 38 37 36 99 25 26 27 28 29 9.58 131 9.58 162 9.58 192 * 9.58 223 9.58 253 9.61 544 9.61 579 9.61 615 9.61 651 9.61 687 0.38 456 0.38 421 0.38 385 0.38 349 0.38 313 9.96 588 9.96 582 9.96 577 9.96 572 9.96 567 35 34 33 32 31 30 31 32 33 34 9.58 284 9.58 314 9.58 345 9.58 375 9.58 406 9.61 722 9.61 758 9.61 794 9.61 830 9.61 865 0.38 278 0.38 242 0.38 206 0.38 170 0.38 135 9.96 562 9.96 556 9.96 551 9.96 546 9.96 541 30 29 28 27 26 35 36 37 38 39 9.58 436 9.58 467 9.58 497 9.58 527 9.58 557 9.61 901 9.61 936 9.61 972 9.62 008 9.62 043 0.38 099 0.38 064 0.38 028 0.37 992 0.37 957 9.96 535 9.96 730 9.96 525 9.96 520 9.96 514 25 24 23 22 21 40 41 42 43 44 9.58 588 9.58 618 9.58 648 9.58 678 9.58 709 9.62 079 9.62 114 9.62 150 9.62 185 9.62 221 0.37 921 0.37 886 0.37 850 0.37 815 0.37 779 9.96 509 9.96 504 9.96 498 9.96 493 9.96 488 20 19 18 17 16 45 46 47 48 49 9.58 739 9.58 769 9.58 799 9.58 829 9.58 859 9.62 256 9.62 292 9.62 327 9.62 362 9.62 398 0.37 744 0.37 708 0.37 673 0.37 638 0.37 602 9.96 483 9.96 477 9.96 472 9.96 467 9.96 461 15 14 13 12 11 50 51 52 53 54 9.58 889 9.58 919 9.58 949 9.58 979 9.59 009 9.62 433 9.62 468 9.62 504 9.62 539 9.62 574 0.37 567 0.37 532 ' 0.37 496 0.37 461 0.37 426 9.96 456 9.96 451 9.96 445 9.96 440 9.96 435 10 9 8 7 6 55 56 57 58 59 9.59 039 9.59 069 9.59 098 9.59 128 9.59 158 9.62 609 9.62 645 9.62 680 9.62 715 9.62 750 0.37 391 0.37 355 0.37 320 0.37 285 0.37 250 9.96 429 9.96 424 9.96 419 9.96 413 9.96 408 5 4 3 2 1 60 9.59 188 9.62 785 0.37 215 996403 L. Cos. L. Cotg. L. Tang. L. Sin. / [64] r L. Sin. L. Tang. L. Cotg. L. Cos. 66 1 2 3 4 9.59 188 9.59 218 9.59 247 9.59 277' 9.59 307 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 0.37 215 0.37 180 0.37 145 0.37 110 0.37 074 9.96 403 9.96 397 9.96 392 9.96 387 9.96 381 60 59 58 57 56 5 6 7 8 9 9.59 336 9.59 366 9.59 396 9.59 425 9.59 455 9.62 961 9.62 996 9.63 031 9.63 066 9.63 101 0.37 039 0.37 004 0.36 969 0.36 934 0.36 899 9.96 376 9.96 370 9.96 365 9.96 360 9.96 354 55 54 53 52 51 10 11 12 13 14 9.59 484 9.59 514 9.59 543 9.59 573 9.59 602 9.63 135 9.63 170 9.63 205 9.63 240 9.63 275 0.36 865 0.36 830 0.36 795 0.36 760 0.36 725 9.96 349 9.96 343 9.96 338 9.96 333 9.96 327 50 49 48 47 46 15 16 17 18 19 9.59 632 9.59 661 9.59 690 9.59 720 9.59 749 9.63 310 9.63 345 9.63 379 9.63 414 9.63 449 0.36 690 0.36 655 0.36 621 0.36 586 0.36 551 9.96 322 9.96 316 9.96 311 9.96 305 9.96 300 45 44 43 42 41 20 21 22 23 24 9.59 778 9.59 808 9.59 837 9.59 866 9.59 895 9.63 484 9.63 519 9.63 553 9.63 588 9.63 623 0.36 516 0.36 481 0.36 447 0.36 412 0.36 377 9.96 294 9.96 289 9.96 284 9.96 278 9.96 273 40 39 38 37 36 23 25 26 27 28 29 9.59 924 9.59 954 9.59 983 9.60 012 9.60 041 9.63 657 9.63 692 9.63 726 9.63 761 9.63 796 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 9.96 267 9.96 262 9.96 256 9.96 251 9.96 245 35 34 33 32 31 30 31 32 33 34 9.60 070 9.60 099 9.60 128 9.60 157 9.60 186 9.63 830 9.63 865 9.63 899 9.63 934 9.63 968 0.36 170 0.36 135 0.36 101 0.36 066 0.36 032 9.96 240 9.96 234 9.96 229 9.96 223 9.96 218 30 29 28 27 26 35 36 37 38 39 9.60 215 9.60 244 9.60 273 9.60 302 9.60 331 9.64 003 9.64 037 9.64 072 9.64 106 9.64 140 0.35 997 0.35 963 0.35 928 0.35 894 0.35 860 9.96 212 9.96 207 9.96 201 9.96 196 9.96 190 25 24 23 22 21 40 41 42 43 44 9.60 359 9.60 388 9.60 417 9.60 446 9.60 474 9.64 175 9.64 209 9.64 243 9.64 278 9.64 312 0.35 825 0.35 791 0.35 757 0.35 722 0.35 688 9.96 185 9.96 179 9.96 174 9.96 168 9.96 162 20 19 18 17 16 45 46 47 48 49 9.60 503 9.60 532 9.60 561 9.60 589 9.60 618 9.64 346 9.64 381 9.64 415 9.64 449 9.64 483 0.35 654 0.35 619 0.35 585 0.35 551 0.35 517 9.96 157 9.96 151 9.96 146 9.96 140 9.96 135 15 14 13 12 11 50 51 52 53 54 9.60 646 9.60 675 9.60 704 9.60 732 9.60 761 9.64 517 9.64 552 9.64 586 9.64 620 9.64 654 0.35 483 0.35 448 0.35 414 0.35 380 0.35 346 9.96 129 9.96 123 9.96 118 9.96 112 9.96 107 10 9 8 7 6 55 56 57 58 59 9.60 789 9.60 818 9.60 846 9.60 875 9.60 903 9.64 688 9.64 722 9.64 756 9.64 790 9.64 824 0.35 312 0.35 278 0.35 244 0.35 210 0.35 176 9.96 101 9.96 095 9.96 090 9.96 084 9.96 079 5 4 3 2 1 60 9.60 931 9.64 858 0.35 142 9.96 073 L. Cos. L. Cotg. L. Tang. L. Sin. ; [65] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 65 1 2 3 4 9.60 931 9.60 960 9.60 988 9.61 016 9.61 045 9.64 858 9.64 892 9.64 926 9.64 960 9.64 994 0.35 142 0.35 108 0.35 074 0.35 040 0.35 006 9.96 073 9.96 067 9.96 062 9.96 056 9.96 050 60 59 58 57 56 5 6 7 8 9 9.61 073 9.61 101 9.61129 9.61 158 9.61 186 9.65 028 9.65 062 9.65 096 9.65 130 9.65 164 0.34 972 0.34 938 0.34 904 0.34 870 0.34 836 9.96 045 9.96 039 9.96 034 9.96 028 9.96 022 55 54 53 52 51 10 11 12 13 14 9.61 214 9.61 242 9.61 270 9.61 298 9.61 326 9.65 197 9.65 231 9.65 265 9.65 299 9.65 333 0.34 803 0.34 769 0.34 735 0.34 701 0.34 667 9.96 017 9.96 Oil 9.96 005 9.96 000 9.95 994 50 49 48 47 46 15 16 17 18 19 9.61 354 9.61 382 9.61 411 9.61 438 9.61 466 9.65 366 9.65 400 9.65 434 9.65 467 9.65 501 0.34 634 0.34 600 0.34 566 0.34 533 0.34 499 9.95 988 9.95 982 9.95 977 9.95 971 9.95 965 45 44 43 42 41 20 21 22 23 24 9.61 494 9.61 522 9.61 550 9.61 578 9.61 606 9.65 535 9.65 568 9.65 602 9.65 636 9.65 669 0.34 465 0.34 432 0.34 398 0.34 364 0.34 331 9.95 960 9.95 954 9.95 948 9.95 942 9.95 937 40 39 38 37 36 24 25 26 27 28 29 9.61 634 9.61 662 9.61 689 9.61 717 9.61 745 9.65 703 9.65 736 9.65 770 9.65 803 9.65 837 0.34 297 0.34 264 0.34 230 0.34 197 0.34 163 9.95 931 9.95 925 9.95 920 9.95 914 9.95 908 35 34 33 32 31 30 31 32 33 34 9.61 773 9.61 800 9.61 828 9.61 856 9.61 883 9.65 870 9.65 904 9.65 937 9.65 971 9.66 004 0.34 130 0.34 096 0.34 063 0.34 029 0.33 996 9.95 902 9.95 897 9.95 891 9.95 885 9.95 879 30 29 28 27 26 35 36 37 38 39 9.61 911 9.61 939 9.61 966 9.61 994 9.62 021 9.66 038 9.66 071 9.66 104 9.66 138 9.66 171 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 9.95 873 9.95 868 9.95 862 9.95 856 9.95 850 25 24 23 22 21 40 41 42 43 44 9.62 049 9.62 076 9.62 104 9.62 131 9.62 159 9.66 204 9.66 238 9.66 271 9.66 304 9.66 337 0.33 796 0.33 762 0.33 729 0.33 696 0.33 663 9.95 844 9.95 839 9.95 833 9.95 827 9.95 821 20 19 18 17 16 45 46 47 48 49 9.62 186 9.62 214 9.62 241 9.62 268 9.62 296 9.66 371 9.66 404 9.66 437 9.66 470 9.66 503 0.33 629 0.33 596 0.33 563 0.33 530 0.33 497 9.95 815 9.95 810 9.95 804 9.95 798 9.95 792 15 14 13 12 11 50 51 52 53 54 9.62 323 9.62 350 9.62 377 9.62 405 9.62 432 9.66 537 9.66 570 9.66 603 9.66 636 9.66 669 0.33 463 0.33 430 0.33 397 0.33 364 0.33 331 9.95 786 9.95 780 9.95 775 9.95 769 9.95 763 10 9 8 7 6 55 56 57 58 59 9.62 459 9.62 486 9.62 513 9.62 541 9.62 568 9.66 702 9.66 735 9.66 768 9.66 801 9.66 834 0.33 298 0.33 265 0.33 232 0.33 199 0.33 166 9.95 757 9.95 751 9.95 745 9.95 739 9.95 733 5 4 3 2 1 60 9.62 595 9.66 867 0.33 133 9.95 728 L. Cos. L. Cotg. L. Tang. L. Sin. / [66] / L. Sin. L. Tang. L. Cotg. L. Cos. 64 1 2 3 4 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 9.66 867 9.66 900 9.66 933 9.66 966 9.66 999 0.33 133 0.33 100 0.33 067 0.33 034 0.33 001 9.95 728 9.95 722 9.95 716 9.95 710 9.95 704 60 59 53 57 56 5 6 7 8 9 9.62 730 9.62 757 9.62 784 9.62 811 9.62 838 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 0.32 968 0.32 935 0.32 902 0.32 869 0.32 837 9.95 698 9.95 692 9.95 686 9.95 680 9.95 674 55 54 53 52 51 10 11 12 13 14 9.62 865 9.62 892 9.62 918 9.62 945 9.62 972 9.67 196 9.67 229 9.67 262 9.67 295 9.67 327 0.32 804 0.32 771 0.32 738 0.32 705 0.32 673 9.95 668 9.95 663 9.95 657 9.95 651 9 95 645 50 49 48 47 46 15 16 17 18 19 9.62 999 9.63 026 9.63 052 9.63 079 9.63 106 9.67 360 9.67 393 9.67 426 9.67 458 9.67 491 0.32 640 0.32 607 0.32 574 0.32 542 0.32 509 9.95 639 9.95 633 9.95 627 9.95 621 9.95 615 45 44 43 42 41 20 21 22 23 24 9.63 133 9.63 159 9.63 186 9.63 213 9.63 239 9.67 524 9.67 556 9.67 589 9.67 622 9.67 654 0.32 476 0.32 444 0.32 411 0.32 378 0.32 346 9.95 609 9.95 603 9.95 597 9.95 591 9.95 585 40 39 38 37 36 25 25 26 27 28 29 9.63 266 9.63 292 9.63 319 9.63 345 9.63 372 9.67 687 9.67 719 9.67 752 9.67 785 9.67 817 0.32 313 0.32 281 0.32 248 0.32 215 0.32 183 9.85 579 9.95 573 9.95 567 9.95 561 9.95 555 35 34 33 32 31 30 31 32 33 34 9.63 398 9.63 425 9.63 451 9.63 478 9.63 504 9.67 850 9.67 882 9.67 915 9.67 947 9.67 980 0.32 150 0.32 118 0.32 085 0.32 053 0.32 020 9.95 549 9.95 543 9.95 537 9.95 531 9.95 525 30 29 28 27 26 35 36 37 38 39 9.63 531 9.63 557 9.63 583 9.63 610 9.63 636 9.68 012 9.68 044 9.68 077 9.68 109 9.68 142 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 9.95 519 9.95 513 9.95 507 9.95 500 9.95 494 25 24 23 22 21 40 41 42 43 44 9.63 662 9.63 689 9.63 715 9.63 741 9.63 767 9.68 174 9.68 206 9.68 239 9.68 271 9.68 303 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 9.95 488 9.95 482 9.95 476 9.95 470 9.95 464 20 19 18 17 16 45 46 47 48 49 9.63 794 9.63 820 9.63 846 9.63 872 9.63 898 9.68 336 9.68 368 9.68 400 9.68 432 9.68 465 0.31 664 0.31 632 0.31 600 0.31 568 0.31 535 9.95 458 9.95 452 9.95 446 9.95 440 9.95 434 15 14 13 12 11 60 51 52 53 54 9.63 924 9.63 950 9.63 976 9.64 002 9.64 028 9.68 497 9.68 529 9.68 561 9.68 593 9.68 626 0.31 503 0.31 471 0.31 439 0.31 407 0.31 374 9.95 427 9.95 421 9.95 415 9.95 409 9.95 403 10 9 8 7 6 55 56 57 58 59 9.64 054 9.64 080 9.64 106 9.64 132 9.64 158 9.68 658 9.68 690 9.68 722 9.68 754 9.68 786 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 9.95 397 9.95 391 9.95 384 9.95 378 9.95 372 5 4 3 2 1 60 9.64 184 9.68 818 0.31 182 9.95 366 L. Cos. L. Cotg. L. Tang.- L. Sin. ; [67] 2fi / L. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 9.64 184 9.64 210 9.64 236 9.64 262 9.64 288 9.68 818 9.68 850 9.68 882 9.68 914 9.68 946 0.31 182 0.31 150 0.31 118 0.31 086 0.31 054 9.95 366 9.95 360 9.95 354 9.95 348 9.95 341 60 59 58 57 56 5 6 7 8 9 9.64 313 9.64 339 9.64 365 9.64 391 9.64 417 9.68 978 9.69 010 9.69 042 9.69 074 9.69 106 0.31 022 0.30 990 0.30 958 0.30 926 0.30 894 - 9.95 335 9.95 329 9.95 323 9.95 317 9.95 310 55 54 53 52 51 10 11 12 13 14 9.64 442 9.64 468 9.64 494 9.64 519 9.64 545 9.69 138 9.69 170 9.69 202 9.69 234 9.69 266 0.30 862 0.30 830 0.30 798 0.30 766 0.30 734 9.95 304 9.95 298 9.95 292 9.95 286 9.95 279 50 49 48 47 46 15 16 17 18 19 9.64 571 9.64 596 9.64 622 9.64 647 9.64 673 9.69 298 9.69 329 9.69 361 9.69 393 9.69 425 0.30 702 0.30 671 0.30 639 0.30 607 0.30 575 9.95 273 9.95 267 9.95 261 9.95 254 9.95 248 45 44 43 42 41 20 21 22 23 24 9.64 698 9.64 724 9.64 749 9.64 775 9.64 800 9.69 457 9.69 488 9.69 520 9.69 552 9.69 584 0.30 543 0.30 512 0.30 480 0.30 448 0.30 416 9.95 242 9.95 236 9.95 229 9.95 223 9.95 217 40 39 38 37 36 25 26 27 28 29 9.64 826 9.64 851 9.64 877 9.64 902 9.64 927 9.69 615 9.69 647 9.69 679 9.69 710 9.69 742 0.30 385 0.30 353 0.30 321 0.30 290 0.30 258 9.95 211 9.95 204 9.95 198 9.95 192 9.95 185 35 34 33 32 31 63 \j 30 31 32 33 34 9.64 953 9.64 978 9.65 003 9.65 029 9.65 054 9.69 774 9.69 805 9.69 837 9.69 868 9.69 900 0.30 226 0.30 195 0.30 163 0.30 132 0.30 100 9.95 179 9.95 173 9.95 167 9.95 160 9.95 154 30 29 28 27 26 35 36 37 38 39 9.65 079 9.65 104 9.65 130 9.65 155 9.65 180 9.69 932 9.69 963 9.69 995 9.70 026 9.70 058 0.30 068 0.30 037 0.30 005 0.29 974 0.29 942 9.95 148 9.95 141 9.95 135 9.95 129 9.95 122 25 24 23 22 21 40 41 42 43 44 9.65 205 9.65 230 9.65 255 9.65 281 9.65 306 9.70 089 9.70 121 9.70 152 9.70 184 9.70 215 0.29 911 0.29 879 0.29 848 0.29 816 0.29 785 9.95 116 9.95 110 9.95 103 9.95 097 9.95 090 20 19 18 17 16 45 46 47 48 49 9.65 331 9.65 356 9.65 381 9.65 406 9.65 431 9.70 247 9.70 278 9.70 309 9.70 341 9.70 372 0.29 753 0.29 722 0.29 691 0.29 659 0.29 628 9.95 084 9.95 078 9.95 071 9.95 065 9.95 059 15 14 13 12 11 50 51 52 53 54 9.65 456 9.65 481 9.65 506 9.65 531 9.65 556 9.70 404 9.70 435 9.70 466 9.70 498 9.70 529 0.29 596 0.29 565 0.29 534 0.29 502 0.29 471 9.95 052 9.95 046 9.95 039 9.95 033 9.95 027 10 9 8 7 6 55 56 57 58 59 9.65 580 9.65 605 9.65 630 9.65 655 9.65 680 9.70 560 9.70 592 9.70 623 9.70 654 9.70 685 0.29 440 0.29 408 0.29 377 0.29 346 0.29 315 9.95 020 9.95 014 9.95 007 9.95 001 9.94 995 5 4 3 2 1 60 9.65 705 9.70 717 0.29 283 9.94 988 L. Cos. L. Cotg. L. Tang. L. Sin. / [68] / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.65 705 9.65 729 9.65 754 9.65 779 9.65 804 9.70 717 9.70 748 9.70 779 9.70 810 9.70 841 0.29 283 0.29 252 0.29 221 0.29 190 0.29 159 9.94 988 9.94 982 9.94 975 9.94 969 9.94 962 60 59 58 57 56 5 6 7 8 9 9.65 828 9.65 853 9.65 878 9.65 902 9.65 927 9.70 873 9.70 904 9.70 935 9.70 966 9.70 997 0.29 127 0.29 096 0.29 065 0.29 034 0.29 003 9.94 956 9.94 949 9.94 943 9.94 936 9.94 930 55 54 53 52 51 10 11 12 13 14 9.65 952 9.65 976 9.66 001 9.66 025 9.66 050 9.71 028 9.71 059 9.71 090 9.71 121 9.71 153 0.28 972 0.28 941 0.28 910 0.28 879 0.28 847 9.94 923 9.94 917 9.94 911 9.94 904 9.94 898 50 49 48 47 46 15 16 17 18 19 9.66 075 9.66 099 9.66 124 9.66 148 9.66 173 9.71 184 9.71 215 9.71 246 9.71 277 9.71 308 0.28 816 0.28 785 0.28 754 0.28 723 0.28 692 9.94 891 9.94 885 9.94 878 9.94 871 9.94 865 45 44 43 42 41 20 21 22 23 24 9.66 197 9.66 221 9.66 246 9.66 270 9.66 295 9.71 339 9.71 370 9.71 401 9.71 431 9.71 462 0.28 661 0.28 630 0.28 599 0.28 569 0.28 538 9.94 858 9.94 852 9.94 845 9.94 839 9.94 832 40 39 38 37 36 27 25 26 27 28 29 9.66 319 9.66 343 9.66 368 9.66 392 9.66 416 9.71 493 9.71 524 9.71 555 9.71 586 9.71 617 0.28 507 0.28 476 0.28 445 0.28 414 0.28 383 9.94 826 9.94 819 9.94 813 9.94 806 9.94 799 35 34 33 32 31 62 30 31 32 33 34 9.66 441 9.66 465 9.66 489 9.66 513 9.66 537 9.71 648 9.71 679 9.71 709 9.71 740 9.71 771 0.28 352 0.28 321 0.28 291 0.28 260 0.28 229 9.94 793 9.94 786 9.94 780 9.94 773 9.94 767 30 29 28 27 26 35 36 37 38 39 9.66 562 9.66 586 9.66 610 9.66 634 9.66 658 9.71 802 9.71 833 9.71 863 9.71 894 9.71 925 0.28 198 0.28 167 0.28 137 0.28 106 0.28 075 9.94 760 9.94 753 9.94 747 9.94 740 9.94 734 25 24 23 22 21 40 41 42 43 44 9.66 682 9.66 706 9.66 731 9.66 755 9.66 779 9.71 955 9.71 986 9.72 017 9.72 048 9.72 078 0.28 045 0.28 014 0.27 983 0.27 952 0.27 922 9.94 727 9.94 720 9.94 714 9.94 707 9.94 700 20 19 18 17 16 45 46 47 48 49 9.66 803 9.66 827 9.66 851 9.66 875 9.66 899 9.72 109 9.72 140 9.72 170 9.72 201 9.72 231 0.27 891 0.27 860 0.27 830 0.27 799 0.27 769 ' 9.94 694 9.94 687 9.94 680 9.94 674 9.94 667 15 14 13 12 11 50 51 52 53 54 9.66 922 9.66 946 9.66 970 9.66 994 9.67 018 9.72 262 9.72 293 9.72 323 9.72 354 9.72 384 . 0.27 738 0.27 707 0.27 677 0.27 646 0.27 616 9.94 660 9.94 654 9.94 647 9.94 640 9.94 634 10 9 8 7 6 55 56 57 58 59 9.67 042 9.67 066 9.67 090 9.67 113 9.67 137 9.72 415 9.72 445 9.72 476 9.72 506 9.72 537 0.27 585 0.27 555 0.27 524 0.27 494 0.27 463 9.94-627 9.94 620 9.94 614 9.94 607 9.94 600 5 4 3 2 1 60 9.67 161 9.72 567 0.27 433 9.94 593 L. Cos. L. Cotg. L. Tang. L. Sin. / [69] i I. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 9.67 161 9.67 185 9.67 208 9.67 232 9.67 256 9.72 567 9.72 598 9.72 628 9.72 659 9.72 689 0.27 433 0.27 402 0.27 372 0.27 341 0.27311 9.94 593 9.94 587 9.94 580 9.94 573 9.94 567 60 59 58 57 56 5 6 7 8 9 9.67 280 9.67 303 9.67 327 9.67 350 9.67 374 9.72 720 9.72 750 9.72 780 9.72 811 9.72 841 0.27 280 0.27 250 0.27 220 0.27 189 0.27 159 9.94 560 9.94 553 9.94 546 9.94 540 9.94 533 55 54 53 52 51 10 11 12 13 14 9.67 398 9.67 421 9.67 445 9.67 468 9.67 492 9.72 872 9.72 902 9.72 932 9.72 963 9 72 993 0.27 128 0.27 098 0.27 068 0.27 037 0.27 007 9.94 526 9.94 519 9.94 513 9.94 506 9.94 499 50 49 48 47 46 15 16 17 18 19 9.67 515 9.67 539 9.67 562 9.67 586 9.67 609 9.73 023 9.73 054 9.73 084 9.73 114 9.73 144 0.26 977 0.26 946 0.26 916 0.26 886 0.26 856 9.94 492 9.94 485 9.94 479 9.94 472 9.94 465 45 44 43 42 41 61 20 21 22 23 24 9.67 633 9.67 656 9.67 680 9.67 703 9.67 726 9.73 175 9.73 205 9.73 235 9.73 265 9.73 295 0.26 825 0.26 795 0.26 765 0.26 735 0.26 705 9.94 458 9.94 451 9.94 445 9.94 438 9.94 431 40 39 38 37 36 28 25 26 27 28 29 9.67 750 9.67 773 9.67 796 9.67 820 9.67 843 9.73 326 9.73 356 9.73 386 9.73 416 9.73 446 0.26 674 0.26 644 0.26 614 0.26 584 0.26 554 9.94 424 9.94 417 9.94 410 9.94 404 9.94 397 35 34 33 32 31 30 31 32 33 34 9.67 866 9.67 890 9.67 913 9.67 936 9.67 959 9.73 476 9.73 507 9.73 537 9.73 567 9.73 597 0.26 524 0.26 493 0.26 463 0.26 433 0.26 403 9.94 390 9.94 383 9.94 376 9.94 369 9.94 362 30 29 28 27 26 35 36 37 38 39 9.67 982 9.68 006 9.68 029 9.68 052 9.68 075 9.73 627 9.73 657 9.73 687 9.73 717 9.73 747 0.26 373 0.26 343 0.26 313 0.26 283 0.26 253 9.94 355 9.94 349 9.94 342 9.94 335 9.94 328 25 24 23 22 21 40 41 42 43 44 9.68 098 9.68 121 9.68 144 9.68 167 9.68 190 9.73 777 9.73 807 9.73 837 9.73 867 9.73 897 0.26 223 0.26 193 0.26 163 0.26 133 0.26 103 9.94 321 9.94 314 9.94 307 9.94 300 9.94 293 20 19 18 17 16 45 46 47 48 49 9.68 213 9.68 237 9.68 260 9.68 283 9.68 305 9.73 927 9.73 957 9.73 987 9.74 017 9.74 047 0.26 073 0.26 043 0.26 013 0.25 983 0.25 953 9.94 286 9.94 279 9.94 273 9.94 266 9.94 259 15 14 . 13 12 11 50 51 52 53 54 9.68 328 9.68 351 9.68 374 9.68 397 9.68 420 9.74 077 9.74 107 9.74 137 9.74 166 9.74 196 0.25 923 0.25 893 0.25 863 0.25 834 0.25 804 9.94 252 9.94 245 9.94 238 9.94 231 9.94 224 10 9 8 7 6 55 56 57 58 59 9.68 443 9.68 466 9.68 489 9.68 512 9.68 534 9.74 226 9.74 256 9.74 286 9.74 316 9.74 345 0.25 774 0.25 744 0.25 714 0.25 684 0.25 655 9.94 217 9.94 210 9.94 203 9.94 196 9.94 189 5 4 3 2 1 60 9.68 557 9.74 375 0.25 625 9.94 182 L. Cos. L. Cotg. L. Tang. L. Sin. / [70] 29 L. Sin. L. Tang. L. Cotg. L. Cos. r 60 i 2 3 4 9.68 557 9.68 580 9.68 603 9.68 625 9.68 648 9.74 375 9.74 405 9.74 435 9.74 465 9.74 494 0.25 625 0.25 595 0.25 565 0.25 535 0.25 506 9.94 182 9.94 175 9.94 168 9.94 161 9.94 154 60 59 58 57 56 5 6 7 8 9 9.68 671 9.68 694 9.68 716 9.68 739 9.68 762 9.74 524 9.74 554 9.74 583 9.74 613 9.74 643 0.25 476 0.25 446 0.25 417 0.25 387 0.25 357 9.94 147 9.94 140 9.94 133 9.94 126 9.94 119 55 54 53 52 51 10 11 12 13 14 9.68 784 9.68 807 9.68 829 9.68 852 9.68 875 9.74 673 9.74 702 9.74 732 9.74 762 9.74 791 0.25 327 0.25 298 0.25 268 0.25 238 0.25 209 9.94 112 9.94 105 9.94 098 9.94 090 9.94 083 50 49 48 47 46 15 16 17 18 19 9.68 897 9.68 920 9.68 942 9.68 965 9.68 987 9.74 821 9.74 851 9.74 880 9.74 910 9.74 939 0.25 179 0.25 149 0.25 120 0.25 090 0.25 061 9.94 076 9.94 069 9.94 062 9.94 055 9.94 048 45 44 43 42 41 20 21 22 23 24 9.69 010 9.69 032 9.69 055 9.69 077 9.69 100 9.74 969 9.74 998 9.75 028 9.75 058 9.75 087 0.25 031 0.25 002 0.24 972 0.24 942 0.24 913 9.94 041 9.94 034 9.94 027 9.94 020 9.94 012 40 39 38 37 36 25 26 27 28 29 9.69 122 9.69 144 9.69 167 9.69 189 9.69 212 9.75 117 9.75 146 9.75 176 9.75 205 9 75 235 0.24 883 0.24 854 0.24 824 0.24 795 0.24 765 9.94 005 9.93 998 9.93 991 9.93 984 9.93 977 35 34 33 32 31 30 31 32 33 34 9.69 234 9.69 256 9.69 279 9.69 301 9.69 323 9.75 264 9.75 294 9.75 323 9.75 353 9.75 382 0.24 736 0.24 706 0.24 677 0.24 647 0.24 618 9.93 970 . 9.93 963 9.93 955 9.93 948 9.93 941 30 29 28 27 26 35 36 . 37 38 39 9.69 345 9.69 368 9.69 390 9.69 412 9.69 434 9.75 411 9.75 441 9.75 470 9.75 500 9.75 529 0.24 589 0.24 559 0.24 530 0.24 500 0.24 471 9.93 934 9.93 927 9.93 920 9.93 912 9.93 905 25 24 23 22 21 f 42 : 43 -44 9.69 356 9.69 479 9.69 501 9.69 523 9.69 545 9.75 558 9.75 588 9.75 617 9.75 647 9.75 676 0.24 442 0.24 412 0.24 383 0.24 353 0.24 324 9.93 898 9.93 891 9.93 884 9.93 876 9.93 869 20 19 18 17 16 45 46 "47 48 49 9.69 567 9.69 589 9.69 611 9.69 633 9.69 655 9.75 705 9.75 735 9.75 764 9.75 793 9.75 822 0.24 295 0.24 265 0.24 236 0.24 207 0.24 178 9.93 862 9.93 855 9.93 847 9.93 840 9.93 833 15 14 13 12 11 50 51 52 53 54 9.69 677 9.69 699 9.69 721 9.69 743 9.69 765 9.75 852 9.75 881 9.75 910 9.75 939 9.75 969 0.24 148 0.24 119 0.24 090 0.24 061 0.24 031 9.93 826 9.93 819 9.93 811 9.93 804 9.93 797 10 9 8 7 6 55 56 57 58 59 9.69 787 9.69 809 9.69 831 9.69 853 9.69 875 9.75 998 9.76 027 9.76 056 9.76 086 9.76 115 0.24 002 0.23 973 0.23 944 0.23 914 0.23 885 9.93 789 9.93 782 9.93 775 9.93 768 9.93 760 5 4 3 2 1 60 9.69 897 9.76 144 0.23 856 9.93 753 L. Cos. L. Cotg. L. Tang. L. Sin. r [71] ' L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.69 897 9.69 919 9.69 941 9.69 963 9.69 984 9.76 144 9.76 173 9.76 202 9.76 231 9.76 261 0.23 856 0.23 827 0.23 798 0.23 769 0.23 739 9.93 753 9.93 746 9.93 738 9.93 731 9.93 724 60 1 59 58 1 5 6 7 8 9 9.70 006 9.70 028 9.70 050 9.70 072 9.70 093 9.76 290 9.76 319 9.76 348 9.76 377 9.76 406 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 9.93 717 9.93 709 9.93 702 9.93 695 9.93 687 55 54 53 52 51 10 11 12 13 14 9.70 115 9.70 137 9.70 159 9.70 180 9.70 202 9.76 435 9.76 464 9.76 493 9.76 522 9.76 551 0.23 565 0.23 536 0.23 507 0.23 478 0.23 449 9.93 680 9.93 673 9.93 665 9.93 658 9.93 650 50 49 48 47 46 15 16 17 18 19 9.70 224 9.70 245 9.70 267 9.70 288 9.70 310 9.76 580 9.76 609 9.76 639 9.76 668 9.76 697 0.23 420 0.23 391 0.23 361 0.23 332 0.23 303 9.93 643 9.93 636 9.93 628 9.93 621 9.93 614 45 44 43 42 41 20 21 22 23 24 9.70 332 9.70 353 9.70 375 9.70 396 9.70 418 9.76 725 9.76 754 9.76 783 9.76 812 9.76 841 0.23 275 0.23 246 0.23 217 0.23 188 0.23 159 9.93 606 9.93 599 9.93 591 9.93 584 9.93 577 40 39 38 37 36 OA 25 26 27 28 29 9.70 439 9.70 461 9.70 482 9.70 504 9.70 525 9.76 870 9.76 899 9.76 928 9.76 957 9.76 986 0.23 130 0.23 101 0.23 072 0.23 043 0.23 014 9.93 569 9.93 562 9.93 554 9.93 547 9.93 539 35 34 33 32 31 Ovf 30 31 32 33 34 9.70 547 9.70 568 9.70 590 9.70 611 9.70 633 9.77 015 9.77 044 9.77 073 9.77 101 9.77 130 0.22 985 22 956 0.22 927 0.22 899 0.22 870 9.93 532 9.93 525 9.93 517 9.93 510 9.93 502 29 28 27 26 35 36 37 38 39 9.70 654 9.70 675 9.70 697 9.70 718 9.70 739 9.77 159 9.77 188 9.77 217 9.77 246 9.77 274 0.22 841 0.22 812 0.22 783 0.22 754 0.22 726 9.93 495 9.93 487 9.93 480 9.93 472 9.93 465 25 . 24 23 22 21 40 41 42 43 44 9.70 761 9.70 782 9.70 803 9.70 824 9.70 846 9.77 303 9.77 332 9.77 361 9.77 390 9.77 418 0.22 697 0.22 668 ' 0.22 639 0.22 610 0.22 582 9.93 457 9.93 450 9.93 442 9.93 435 9.93 427 20^ inj 45 46 47 48 49 9.70 867 9.70 888 9.70 909 9.70 931 9.70 952 9.77 447 9.77 476 9.77 505 9.77 533 9.77 562 0.22 553 0.22 524 0.22 495 0.22 467 0.22 438 9.93 420 9.93 412 9.93 405 9.93 397 9.93 390 14 13 12 11 50 51 52 53 54 9.70 973 9.70 994 9.71 015 9.71 036 9.71 058 9.77 591 9.77 619 9.77 648 9.77 677 9.77 706 0.22 409 0.22 381 0.22 352 0.22 323 0.22 294 9.93 382 9.93 375 9.93 367 9.93 360 9.93 352 10 8 7 6 55 56 57 58 59 9.71 079 9.71 100 9.71 121 9.71 142 9 71 163 9.77 734 9.77 763 9.77 791 9.77 820 9.77 849 0.22 266 0.22 237 0.22 209 0.22 180 0.22 151 9.93 344 9.93 337 9.93 329 9.93 322 9.93 314 5 4 3 2 1 60 9.71-184 9.77. 877 0.22 123 9.93 307 L. Cos. L. Cotg. L. Tang. L. Sin. r [72] L. Sin. 9.71 184 9.71 205 9.71 226 9.71 247 ).71 268 9.71 289 9.71 310 9.71 331 9.71 352 9.71 373 9.71 393 9.71 414 9.71 435 9.71 456 9.71 477 9.72 421 L. Cos. L. Tang. 9.77 877 9.77 906 9.77 935 9.77 963 9.77 992 9.78 020 9.78 049 9.78 077 9.78 106 9.78 135 9.78 163 9.78 192 9.78 220 9.78 249 9.78 277 9.78 306 9.78 334 9.78 363 9.78 391 9.78 419 9.78 448 9.78 476 9.78 505 9.78 533 9.78 562 9.78 590 9.78 618 9.78 647 9.78 675 9.78 704 .71 498 9.71 519 9.71 539 9.71 560 9.71 581 9.71 602 9.71 622 9.71 643 9.71 664 9.71 685 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.71 809 .71 829 9.71 850 9.71 870 9.71 891 9.71 911 9.71 932 9.71 952 9.71 973 9.71 994 9.72 014 9.72 034 9.72 055 9.72 075 9.72 096 9.79 015 9.79 043 9.79 072 9.79 100 9.79 128 9.72 116 9.72 137 9.72 157 9.72 177 9.72 198 9.79 156 9.79 185 9.79 213 9.79 241 9.79 269 9.72 218 9.72 238 9.72 259 9.72 279 9.72 299 9.72 320 9.72 340 9.72 360 9.72 381 9.72 401 9.78 732 9.78 760 9.78 789 9.78 817- 9.78 845 ).78 874 9.78 902 9.78 930 9.78 959 9.78 987 9.79 297 9.79 326 9.79 354 9.79 382 9.79 410 9.79 438 9.79 466 9.79 495 9.79 523 9.79 551 9.79 579 L. Cotg. L. Cotg. 0.22 123 0.22 094 0.22 065 0.22 037 0.22 008 0.21 980 0.21 951 0.21 923 0.21 894 0.21 865 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 0.21 694 0.21 666 0.21 637 0.21 609 0.21 581 0.21 552 0.21 524 0.21 495 0.21 467 0.21 438 0.21 410 0.21 382 0.21 353 0.21 325 0.21 296 0.21 268 0.21 240 0.21 211 0.21 183 0.21 155 0.21 126 0.21 098 0.21 070 0.21 041 0.21 013 0.20 985 0.20 957 0.20 928 0.20 900 0.20 872 0.20 844 0.20 815 0.20 787 0.20 759 0.20 731 0.20 703 0.20 674 0.20 646 0.20 618 0.20 590 0.20 562 0.20 534 0.20 505 0.20 477 0.20 449 0.20 421 L. Tang. [73] L. Cos. 9.93 307 9.93 299 9.93 291 9.93 284 9.93 276 9.93 269 9.93 261 9.93 253 9.93 246 9.93 238 9.93 230 9.93 223 9.93 215 9.93 207 9.93 200 9.93 192 9.93 184 9.93 177 9.93 169 9.93 161 9.93 154 9.93 146 9.93 138 9.93 131 9.93 123 9.93 115 9.93 108 9.93 100 9.93 092 9.93 084 9.93 077 9.93 069 9.93061 9.93 053 9.93 046 9.93 038 9.93 030 9.93 022 9.93 014 9.93 007 9.92 999 9.92 991 9.92 983 9.92 976 9.92 968 9.92 960 9.92 952 9.92 944 9.92 936 9.92 929 9.92 921 9.92 913 9.92 905 9.92 897 9.92 889 9.92 881 9.92 874 9.92 866 9.92 858 9.92 850 9.92 842 L. Sin. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 58 ' L. Sin. L. Tang. L. Cotg. L. Cos. 57 i 2 3 4 9.72 421 9.72 441 9.72 461 9.72 482 9.72 502 9.79 579 9.79 607 9.79 635 9.79 663 9.79 691 0.20 421 0.20 393 0.20 365 0.20 337 0.20 309 9.92 842 9.92 834 9.92 826 9.92 818 9.92 810 60 59 58 57 56 5 6 7 8 9 9.72 522 9.72 542 9.72 562 9.72 582 9.72 602 9.79 719 9.79 747 9.79 776 9.79 804 9.79 832 0.20 281 0.20 253 0.20 224 0.20 196 0.20 168 9.92 803 9.92 795 9.92 787 9.92 779 9.92 771 55 54 53 52 51 10 11 12 13 14 9.72 622 9.72 643 9.72 663 9.72 683 9.72 703 9.79 860 9.79 888 9.79 916 9.79 944 9.79 972 0.20 140 0.20 112 0.20 084 0.20 056 0.20 028 9.92 763 9.92 755 9.92 747 9.92 739 9.92 731 50 49 48 47 46 15 16 17 18 19 9.72 723 9.72 743 9.72 763 9.72 783 9.72 803 9.80 000 9.80 028 9.80 056 9.80 084 9.80 112 0.20 000 0.19 972 0.19 944 0.19 916 ' 0.19 888 9.92 723 9.92 715 9.92 707 9.92 699 9.92 691 45 44 43 42 41 20 21 22 23 24 9.72 823 9.72 843 9.72 863 9.72 883 9.72 902 9.80 140 9.80 168 9.80 195 9.80 223 9.80 251 0.19 860 0.19 832 0.19 805 0.19 777 0.19 749 9.92 683 9.92 675 9.92 667 9.92 659 9.92 651 40 39 38 37 36 32 25 26 27 28 29 9.72 922 9.72 942 9.72 962 9.72 982 9.73 002 9.80 279 9.80 307 9.80 335 9.80 363 9.80 391 0.19 721 0.19 693 0.19 665 0.19 637 0.19 609 9.92 643 9.92 635 9.92 627 9.92 619 9.92 611 35 34 33 32 31 30 31 32 33 34 9.73 022 9.73 041 9.73 061 9.73 081 9.73 101 9.80 419 9.80 447 9.80 474 9.80 502 9.80 530 0.19 581 0.19 553 0.19 526 0.19 498 0.19 470 9.92 603 9.92 595 9.92 587-- 9.92 579 9.92 571 30 29 28 27 26 CO CO CO CO CO CO OO -< CO Ol V 9.73 121 9.73 140 9.73 160 9.73 180 9.73 200 9.80 558 9.80 586 9.80 614 9.80 642 9.80 669 0.19 442 0.19 414 0.19 386 0.19 358 0.19 331 9.92 563 9.92 555 9.92 546 9.92 538 9.92 530 25 24 23 22 21 40 41 42 43 44 9.73 219 9.73 239 9.73 259 9.73 278 9.73 298 9.80 697 9.80 725 9.80 753 9.80 781 9.80 808 0.19 303 0.19 275 0.19 247 0.19 219 0.19 192 9.92 522 9.92 514 9.92 506 9.92 498 9.92 490 20 19, 18 17 45 46 47 48 49 9.73 318 9.73 337 9.73 357 9.73 377 9.73 396 9.80 836 9.80 864 9.80 892 9.80 919 9.80 947 0.19 164 0.19 136 0.19 108 0.19 081 0.19 053 9.92 482 9.92 473 9.92 465 9.92 457 9.92 449 15 14 13 12 11 50 51 52 53 54 9.73 416 9.73 435 9.73 455 9.73 474 9.73 494 9.80 975 9.81 003 9.81 030 9.81 058 9.81 086 0.19 025 0.18 997 0.18 970 0.18 942 0.18 914 9.92 441 9.92 433 9.92 425 9.92 416 9.92 408 10 9 8 7 6 55 56 57 58 59 9.73 513 9.73 533 9.73 552 9.73 572 9.73 591 9.81 113 9.81 141 9.81 169 9.81 196 9.81 224 0.18 887 0.18 859 0.18 831" 0.18 804 0.18 776 9.92 400 9.92 392 9.92 384 9.92 376 9.92 367 5 4 3 2 1 60 9.73 611 9.81 252 0.18 748 9.92 359 L. Cos. L. Cotg. L. Tang. L. Sin. t i L. Siii. L. Tang. L. Cotg. L. Cos. i 2 3 4 9.73 611 9.73 630 9.73 650 9.73 669 9.73 689 9.81 252 9.81 279 9.81 307 9.81 335 9.81 362 0.18 748 0.18 721 0.18 693 0.18 665 0.18 638 9.92 359 9.92 351 9.92 343 9.92 335 9.92 326 2 58 57 56 5 6 7 8 9 9.73 708 9.73 727 9.73 747 9.73 766 9.73 785 9.81 390 9.81 418 9.81 445 9.81 473 9.81 500 0.18 610 0.18 582 0.18 555 0.18 527 0.18 500 9.92 318 9.92 310 9.92 302 9.92 293 9.92 285 55 54 53 52 51 10 11 12 13 14 9.73 805 9.73 824 9.73 843 9.73 863 9.73 882 9.81 528 9.81 556 9.81 583 9.81 611 9.81 638 0.18 472 0.18 444 0.18 417 0.18 389 0.18 362 9.92 277 9.92 269 9.92 260 9.92 252 9.92 244 50 l^tol 46* 15 16 17 18 19 9.73 901 9.73 921 9.73 940 9.73 959 9.73 978 9.81 666 9.81 693 9.81 721 9.81 748 9.81 776 0.18 334 0.18 307 0.18 279 0.18 252 0.18 224 9.92 235 9.92 227 9.92 219 9.92 211 9.92 202 45 A 44 43 42 41 '| 20 21 22 23 24 9.73 997 9.74 017 9.74 036 9.74 055 9.74 074 9.81 803 9.81 831 9.81 858 9.81 886 9.81 913 0.18 197 0.18 169 0.18 142 0.18 114 0.18 087 9.92 194 9.92 186 9.92 177 9.92 169 9.92 161 40 39 38 37 36 | 33 25 26 27 28 29 9.74 093 9.74 113 9.74 132 9.74 151 9.74 170 9.81 941 9.81 968 9.81 996 9.82 023 9.82 051 0.18 059 0.18 032 0.18 004 0.17 977 0.17 949 9.92 152 9.92 144 9.92 136 9.92 127 9.92 119 35 34 33 32 31 56 30 31 32 33 34 9.74 189 9.74 208 9.74 227 9.74 246 9.74 265 9.82 078 9.82 106 9.82 133 9.82 161 9.82 188 0.17 922 0.17 894 0.17 867 0.17 839 0.17 812 9.92 111 9.92 102 9.92 094 9.92 086 9.92 077 30 Y 29 28 27 26 | 35 36 37 38 39 9.74 284 9.74 303 9.74 322 9.74 341 9.74 360 9.82 215 9.82 243 9.82 270 9.82 298 9.82 325 0'.17 785 0.17 757 0.17 730 0.17 702 0.17 675 9.92 069 9.92 060 9.92 052 9.92 044 9.92 035 25 24 23 22 21 40 41- 42 43 44 9.74 379 9.74 398 9.74 417 9.74 436 9.74 455 9.82 352 9.82 380 9.82 407 9.82 435 9.82 462 0.17 648 0.17 620 0.17 593 0.17 565 0.17 538 9.92 027 9.92 018 9.92 010 9.92 002 9.91 993 20 19 18 17 16 | 45 46 47, 48 49 9.74 474 9.74 493 9.74 512 9.74 531 9.74 549 9.82 489 9.82 517 9.82 544 9.82 571 9.82 599 0.17 511 0.17 483 0.17 456 0.17 429 0.17401 9.91 985 9.91 976 9.91 968 9.91 959 9.91 951 15 14 13 12 11 50 51 52 53 54 9.74 568 9.74 587 9.74 606 9.74 625 9.74 644 9.82 626 9.82 653 9.82 681 9.82 708 9.82 735 0.17 374 0.17 347 0.17 319 0.17 292 0.17 265 9.91 942 9.91 934 9.91 925 9.91 917 9.91 908 10 8 I 55 56 57 58 59 9.74 662 9.74 681 9.74 700 9.74 719 9.74 737 9.82 762 9.82 790 9.82 817 9.82 844 9.82 871 0.17 238 0.17 210 0.17 183 0.17 156 0.17 129 9.91 900 9.91 891 9.91 883 9.91 874 9.91 866 3 5 ] 60 9.74 756 9.82 899 0.17 101 9.91 857 L. Cos. L. Cotg. L. Tang. L. Sin. ; [75] 31 / L. Sin. L. Tang. L. Cotg. L. Cos. 55 i 2 3 4 9.74 756 9.74 775 9.74 794 9.74 812 9.74 831 9.82 899 9.82 926 9.82 953 9.82 980 9.83 008 0.17 101 0.17 074 0.17 047 0.17 020 0.16 992 9.91 857 9.91 849 9.91 840 9.91 832 9.91 823 60 59 58 57 56 5 6 7 8 9 9.74 850 9.74 868 9.74 887 9.74 906 9.74 924 9.83 035 9.83 062 9.83 089 9.83 117 9.83 144 0.16 965 0.16 938 0.16 911 0.16 883 0.16 856 9.91 815 9.91 806 9.91 798 9.91 789 9.91 781 55 54 53 52 51 10 11 12 13 14 9.74 943 9.74 961 9.74 980 9.74 999 9.75 017 9.83 171 9.83 198 9.83 225 9.83 252 9.83 280 0.16 829 0.16 802 0.16 775 0.16 748 0.16 720 9.91 772 9.91 763 9.91 755 9.91 746 9.91 738 50 49 48 47 46 15 16 17 18 19 9.75 036 9.75 054 9.75 073 9.75 091 9.75 110 9.83 307 9.83 334 9.83 361 9.83 388 9.83 415 0.16 693 0.16 666 0.16 639 0.16 612 0.16 585 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 45 44 43 42 41 20 21 22 23 24 9.75 128 9.75 147 9.75 165 9.75 184 9.75 202 9.83 442 9.83 470 9.83 497 9.83 524 9.83 551 0.16 558 0.16 530 0.16 503 0.16 476 0.16 449 9.91 686 9.91 677 9.91 669 9.91 660 9.91 651 40 39 38 37 36 25 26 27 28 29 9.75 221 9.75 239 9.75 258 9.75 276 9.75 294 9.83 578 9.83 605 9.83 632 9.83 659 9.83 686 0.16 422 0.16 395 0.16 368 0.16 341 0.16 314 9.91 643 9.91 634 9.91 625 9.91 617 9.91 608 35 34 33 32 31 30 31 32 33 34 9.75 313 9.75 331 9.75 350 9.75 368 9.75 386 9.83 713 9.83 740 9.83 768 9.83 795 9.83 822 0.16 287 0.16 260 0.16 232 0.16 205 0.16 178 9.91 599 9.91 591 9.91 582 9.91 573 9.91 565 30 29 28 27 26 35 36 37 38 39 9.75 405 9.75 423 9.75 441 9.75 459 9.75 478 9.83 849 9.83 876 9.83 903 9.83 930 9.83 957 0.16 151 0.16 124 0.16 097 0.16 070 0.16 043 9.91 556 9.91 547 9.91 538 9.91 530 9.91 521 25 24 23 22 21 40 41 42 43 44 9.75 496 9.75 514 9.75 533 9.75 551 9.75 569 9.83 984 9.84 Oil 9.84 038 9.84 065 9.84 092 0.16 016 0.15 989 0.15 962 0.15 935 0.15 908 9.91 512 9.91 504 9.91 495 9.91 486 9.91 477 20 19 18 17 16 45 46 47 48 49 9.75 587 9.75 605 9.75 624 9.75 642 9.75 660 9.84 119 9.84 146 ' 9.84 173 9.84 200 9.84 227 0.15 881 0.15 854 0.15 827 0.15 800 0.15 773 9.91 469 9.91 460 9.91 451 9.91 442 9.91 433 15 14 13 12 11 50 51 52 53 54 9.75 678 9.75 696 9.75 714 9.75 733 9.75 751 9.84 254 9.84 280 9.84 307 9.84 334 9.84 361 0.15 746 0.15 720 0.15 693 0.15 666 0.15 639 9.91 425 9.91 416 9.91 407 9.91 398 9.91 389 10 9 8 7 6 55 56 57 58 59 9.75 769 9.75 787 9.75 805 9.75 823 9.75 841 9.84 388 9.84 415 9.84 442 9.84 469 9.84 496 0.15 612 0.15 585 0.15 558 0.15 531 0.15 504 9.91 381 9.91 372 9.91 363 9.91 354 9.91 345 5 4 3 2 1 60 9.75 859 9.84 523 0.15 477 9.91 336 L. Cos. L. Cotg. L. Tang. L. Sin. t [76] / L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.75 859 9.75 877 9.75 895 9.75 913 9.75 931 9.84 523 9.84 550 9.84 576 9.84 603 9.84 630 0.15 477 0.15 450 0.15 424 ' 0.15 397 0.15 370 9.91 336 9.91 328 9.91 319 9.91 310 9.91 301 60 59 58 57 56 5 6 7 8 9 9.75 949 9.75 967 9.75 985 9.76 003 9.76 021 9.84 657 9.84 684 9.84 711 9.84 738 9.84 764 0.15 343 0.15 316 0.15 289 0.15 262 0.15 236 9.91 292 9.91 283 9.91 274 9.91 266 9.91 257 55 54 53 52 51 54 10 11 12 13 14 9.76 039 9.76 057 9.76 075 9.76 093 9.76 111 9.84 791 9.84 818 9.84 845 9.84 872 9.84 899 0.15 209 0.15 182 0.15 155 0.15 128 0.15 101 9.91 248 9.91 239 9.91 230 9.91 221 9.91 212 50 49 48 47 46 15 16 17 18 19 9.76 129 9.76 146 9.76 164 9.76 182 9.76 200 9.84 925 9.84 952 9.84 979 9.85 006 9.85 033 0.15 075 0.15 048 0.15 021 0.14 994 0.14 967 9.91 203 9.91 194 9.91 185 9.91 176 9.91 167 45 44 43 42 41 35 20 21 22 23 24 9.76 218 9.76 236 9.76 253 9.76 271 9.76 289 9.85 059 9.85 086 9.85 113 9.85 140 9.85 166 0.14 941 0.14 914 0.14 887 0.14 860 0.14 834 9.91 158 9.91 149 9.91 141 9.91 132 9.91 123 40 39 38 37 36 25 26 27 28 29 9.76 307 9.76 324 9.76 342 9.76 360 9.76 378 9.85 193 9.85 220 9.85 247 9.85 273 9.85 300 0.14 807 0.14 780 0.14 753 0.14 727 0.14 700 9.91 114 9.91 105 9.91 096 9.91 087 9.91 078 35 34 33 32 31 30 31 32 33 34 9.76 395 9.76 413 9.76 431 9.76 448 9.76 466 9.85 327 9.85 354 9.85 380 9.85 407 9.85 434 0.14 673 0.14 646 0.14 620 0.14 593 0.14 566 9.91 069 9.91 060 9.91 051 9.91 042 9.91 033 30 29 28 27 26 35 36 37 38 39 9.76 484 9.76 501 9.76 519 9.76 537 9.76 554 9.85 460 9.85 487 9.85 514 9.85 540 9.85 567 0.14 540 0.14 513 0.14 486 0.14 460 0.14 433 9.91 023 9.91 014 9.91 005 9.90 996 9.90 987 25 24 23 22 21 40 41 42 43 44 9.76 572 9.76 590 9.76 607 9.76 625 9.76 642 9.85 594 9.85 620 9.85 647 9.85 674 9.85 700 0.14 406 0.14 380 0.14 353 0.14 326 0.14 300 9.90 978 9.90 969 9.90 960 9.90 951 9.90 942 20 19 18 17 16 45 46 47 48 49 9.76 660 9.76 677 9.76 695 9.76 712 9.76 730 9.85 727 9.85 754 9.85 780 9.85 807 9.85 834 0.14 273 0.14 246 0.14 220 0.14 193 0.14 166 9.90 933 9.90 924 9.90 915 9.90 906 9.90 896 15 14 13 12 11 50 51 52 53 54 9.76 747 9.76 765 9.76 782 9.76 800 9.76 817 9.85 860 9.85 887 9.85 913 9.85 940 9.85 967 0.14 140 0.14 113 0.14 087 0.14 060 0.14 033 9.90 887 9.90 878 9.90 869 9.90 860 9.90 851 10 9 8 7 6 55 56 57 58 59 9.76 835 9.76 852 9.76 870 9.76 887 9.76 904 9.85 993 9.86 020 9.86 046 9.86 073 9.86 100 0.14 007 0.13 980 0.13 954 0.13 927 0.13 900 9.90 842 9.90 832 9.90 823 9.90 814 9.90 805 5 4 3 2 1 60 9.76 922 9.86 126 0.13 874 9.90 796 L. Cos. L. Cotg. L. Tang. L. Sin. / [77] 1 L. Sin. L. Tang. L. Cotg. L. Cos. 53 1 2 3 4 9.76 922 9.76 939 9.76 957 9.76 974 9.76 991 9.86 126 9.86 153 9.86 179 9.86 206 9.86 232 0.13 874 0.13 847 0.13 821 0.13 794 0.13 768 9.90 796 9.90 787 9.90 777 9.90 768 9.90 759 60 59 58 57 56 5 6 7 8 9 9.77 009 9.77 026 9.77 043 9.77 061 9.77 078 9.86 259 9.86285 9.86 312 9.86 338 9.86 365 0.13 741 0.13715 0.13 688 0.13 662 0.13 635 9.90 750 9.90 741 9.90 731 9.90 722 9.90 713 55 54 53 52 51 10 11 12 13 14 9.77 095 9.77 112 9.77 130 9.77 147 9.77 164 9.86 392 9.86 418 9.86 445 9.86 471 9.86 498 0.13 608 0.13 582 0.13 555 0.13 529 0.13 502 9.90 704 9.90 694 9.90 685 9.90 676 9.90 667 50 49 48 47 46 15 16 17 18 19 9.77 181 9.77 199 9.77 216 9.77 233 9.77 250 9.86 524 9.86 551 9.86 577 9.86 603 9.86 630 0.13 476 0.13 449 0.13 423 0.13 397 0.13 370 9.90 657 9.90 648 9.90 639 9.90 630 9.90 620 45 44 43 42 41 20 21 22 23 24* 9.77 268 9.77 285 9.77 302 9.77 319 9.77 336 9.86 656 9.86 683 9.86 709 9.86 736 9.86 762 0.13 344 0.13 317 0.13 291 0.13 264 0.13 238 9.90 611 9.90 602 9.90 592 9.90 583 9.90 574 40 39 38 37 36 36 25 26 27 28 29 9.77 353 9.77 370 9.77 387 9.77 405 9.77 422 9.86 789 9.86 815 9.86 842 9.86 868 9.86 894 0.13 211 0.13 185 0.13 158 0.13 132 0.13 106 9.90 565 9.90 555 9.90 546 9.90 537 9.90 527 35 34 33 32 31 30 31 32 33 34 9.77 439 9.77 456- 9.77 473 9.77 490 9.77 507 9.86 921 9.86 947 9.86 974 9.87 000 9.87 027 0.13 079 0.13 053 0.13 026 0.13 000 0.12 973 9.90 518 9.90 509 9.90 499 9.90 490 9.90 480 30 29 28 27 26 35 36 37 38 39 9.77 524 9.77 541 9.77 558 9.77 575 9.77 592 9.87 053 9.87 079 9.87 106 9.87 132 9.87 158 0.12 947 0.12 921 0.12 894 0.12 868 0.12 842 9.90 471 9.90 462 9.90 452 9.90 443 9.90 434 25 24 23 22 21 40 41 42 43 44 9.77 609 9.77 626 9.77 643 9.77 660 9.77 677 9.87 185 9.87 211 9.87 238 9.87 264 9.87 290 0.12 815 0.12 789 0.12 762 0.12 736 0.12 710 9.90 424 9.90 415 9.90 405 9.90 396 9.90 386 20 19 18 17 16 45 46 47 48 49 9.77 694 9.77 711 9.77 728 9.77 744 9.77 761 9.87 317 9.87 343 9.87 369 9.87 396 9.87 422 0.12 683 0.12 657 0.12 631 0.12 604 0.12 578 9.90 377 9.90 368 9.90 358 9.90 349 9.90 339 15 14 13 12 11 50 51 52 53 54 9.77 778 9.77 795 9.77 812 9.77 829 9.77 846 9.87 448 9.87 475 9.87 501 9.87 527 9.87 554 0.12 552 0.12 525 0.12 499 0.12 473 0.12 446 9.90 330 9.90 320 9.90 311 9.90 301 9.90 292 10 9 8 7 6 55 56 57 58 59 9.77 862 9.77 879 9.77 896 9.77 913 9.77 930 9.87 580 9.87 606 9.87 633 9.87 659 9.87 685 0.12 420 0.12 394 0.12 367 0.12 341 0.12 315 9.90 282 9.90 273 9.90 263 9.90 254 9.90 244 5 4 3 2 1 60 977 946 9.87 711 0.12 289 9.90 235 L. Cos. L. Cotg. L. Tang. L. Sin. r [78] 37 / L. Sin. L. Tang. L. Cotg. L. Cos. 52 1 2 3 4 9.77 946 9.77 963 9.77 980 9.77 997 9.78 013 9.87 711 9.87 738 9.87 764 9.87 790 9.87 817 0.12 289 0.12 262 0.12 236 0.12 210 0.12 183 9.90 235 9.90 225 9.90 216 9.90 206 9.90 197 60 59 58 57 56 5 6 7 8 9 9.78 030 9.78 047 9.78 063 9.78 080 9.78 097 9.87 843 9.87 869 9.87 895 9.87 922 9.87 948 0.12 157 0.12 131 0.12 105 0.12 078 0.12 052 9.90 187 9.90 178 9.90 168 9.90 159 9.90 149 55 54 53 52 51 10 11 12 13 14 9.78 113 9.78 130 9.78 147 9.78 163 9.78 180 9.87 974 9.88 000 9.88 027 9.88 053 9.88 079 0.12 026 0.12 000 0.11 973 0.11 947 0.11 921 9.90 139 9.90 130 9.90 120 9.90 111 9.90 101 50 49 48 47 46 15 16 17 18 19 9.78 197 9.78 213 9.78 230 9.78 246 9.78 263 9.88 105 9.88 131 9.88 158 9.88 184 9.88 210 0.11 895 0.11 869 0.11 842 0.11 816 0.11 790 9.90 091 9.90 082 9.90 072 9.90 063 9.90 053 45 44 43 42 41 20 21 22 23 24 9.78 280 9.78 296 9.78 313 9.78 329 9.78 346 9.88 236 9.88 262 9.88 289 9.88 315 9.88 341 0.11 764 0.11 738 0.11 711 0.11 685 0.11 659 9.90 043 9.90 034 9.90 024 9.90 014 9.90 005 40 39 38 37 36 25 26 27 28 29 9.78 362 9.78 379 9.78 395 9.78 412 9.78 428 9.88 367 9.88 393 9.88 420 9.88 446 9.88 472 0.11 633 0.11 607 0.11 580 0.11 554 0.11 528 9.89 995 9.89 985 9.89 976 9.89 966 9.89 956 35 34 33 32 31 30 31 32 33 34 9.78 445 9.78 461 9.78 478 9.78 494 9.78 510 9.88 498 9.88 524 9.88 550 9.88 577 9.88 603 0.11 502 0.11 476 0.11 450 0.11 423 0.11 397 9.89 947 9.89 937 9.89 927 9.89 918 9.89 908 30 29 28 27 26 35 36 37 38 39 9.78 527 9.78 543 9.78 560 9.78 576 9.78 592 9.88 629 9.88 655 9.88 681 9.88 707 9.88 733 0.11 371 0.11 345 0.11 319 0.11-293 0.11 267 9.89 898 9.89 888 9.89 879 9.89 869 9.89 859 25 24 23 22 21 40 41 42 43 44 9.78 609 9.78 625 9.78 642 9.78 658 9.78 674 9.88 759 9.88 786 9.88 812 9.88 838 9.88 864 0.11 241 0.11 214 0.11 188 0.11 162 ' 0.11 136 9.89 849 9.89 840 9.89 830 9.89 820 9.89 810 20 19 18 17 16 45 46 47 48 49 9.78 691 9.78 707 9.78 723 9.78 739 9.78 756 9.88 890 9.88 916 9.88 942 9.88 968 9.88 994 0.11 110 0.11 084 0.11 058 0.11 032 0.11 006 9.89 801 9.89 791 9.89 781 9.89 771 9.89 761 15 14 13 12 11 50 51 52 53 54 9.78 772 9.78 788 9.78 805 9.78 821 9.78 837 9.89 020 9.89 046 9.89 073 9.89 099 9.89 125 0.10 980 0.10 954 0.10 927 0.10 901 0.10 875 9.89 752 9.89 742 9.89 732 9.89 722 9.89 712 10 9 8 7 6 55 56 57 58 59 9.78 853 9.78 869 9.78 886 9.78 902 9.78 918 9.89 151 9.89 177 9.89 203 9.89 229 9.89 255 0.10 849 0.10 823 0.10 797 0.10 771 0.10 745 9.89 702 9.89 693 9.89 683 9.89 673 9.89 663 5 4 3 2 1 60 9.78 934 9.89 281 0.10 719 9.89 653 L. Cos. L. Cotg. L. Tang. L. Sin. [79] i L. Sin. L. Tang. L. Cotg. L* Cos* fJ1 i 2 3 4 9.78 934 9.78 950 9.78 967 9.78 983 9.78 999 9.89 281 9.89 307 9.89 333 9.89 359 9.89 385 0.10 719 0.10 693 0.10 667 0.10 641 0.10 615 9.89 653 9.89 643 9.89 633 9.89 624 9.89 614 60 59 58 57 56 5 6 7 8 9 9.79 015 9.79 031 9.79 047 9.79 063 9.79 079 9.89 411 9.89 437 9.89 463 9.89 489 9.89 515 ' 0.10 589 0.10 563 0.10 537 0.10 511 0.10 485 9.89 604 9.89 594 9.89 584 9.89 574 9.89 564 55 54 53 52 51 10 11 12 13 14 9.79 095 9.79 111 9.79 128 9.79 144 9.79 160 9.89 541 9.89 567 9.89 593 9.89 619 9.89 645 0.10 459 0.10 433 0.10 407 0.10 381 0.10 355 ' 9.89 554 9.89 544 9.89 534 9.89 524 9.89 514 50 49 48 47 46 15 16 17 18 19 9.79 176 9.79 192 9.79 208 9.79 224 9.79 240 9.89 671 9.89 697 9.89 723 9.89 749 9.89 775 0.10 329 0.10 303 0.10 277 0.10 251 0.10 225 9.89 504 9.89 495 9.89 485 9.89 475 9.89 465 45 44 43 42 41 20 21 22 23 24 9.79 256 9.79 272 9.79 288 9.79 304 9.79 319 9.89 801 9.89 827 9.89 853 9.89 879 9.89 905 0.10 199 0.10 173 0.10 147 0.10 121 0.10 095 9.89 455 9.89 445 9.89 435 9.89 425 9.89 415 40 39 38 37 36 38 25 26 27 28 29 9.79 335 9.79 351 9.79 367 9.79 383 9.79 399 9.89 931 9.89 957 9.89 983 9.90 009 9.90 035 0.10 069 0.10 043 0.10 017 0.09 991 0.09 965 9.89 405 9.89 395 9.89 385 9.89 375 9.89 364 35 34 33 32 31 30 31 32 33 34 9.79 415 9.79 431 9.79 447 9.79 463 9.79 478 9.90 061 9.90 086 9.90 112 9.90 138 9.90 164 0.09 939 0.09 914 0.09 888 0.09 862 0.09 836 9.89 354 9.89 344 9.89 334 9.89 324 9.89 314 30 29 28 27 26 tf J. 35 36 37 38 39 9.79 494 9.79 510 9.79 526 9.79 542 9.79 558 9.90 190 9.90 216 9.90 242 9.90 268 9.90 294 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 9.89 304 9.89 294 9.89 284 9.89 274 9.89 264 25 24 23 22 21 40 41 42 43 44 9.79 573 9.79 589 9.79 605 9.79 621 9.79 636 9.90 320 9.90 346 9.90 371 9.90 397 9.90 423 0.09 680 0.09 654 0.09 629 0.09 603 0.09 577 9.89 254 9.89 244 9.89 233 9.89 223 9.89 213 20 19 18 17 16 45 46 47 48 49 9.79 652 9.79 668 9.79 684 9.79 699 9.79 715 9.90 449 9.90 475 9.90 501 9.90 527 9.90 553 0.09 551 0.09 525 0.09 499 0.09 473 0.09447 9.89 203 9.89 193 9.89 183 9.89 173 9.89 162 15 14 13 12 11 60 51 52 53 54 9.79 731 9.79 746 9.79 762 9.79 778 9.79 793 9.90 578 9.90 604 9.90 630 9.90 656 9.90 682 0.09 422 0.09 396 0.09 370 0.09 344 0.09 318 9.89 152 9.89 142 9.89 132 9.89 122 9.89 112 10 9 8 7 6 55 56 57 58 59 9.79 809 9.79 825 9.79 840 9.79 856 9.79 872 9.90 708 9.90 734 9.90 759 9.90 785 9.90 811 0.09 292 0.09 266 0.09 241 0.09 215 0.09 189 9.89 101 9.89 091 9.89 081 9.89 071 9.89 060 5 4 3 2 1 60 9.79 887 9.90 837 0.09 163 9.89 050 L. Cos. L. Cotg. L. Tang. L. Sin. F [80] / L. Sin. L. Tang. L. Cotg. L. Cos. 50 i 2 3 4 9.79 887 9.79 903 9.79 918 9.79 934 9.70 950 9.90 837 9.90 863 9.90 889 9.90 914 9.90 940 0.09 163 0.09 137 0.09 111 0.09 086 0.09 060 9.89 050 9.89 040 9.89 030 9.89 020 9.89 009 60 59 58 57 56 5 6 7 ' 8 9 9.79 965 9.79 981 9.79 996 9.80 012 9.80 027 9.90 966 9.90 992 9.91 018 9.91 043 9.91 069 0.09 034 0.09 008 0.08 982 0.08 957 0.08 931 9.88 999 9.88 989 9.88 978 9.88 968 9.88 958 55 54 53 52 51 . 10 11 12 13 14 9.80 043 9.80 058 9.80 074 9.80 089 9.80 105 9.91 095 9.91 121 9.91 147 9.91 172 9.91 198 0.08 905 0.08 879 0.08 853 0.08 828 0.08 802 9.88 948 9.88 937 9.88 927 9.88 917 9.88 906 50 49 48 47 46 15 16 17 18 19 9.80 120 9.80 136 9.80 151 9.80 166 9.80 182 9.91 224 9.91 250 9.91 276 9.91 301 9.91 327 0.08 776 0.08 750 0.08 724 0.08 699 0.08 673 9.88 896 9.88 886 9.88 875 9.88 865 9.88 855 45 44 43 42 41 39 20 21 22 23 24 9.80 197 9.80 213 9.80 228 9.80 244 9.80 259 9.91 353 9.91 379 9.91 404 9.91 430 9.91 456 0.08 647 0.08 621 0.08 596 0.08 570 0.08 544 9.88 844 9.88 834 9.88 824 9.88 813 9.88 803 40 39 38 37 36 25 26 27 28 29 9.80 274 9.80 290 9.80 305 9.80 320 9.80 336 9.91 482 9.91 507 9.91 533 9.91 559 9.91 585 0.08 518 0.08 493 0.08 467 0.08 441 0.08 415 9.88 793 9.88 782 9.88 772 9.88 761 9.88 751 35 34 33 32 31 30 31 32 33 34 9.80 351 9.80 366 9.80 382 9.80 397 9.80 412 9.91 610 9.91 636 9.91 662 9.91 688 9.91 713 0.08 390 0.08 364 0.08 338 0.08 312 0.08 287 9.88 741 9.88 730 9.88 720 9.88 709 9.88 699 30 29 28 27 26 35 36 37 38 39 9.80 428 9.80 443 9.80 458 9.80 473 9.80 489 9.91 739 9.91 765 9.91 791 9.91 816 9.91 842 0.08 261 0.08 235 0.08 209 0.08 184 0.08 158 9.88 688 9.88 678 9.88 668 9.88 657 9.88 647 25 24 23 22 21 40 41 42 43 44 9.80 504 9.80 519 9.80 534 9.80 550 9.80 565 9.91 868 9.91 893 9.91 919 9.91 945 9.91 971 0.08 132 0.08 107 0.08 081 0.08 055 0.08 029 9.88 636 9.88 626 9.88 615 9.88 605 9.88 594 20 19 18 17 16 45 46 47 48 49 9.80 580 9.80 595 9.80 610 9.80 625 9.80 641 9.91 996 9.92 022 9.92 048 9.92 073 9.92 099 0.08 004 0.07 978 0.07 952 0.07 927 0.07 901 9.88 584 9.88 573 9.88 563 9.88 552 9.88 542 15 14 13 12 11 50 51 52 53 54 9.80 656 9.80 671 9.80 686 9.80 701 9.80 716 9.92 125 9.92 150 9.92 176 9.92 202 9.92 227 0.07 875 0.07 850 0.07 824 0.07 798 0.07 773 9.88 531 9.88 521 9.88 510 9.88 499 9.88 489 10 9 8 7 6 55 56 57 58 59 9.80 731 9.80 746 9.80 762 9.80 777 9.80 792 9.92 253 9.92 279 9.92 304 9.92 330 9.92 356 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 9.88 478 9.88 468 9.88 457 9.88 447 9.88 436 5 4 3 2 1 60 9.80 807 9.92 381 0.07 619 9.88 425 L. Cos. L. Cotg. L. Tang. L. Sin. i [81] i L. Sin. L. Tang. L. Cotg. L. Cos. i 2 3 4 9.80 807 9.80 822 9.80 837 9.80 852 9.80 867 9.92 381 9.92 407 9.92 433 9.92 458 9.92 484 0.07 619 0.07 593 0.07 567 0.07 542 0.07 516 9.88 425 9.88 415 9.88 404 9.88 394 9.88 383 60 59 58 57 56 5 6 7 8 9 9.80 882 9.80 897 9.80 912 9.80 927 9.80 942 9.92 510 9.92 535 9.92 561 9.92 587 9.92 612 0.07 490 0.07 465 0.07 439 0.07 413 0.07 388 9.88 372 9.88 362 9.88 351 9.88 340 9.88 330 55 54 53 52 51 10 11 12 13 14 9.80 957 9.80 972 9.80 987 9.81 002 9.81 017 9.92 638 9.92 663 9.92 689 9.92 715 9.92 740 0.07 362 0.07 337 0.07 311 0.07 285 0.07 260 9.88 319 9.88 308 9.88 298 9.88 287 9.88 276 50 49 48 47 46 15 16 17 18 19 9.81 032 9.81 047 9.81 061 9.81 076 9.81 091 9.92 766 9.92 792 9.92 817 9.92 843 9.92 868 0.07 234 0.07 208 0.07 183 0.07 157 0.07 132 9.88 266 9.88 255 9.88 244 9.88 234 9.88 223 45 44 43 42 41 20 21 22 23 24 9.81 106 9.81 121 9.81 136 9.81 151 9.81 166 9.92 894 9.92 920 9.92 945 9.92 971 9.92 996 0.07 106 0.07 080 0.07 055 0.07 029 0.07 004 9.88 212 9.88 201 9.88 191 9.88 180 9.88 169 40 39 38 37 36 40 25 26 27 28 29 9.81 180 9.81 195 9.81 210 9.81 225 9.81 240 9.93 022 9.93 048 9.93 073 9.93 099 9.93 124 0.06 978 0.06 952 0.06 927 0.06 901 0.06 876 9.88 158 9.88 148 9.88 137 9.88 126 9.88 115 35 34 33 32 31 49 30 31 32 33 34 9.81 254 9.81 269 9.81 284 9.81 299 9.81 314 9.93 150 9.93 175 9.93 201 9.93 227 9.93 252 0.06 850 0.06 825 0.06 799 0.06 773 0.06 748 9.88 105 9.88 094 9.88 083 9.88 072 9.88 061 30 29 28 27 26 35 36 37 38 39 9.81 328 9.81 343 9.81 358 9.81 372 9.81 387 9.93 278 9.93 303 9.93 329 9.93 354 9.93 380 0.06 722 0.06 697 0.06 671 0.06 646 0.06 620 9.88 051 9.88 040 9.88 029 9.88 018 9.88 007 25 24 23 22 21 40 41 42 43 44 9.81 402 9.81 417 9.81 431 9.81 446 9.81 461 9.93 406 9.93 431 9.93 457 9.93 482 9.93 508 0.06 594 0.06 569 0.06 543 0.06 518 0.06 492 9.87 996 9.87 985 9.87 975 9.87 964 9.87 953 20 19 18 17 16 45 46 47 48 49 9.81 475 9.81 490 9.81 505 9.81 519 9.81 534 9.93 533 9.93 559 9.93 584 9.93 610 9.93 636 0.06 467 0.06 441 06 416 0.06 390 0.06 364 9.87 942 9.87 931 9.87 920 9.87 909 9.87 898 15 14 13 12 11 50. 51 52 53 54 9.81 549 9.81 563 9.81 578 9.81 592 9.81 607 9.93 661 9.93 687 9.93 712 9.93 738 9.93 763 0.06 339 0.06 313 0.06 288 0.06 262 0.06 237 9.87 887 9.87 877 9.87 866 9.87 855 9.87 844 10 9 8 7 6 55 56 57 58 59 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 9.93 789 9.93 814 9.93 840 9.93 865 9.93 891 0.06 211 0.06 186 0.06 160 0.06 135 0.06 109 9.87-833 9.87 822 9.87 811 9.87 800 9.87 789 5 4 3 2 1 60 9.81 694 9.93 916 0.06 084 9 87 778 L. Cos. L. Cotg. L. Tang. L. Sin. / [82] I L. Sin. L. Tang. L. Cotg. L. Cos. 48 1 2 3 4 9.81 694 9.81 709 9.81 723 9.81 738 9.81 752 9.93 916 9.93 942 9.93 967 9.93 993 9.94 018 0.06 084 0.06 058 0.06 033 0.06 007 0.05 982 9.87 778 9.87 767 9.87 756 9.87 745 9.87 734 60 59 58 57 56 5 6 7 8 9 9.81 767 9.81 781 9.81 796 9.81 810 9.81 825 9.94 044 9.94 069 9.94 095 9.94 120 9.94 146 0.05 956 0.05 931 0.05 905 0.05 880 0.05 854 9.87 723 9.87 712 9.87 701 9.87 630 9.87 679 55 54 53 52 51 10 11 12 13 14 9.81 839 9.81 854 9.81 868 9.81 882 9.81 897 9.94 171 9.94 197 9.94 222 9.94 248 9.94 273 0.05 829 0.05 803 0.05 778 0.05 752 0.05 727 9.87 668 9.87 657 9.87 646 9.87 635 9.87 624 50 49 48 47 46 15 16 17 18 19 9.81 911 9.81 926 9.81 940 9.81 955 9.81 969 9.94 299 9.94 324 9.94 350 9.94 375 9.94 401 0.05 701 0.05 676 0.05 650 0.05 625 0.05 599 9.87 613 9.87 601 9.87 590 9.87 579 9.87 568 45 44 43 42 41 20 21 22 23 24 9.81 983 9.81 998 9.82 012 9.82 026 9.82 041 9.94 426 9.94 452 9.94 477 9.94 503 9.94 528 0.05 574 0.05 548 0.05 523 0.05 497 0.05 472 9.87 557 9.87 546 9.87 535 9.87 524 9.87 513 40 39 38 37 36 41 25 26 27 28 29 9.82 055 9.82 069 9.82 084 9.82 098 9.82 112 9.94 554 9.94 579 9.94 604 9.94 630 9.94 655 0.05 446 0.05 421 0.05 396 0.05 370 0.05 345 9.87 501 9.87 490 9.87 479 9.87 468 9.87 457 35 34 33 32 31 30 31 32 33 34 9.82 126 9.82 141 9.82 155 9.82 169 9.82 184 9.94 681 9.94 706 9.94 732 9.94 757 9.94 783 0.05 319 0.05 294 0.05 268 0.05 243 0.05 217 9.87 446 9.87 434 9.87 423 9.87 412 ' 9.87401 30 29 28 27 26 35 36 37 38 39 9.82 198 9.82 212 9.82 226 9.82 240 9.82 255 9.94 808 9.94 834 9.94 859 9.94 884 9.94 910 0.05 192 0.05 166 O.*05 141 0.05 116 0.05 090 9.87 390 9.87 378 9.87 367 9.87 356 9.87 345 25 24 23 22 21 40 41 42 43 44 9.82 269 9.82 283 9.82 297 9.82 311 9.82 326 9.94 935 9.94 961 9.94 986 9.95 012 9.95 037 0.05 065 0.05 039 0.05 014 0.04 988 0.04 963 9.87 334 9.87 322 9.87 311 9.87 300 9.87 288 20 19 18 17 16 45 46 47 48 49 9.82 340 9.82 354 9.82 368 9.82 382 9.82 396 9.95 062 9.95 088 9.95 113 9.95 139 9.95 164 0.04 938 0.04 912 0.04 887 0.04 861 0.04 836 9.87 277 9.87 266 9.87 255 9.87 243 9.87 232 15 14 13 12 11 50 51 52 53 54 9.82 410 .82 424 9.82 439 9.82 453 9.82 467 9.95 190 9.95 215 9.95 240 9.95 266 9.95 291 0.04 810 0.04 785 0.04 760 0.04 734 0.04 709 9.87 221 9.87 209 9.87 198 9.87 187 9.87 175 10 9 8 7 6 55 56 57 58 59 9.82 481 9.82 495 9.82 509 9.82 523 9.82 537 9.95 317 9.95 342 9.95 368 9.95 393 9.95 418 0.04 683 0.04 658 0.04 632 0.04 607 0.04 582 9.87 164 9.87 153 9.87 141 9.87 130 9.87 119 5 4 3 2 1 60 9.82 551 9.95 444 0.04 556 9.87 107 L. Cos. L. Cotg. L. Tang. L. Sin. / [83] r L. Sin. L. Tang. L. Cotg. L. Cos. 1 2 3 4 9.82 551 9.82 565 9.82 579 9.82 593 9.82 607 9.95 444 9.95 469 9.95 495 9.95 520 9.95 545 0.04 556 0.04 531 0.04 505 0.04 480 0.04 455 9.87 107 9.87 096 9.87 085 9.87 073 9.87 062 60 59 58 57 56 5 6 7 8 9 9.82 621 9.82 635 9.82 649 9.82 663 9.82 677 9.95 571 9.95 596 9.95 622 9.95 647 9.95 672 0.04 429 0.04 404 0.04 378 0.04 353 0.04 328 9.87 050 9.87 039 9.87 028 9.87 016 9.87 005 55 54 53 52 51 10 11 12 13 14 9.82 691 9.82 705 9.82 719 9.82 733 9.82 747 9.95 698 9.95 723 9.95 748 9.95 774 9.95 799 0.04 302 0.04 277 0.04 252 0.04 226 0.04 201 9.86 993 9.86 982 9.86 970 9.86 959 9.86 947 50 49 48 47 46 15 16 17 18 19 9.82 761 9.82 775 9.82 788 9.82 802 9.82 816 9.95 825 9.95 850 9.95 875 9.95 901 9.95 926 0.04 175 0.04 150 0.04 125 0.04 099 0.04 074 9.86 936 9.86 924 9.86 913 9.86 902 9.86 890 45 44 43 42 41 20 21 22 23 24 9.82 830 9.82 844 9.82 858 9.82 872 9.82 885 9.95 952 9.95 977 9.96 002 9.96 028 9.96 053 0.04 048 0.04 023 0.03 998 0.03 972 0.03 947 9.86 879 9.86 867 9.86 855 9.86 844 9.86 832 40 39 38 37 36 42 C 25 26 27 28 29 9.82 899 9.82 913 9.82 927 9.82 941 9.82 955 9.96 078 9.96 104 9.96 129 9.96 155 9.96 180 0.03 922 0.03 896 0.03 871 0.03 845 0.03 820 9.86 821 9.86 809 9.86 798 9.86 786 9.86 775 35 34 33 32 31 47 30 31 32 33 34 9.82 968 9.82 982 9.82 996 9.83 010 9.83 023 9.96 205 9.96 231 9.96 256 9.96 281 9.96 307 0.03 795 0.03 769 0.03 744 0.03 719 0.03 693 9.86 763 9.86 752 9.86 740 9.86 728 9.86 717 30 29 28 27 26 35 36 37 38 39 9.83 037 9.83 051 9.83 065 9.83 078 9.83 092 9.96 332 9.96 357 9.96 383 9.96 408 9.96 433 0.03 668 0.03 643 0.03 617 0.03 592 0.03 567 9.86 705 9.86 694 9.86 682 9.86 670 9.86 659 25 24 23 22 21 40 41 42 43 44 9.83 106 9.83 120 9.83 133 9.83 147 9.83 161 9.96 459 9.96 484 9.96 510 9.96 535 9.96 560 0.03 541 0.03 516 0.03 490 .0.03 465 0.03 440 9.86 647 9.86 635 9.86 624 9.86 612 9.86 600 20 19 18 17 16 45 46 47 48 49 9.83 174 9.83 188 9.83 202 9.83 215 9.83 229 9.96 586 9.96 611 9.96 636 9.96 662 9.96 687 0.03 414 0.03 389 0.03 364 0.03 338 0.03 313 9.86 589 9.86 577 9.86 565 9.86 554 9.86 542 15 14 13 12 11 50 51 52 53 54 9.83 242 9.83 256 9.83 270 9.83 283 9.83 297 9.96 712 9.96 738 9.96 763 9.96 788 9.96 814 0.03 288 0.03 262 0.03 237 0.03 212 0.03 186 9.86 530 9.86 518 9.86 507 9.86 495 9.86 483 10 9 8 7 6 55 56 57 58 59 9.83 310 9.83 324 9.83 338 9.83 351 9.83 365 9.96 839 9.96 864 9.96 890 9.96 915 9.96 940 0.03 161 0.03 136 0.03 110 0.03 085 0.03 060 9-86472 9.86 460 9.86 448 9.86 436 9.86 425 5 4 3 2 1 60 9.83 378 9.96 966 0.03 034 9.86 413 L. Cos. L. Cotg. L. Tang. L. Sin. / t L. Sin. L. Tang. L. Cotg. L. Cos. 46 i 2 3 4 9.83 378 9.83 392 9.83 405 9.83 419 9.83 432 9.96 966 9.96 991 9.97 016 9.97 042 9.97 067 0.03 034 0.03 009 0.02 984 0.02 958 0.02 933 9.86 413 9.86 401 9.86 389 9.86 377 9.86 366 60 59 58 57 56 5 6 7 8 9 9.83 446 9.83 459 9.83 473 9.83 486 9.83 500 9.97 092 9.97 118 9.97 143 9.97 168 9.97 193 0.02 908 0.02 882 0.02 857 0.02 832 0.02 807 9.86 354 9.86 342 9.86 330 9.86 318 9.86 306 55 54 53 52 51 10 11 12 13 14 9.83 513 9.83 527 9.83 540 9.83 554 9.83 567 9.97 219 9.97 244 9.97 269 9.97 295 9.97 320 '0.02 781 0.02 756 0.02 731 0.02 705 0.02 680 9.86 295 9.86 283 9.86 271 9.86 259 9.86 247 50 49 48 47 46 15 16 17 18 19 9.83 581 9.83 594 9.83 608 9.83 621 9.83 634 9.97 345 9.97 371 9.97 396 9.97 421 9.97 447 0.02 655 0.02 629 0.02 604 0.02 579 0.02 553 9.86 235 9.86 223 9.86 211 9.86 200 9.86 188 45 44 43 42 41 43 20 21 22 23 24 9.83 648 9.83 661 9.83 674 9.83 688 9.83 701 9.97 472 9.97 497 9.97 523 9.97 548 9.97 573 0.02 528 0.02 503 0.02 477 0.02 452 0.02 427 9.86 176 9.36 164 . 9.86 152 9.86 140 9.86 128 40 39 38 37 36 25 26 27 28 29 9.83 715 9.83 728 9.83 741 9.83 755 9.83 768 9.97 598 9.97 624 9.97 649 9.97 674 9.97 700 0.02 402 0.02 376 0.02 351 0.02 326 0.02 300 9.86 116 9.8G 104 9.83 092 9.86 080 9.86 068 35 34 33 32 31 30 31 32 33 34 9.83 781 9.83 595 9.83 808 9.83 821 9.83 834 9.97 725 9.97 750 9.97 776 9.97 801 9.97 826 0.02 275 0.02 250 0.02 224 0.02 199 0.02 174 9.8G 056 9.8G 044 9.86 032 9.86 020 9.86 008 30 29 28 27 26 35 36 37 38 39 9.83 848 9.83 861 9.83 874 9.83 887 9.83 901 9.97 851 97 877 9.97 902 9.97 927 9.97 953 0.02 149 0.02 123 0.02 098 0.02 073 0.02 047 9.85 996 9.85 984 9.85 972 9.85 960 9.85 948 25 24 23 22 21 40 41 42 43 44 9.83 914 9.83 927 9.83 940 9.83 954 9.83 967 9.97 978 ' 9.98 003 9.98 029 9.98 054 9.98 079 0.02 022 0.01 997 0.01 971 0.01 946 0.01 921 9.85 936 9.85 924 9.85 912 9.85 900 9.85 888 20 19 18 17 16 45 46 47 48 49 9.83 980 9.83 993 9.84 006 9.84 020 9.84 033 9.98 104 9.98 130 9.98 155 9.98 180 9.98 206 0.01 896 0.01 870 0.01 845 0.01 820 0.01 794 9.85 876 9.85 864 9.85 851 9.85 839 9.85 827 15 14 13 12 11 50 51 52 53 54 9.84- 046 9.84 059 9.84 072 9.84 085 9.84 098 9.98 231 9.98 256 9.98 281 9.98 307 9.98 332 0.01 769 0.01 744 0.01 719 0.01 693 0.01 668 9.85 815 9.85 803 9.85 791 9.85 779 9.85 766 10 9 8 7 6 55 56 57 58 59 9.84 112 9.84 125 9.84 138 9.84 151 9.84 164 9.98 357 9.98 383 9.98 408 9.98 433 9.98 458 0.01 643 0.00 617 0.01 592 0.01 567 0.01 542 9.85 754 9.85 742 9.85 730 9.85 718 9.85 706 5 4 3 2 1 60 9.84 177 9.98 484 0.01 516 9.85 693 L. Cos. L. Cotg. L. Tang. L. Sin. f [85] / L. Sin. L. Tang. L. Cotg. L. Cos. 45 i 2 3 4 9.84 177 9.84 190 9.84 203 9.84 216 9.84 229 9.98 484 9.98 509 9.98 534 9.98 560 9.98 585 0.01 516 0.01 491 0.01 466 0.01 440 0.01 415 9.85 693 9.85 681 9.85 669 9.85 657 9.85 645 60 59 58 57 56 5 6 7 8 9 9.84 242 9.84 255 9.84 269 9.84 282 9.84 295 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 0.01 390 0.01 365 0.01 339 0.01 314 0.01 289 9.85 632 9.85 620 9.85 608 9.85 596 9.85 583 55 54 53 52 51 10 11 12 13 14 9.84 308 9.84 321 9.84 334 9.84 347 9.84 360 9.98 737' 9.98 762 9.98 787 9.98 812 9.98 838 0.01 263 0.01 238 0.01 213 0.01 188 0.01 162 9.85 571 9.85 559 9.85 547 9.85 534 9.85 522 50 49 48 47 46 15 16 17 18 19 9.84 373 9.84 385 9.84 398 9.84 411 9.84 424 9.98 863 9.98 888 9.98 913 9.98 939 9.98 964 0.01 137 0.01 112 0.01 087 0.01 061 0.01 036 9.85 510 1 9.85 497 9.85 485 9.85 473 9.85 460 45 44 43 42 41 20 21 22 23 24 9.84 437 9.84 450 . 9.84 463 9.84 476 9.84 489 9.98 989 9.99 015 9.99 040 9.99 065 9.99 090 0.01 Oil 0.00 985 0.00 960 0.00 935 0.00 910 9.85 448 9.85436 9.85 423 9.85 411 9.85 399 40 39 38 37 36 44 25 26 27 28 29 9.84 502 9.84 515 9.84 528 9.84 540 9.84 553 9.99 116 9.99 141 9.99 166 9.99 191 9.99 217 0.00 884 0.00 859 0.00 834 0.00 809 0.00 783 9.85 386 9.85 374 9.85 361 9.85 349 9.85 337 35 34 33 32 31 30 31 32 33 34 9.84 566 9.84 579 9.84 592 9.84 605 9.84 618 9.99 242 9.99 267 9.99 293 9.99 318 9.99 343 0.00 758 0.00 733 0.00 707 0.00 682 0.00 657 9.85 324 9.85 312 9.85 299 9.85 287 9.85 274 30 29 28 27 26 35 36- 37 38 39 9.84 630 9.84 643 9.84 656 9.84 669 9.84 682 9.99 368 9.99 394 9.99 419 9.99 444 9.99 469 0.00 632 0.00 606 0.00 581 0.00 556 0.00 531 9.85 262 9.85'250 9.85 237 9.85 225 9.85 212 25 24 23 22 21 40 41 42 43 44 9.84 694 9.84 707 9.84 720 9.84 733 9.84 745 9.99 495 9.99 520 9.99 545 9.99 570 9.99 596 0.00 505 0.00 480 0.00 455 0.00 430 0.00 404 9.85 200 9.85 187 9.85 175 9.85 162 9.85 150 20 19 18 17 16 45 46 47 48 49 9.84 758 9.84 771 9.84 784 9.84 796 9.84 809 9.99 621 9.99 646 9.99 672 9.99 697 9.99 722 0.00 379 0.00 354 0.00 328 0.00 303 0.00 278 9.85 137 9.85 125 9.85 112 9.85 100 9.85 087 15 14 13 12 11 50 51 52 53 54 9.84 822 9.84 835 9.84 847 9.84 860 9.84 873 9.99 747 9.99 773 9.99 798 9.99 823 9.99 848 0.00 253 0.00 227 0.00 202 0.00 177 0.00 152 9.85-074 9.85 062 9.85 049 9.85 037 9.85 024 10 9 8 7 6 - 55 56 57 58 59 9.84 885 9.84 898 9.84 911 9.84 923 9.84 936 9.99 874 9.99 899 9.99 924 9.99 949 9.99 975 0.00 126 0.00 101 0.00 076 0.00 051 0.00 025 9.85 012 9.84 999 9.84 986 9.84 974 9.84 961 5 . 4 3 2 1 60 9.84 949 0.00 000 0.00 000 9.84 949 L. Cos. L, Cotg. L. Tang. L. Sin. ; [86] TABLE IV AUXILIARY FIVE-PLACE TABLE FOR SMALL ANGLES [87] // i 8 T w T' L. Sin. 60 120 180 240 1 2 3 4 4.68557 .68557 .68557 .68557 .68557 4.68557 .68557 .68557 .68557 .68558 5.31443 .31443 .31443 .31443 .31443 5.31443 .31443 .31443 .31443 .31442 6.46373 .76476 .94085 7.06579 300 360 420 480 540 5 6 7 8 9 4.68557 .68557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68558 5.31443 .31443 .31443 .31443 .31443 5.31442 .31442 .31442 .31442 .31442 7.16270 .24188 .30882 .36682 .41797 600 660 720 780 840 10 11 12 13 14 4.68557 .68557 .68557 .68557 .68557" 4.68558 .68558 .68558 .68558 .68558 5.31443 .31443 .31443 .31443 .31443 5.31442 .31442 .31442 .31442 .31442 7.46373 .50512 .54291 .57767 .60985 900 960 1020 1080 1140 15 16 17 18 19 4.68557 .68557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68558 5.31443 .31443 .31443 .31443 .31443 5.31442 .31442 .31442 .31442 .31442 7.63982 .66784 .69417 .71900 .74248 1200 1260 1320 1380 1440 20 21 22 23 24 4.68557 .6-8557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68558 5.31443 ^.31443 .31443 .31443 .31443 5.31442 .31442 .31442 .31442 .31442 7.76475 .78594 .80615 .82545 .84393 1500 1560 1620 1680 1740 25 26 27 28 29 4.68557 .68557 .68557 .68557 .68557 4.68558 .68558 .68558 .68558 .68559 5.31443 .31443 .31443 .31443 .31443 5.31442 .31442 .31442 .31442 .31441 7.86166 .87870 .89509 .91088 .92612 1800 1860 1920 1980 2040 30 31 32 33 34 4.68557 .68557 .68557 .68557 .68557 4.68559 .68559 .68559 .68559 .68559 5.31443 .31443 .31443 .31443 .31443 5.31441 .31441 .31441 .31441 .31441 7.94084 .95508 .96887 .98223 .99520 2100 2160 2220 2280 2340 35 36 37 38 39 4.68557 .68557 .68557 .68557 .68557 4.68559 .68559 .68559 .68559 .68559 5.31443 .31443 .31443 .31443 .31443 5.31441 .31441 .31441 .31441 .31441 8.00779 .02002 .03192 .04350 .05478 2400 2460 2520 2580 2640 40 41 42 43 44 4.68557 .68556 .68556 .68556 .68556 4.68559 .68560 .68560 .68560 .68560 5.31443 .31444 .31444 .31444 ".31444 5.31441 .31440 .31440 .31440 .31440 8.06578 .07650 .08696 .09718 .10717 2700 2760 2820 2880 2940 45 46 47 48 49 4.68556 .68556 .68556 .68556 .68556 4.68560 .68560 .68560 .68560 .68560 f!444 1444 .31444 .31444 .31444 . 5.31440 .31440 .31440 .31440 .31440 8.11693 .12647 .13581 .14495 .15391 3000 3060 3120 3180 3240 50 51 52 53 54 4.68556 .68556 .68556 .68556 .68556 4.68561 .68561 .68561 .68561 .68561 5.31444 .31444 .31444 .31444 .31444 5.31439 .31439 .31439 .31439 .31439 8.16268 .17128 .17971 .18798 .19610 3300 3360 3420 3480 3540 55 56 57 58 .59 4.68556 .68556 .68555 .68555 .68555 4.68561 .68561 .68561 .68562 .68562 5.31444 .31444 .31445 .31445 .31445 5.31439 .31439 .31439 .31438 .31438 8.20407 .21189 .21958 .22713 .23456 3600 60 4.68555 4.68562 5.31445 5.31438 8.24186 [88] // / S T S' T' L. Sin. 3600 3660 3720 3780 3840 1 2 3 4 4.68555 .68555 .68555 .68555 .68555 4.68562 .68562 .68562 .68562 .68663 5.31445 .31445 .31445 .31445 .31445 5.31438 .31438 .31438 .31438 .31437 8.24186 .24903 .25609 .26304 .26988 3900 3960 4020 4080 4140 5 6 7 8 9 4.68555 .68555 .68555 .68555 .68555 4.68563 .68563 .68563 .68563 .68563 5.31445 .31445 .31445 .31445 .31445 5.31437 .31437 .31437 .31437 .31437 8.27661 .28324 .28977 .29621 .30255 4200 4260 4320 4380 4440 10 11 12 13 14 4.68554 .68554 .68554 .68554 .68554 4.68563 .68564 .68564 .68564 .68564 5.31446 .31446 .31446 .31446 .31446 5.31437 .31436 .31436 .31436 .31436 8.30879 .31495 .32103 .32702 .33292 4500 4560 4620 4680 4740 15 16 17 18 19 4.68554 .68554 .68554 .68554 .68554 4.68564 .68565 .68565 .68565 .68565 5.31446 .31446 .31446 .31446 .31446 5.31436 .31435 .31435 .31435 .31435 8.33875 .34450 .35018 .35578 ' .36131 1 4800 4860 4920 4980 5040 20 21 22 23 24^ 4.68554 .68553 .68553 .68553 .68553 4.68565 .68566 .68566 .68566 .68566- 5.31446 .31447 .31447 .31447 .31447 5.31435 .31434 .31434 .31434 .31434 8.36678 .37217 .37750 .38276 .38794 5100 5160 5220 5280 5340 25- 26, 27 28 29 4.68553 .68553. .685984 .68553 .68553 4.68566 .68567 .68567 .68567 .68567 5.31447- .31447 .31447 .31447 .31447 5.31434" .31433 .31433 .31433 .31433 8.393,trf .39818 .40320 .40816 .41307 5400 5460 5520 5580 5640 30 31 32 33 34 4.68553 .68552 .68552 .68552 .68552 4.68567 .68568 .68568 .68568 .68568 5.31447 .31448 .31448 .31448 .31448 5.31433 .31432 .31432 .31432 .31432 8.41792 .42272 .42746 .43216 .43680 5700 5760 5820 5880 5940 35 36 37 38 39 4.68552 .68552 .68552 .68552 .68551 4.68569 .68569 .68569 .68569 .68569 5.31448 .31448 .31448 .31448 .31449 5.31431 .31431 .31431 .31431 .31431 8.44139 .44594 .45044 .45489 .45930 6000 6060 6120 6180 6240 40 41 42 43 44 4.68551 .68551 .68551 .68551 .68551 4.68570 .68570 .68570 .68570 .68571 5.31449 .31449 .31449 .31449 ..31449 5.31430 .31430 .31430 .31430 .31429 8.46366 .46799 .47226 .47650 .48069 6300 6360 6420 6480 6540 45 46 47 48 49 4.68551 .68551 .68550 .68550 .68550 4.68571 .68571 .68572 .68572 .68572 5.31449 .31449 .31450 .31450 .31450 5.31429 .31429 .31428 .31428 .31428 8.48485 .48896 .49304 .49708 .50108 6600 6660 6720 6780 6840 50 51 52 53 54 4.68550 .68550 .68550 .68550 .68550 4.68572 .68573 .68573 .68573 .68573 5.31450 .31450 .31450 .31450 .31450 5.31428 .31427 .31427 .31427 .31427 8.50504 .50897 .51287 .51673 .52055 6900 6960 7020 7080 7140 55 56 57 58 59 4.68549 .68549 .68549 .68549 .68549 4.68574 .68574 .68574 .68575 .68575 5.31451 .31451 .31451 .31451 .31451 5.31426 .31426 .31426 .31425 .31425 8.52434 .52810 .53183 .53552 .53919 7200 60 4.68549 4.68575 5.31451 5.31425 8.54282 [89] TABLE V FOUR-PLACE TABLE OF THE NATURAL SINE, COSINE, TANGENT, AND COTANGENT EVERY 10' OF THE QUADRANT [91] o 1 N. Sin. N. Tan. N. Cot. N. Cos. 00 10 20 30 40 50 .0000 .0029 .0058 .0087 .0116 .0145 .0000 .0029 .0058 .0087 .0116 .0145 oo 343.77 171.89 114.59 85.940 68.750 1.0000 1.0000 1.0000 1.0000 .9999 .9999 00 90 50 40 30 20 . -.10 1 00 10 20 ^30 s4J3 50 .0175 .0204 .0233 Tff262 ,291 .0320 .0175 .0204 .0233 .0262 .0291 .0320 57.290 49.104 42.964 38.188 34.368 31.242 .9998 .9998 .9997 .9997 .9996 .9995 00 89 50 40 30 20 10 2 00 10 20 30 40 50 .0349 .0378 .0407 .0436 .0465 .0494 .0349 .0378 .0407 .0437 .0466 .0495 28.636 26.432 24.542 22.904 21.470 20.206 .9994 .9993 .9992 .9990 .9989 .9988 00 88 50 40 30 20 10 3 00 10 20 30 40 50 .0523 .0552 .0581 .06,10 .0640 .0669 .0524 .0553 .0582 .0612 .0641 .0670 19.081 18.075 17.169 16.350 15.605 14.924 .9986 .9985 .9983 .9981 .9980 .9978 00 87 50 40 30 20 10 4 00 10 20 30 40 50 .0698 .0727 .0756 .0785 .0814 .0843 .0699 .0729 .0758 .0787 .0816 0846 14.301 13.727 13.197 12.706 12.251 11.826 .9976 .9974 .9971 .9969 .9967 .9964 00 86 50 40 30 20 10 5 00 10 20 30 40 50 .0872 .0901 .0929 .0958 .0987 .1016 .0875 .0904 .0934 .0963 .0992 .1022 11.430 11.059 10.712 10.385 10.078 9.7882 .9962 .9959 .9957 .9954 .9951 .9948 00 85 50 40 30 20 10 6 00 10 20 30 40 50 .1045 .1074 .1103 .1132 .1161 .1190 .1051 .1080 .1110 .1139 .1169 .1198 9.5144 9.2553 9.0098 8.7769 8.5555 8.3450 .9945 .9942 .9939 .9936 .9932 .9929 00 84 50. 40 30 20 10 7 00 10 20 30 40 50 .1219 .1248 .1276 .1305 .1334 .1363 .1228 .1257 .1287 .1317 .1346 .1376 8.1443' 7.9530 7.7704 7.5958 7.4287 7.2687 .9925 .9922 .9918 .9914 .9911 .9907 00 83 50 40 30 20 10 8 00 10 20 30 40 50 .1392 .1421 .1449 .1478 .1507 .1536 .1405 .1435 .1465 .1495 .1524 .1554 7.1154 6.9682 6.8269 6.6912 6.5606 6.4348 .99j03 .9899 .9^4 .9890 .9886 .9881 00 82 50 40 30 20 10 9 00 .1564 .1584 6.3138 .9877 00 81 N. Cos.' N. Cot. If. Tan. N.Sin. ' r o [92] o ; N. Sin. N. Tan. N. Cot. N. Cos. 9 00' 10 20 30 40 50 .1564 .1593 .1622 .1650 .1679 .1708 .1584 .1614 .1644 .1673 .1703 .1733 6.3138 6.1970 6.0844 5.9758 5.8708 5.7694 .9877 .9872 .9868 .9863 .9858 .9853 00 81 50 40 30 20 10 10 00 10 20 30 40 50 .1736 .1765 .1794 . .1822 / .1851 .1880 .1763 .1793 .1823 .1853 .1883 .1914 5.6713 ' 5.5764 5.4845 5.3995 5.3093 5.2257 .9848 .9843 .9838 .9833 .9827 .9822 00 80 50 40 30 20 10 11 00 10 20 30 40 50 .1908 .1937 .1965 .1994 .2022 .2051 .1944 .1974 .2004 .2035 .2065 .2095 5.1446 5.0658 4.9894 4.9152 4.8430 4.7729 .9816 .9811 .9805 .9799 .9793 . .9787 00 79 50 40 30 20 10 12 00 10 . 20 30 40 50 .2079 .2108 .2136 .2164 .2193 .2221 .2126 .2156 .2186 .2217 .2247 .2278 4.7046 4.6382 4.5736 4.5107 4.4494 4.3897 .9781 .9775 .9769 .9763 .9757 .9750 00 78 40 30 20 10 13 00 10 20 30 40 50 .2250 .2278 .2306 .2334 .2363 .2391 .2309 .2339 .2370 .2401 .2432 .2462 4.3315 4.2747 4.2193 4.1653 4.1126 4.0611 .9744 .9737 .9730 .9724 .9717 .9710 00 77 50 40 30 20 10 -. 14 00 10 20 30 40 50 .2419 .2447 .2476 .2504 .2532 .2560 .2493 .2524 .2555 .2586 .2617 .2648 4.0108 3.9617 3.9136 3.8667 3.8208 3.7760 .9703 .9696- .9689 .9681 .9674 .9667 00 76 50 40 30 20 10 15 00 10 20 30 40 50 .2588 .2616 .2644 .2672 .2700 .2728 .2679 ^ .2711 .2742 .2773 .2805 .2836 3.7321 3.6891. 3.6470 3.6059 3.5656 3.5261 .9659 - .9652 .9644 .9636 .9628 .9621 00 75 50 40 30 20 10 16 00 10 20 30 40 50 .2756 .2784 .2812 .2840 .2868 .2896 .2867 .2899 .2931 .2962 .2994 .3026 3.4874 3.4495 3.4124 3.3759 33402 3.3052 .9613 .9605 .9596 .9588 .9580 .9572 00 74 50 40 30 20 10 17 00 10 20 30 40 50 .2924 .2952 .2979 .3007 .3035 .3062 .3057 .3089 .3121 .3153 .3185 .3217 3.2709 3.2371 3.2041 3.1716 3.1397 3.1084 .9563 .9555 .9546 .9537 .9528 .9520 00 73 50 40 30 20 10 18 00 .3090 .3249 3.0777 .9511 00 72 N. Cos. N. Cot. S. Tan. N. Sin. r [93] f N. Sin. N. Tan. N. Cot. N. Cos. 18 00 10 20 30 40 50 .3090 .3118 .3145 .3173 .3201 .3228 .3249 .3281 .3314 .3346 .3378 .3411 3.0777 3.0475 3.0178 2.9887 2.9600 2.9319 .9511 .9502 .9492 .9483 .9474 .9465 00 72 50 40 30 20 10 19 00 10 20 30 40 ^ 50 .3256 .3283 .3311 .3338 .3365 .3393 .3443 .3476 .3508 .3541 .3574 .3607 2.9042 2.8770 2.8502 2.8239 2.7980 2.7725 .9455 .9446 .9436 .9426 .9417 .9407 00 71 50 40 30 20 10 20 00 10 20 30 40 50 .3420 .3448 " .3475 .3502 .3529 .3557 .3640 .3673 .3706 .3739 .3772 .3805 2.7475 2.7228 2.6985 2.6746 2.6511 2.6279 .9397 .9387 .9377 .9367 .9356 .9346 00 70 50 40 30 20 10 21 00 10 20 30 40 50 .3584 .3611 .3638 .3665 .3692 .3719 .3839 .3872 .3906 .3939 .3973 .4006 2.6051 2.5826 2.5605 2.5386 2.5172 2.4960 .9336 .9325 .9315 .9304 .9293 .9283 00 69 50 40 30 20 10 22 00 10 20 30 40 50 .3746 .3773 .3800 .3827 .3854 .3881 .4040 .4074 .4108 .4142 .4176 .4210 2.4751 2.4545 2.4342 2.4142 2.3945 2.3750 .9272 .9261 .9250 .9239 . .9228 .9216 00 68 50 40 30 20 10 23 00 10 20 30 40 50 .3907 .3934 .3961 .3987 .4014 .4041 .4245 .4279 .4314 .4348 .4383 .4417 2.3559 2.3369 2.3183 2.2998 2.2817 2.2637 .9205 .9194 .9182 .9171 .9159 .9147 00 67 50 40 30 20 10 24 00 10 20 30 40 50 .4067 .4094 .4120 .4147 .4173 .4200 .4452 .4487 .4522 .4557 .4592 .4628 2.2460 2.2286 2.2113 2.1943 2.1775 2.1609 .9135 .9124 .9112 .9100 .9088 .9075 00 66 50 40 30 20 10 25 00 10 20 30 40 50 .4226 .4253 .4279 .4305 .4331 .4358 .4663 .4699 .4734 4770 .4806 .4841 2.1445 2.1283 2.1123 2.0965 2.0809 2.0655 .9063 .9051 .9038 .9026 .9013 .9001 ' 00 65 50 40 30 20' 10 26 00 10 20 30 40 50 .4384 .4410 .4436 .4462 .4488 .4514 .4877 .4913 .4950 .4986 .5022 .5059 2.0503 2.0353 2.0204 2.0057 1.9912 1.9768 .8988 .8975 .8962 .8949 .8936 .8923 00 64 50 40 30 20 10 27 00 .4540 .5095 1.9626 .8910 00 63 5. Cos. X. Cot. N. Tan. N. Sin. r o [94] o / N. Sin. N. Tan. N. Cot. N. Cos. 27 00 10 20 30 40 50 .4540 .4566 .4592 .4617 .4643 .4669 .5095 .5132 .5169 .5206 .5243 .5280 ' 1.9626 1.9486 1.9347 1.9210 1.9074 1.8940 .8910 .8897 .8884 .8870 .8857 .8843 00 63 50 40 30 20 10 28 00 10 20 30 40 50 .4695 .4720 .4746 .4772 .4797 .4823 .5317 .5354 .5392 .5430 .5467 .5505 1.8807 1.8676 1.8546 1.8418 1.8291 1.8165 .8829 .8816 .8802 .8788 .8774 .8760 00 62 50 40 30 20 10 29 00 10 20 30 40 50 .4848 .4874 .4899 .4924 .4950 .4975 .5543 .5581 .5619 .5658 .5696 .5735 1.8040 1.7917 1.7796 1.7675 1.7556 1.7437 .8746 .8732 .8718 .8704 .8689 .8675 00 61 50 40 30 20 10 30 00 10 20 30 40 50 .5000 .5025 .5050 .5075 .5100 .5125 .5774 .5812 .5851 .5890 .5930 .5969 1.7321 1.7205 1.7090 1.6977 1.6864 1.6753 .8660 . , .8646 .8631 .8616 .8601 .8587 00 60 50 40 30 20 10 31 00 10 20 30 40 50 .5150 .5175 .5200 .5225 .5250 .5275 .6009 .6048 .6088 .6128 .6168 .6208 1.6643 1.6534 1.6426 1.6319 .1.6212 1.6107 .8572 .8557 .8542 .8526 .8511 .8496 00 59 50 40 30 20 10 32 00 10 20 30 40 50 .5299 .5324 .5348 .5373 .5398 .5422 .6249 .6289 .6330 .6371 .6412 .6453 1.6003 1.5900 1.5798 1.5697 1.5597 1.5497 .8480 .8465 .8450 .8434 .8418 .8403 00 58 50 40 " 30 20 10 33 00 10 20 30 40 50 .5446 .5471 .5495 .5519 ' .5544 .5568 .6494 .6536 .6577 .6619 .6661 .6703 1.5399 1.5301 1.5204 1.5108 1.5013 1.4919 .8387 .8371 .8355 .8339 .8323 .8307 00 57 50 40 30 20 10 34 00 10 20 30 40 50 .5592 .5616 .5640 .5664 .5688 .5712 .6745 .6787 .6830 .6873 .6916 .6959 1.4826 1.4733 1.4641 1.4550 1.4460 1.4370 .8290 .8274 .8258 .8241 .8225 .8208 00 56 50 40 30 20 10 35 00 10 20 30 40 50 .5736 .5760 .'5783 .5807 .5831 .5854 .7002 .7046 .7089 .7133 .7177 .7221 14281 1.4193 1.4106 . 1.4019 1.3934 1.3848 .8192 .8175 .8158 .8141 .8124 .8107 00 55 50 40 30 20 10 36 00 .5878 .7265 1.3764 .8090 00 54 N. Cos. N. Cot. N. Tan. N. Sin. t [95] o f N. Sin. N. Tan. N. Cot. N. Co& 36 00 10 20 30 40 50 .5878 .5901 .5925 .5948 .5972 .5995 .7265 .7310 .7355 .7400 .7445 .7490 1.3764 1.3680 1.3597 1.3514 1.3432 1.3351 .8090 .8073 .8056 .8039 .8021 .8004 00 54 50 40 30 20 10 37 00 10 20 30 40 50 .6018 .6041 .6065 .6088 .6111 .6134 .7536 .7581 .7627 .7673 .7720 .7766 1.3270 1.3190 1.3111 1.3032 1.2954 1.2876 .7986 .7969^ .7951 .7934 .7916 .7898 00 53 50 40 30 20 10 38 00 10 20 30 40 50 .6157 .6180 .6202 .6225 .6248 .6271 .7813 .7860 .7907 .7954 .8002 .8050 1.2799 1.2723 1.2647 1.2572 1.2497 1.2423 .7880 .7862 .7844 .7826 .7808 .7790 00 52 50 40 30 20 10 39 00 10 20 30 40 50 .6293 .6316 .6338 .6361 .6383 .6406 .8098 .8146 .8195 .8243 .8292 .8342 1.2349 1.2276 1.2203 1.2131 1.2059 1.1988 .7771 .7753 .7735 .7716 .7698 .7679 00 51 50 40 30 20 10 40 00 10 20 30 40 50 .6428 .6450 .6472 .6494 .6517 .6539 .8391 .8441 .8491 .8541 .8591 .8642 1.1918 1.1847 1.1778 1.1708 1.1640 1.1571 .7660 .7642 .7623 .7604 .7585 .7566 00 50 50 40 30 20 10 41 00 10 20 30 40 50 .6561 .6583 .6604 .6626 .6648 .b670 .8693 .8744 .8796 .8847 .8899 .8952 1.1504 1.1436 1.1369 ' 1.1303 1.1237 1.1171 .7547 .7528 .7509 .7490 .7470 .7451 00 49 50 40 30 20 10 42 00 10 20 30 40 50 .6691 .6713 .6734 .6756 .6777 .6799 .9004 .9057 .9110 .9163 .9217 .9271 1.110.6 1.1041 1.0977 1.0913 1.0850 1.0786 ?431 .7412 .7392 .7373 .7353 .7333 00' 48 50 40 30 20 10 43 00 10 20 30 40 50 .6820 .6841 .6862 .6884 .6905 .6926 .9325 .9380 .9435 .9490 .9545 .9601 1.0724 1.0661 1,0599 1.0538 1.0477 1.0416 .7314 .7294 .7274 .7254 .7234 .7214 00 47 50 40 30 20 10 44 00 10 20 30 40 50 .6947 .6967 .6988 .7009 .7030 .7050 .9657 .9713 .9770 .9827 .9884 .9942 1.0355 1.0295 1.0235 1.0176 1.0117 1.0058 .7193 .7173 .7153 .7133 .7112 .7092 00 46 50 40 30 20 10 45 00 .7071 1.0000 1.0000 .7071 00 45 N. Cos. ir. Cot. N. Tan. N. Sin. / o [96] TABLE VI FOUR-PLACE LOGARITHMS OF NUMBERS 1-2000 N. O 1 2 3 4 5 6 7 8 9 1 2 3 0000 0000 3010 4771 6021 6990 7782 8451 9031 9542 0000 3010 4771 0414 3222 4914 0792 3424 5051 1139 3617 5185 1461 3802 5315 1761 3979 5441 2041 4150 5563 2304 4314 5682 2553 4472 5798 2788 4624 5911 4 5 6 6021 6990 7782 6128 7076 7853 6232 7160 7924 6335 7243 7993 6435 7324 8062 6532 7404 8129 6628 7482 8195 6721 7559 8261 6812 7634 8325 6902 7709 8388 7 8 9 10 11 12 13 8451 9031 9542 8513 9085 9590 8573 9138 9638 8633 9191 9685 8692 9243 9731 8751 9294 9777 8808 9345 9823 8865 9395 9868 8921 9445 9912 8976 9494 9956 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 0414 0792 1139 0453 0828 1173 0492 0864 1206 0531 0899 1239 0569 0934 1271 0607 0969 1303 0645 1004 1335 0682 1038 1367 0719 1072 1399 0755 1106 1430 14 15 16 1461 1761 2041 1492 1790 2068 1523 1818 2095 1553 1847 2122 1584 1875 2148 1614 1903 2175 1644 1931 2201 1673 1959 2227 1703 1987 2253 1732 2014 2279 17 18 19 20 21 22 23 2304 2553 2788 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 2900 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 3222 3424 3617 3243 3444 3636 3263 3464 3655 3284 3483 3674 3304 3502 3692 3324 3522 3711 3345 3541 3729 3365 3560 3747 3385 3579 3766 3404 3598 3784 24 25 26 3802 3979 4150 3820 3997 4166 3838 4014 4183 3856 4031 4200 3874 4048 4216 3892 4065 4232 3909 4082 4249 3927 4099 4265 3945 4116 4281 3962 4133 4298 27 28 29 30 31 32 33 4314 4472 4624 4330 4487 4639 4346 4502 4654 4362 4518 4669 4378 4533 4683 4393 4548 4698 4409 4564 4713 4425 4579 4728 4440 4594 4742 4456 4609 4757 4771 4786 4800 1 4814 4829 4843 4857 4871 4886 4900 4914 5051 5185 4928 5065 5198 4942 4955 5079 1 5092 5211 5224 4969 5105 5237' 4983 5119 5250 4997 5132 5263 5011 5145 5276 5024 5159 5289 5038 5172 5302 34 35 36 5315 5441 5563 5328 5453 5575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 5502 5623 5391 5514 5635 5403 5527 5647 5416 5539 5658 5428 5551 5670 37 38 39 40 41 42 43 5682 5798 5911 5694 5809 5922 5705. 582 ll 5933^ L5717 K832 ~5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5775 5888 5999 5786 5900 6010 6021 6031 6042 | 6053 6064 6075 6085 6096 6107 6117 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 6325 6425 44 45 46 6435 6532 6628 6444 6542 6637 6454 6551 6646 6464 6561 6656 6474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6599 6693 6513 6609 6702 6522 6618 6712 47 48 49 50 6721 6812 6902 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 6875 6964 6794 6884 6972 6803 6893 6981 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 N. 1 2 3 4 5 6 7 8 9 [98] N. O 1 2 3 4 5 6 7 8 9 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973, 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 . 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8803 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 90G9 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9513 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9866 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9323 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 '9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 N. O 1 2 3 | 4 5 6 7 8 9 [99] N. 1 2 3 4 5 6 7 8 9 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 101 0043 0048 0052 0056 0060 0065 0069 0073 0077 0082 102 0086 0090 0095 0099 0103 0107 0111 0116 0120 0124 103 0128 0133 0137 0141 0145 0149 0154 0158 0162 0166 104 0170 0175 0179 0183 0187 0191 0195 0199 0204 0208 105 0212 0216 0220 0224 0228 0233 0237 0241 0245 0249 106 0253 0257 0261 0265 0269 0273 0278 0282 0286 0290 107 0294 0298 0302 0306 0310 0314 0318 0322 0326 0330 108 0334 0338 0342 0346 0350 0354 0358 0362 0366 0370 109 0374 0378 0382 0386 0390 0394 0398 0402 0406 0410 110 0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 111 0453 0457 0461 0465 0469 0473 0477 0481 0484 0488 112 0492 0496 0500 0504 0508 0512 0515 0519 0523 0527 113 0531 0535 0538 0542 0546 0550 0554 0558 0561 0565 114 0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 115 0607 0611 0615 0618 0622 0626 0630 0633 0637 0641 116 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 117 0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 118 0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 119 0755 0759 0763 0766 0770 0774 0777 0781 0785 0788 120 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 121 0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 122 0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 123 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 124 0934 0938 0941 0945 0948 0952 0955 0959 0962 0966 125 0969 0973 0976 0980 0983 0986 0990 0993 0997 1000 126 1004 1007 1011 1014 1017 1021 1024 1028 1031 1035 127 1038 1041 1045 1048 1052 1055 1059 1062 1065 1069 128 1072 1075 1079 1082 1086 1089 1092 1096 1099 1103 129 1106 1109 1113 1116 1119 1123 1126 1129 1133 1136 130 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 131 1173 1176 1179 1183 1186 4189 1193 1196 1199 1202 132 1206 1209 1212 1216 1219 1222 1225 1229 1232 1235 133 1239 1242 1245 1248 1252 1255 1258 1261 1265 1268 134 1271 1274 1278 1281 1284 1287 1290 1294 1297 1300 135 1303 1307 1310 1313 1316 1319 1323 1326 1329 1332 136 1335 1339 1342 1345 1348 1351 1355 1358 1361 1364 137 1367 1370 1374 1377 1380 1383 1386 1389 1392 1396 138 1399 1402 1405 1408 1411 1414 1418 1421 1424 1427 139 1430 1433 1436 1440 1443 1446 1449 1452 1455 1458 140 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 141 1492 1495 1498 1501 1504 1508 1511 1514 1517 1520 142 1523 1526 1529 1532 1535 1538 1541 1544 1547 1550 143 1553 1556 1559 1562 1565 1569 1572 1575 1578 1581 144 1584 1587 1590 1593 1596 1599 1602 1605 1608 1611 145 1614 1617 1620 1623 1626 1629 1632 1635 1638 1641 146 1644 1647 1649 1652 1655 1658 1661 1664 1667 1670 147 1673 1676 1679 1682 1685 1688 1691 1694 1697 1700 148 1703 1706 1708 1711 1714 1717 1720 1723 1726 1729 149 1732 1735 1738 1741 1744 1746 1749 1752 1755 1758 150 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 N. 1 2 3 4 5 6 7 8 9 [100] N. 1 2 3 4 5 6 7 8 9 150 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 151 1790 1793 1796 1798 1801 1804 1807 1810 1813 1816 152 1818 , 1821 1824 1827 1830 1833 1836 1838 1841 1844 153 1847 1850 1853 1855 1858 1861 1864 1867 1870 1872 154 1875 1878 1881 1884 1886 1889 1892 1895 1898 1901 155 1903 1906 1909 1912 1915 1917 1920 1923 1926 1928 156 1931 1934 1937 1940 1942 1945 1948 1951 1953 1956 157 1959 1962 1965 1967 1970 1973 1976 1978 1981 1984 158 1987 1989 1992 1995 1998 2000 2003 2006 2009 2011 159 2014 2017 2019 2022 2025 2028 2030 2033 2036 2038 160 2041 2044 2047 2049 2052 2055 2057 2060 2063 2066 161 2068 2071 2074 2076 2079 2082 2084 2087 2090 2092 162 2095 2098 2101 2103 2106 2109 2111 2114 2117 2119 163 2122 2125 2127 2130 2133 2135 2138 2140 2143 2146 164 2148 2151 2154 2156 2159 2162 2164 2167 2170 2172 165 2175 2177 2180 2183 2185 2188 2191 2193 2196 2198 166 2201 2204 2206 2209 2212 2214 2217 2219 2222 2225 167 2227 2230 2232 2235 2238 2240 2243 2245 2248 2251 168 2253 2256 2258 2261 2263 2266 2269 2271 2274 2276 169 2279 2281 2284 2287 2289 2292 2294 2297 2299 2302 170 2304 2307 2310 2312 2315 2317 2320 2322 2325 2327 171 2330 2333 2335 2338 2340 2343 2345 2348 2350 2353 172 2355 2358 2360 2363 2365 2368 2370 2373 2375 2378 173 2380 2383 2385 2388 2390 2393 2395 2398 2400 2403 174 2405 2408 2410 2413 2415 2418 2420 2423 2425 2428 175 2430 2433 2435 2438 2440 2443 2445 2448 2450 2453 176 2455 2458 2460 2463 2465 2467 2470 2472 2475 2477 177 2480 2482 2485 2487 2490 2492 2494 2497 2499 2502 178 2504 2507 2509 2512 2514 2516 2519 2521 2524 2526 179 2529 2531 2533 2536 2538 2541 2543 2545 2548 2550 180 2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 181 2577 2579 2582 2584 2586 2589 2591 2594 2596 2598 182 2601 2603 2605 2608 2610 2613 2615 2617 2620 2622 183 2625 2627 2629 2632 2634 2636 2639 2641 2643 2646 184 2648 2651 2653 2655 2658 2660 2662 2665 2667 2669 185 2672 2674 2676 2679 2681 2683 2686 2688 2690 2693 186 2695 2697 2700 2702 2704 2707 2709 2711 2714 2716 187 2718 2721 2723 2725 2728 2730 2732 2735 2737 2739 188 2742 2744 2746 2749 2751 2753 2755 2758 2760 2762 189 2765 .2767 2769 2772 2774 2776 2778 2781 2783 2785 190 2788 2790 2792 2794 2797 2799 2801 2804 2806 2808 191 2810 2813 2815 2817 2819 2822 2824 2826 2828 2831 192 2833 2835 2838 2840 2842 2844 2847 2849 2851 2853 193 2856 2858 2860 2862 2865 2867 2869 2871 2874 2876 194 2878 2880 2883 2885 2887 2889 2891 2894 2896 2898 195 2900 2903 2905 2907 2909 2911 2914 2916 2918 2920 196 2923 2925 2927 2929 2931 2934 2936 2938 2940 2942 197 2945 2947 2949 2951 2953 2956 2958 2960 2962 2964 198 2967 2969 2971 2973 2975 2978 2980 2982 2984 2986 199 2981? 2991 2993 2995 2997 2999 3002 3004 3006 3008 200 3010 3012 3015 3017 3019 3021 3023 3025 3028 3030 N. 1 . _ I^BM 2 3 4 m [101] TABLE VII FOUR-PLACE LOGARITHMS OP THE TRIGONOMETRIC FUNCTIONS FOR THE DECIMALLY DIVIDED DEGREE [103] L. Sin. 1 2 3 4 5 6 7 8 9 ^ o.o H^HKXMB 00 6.2419 5429 7190 8439 9408 *0200 *0870 *1450 *1961 *2419 89.9 0.1 7.2419 2833 3211 3558 3880 4180 4460 4723 4971 5206 5429 89.8 0.2 7.5429 5641 5843 6036 6221 6398 6568 6732 6890 7043 7190 89.7 0.3 7.7190 7332 7470 7604 7734 7859 7982 8101 8217 8329 8439 89.6 0.4 7.8439 8547 8651 8753 8853 8951 9046 9140 9231 9321 9408 89.5 0.5 7.9408 9494 9579 9661 9743 9822 9901 9977 *0053 *0127 *0200 89.4 0.6 8.0200 0272 0343 0412 0480 0548 0614 0679 0744 0807 0870 89.3 0.7 8.0870 0931 0992 1052 1111 1169 1227 1284 1340 1395 1450 89.2 0.8 8.1450 1503 1557 1609 1661 1713 1764 1814 1863 1912 1961 89.1 0.9 8.1961 2009 2056 2103 2150 2196 2241 2286 2331 2375 2419 89.0 1.0 8.2419 2462 2505 2547 2589 2630 2672 2712 2753 2793 2832 88.9 1.1 8.2832 2872 2911 2949 2988 3025 3063 3100 3137 3174 3210 88.8 1.2 8.3210 3246 3282 3317 3353 3388 3422 3456 3491 3524 3558 88.7 1.3 8.3558 3591 3624 3657 3689 3722 3754 3786 3817 3848 3880 88.6 1.4 8.3880 3911 3941 3972 4002 4032 4062 4091 4121 4150 4179 88.5 1.5 8.4179 4208 4237 4265 4293 4322 4349 4377 4405 4432 4459 88.4 1.6 8.4459 4486 4513 4540 4567 4593 4619 4645 4671 4697 4723 88.3 1.7 8.4723 4748 4773 4799 4824 4848 4873 4898 4922 4947 4971 88.2 1.8 8.4971 4995 5019 5043 5066 5090 5113 5136 5160 5183 5206 88.1 1.9 8.5206 5228 5251 5274 5296 5318 5340 5363 5385 5406 5428 88.0 2.0 8.5.428 5450 5471 5493 5514 5535 5557 5578 5598 5619 5640 87.9 2.1 8.5640 5661 5681 5702 5722 5742 5762 5782 5802 5822 5842 87.8 2.2 8.5842 5862 5881 5901 5920 5939 5959 5978 5997 6016 6035 87.7 2.3 8.6035 6054 6072 6091 6110 6128 6147 6165 6183 6201 6220 87.6 2.4 8.6220 6238 6256 6274 6291 6309 6327 6344 6362 6379 6397 87.5 2.5 8.6397 6414 6431 6449 6466 6483 6500 6517 6534 6550 6567 87.4 2.6 8.6567 6584 6600 6617 6633 6650 6666 6682 6699 6715 6731 87.3 2.7 8.6731 6747 6763 6779 6795 6810 6826 6842 6858 6873 6889 87.2 2.8 8.6889 6904 6920 6935 6950 6965 6981 6996 7011 7026 7041 87.1 2.9 8.7041 7056 7071 7086 7100 7115 7130 7144 7159 7174 7188 87.0 3.0 8.7188 7202 7217 7231 7245 7260 7274 7288 7302 7316 7330 86.9 3.1 8.7330 7344 7358 7372 7386 7400 7413 7427 7441 7454 7468 86.8 3.2 8.7468 7482 7495 7508 7522 7535 7549 7562 7575 7588 7602 86.7 3.3 8.7602 7615 7628 7641 7654 7667 7680 7693 7705 7718 7731 86.6 3.4 8.7731 7744 7756 7769 7782 7794 7807 7819 7832 7844 7857 86.5 3.5 8.7857 7869 7881 7894 7906 7918 7930 7943 7955 7967 7979 86.4 3.6 8.7979 7991 8003 8015 8027 8039 8051 8062 8074 8086 8098 86.3 3.V 8.8098 8109 8121 8133 8144 8156 8168 8179 8191 8202 8213 86.2 3.8 8.8213 8225 8236 8248 8259 8270 8281 8293 8304 8315 8326 86.1 3.9 8.8326 8337 8348 8359 8370 8381 8392 8403 8414 8425 8436 86.0 4.0 8.8436 8447 8457 8468 8479 8490 8500 8511 8522 8532 8543 85.9 4.1 8.8543 8553 8564 8575 8585 8595 8606 8616 8627 8637 8647 85.8 4.2 8.8647 8658 8668 8678 8688 8699 8709 8719 8729 8739 8749 85.7 4.3 8.8749 8759 8769 8780 8790 8799 8809 8819 8829 8839 8849 85.6 4.4 8.8849 8859 8869 8878 8888 8898 8908 8917 8927 8937 8946 85.5 I 4.5 8.8946 8956 8966 8975 8985 8994 9004 9013 9023 9032 9042 85.4 4.6 8.9042 9051 9060 9070 9079 9089 9098 9107 9116 9126 9135 85.3 4.7 8.9135 9144 9153 9162 9172 9181 9190 9199 9208 9217 9226 85.2 4.8 8.9226 9235 9244 9253 9262 9271 9280 9289 9298 9307 9315 85.1 4.9 8.9315 MHHIHB 9324 ^i^MM 9333 9342 9351 9359 9368 9377 9386 9394 9403 85.O 9 8 7 6 5 4 3 2 1 L. Cos. [104] L. Sin. O 1 2 3 4 5 6 7 8 9 5.0 8.9403 9412 9420 9429 9437 9446 9455 9463 9472 9480 9489 84.9 5.1 8.9489 9497 9506 9514 9523 9531 9539 9548 9556 9565 9573 84.8 5.2 8.9573 9581 9589 9598 9606 9614 9623 9631 9639 9647 9655 84.7 5.3 8.9655 9664 9672 9680 9688 9696 9704 9712 9720 9728 9736 84.6 5.4 8.9736 9744 9752 9760 9768 9776 9784 9792 9800 9808 9816 84.5 5.5 8.9816 9824 9831 9839 9847 9855 9863 9870 9878 9886 9894 84.4 5.6* 8.9894 9901 9909 9917 9925 9932 9940 9948 9955 9933 9970 84.3 5.7 8.9970 9978 9986 9993 *0001 *0008 *0016 *0023 *0031 *0038 *0046 84.2 5.8 9.0046 0053 0061 0068 0075 0083 0090 0098 0105 0112 0120 84.1 5.9 9.0120 0127 0134 0142 0149 0156 0163 0171 0178 0185 0192 84.O 6.0 9.0192 0200 0207 0214 0221 0228 0235 0243 0250 0257 0264 83.9 6.1 9.0264 0271 0278 0285 0292 0299 0306 0313 0320 0327 0334 83.8 6.2 9.0334 0341 0348 0355 0362 0369 0376 0383 0390 0397 0403 83.7 6.3 9.0403 0410 0417 0424 0431 0438 0444 0451 0458 0465 0472 83.6 6.4 9.0472 0478 0485 0492 0498 0505 0512 0519 0525 0532 0539 83.5 6.5 9.0539 0545 0552 0558 0565 0572 0578 0585 0591 0598 0605 83.4 6.6 9.0605 0311 0618 0624 0631 0637 0644 0650 0657 0663 0670 83.3 6.7 9.0670 0676 0683 0689 0695 0702 0708 0715 0721 0727 0734 83.2 6.8 9.0734 0740 0746 0753 0759 0765 0772 0778 0784 0790 0797 83.1 6.9 9.0797 0803 0809 0816 0822 0828 0834 0840 0847 0853 0859 83.0 7.0 9.0859 0865 0871 0877 0884 0890 0896 0902 0908 0914 0920 82.9 7.1 9.0920 0926 0932 0938 0945 0951 0957 0933 0969 0975 0981 82.8 7.2 9.0981 0987 0993 0999 1005 1011 1017 1022 1028 1034 1040 82.7 7.3 9.1040 1046 1052 1058 1064 1070 1076 1031 1087 1093 1099 82.6 7.4 9.1099 1105 1111 1116 1122 1128 1134 1140 1145 1151 1157 82.5 7.5 9.1157 1163 1168 1174 1180 1186 1191 "1197 1203 1208 1214 82.4 7.6 9.1214 1220 1226 1231 1237 1242 1248 1254 1259 1265 1271 82.3 7.7 9.1271 1276 1282 1287 1293 1299 1304 1310 1315 1321 1326 82.2 7.8 9.1326 1332 1337 1343 1348 1354 1359 1365 1370 1376 1381 82.1 7.9 9.1381 1387 1392 1398 1403 1409 1414 1419 1425 1430 1436 82.0 8.0 9.1436 1441 1446 1452 1457 1462 1468 1473 1478 1484 1489 81.9 8.1 9.1489 1494 1500 1505 1510 1516 1521 1526 1532 1537 1542 81.8 8.2 9.1542 1547 1553 1558 1563 1568 1574 1579 1584 1589 1594 81.7 8.3 9.1594 1600 1605 1610 1615 1620 1625 1631 1636 1641 1646 81.6 8.4 9.1646 1651 1656 1661 1666 1672 1677 1682 1687 1692 1697 81.5 8.5 9.1697 1702 1707 1712 1717 1722 1727 1732 1737 1742 1747 81.4 8.6 9.1747 1752 1757 1762 1767 1772 1777 1782 1787 1792 1797 81.3 8.7 9.1797 1802 1807 1812 1817 1822 1827 1832 1837 1842 1847 81.2 8.8 9.1847 1851 1856 1861 1866 1871 1876 1881 1886 1890 1895 81.1 8.9 9.1895 1900 1905 1910 1915 1919 1924 1929 1934 1939 1943 81.O 9.0 9.1943 1948 1953 1958 1962 1967 1972 1977 1981 1986 1991 80.9 9.1 9.1991 1996 2000 2005 2010 2015 2019 2024 2029 2033 2038 80.8 9.2 9.2038 2043 2047 2052" 2057 2061 2066 2071 2075 2080 2085 80.7 9.3 9.2085 2089 2094 2098 2103 2108 2112 2117 2121 2126 2131 80.6 9.4 9.2131 2135 2140 2144 2149 2153 2158 2162 2167 2172 2176 80.5 9.5 9.2176 2181 2185 2190 2194 2199 2203 2208 2212 2217 2221 80.4 9.6 9.2221 2226 2230 2235 2239 2243 2248 2252 2257 2261 2266 80.3 9.7 9.2266 2270 2275 2279 2283 2288 2292 2297 2301 2305 2310 80.2 9.8 9.2310 2314 2319 2323 2327 2332 2336 2340 2345 2349 2353 80.1 9.9 9.2353 Ml 2358 2362 2367 2371 2375 2379 2384 2388 2392 2397 8O.0 9 8 7 6 5 4 3 2 1 L. Cos. [105] L. Sin. O 1 2 3 4 5 6 7 8 9 HHBm 00 90 -00 7.2419 5429 7190 8439 9408 *0200 *0870 *1450 *1961 *2419 89 1 8.2419 2832 3210 3558 3880 4179 4459 4723 4971 5206 5428 88 2 8.5428 5640 5842 6035 6220 6397 6567 6731 6889 7041 7188 87 3 8.7188 7330 7468 7602 7731 7857 7979 8098 8213 8326 8436 86 4 8.8436 8543 8647 8749 8849 8946 9042 9135 9226 9315 9403 85 5 8.9403 9489 9573 9655 9736 9816 9894 9970 *0046 *0120 *0192 84 6 9.0192 0264 0334 0403 0472 0539 0605 0670 0734 0797 0859 83 7 9.0859 0920 0981 1040 1099 1157 1214 1271 1326 1381 1436 82 8 9.1436 1489 1542 1594 1646 1697 1747 1797 1847 1895 1943 81 9 9.1943 1991 2038 2085 2131 2176 2221 2266 2310 2353 2397 80 10 9.2397 2439 2482 2524 2565 2606 2647 2687 2727 2767 2806 79 11 9.2806 2845 2883 2921 2959 2997 3034 3070 3107 3143 3179 78 12 9.3179 3214 3250 3284 3319 3353 3387 3421 3455 3488 3521 77 13 9.3521 3554 3586 3618 3650 3682 3713 3745 3775 3806 3837 76 14 9.3837 3867 3897 3927 3957 3986 4015 4044 4073 4102 4130 75 15 9.4130 4158 4186 4214 4242 4269 4296 4323 4350 4377 4403 74 16 9.4403 4430 4456 4482 4508 4533 4559 4584 4609 4634 4659 73 17 9.4659 4684 4709 4733 4757 4781 4805 4829 4853 4876 4900 72 18 9.4900 4923 4946 4969 4992 5015 5037 5060 5082 5104 5126 71 19 9.5126 5148 5170 5192 5213 5235 5256 5278 5299 5320 5341 7O 20 9.5341 5361 5382 5402 5423 5443 5463 5484 5504 5523 5543 69 21 9.5543 5563 5583 5602 5621 5641 5660 5679 5698 5717 5736 68 22 9.5736 5754 5773 5792 5810 5828 5847 5865 5883 5901 5919 67 23 9.5919 5937 5954 5972 5990 6007 6024 6042 6059 6076 6093 66 24 9.6093 6110 6127 6144 6161 6177 6194 6210 6227 6243 6259 65 25 9.6259 6276 6292 6308 6324 6340 6356 6371 6387 6403 6418 64 26 9.6418 6434 6449 6465 6480 6495 6510 6526 6541 6556 6570 63 27 9.6570 6585 6600 6615 6629 6644 6659 6673 6687 6702 6716 62 28 9.6716 6730 6744 6759 6773 6787 6801 6814 6828 6842 6856 61 29 9.6856 6869 6883 6896 6910 6923 6937 6950 6963 6977 6990 60 30 9.6990 7003 7016 7029 7042 7055 7068 7080 7093 7106 7118 59 31 9.7118 7131 7144 7156 7168 7181 7193 7205 7218 7230 7242 58 32 9.7242 7254 7266 7278 7290 7302 7314 7326 7338 7349 7361 57 33 9.7361 7373 7384 7396 7407 7419 7430 7442 7453 7464 7476 56 34 9.7476 7487 7498 7509 7520 7531 7542 7553 7564 7575 7586 55 35 9.7586 7597 7607 7618 7629 7640 7650 7661 7671 7682 7692 54 36 9.7692 7703 7713 7723 7734 7744 7754 7764 7774 7785 7795 53 37 9.7795 7805 7815 7825 7835 7844 7854 7864 7874 7884 7893 52 38 9.7893 7903 7913 7922 7932 7941 7951 7960 7970 7979 7989 51 39 9.7989 7998 8007 8017 8026 8035 8044 8053 8063 8072 8081 50 40 9.8081 8090 8099 8108 8117 8125 8134 8143 8152 8161 8169 49 41 9.8169 8178 8187 8195 8204 8213 8221 8230 8238 8247 8255 48 42 9.8255 8264 8272 8280 8289 8297 8305 8313 8322 8330 8338 47 43 9.8338 8346 8354 8362 8370 8378 8386 8394 8402 8410 8418 46 44 9.8418 8426 8433 8441 8449 8457 8464 8472 8480 8487 8495 45- 45 9.8495 9 8 7 6 5 4 3 2 1 L. Cos. [106] L. Sin. O 1 2 3 4 5 6 7 8 9 " 9.8495 45 45 9.8495 8502 8510 8517 8525 8532 8540 8547 8555 8562 8569 44 46 9.8569 8577 8584 8591 8598 8606 8613 8620 8627 8634 8641 43 47 9.8641 8648 8655 8662 8669 8676 8683 8690 8697 8704 8711 42 48 9.8711 8718 8724 8731 8738 8745 8751 8758 8765 8771 8778 41 49 9.8778 8784 8791 8797 8804 8810 8817 8823 8830 8836 8843 40 50 9.8843 8849 8855 8862 8868 8874 8880 8887 8893 8899 8905 39 51 9.8905 8911 8917 8923 8929 8935 8941 8947 8953 8959 8965 38 52 9.8965 8971 8977 8983 8989 8995 9000 9006 9012 9018 9023 37 53 9.9023 9029 9035 9041 9046 9052 9057 9063 9069 9074 9080 36 54 9.9080 9085 9091 9096 9101 9107 9112 9118 9123 9128 9134 35 55 9.9134 9139 9144 9149 9155 9160 9165 9170 9175 9181 9186 34 56 57 9.9186 9.9236 9191 9241 9196 9246 9201 9251 9206 9255 9211 9260 9216 9265 9221 9270 9226 9275 9231 9279 9236 9284 33 32 58 9.9284 9289 9294 9298 9303 9308 9312 9317 9322 9326 9331 31 59 9.9331 9335 9340 9344 9349 9353 9358 9362 9367 9371 9375 30 60 9.9375 9380 9384 9388 9393 9397 9401 9406 9410 9414 9418 29 61 9.9418 9422 9427 9431 9435 9439 9443 9447 9451 9455 9459 28 62 9.9459 9463. 9467 9471 9475 9479 9483 9487 9491 9495 9499 27 63 9.9499 9503 9506 9510 9514 9518 9522 9525 9529 9533 9537 26 64 9.9537 9540 9544 9548 9551 9555 9558 9562 9566 9569 9573 25 65 9.9573 9576 9580 9583 9587 9590 9594 9597 9601 9604 9607 24 66 9.9607 9611 9614 9617 9621 9624 9627 9631 9634 9637 9640 23 67 9.9640 9643 9647 9650 9653 9656 9659 9662 9666 9669 9672 22 68 9.9672 9675 9678 9681 9684 9687 9690 9693 9696 9699 9702 21 69 9.9702 9704 9707 9710 9713 9716 9719 9722 9724 9727 9730 20 70 9.9730 9733 9735 9738 9741 9743 9746 9749 9751 9754 9757 19 71 9.9757 9759 9762 9764 9767 9770 9772 9775 9777 9780 9782 18 72 9.9782 9785 9787 9789 9792 9794 9797 9799 9801 9804 9806 17 73 9.9806 9808 9811 9813 9815 9817 9820 9822 9824 9826 9828 16 74 9.9828 9831 9833 9835 9837 9839 9841 9843 9845 9847 '9849 15 75 9.9849 9851 9853 9855 9857 9859 9861 9863 9865 9867 9869 14 76 9.9869 9871 9873 9875 9876 9878 9880 9882 9884 9885 9887 13 77 9.9887 9889 9891 9892 9894 9896 9897 9899 9901 9902 9904 12 78 9.9904 9906 9907 9909 9910 9912 9913 9915 9916 9918 9919 11 79 9.9919 9921 9922 9924 9925 9927 9928 9929 9931 9932 9934 10 80 9.9934 9935 9936 9937 9939 9940 9941 9943 9944 9945 9946 9 81 9.9946 9947 9949 9950 9951 9952 9953 9954 9955 9956 9958 8 82 9.9958 9959 9960 9961 9962 9963 9964 9965 9966 9967 9968 7 83 9.9968 9968 9969 9970 9971 9972 9973 9974 9975 9975 9976 6 84 9.9976 9977 9978 9978 9979 9980 9981 9981 9982 9983 9983 5 85 9.9983 9984 9985 9985 9986 9987 9987 9988 9988 9989 9989 4 86 9.9989 9990 9990 9991 9991 9992 9992 9993 9993 9994 9994 3 87 9.9994 9994 9995 9995 9996 9996 9996 9996 9997 9997 9997 2 88 9.9997 9998 9998 9998 9998 9999 9999 9999 9999 9999 9999 1 89 9.9999 9999 *0000 *0000 *0000 *0000 *0000 *0000 *0000 0000 *0000 90 0.0000 9 8 7 6 5 4 3 2 1 L. Cos. [107] L. Tang. O 1 2 3 4 5 6 7 8 9 __ o.o OB^HBMK 00 6.2419 5429 7190 8439 9408 *0200 *0870 *1450 *1961 *2419 89.9 0.1 7.2419 2833 3211 3558 3880 4180 4460 4723 4972 5206 5429 89.8 0.2 7.5429 5641 5843 6036 6221 6398 6569 6732 6890 7043 7190 89.7 0.3 7.7190 7332 7470 7604 7734 7860 7982 8101 8217 8329 8439 89.6 0.4 7.8439 8547 8651 8754 8853 8951 9046 9140 9231 9321 9409 89.5 0.5 7.9409 9495 9579 9662 9743 9823 9901 9978 *0053 *0127 *0200 89.4 0.6 8.0200 0272 0343 0412 0481 0548 0614 0680 0744 0807 087C 89.3 0.7 8.0870 0932 0992 1052 1111 1170 1227 1284 1340 1395 14 5C 89.2 0.8 8.1450 1504 1557 1610 1662 1713 1764 1814 1864 1913 1962 89.1 0.9 8.1962 2010 2057 2104 2150 2196 2242 2287 2331 2376 2419 89.0 1.0 8.2419 2462 2505 2548 2590 2631 2672 2713 2754 2794 2833 88.9 1.1 8.2833 2873 2912 2950 2988 3026 3064 3101 3138 3175 3211 88.8 1.2 8.3211 3247 3283 3318 3354 3389 3423 3458 3492 3525 3559 88.7 1.3 8.3559 3592 3625 3658 3691 3723 3755 3787 3818 3850 3881 88.6 1.4 8.3881 3912 3943 3973 4003 4033 4063 4093 4122 4152 4181 88.5 1.5 8.4181 4210 4238 4267 4295 4323 4351 4379 4406 4434 4461 88.4 1.6 8.4461 4488 4515 4542 4568 4595 4621 4647 4673 4699 4725 88.3 1.7 8.4725 4750 4775 4801 4826 4851 4875 4900 4924 4949 4973 88.2 1.8 8.4973 4997 5021 5045 5068 5092 5115 5139 5162 5185 5208 88.1 1.9 8.5208 5231 5253 5276 5298 5321 5343 5365 5387 5409 5431 88.O 2.0 8.5431 5453 5474 5496 5517 5538 5559 5580 5601 5622 5643 87.9 2.1 8.5643 5664 5684 5705 5725 5745 5765 5785 5805 5825 5845 87.8 2.2 8.5845 5865 5884 5904 5923 5943 5962 5981 6000 6019 6038 87.7 2.3 8.6038 6057 6076 6095 6113 6132 6150 6169 6187 6205 6223 87.6 2.4 8.6223 6242 6260 6277 6295 6313 6331 6348 6366 6384 6401 87.5 2.5 8.6401 6418 6436 6453 6470 6487 6504 6521 6538 6555 6571 87.4 2.6 8.6571 6588 6605 6621 6638 6654 6671 6687 6703 6719 6736 87.3 2.7 8.6736 6752 6768 6784 6800 6815 6831 6847 6863 6878 6894 87.2 2.8 8.6894 6909 6925 6940 6956 6971 6986 7001 7016 7031 7046 87.1 2.9 8.7046 7061 7076 7091 7106 7121 7136 7150 7165 7179 7194 87.O 3.0 8.7194 7208 7223 7237 7252 7266 7280 7294 7308 7323 7337 86.9 3.1 8.7337 7351 7365 7379 7392 7406 7420 7434 7448 7461 7475 86.8 3.2 8.7475 7488 7502 7515 7529 7542 7556 7569 7582 7596 7609 86.7 3.3 8.7609 7622 7635 7648 7661 7674 7687 7700 7713 7726 7739 86.6 3.4 8.7739 7751 7764 7777 7790 7802 7815 7827 7840 7852 7865 86.5 3.5 8.7865 7877 7890 7902 7914 7927 7939 7951 7963 7975 7988 86.4 3.6 8.7988 8000 8012 8024 8036 8048 8059 8071 8083 8095 8107 86.3 3.7 8.8107 8119 8130 8142 8154 8165 8177 8188 8200 8212 8223 86.2 3.8 8.8223 8234 8246 8257 8269 8280 8291 8302 8314 8325 8336 86.1 3.9 8.8336 8347 8358 8370 8381 8392 8403 8414 8425 8436 8446 86.0 4.0 8.8446 8457 8468 8479 8490 8501 8511 8522 8533 8543 8554 85.9 4.1 8.8554 8565 8575 8586 8596 8607 8617 8628 8638 8649 8659 85.8 4.2 8.8659 8669 8680 8690 8700 8711 8721 8731 8741 8751 8762 85.7 4.3 8.8762 8772 8782 8792 8802 8812 8822 8832 8842 8852 8862 85.6 4.4 8.8862 8872 8882 8891 8901 8911 8921 8931 8940 8950 8960 85.5 4.5 8.8960 8970 8979 8989 8998 9008 9018 9027 9037 9046 9056 85.4 4.6 8.9056 9065 9075 9084 9093 9103 9112 9122 9131 9140 9150 85.3 4.7 8.9150 9159 9168 9177 9186 9196 9205 9214 9223 9232 9241 85.2 4.8 8.9241 9250 9260 9269 9278 9287 9296 9305 9313 9322 9331 85.1 4.9 8.9331 9340 9349 9358 9367 9376 9384 9393 """ 9402 9411 9420 85.0 ^M 9 8 7 6 5 4 3 2 1 O L. Cot. [108] L. Tang. O 1 2 3 4 5 6 7 8 9 5.0 8.9420 9428 9437 9446 9454 9463 9472 9480 9489 9497 ^^i 9506 84.9 5.1 8.9506 9515 9523 9532 9540 9549 9557 9565 9574 9582 9591 84.8 5.2 8.9591 9599 9608 9616 9624 9633 9641 9649 9657 9666 9674 84.7 5.3 8.9674 9682 9690 9699 9707 9715 9723 9731 9739 9747 9756 84.6 5.4 8.9756 9764 9772 9780 9788 9796 9804 9812 9820 9828 9836 84.5 5.5 8.9836 9844 9852 9860 9867 9875 9883 9891 9899 9907 9915 84.4 5.6 8.9915 9922 9930 9938 9946 9953 9961 9969 9977 9984 9992 84.3 5.7 8.9992 *0000 *0007 *0015 *0022 *0030 *0038 *0045 *0053 *0060 *0068 84.2 5.8 9.0068 0075 0083 0090 0098 0105 0113 0120 0128 0135 0143 84.1 5.9 9.0143 0150 0157 0165 0172 0180 0187 0194 0202 0209 0216 84.0 6.0 9.0216 0223 0231 0238 0245 0253 0260 0267 0274 0281 0289 83.9 6.1 9.0289 0296 0303 0310 0317 0324 0331 0338 0346 0353 0360 83.8 6.2 9.0360 0367 0374 0381 0388 0395 0402 0409 0416 0423 0430 83.7 6.3 9.0430 0437 0444 0451 0457 0464 0471 0478 0485 0492 0499 83.6 6.4 9.0499 0506 0512 0519 0526 0533 0540 0546 0553 0560 0567 83.5 6.5 9.0567 0573 0580 0587 0593 0600 0607 0614 0620 0627 0633 83.4 6.6 9.0633 0640 0647 0653 0660 0667 0673 0680 0686 0693 0699 83.3 6.7 9.0699 0706 0712 0719 0725 0732 0738 0745 0751 0758 0764 83.2 6.8 9.0764 0771 0777 0784 0790 0796 0803 0809 0816 0822 0828 83.1 6.9 90828 0835 0841 0847 0854 0860 0866 0873 0879 0885 0891 83.0 7.0 9.0891 0898 0904 0910 0916 0923 0929 0935 0941 0947 0954 82.9 7.1 9.0954 0960 0966 0972 0978 0984 0991 0997 1003 1009 1015 82.8 7.2 9.1015 1021 1027 1033 1039 1045 1051 1058 1064 1070 1076 82.7 7.3 9.1076 1082 1088 1094 1100 1106 1112 1117 1123 1129 1135 82.6 7.4 9.1135 1141 1147 1153 1159 1165 1171 1177 1183 1188 1194 82.5 7.5 9.1194 1200 1206 1212 1218 1223 1229 1235 1241 1247 1252 82.4 7.6 9.1252 1258 1264 1270 1276 1281 1287 1293 1299 1304 1310 82.3 7.7 9.1310 1316 1321 1327 1333 1338 1344 1350 1355 1361 1367 82.2 7.8 9.1367 1372 1378 1384 1389 1395 1400 1406 1412 1417 1423 82.1 7.9 9.1423 1428 1434 1439 1445 1450 1456 1461 1467 1473 1478 82.0 8.0 9.1478 1484 1489 1494 1500 1505 1511 1516 1522 1527 1533 81.9 8.1 9.1533 1538 1544 1549 1554 1560 1565 1571 1576 1581 1587 81.8 8.2 9.1587 1592 1597 1603 1608 1613 1619 1624 1629 1635 1640 81.7 8.3 9.1640 1645 1651 1656 1661 1667 1672 1677 1682 1688 1693 81.6 8.4 9.1693 1698 1703 1709 1714 1719 1724 1729 1735 1740 1745 81.5 8.5 9.1745 1750 1755 1761 1766 1771 1776 1781 1786 1791 1797 81.4 8.6 9.1797 1802 1807 1812 1817 1822 1827 1832 1837 1842 1848 81.3 8.7 9.1848 1853 1858 1863 1868 1873 1878 1883 1888 1893 1898 81.2 8.8 9.1898 1903 1908 1913 1918 1923 1928 1933 1938 1943 1948 81.1 8.9 9.1948 1953 1958 1963 1968 1973 1977 1982 1987 1992 1997 81.0 9.0 9.1997 2002 2007 2012 2017 2022 2026 2031 2036 2041 2046 80.9 9.1 9.2046 2051 2056 2060 2065 2070 2075 2080 2085 2089 2094 80.8 9.2 9.2094 2099 2104 2109 2113 2118 2123 2128 2132 2137 2142 80.7 9.3 9.2142 2147 2151 2156 2161 2166 2170 2175 2180 2185 2189 80.6 9.4 9.2189 2194 2199 2203 2208 2213 2217 2222 2227 2231 2236 80.5 9.5 9.2236 2241 2245 2250 2255 2259 2264 2269 2273 2278 2282 80.4 9.6 9.2282 2287 2292 2296 2301 2305 2310 2315 2319 2324 2328 80.3 . 9.7 9.2328 2333 2337 2342 2346 2351 2356 2360 2365 2369 2374 80.2 9.8 9.2374 2378 2383 2387 2392 2396 2401 2405 2410 2414 2419 80.1 9.9 9.2419 IMHI^HMHM 2423 2428 2432 2437 2441 2445 2450 2454 2459 2463 80.0 9 8 7 6 5 4 3 2 1 O L. Cot. [109] iL.Tang. I ^^^^ 1 2 5 6 7 8 9 1 mmmmmmmmim "" ^BBHBBHHI ~ ^^ -oo 90 oo 7.2419 5429 7190 8439 9409 *0200 *0870 *1450 *1962 *2419 89 1 8.2419 2833 3211 3559 3881 4181 4461 4725 4973 5208 5431 88 2 8.5431 5643 5845 6038 6223 6401 6571 6736 6894 7046 7194 87 3 8.7194 7337 7475 7609 7739 7865 7988 8107 8223 8336 8446 86 4 8.8446 8554 8659 8762 8862 8960 9056 9150 9241 9331 9420 85 5 8.9420 9506 9591 9674 9756 9836 9915 9992 *0068 *0143 *0216 84 6 9.0216 0289 0360 0430 0499 0567 0633 0699 0764 0828 0891 83 7 9.0891 0954 1015 1076 1135 1194 1252 1310 1367 1423 1478 82 8 9.1478 1533 1587 1640 1693 1745 1797 1848 1898 1948 1997 81 9 9.1997 2046 2094 2142 2189 2236 2282 2328 2374 2419 2463 80 10 9.2463 2507 2551 2594 2637 2680 2722 2764 2805 2846 2887 79 11 9.2887 2927 2967 3006 3046 3085 3123 3162 3200 3237 3275 78 12 9.3275 3312 3349 3385 3422 3458 3493 3529 3564 3599 3634 77 13 9.3634 3668 3702 3736 3770 3804 3837 3870 3903 3935 3968 76 14 9.3968 4000 4032 4064 4095 4127 4158 4189 4220 4250 4281 75 15 9.4281 4311 4341 4371 4400 4430 4459 4488 4517 4546 4575 74 16 9.4575 4603 4632 4660 4688 4716 4744 4771 4799 4826 4853 73 17 9.4853 4880 4907 4934 4961 4987 5014 5040 5066 5092 5118 72 18 9.5118 5143 5169 5195 5220 5245 5270 5295 5320 5345 5370 71 19 9.5370 5394 5419 5443 5467 5491 5516 5539 5563 5587 5611 70 20 9.5611 5634 5658 5681 5704 5727 5750 5773 5796 5819 5842 69 21 9.5842 5864 5887 5909 5932 5954 5976 5998 6020 6042 6064 68 22 9.6064 6086 6108 6129 6151 6172 6194 6215 6236 6257 6279 67 23 9.6279 6300 6321 6341 6362 6383 6404 6424 6445 6465 6486 66 24 9.6486 6506 6527 6547 6567 6587 6607 6627 6647 6667 6687 65 25 9.6687 6706 6726 6746 6765 6785 6804 6824 6843 6863 6882 64 26 9.6882 6901 6920 6939 6958 6977 6996 7015 7034 7053 7072 63 27 9.7072 7090 7109 7128 7146 7165 7183 7202 7220 7238 7257 62 28 9.7257 7275 7293 7311 7330 7348 7366 7384 7402 7420 7438 61 29 9.7438 7455 7473 7491 7509 7526 7544 7562 7579 7597 7614 60 30 9.7614 7632 7649 7667 7684 7701 7719 7736 7753 7771 7788 59 31 9.778*8 7805 7822 7839 7856 7873 7890 7907 7924 7941 7958 58 32 9.7958 7975 7992 8008 8025 8042 8059 8075 8092 8109 8125 57 33 9.8125 8142 8158 8175 8191 8208 8224 8241 8257 8274 8290 56 34 9.8290 8306 8323 8339 8355 8371 8388 8404 8420 8436 8452 55 35 9.8452 8468 8484 8501 8517 8533 8549 8565 8581 8597 8613 54 36 9.8613 8629 8644 8660 8676 8692 8708 8724 8740 8755 8771 53 37 9.8771 8787 8803 8818 8834 8850 8865 8881 8897 8912 8928 52 38 9.8928 8944 8959 8975 8990 9006 9022 9037 9053 9068 9084 51 39 9.9084 9099 9115 9130 9146 9161 9176 9192 9207 9223 9238 50 40 9.9238 9254 9269 9284 9300 9315 9330 9346 9361 9376 9392 49 41 9.9392 9407 9422 9438 9453 9468 9483 9499 9514 9529 9544 48 42 9.9544 9560 9575 9590 9605 9621 9636 9651 9666 9681 9697 47 43 9.9697 9712 9727 9742 9757 9772 9788 9803 9818 9833 9848 46 44 9.9848 9864 9879 9894 9909 9924 9939 9955 9970 9985 *0000 45 45 0.0000 9 8 7 6 5 4 3 2 1 L. Cot. [110] L. Tang. O 1 2 3 4 5 6 7 8 9 0.0000 45 45 0.0000 0015 0030 0045 0061 0076 0091 0106 0121 0136 0152 44 46 0152 0167 0182 0197 0212 0228 0243 0258 0273 0288 0303 43 47 0303 0319 0334 0349 0364 0379 0395 0410 0425 0440 0456 42 48 0456 0471 0486 0501 0517 0532 0547 0562 0578 0593 0608 41 49 0608 0624 0639 0654 0670 0685 0700 0716 0731 0746 0762 40 50 0.0762 0777 0793 0808 0824 0839 0854 0870 0885 0901 0916 39 51 0916 0932 0947 0963 0978 0994 1010 1025 1041 1056 1072 38 52 1072 1088 1103 1119 1135 1150 1166 1182 1197 1213 1229 37 53 1229 1245 1260 1276 1292 1308 1324 1340 1356 1371 1387 36 54 1387 1403 1419 1435 1451 1467 1483 1499 1516 1532 1548 35 55 1548 1564 1580 1596 1612 1629 1645 1661 1677 1694 1710 34 56 1710 1726 1743 1759 1776 1792 1809 1825 1842 1858 1875 33 57 1875 1891 1908 1925 1941 1958 1975 1992 2008 2025 2042 32 58 2042 2059 2076 2093 2110 2127 2144 2161 2178 2195 2212 31 59 2212 2229 2247 2264 2281 2299 2316 2333 2351 2368 2386 30 60 0.2386 2403 2421 2438 2456 2474 2491 2509 2527 2545 2562 29 61 2562 2580 2598 2616 2634 2652 2670 2689 2707 2725 2743 28 62 2743 2762 2780 2798 2817 2835 2854 2872 2891 2910 2928 27 63 2928 2947 2966 2985 3004 3023 3042 3061 3080 3099 3118 26 64 3118 3137 3157 3176 3196 3215 3235 3254 3274 3294 3313 25 65 3313 3333 3353 3373 3393 3413 3433 3453 3473 3494 3514 24 66 3514 3535 3555 3576 3596 3617 3638 3659 3679 3700 3721 23 67 3721 3743 3764 3785 3806 3828 3849 3871 3892 3914 3936 22 68 3936 3958 3980 4002 4024 4046 4068 4091 4113 4136 4158 21 69 4158 4181 4204 4227 4250 4273 4296 4319 4342 4366 4389 20 70 0.4389 4413 4437 4461 4484 4509 4533 4557 451 4606 4630 19 71 4630 4655 4680 4705 4730 4755 4780 4805 4831 4857 4882 18 72 4882 4908 4934 4960 4986 5013 5039 5066 5093 5120 5147 17 73 5147 5174 5201 5229 5256 5284 5312 5340 5368 5397 5425 16 74 5425 5454 5483 5512 5541 5570 5600 5629 5659 5689 5719 15 75 5719 5750 5780 5811 5842 5873 5905 5936 5968 6000 6032 14 76 6032 6065 6097 6130, 6163 6196 6230 6264 6298 6332 6366 13 77 6366 6401 6436 6471 6507 6542 6578 6615 6651 6688 6725 12 78 6725 6763 6800 6838 6877 6915 6954 6994 7033 7073 7113 11 79 7113 7154 7195 7236 7278 7320 7363 7406 7449 7493 7537 10 80 0.7537 7581 7626 7672 7718 7764 7811 7858 7906 7954 8003 9 81 8003 8052 8102 8152 8203 8255 8307 8360 8413 8467 8522 8 82 8522 8577 8633 8690 8748 8806 8865 8924 8985 9046 9109 7 83 9109 9172 9236 9301 9367 9433 9501 9570 9640 9711 9784 6 84 0.9784 9857 9932 *0008 *0085 *0164 *0244 *0326 *0409 *0494 *0580 5 85 1.0580 0669 0759 0850 0944 1040 1138 1238 1341 1446 1554 4 86 1554 1664 1777 1893 2012 2135 2261 2391 2525 2663 2806 3 87 2806 2954 3106 3264 3429 3599 3777 3962 4155 4357 4569 2 88 4569 4792 5027 5275 5539 5819 6119 6441 6789 7167 7581 1 89 1.7581 8038 8550 9130 9800 *0591 *1561 *2810 *4571 *7581 00 90 CO 9 8 7 6 5 4 3 2 1 L. Cot. [Ill] L. Tang. 1 2 3 4 5 6 7 8 9 80.0 0.7537 7541 7546 7550 7555 7559 7563 7568 7572 7577 7581 9.9 80.1 7581 7586 7590 7595 7599 7604 7608 7613 7617 7622 7626 9.8 80.2 7626 7631 7635 7640 7644 7649 7654 7658 7663 7667 7672 9.7 80.3 7672 7676 7681 7685 7690 7695 7699 7704 7708 7713 7718 9.6 80.4 7718 7722 7727 7731 7736 7741 7745 7750 7755 7759 7764 9.5 80.5 7764 7769 7773 7278 7783 7787 7792 7797 7801 7806 7811 9.4 80.6 7811 7815 7820 7825 7830 7834 7839 7844 7849 7853 7858 9.3 80.7 7858 7863 7868 7872 7877 7882 7887 7891 7896 7901 7906 9.2 80.8 7906 7911 7915 7920 7925 7930 7935 7940 7944 7949 7954 9.1 80.9 7954 7959 7964 7969 7974 7978 7983 7988 7993 7998 8003 9.0 81.0 0.8003 8008 8013 8018 8023 8027 8032 8037 8042 8047 8052 8.9 81.1 8052 8057 8062 8067 8072 8077 8082 8087 8092 8097 8102 8.8 81.2 8102 8107 8112 8117 8122 8127 8132 8137 8142 8147 8152 8.7 81.3 8152 8158 8163 8168 8173 8178 8183 8188 8193 8198 8203 8.6 81.4 8203 8209 8214 8219 8224 8229 8234 8239 8245 8250 8255 8.5 81.5 8255 8260 8265 8271 8276 8281 8286 8291 8297 8302 8307 8.4 81.6 8307 8312 8318 8323 8328 8333 8339 8344 8349 8355 8360 8.3 81.7 8360 8365 8371 8376 8381 8387 8392 8397 8403 8408 8413 8.2 81.8 8413 8419 8424 8429 8435 8440 8446 8451 8456 8462 8467 8.1 81.9 8467 8473 8478 8484 8489 8495 8500 8506 8511 8516 8522 8.0 82.0 0.8522 8527 8533 8539 8544 8550 8555 8561 8566 8572 8577 7.9 82.1 8577 8583 8588 8594 8600 8605 8611 8616 8622 8628 8633 7.8 82.2 8633 8639 8645 8650 6856 8662 8667 8673 8679 8684 8690 7.7 82.3. 8690 8696 8701 8707 8713 8719 8724 8730 8736 8742 8748 7.6 82.4 8748 8753 8759 8765 8771 8777 8782 8788 8794 8800 8806 7.5 . 82.5 8806 8812 8817 8823 8829 8835 8841 8847 8853 8859 8865 7.4 82.6 8865 8871 8877 8883 8888 8894 8900 8906 8912 8918 8924 7.3 82.7 8924 8930 8936 8942 8949 8955 8961 8967 8973 8979 8985 7.2 82.8 8985 8991 8997 9003 9009 9016 9022 9028 9034 9040 9046 7.1 82.9 9046 9053 9059 9065 9071 9077 9084 9090 9096 9102 9109 7.0 83.0 0.9109 9115 9121 9127 9134 9140 9146 9153 9159 9165 9172 6.9 83.1 9172 9178 9184 9191 9197 9204 9210 9216 9223 9229 9236 6.8 83.2 9236 9242 9249 9255 9262 9268 9275 9281 9288 9294 9301 6.7 83.3 9301 9307 9314 9320 9327 9333 9340 9347 9353 9360 9367 6.6 83.4 9367 9373 9380 9386 9393 9400 9407 9413 9420 9427 9433 6.5 83.5 9433 9440 9447 9454 9460 9467 9474 9481 9488 9494 9501 6.4 83.6 9501 9508 9515 9522 9529 9536 9543 9549 9556 9563 9570 6.3 83.7 9570 9577 9584 9591 9598 9605 9612 9619 9626 9633 9640 6.2 83.8 9640 9647 9654 9662 9669 9676 9683 9690 9697 9704 9711 6.1 83.9 9711 9719 9726 9733 9740 9747 9755 9762 9769 9777 9784 6.O 84.0 0.9784 9791 9798 9806 9813 9820 9828 9835 9843 9850 9857 5.9 84.1 9857 9865 9872 9880 9887 9895 9902 9910 9917 9925 9932 5.8 84.2 0.9932 9940 9947 9955 9962 9970 9978 9985 9993 *0000 *0008 5.7 84.3 1.0008 0016 0023 0031 '0039 0047 0054 0062 0070 0078 0085 6.6 84.4 0085 0093 0101 0109 0117 0125 0133 0140 0148 0156 0164 5.5 84.5 0164 0172 0180 0188 0196 0204 0212 0220 0228 0236 0244 5.4 84.6 0244 0253 0261 0269 0277 0285 0293 0301 0310 0318 0326 5.3 84.7 0326 0334 0343 0351 0359 0367 0376 0384 0392 0401 0409 5.2 84.8 0409 0418 0426 0435 0443 0451 0460 0468 0477 0485 0494 5.1 84.9 1.0494 0503 0511 0520 0528 0537 0546 0554 0563 0572 0580 5.O 9 8 7 6 5 4 3 2 1 L. Cot. [112] L. Taiig. O 1 2 3 4 5 6 7 8 9 85.0 1.0580 0589 0598 0607 0616 0624 0633 0642 0651 0660 0669 4.9 85.1 0669 0678 0687 0695 0704 0713 0722 0731 0740 0750 0759 4.8 85.2 0759 0768 0777 0786 0795 0804 0814 0823 0832 0841 0850 4.7 85.3 0850 0860 0869 0878 0888 0897 0907 0916 0925 0935 0944 4.6 85.4 0944 0954 0963 0973 0982 0992 1002 1011 1021 1030 1040 4.5 85.5 1040 1050 1060 1069 1079 1089 1099 1109 1118 1128 1138 4.4 85.6 1138 1148 1158 1168 1178 1188 1198 1208 1218 1228 1238 4.3 85.7 1238 1249 1259 1269 1279 1289 1300 1310 1320 1331 1341 4.2 85.8 1341 1351 1362 1372 1383 1393 1404 1414 1425 1435 1446 4.1 85.9 1446 1457 1467 1478 1489 1499 1510 1521 1532 1543 1554 4.0 86.0 1.1554 1564 1575 1586 1597 1608 1619 1630 1642 1653 1664 3.9 86.1 1664 1675 1686 1698 1709 1720 1731 1743 1754 1766 1777 3.8 86.2 1777 1788 1800 1812 1823 1835 1846 1858 1870 1881 1893 3.7 86.3 1893 1905 1917 1929 1941 1952 1964 1976 1988 2000 2012 3.6 86.4 2012 2025 2037 2049 2061 2073 2086 2098 2110 2123 2135 3.5 86.5 2135 2148 2160 2173 2185 2198 2210 2223 2236 2249 2261 3.4 86.6 2261 2274 2280 2300 2313 2326 2339 2352 2365 2378 2391 3.3 86.7 2391 2404 2418 2431 2444 2458 2471 2485 2498 2512 2525 3.2 86.8 2525 2539 2552 2566 2580 2594 2608 2621 2635 2649 2663 3.1 86.9 2663 2677 2692 2706 2720 2734 2748 2763 2777 2792 2806 3.0 87.0 1.2806 2821 2835 2850 2864 2879 2894 2909 2924 2939 2954 2.9 87.1 2954 2969 2984 2999 3014 3029 3044 30GO 3075 3091 3106 2.8 87.2 3106 3122 3137 3153 3169 3185 3200 3216 3232 3248 3264 2.7 87.3 3264 3281 3297 3313 3329 3346 3362 3379 3395 3412 3429 2.6 87.4 3429 3445 3462 3479 3496 3513 3530 3547 3564 3582 3599 2.5 87.5 3599 3616 3634 3652 3669 3687 3705 3723 3740 3758 3777 2.4 87.6 3777 3795 3813 3831 3850 3868 3887 3905 3924 3943 3962 2.3 87.7 3962 3981 4000 4019 4038 4057 4077 4096 4116 4135 4155 2.2 87.8 4155 4175 4195 4215 4235 4255 4275 4295 4316 4336 4357 2.1 87.9 4357 4378 4399 4420 4441 4462 4483 45CH 4526 4547 4569 2.0 88.0 1.4569 4591 4613 4635 4657 4679 4702 4724 4747 4769 4792 1.9 88.1 4792 4815 4838 4861 4885 4908 4932 4955 4979 5003 5027 1.8 88.2 5027 5051 5076 5100 5125 5149 5174 5199 5225 5250 5275 1.7 88.3 5275 5301 5327 5353 5379 5405 5432 5458 5485 5512 5539 1.6 88.4 5539 5566 5594 5621 5649 5677 5705 5733 5762 5790 5819 1.5 88.5 5819 5848 5878 5907 5937 5967 5997 6027 6057 6088 6119 1.4 88.6 6119 6150 6182 6213 6245 6277 6309 6342 6375 6408 6441 1.3 88.7 6441 6475 6508 6542 6577 6611 664 6 6682 6717 6753 6789 1.2 88.8 6789 6825 6862 6899 6936 6974 7012 7050 7088 7127 7167 1.1 88.9 7167 7206 7246 7287 7328 7369 7410 7452 7495 7538 7581 1.O 89.0 1.7581 7624 7669 7713 7758 7804 7850 7896 7943 7990 8038 0.9 89.1 8038 8087 8136 8186 8236 8287 8338 8390 8443 8496 8550 0.8 89.2 8550 8605 8660 8716 8773 8830 8889 8948 9008 9068 9130 0.7 89.3 9130 9193 9256 9320 9386 9452 9519 9588 9657 9728 9800 0.6 89.4 1.9800 9873 9947 *0022 *0099 *0177 *0257 *0338 *0421 *0505 *0591 0.5 89.5 2.0591 0679 0769 0860 0954 1049 1147 1246 1349 1453 1561 0.4 89.6 1561 1671 1783 1899 2018 2140 2266 2396 2530 2668 2810 0.3 89.7 2810 2957 3110 3268 3431 3602 3779 3964 4157 4359 4571 0.2 89.8 4571 4794 5028 5277 5540 5820 6120 6442 6789 7167 7581 0.1 89.9 2.7581 8039 8550 9130 9800 *0592 *1561 *2810 *4573 *7581 -00 o.o 9 __ 6 5 4 -- L. Cot. 1 [113] TABLE VIII CONVERSION OF f " INTO DECIMAL PARTS OF A DEGREE V 0.016 11' 0.183 21' 0.350 31' 0.516 41' 0.683 51' 0.850 2' .033 12' .200 22' .366 32' .533 42' .700 52' .866 3' .050 13' .216 23' .383 33' .550 43' .716 53' .883 V .066 14' .233 24' .400 34' .566 44' .733 54' .900 5' .083 15' .250 25' .416 35' .583 45' .750 55' .916 6' .100 16' .266 26' .433 36' .600 46' .766 56' .933 7' .116 17' .283 27' .450 37' .616 47' .783 57' .950 8' .133 18' .300 28' .466 38' .633 48' .800 58' .966 9' .150 19' .316 29' .483 39' .650 49' .816 59' .983 10' .166 20' .333 30' .500 40' .666 50' .833 60' 1.000 1" 0.00028 6" 0.00166 10" 0.00277 2" .00056 7" .00194 20" .00555 3" .00083 8" .00222 30" .00833 4" .00111 9" .00250 40" .01111 5" .00138 50" .01388 TABLE IX CONVERSION OF DECIMAL PARTS OF A DEGREE INTO f " 0.01 0' 36" 0.11 6' 36" 0.21 12' 36" 0.31 18' 36" .02 . 1' 12" .12 7' 12" .22 13' 12" .32 19' 12" .03 1' 48" .13 7' 48" .23 13' 48" .33 19' 48" .04 2' 24" .14 8' 24" .24 14' 24" .34 20' 24" .05 3' .15 9' .25 15' .35 21' .06 3' 36" .16 9' 36" .26 15' 36" .36 21' 36" .07 4' 12" .17 10' 12" .27 16' 12" .37 22' 12" .08 4' 48" .18 10' 48" .28 16' 48" .38 22' 48" .09 5' 24" .19 11' 24" .29 17' 24" .39 23' 24" .10 6' .20 12' .30 18' .40 24' 0.41 24' 36" 0.51 30' 36" 0.61 36' 36" 0.71 42' 36" .42 25' 12" .52 31' 12" .62 37' 12" .72 43' 12" .43 25' 48" .53 31' 48" .63 37' 48" .73 43' 48" .44 26' 24" .54 32' 24" .64 38' 24" .74 44' 24" .45 27' .55 33' .65 39' .75 45' .46 27' 36" .56 33' 36" .66 39' 36" .76 45' 36" .47 28' 12" .57 34' 12" .67 40' 12" .77 46' 12" .48 28' 48" .58 34' 48" .68 40' 48" .78 46' 48" .49 29' 24" .59 35' 24" .69 41' 24" .79 47' 24" .50 30' .60 36' .70 42' .80 48' 0.81 48' 36" 0.91 54' 36" 0.001 3.6" .82 49' 12" .92 55' 12" .002 7.2" .83 49' 48" .93 55' 48" .003 10.8" .84 50' 24" .94 56' 24" .004 14.4" .85 51' .95 57' .005 18 " .86 51' 36" .96 57' 36" .006 21.6" .87 52' 12" .97 58' 12" .007 25.2" .88 52' 48" .98 58' 48" .008 28.8" .89 53' 24" .99 59' 24" .009 32.4" .90 54' 1.00 60' [114] ANSWERS Exercise 1 1 . loga 9 = 2. logs 27 = 3. Iog 4 64 = 4. = 4. logic > = 1. log .01 = - 2. 2. Iog 2 32 = 5. log 2l fe=-5. Iog 4 8 = |. 3. 1. 9. 4/64=4. logio .001 = 3. ^4096 = logs | = -2. Iog 8 16 = Exercise 2 1. 2. 3. 2. 5. 0. 7. 0. 9. - 3. 11. 0. 13. -4. 15. 1. 2. 4. 4. 1. 6. -2. 8. 0. 10. -5. 12. 3. 14. 2. 16. 3 = 4. 2 = 3. 5 = 6. 1=2. = 1. 4 = 5. 8-10 = 1. 7-10 = 2. 9 - 10 = 0. Exercise 3 1. 3.88235. 8. 1.82751. 15. 1.93952. 22. 8.27135-10. 2. 3.82737. 9. 0.52410. 16. 9.88081 - 10. 23. 4.51427. 3. 1.91381. 10. 7.82737-10. 17. 6.09691-10. 24. 3.51427. 4. 3.89553. 11. 4.84510-10. 18. 2.00109. 25. 2.51427. 5. 1.87506. 12. 5.60206. 19. 1.24622. 26. 1.51427. 6. 2.19590. 13. 1.16505-10. 20. 1.62325. 27. 0.51427. 7. 4.55965. 14. 7.35550. 21. 4.0000-10. 28. log 200 = 2. 30103. log 3000 = 3.47712. log 50 = 1.69897. log!007r = 2.49715. log 20 = 1 1.30103. log .002 = 7.30103 - 10. log 30 = 1.47712. log .0005 = 6.69897 - 10. log = 8.49715 - 10. log luu .3=9.47712-10. log .2 = 9.30103 - 10. log 10 TT = 1.49715. log 20000 = 4.30103. 29. 1.1028. 35. .0011. 40. 2.9847. 45. 4.4619. 30. 2.8824. 36. 1.3923. 41. 0.1666. 46. 1.2916. 31. 1.6302. 37. 9.0459-10. 42. 0.2462. 47. 9.9358 - 10. 32. .0887. 38. 1.0676. 43. 5.5655-10. 48. 8.0012 - 10. 33. 8.4200 10. 39. 7.1030-10. 44. 7.4213-10. 49. 0.3474. 34. 7.1030- 10. Exercise 4 1. 26.22. 11. 221.705. 20. 25.6. 29. 454.44. 2. 157.6. 12. .01569. 21. 541. 30. .0000022337. 3. 9.627. 13. 10.88375. 22. 1712. 31. 657.166. 4. 48323333.3. 14. .50742. 23. .14277. 32. 201.409. 5. .16719. 15. 1647.3. 24. 107.8. 33. .3625. 6. .00026827. 16. 1008581.4. 25. 10.315. 34. 9.6968-10. 7. 3896545.45. 17. .78488. 26. .0106725. 35. 3.1443. 8. .000055855. 18. 96988. 27. .001309. 36. 49.25. 9. 100925581.4. 19. .69781. 28. .000010044. 37. .2285. 10. .37029. 4 ANSWERS Exercise 5 1. 53295. 4. 8.3552. 7. 1.492. 10. .96518. 13. -.34526. 2. 1383.62. 5. 514.055. 8. .01141. 11. -1.8583. 14. $33945. 3. 211820. 6. 19.033913. 9. 5.3921. 12. - .059439. 15. $491.04. 200 ,oi c |i 100 *" 5 4165 300 x 500 47?46 67 376 58 . 7T 18. 1.3774 A., 3.4435 A., 45.9134 A. 19. 33.38. 21. .4171. 23. 3261. 25. 3.908. 27. .0939. 31. $325.60. 20. 6.727. 22. 2034.3. 24. 1. 16467. 26. 3.413. 30. $213.47. 32. $5874.75. Exercise 6 1. .972. 9. 2.34667. 19. .11069. 29. 6080000. 2. 99.266. 10. - .0447. 20. 2519.6. 30. 4.245. 3. 8.9254. 11. - 1.5793. 21. 7061.67. 31. 17.49. 4. .182916. 12. 24.1394. 22. 65.97 = 66 yr. 32. 1.272. 5. 1602.4 13. 19.85. 23. .5342. 33. .4163. 6. 2.37242. 14. 24.035 24. 1.6167. 34. 12.07. 7. 218.51. 15. 189.66. 25. 1.1377. 35. 5.77. 6.6943. 16. .12246. 26. 22.33. 36. 2316.8. 7.1845. 17. 13306.06. 27. 10695. 8. 500 m. = 1640.5 ft. 7294 m. = 23931.11 ft. 1029.4. 28. .1705. 300 m. = 984.26 ft. Exercise 7 1. 2.544. 6. .65959. 11. - . 4167. 16. -f. 21. 25. 2. 1.2445. 7. -29.78. 12. .29414. 17. -3. 22. /,. 3. 2.495. 8. 5.9837. 13. 3. 18. -4. 23. 32. 4. -.053474. 9. - .46187. 14. 5. 19. 2. 24. 17.677. 5. 1.465. 10. .64509. 15. -2. 20. 81. 11.894. 25. 5%. Exercise 8 1. sin B = b b sec 5=^-, cos5=7 cot 5= csc5=. c' a' a c 2. sin A = f, tan ^4 = |, sec A = |, cos .4 = f, cotJ.= |, csc^4=f. 3. sin A = , tan^l = t, sec^l=f, cos ,4=f, cot .4= |, csc^4.=f. 4. sin A= T 8 y, tan A = -jTf , sec A = \l, cosvl = !f, cotJ.= J/, csc^l= J ^. 5. sin ,4 = if, tan .4=^, sec^=V, cos A = j 5 ^, cot A = r \, 080^=^1. 6. sin .4 = ||, tan4=ff, secvl = -|, cos^4 = f|, cotA = |f, CSC^l = ^f. 7. sin ^4 = 9 T , tan A = %, sec Afa cosA=$, cotA = - 4 ^ ) -, csc ^4.=^-. 8. sin A = Mf, tan^4=ii, secA = \%% , cos.4=^!, cot .4 = iff, CSCu4 = Hf. 9. III. sin 5=|, tan 5=f, sec5=f, cos 5=f, cot 5 = |, csc5=f. IV. sin 5= if, tan 5=^, sec5=Y- cos 5 = yy} cot 5 = r 8 5 , csc5=4;|. V. sin 5 = T 5 3, tan5= T \, sec5=}|, cos 5 = 4~? , cot 5 = - 1 /, csc5= 1 - 5 3 - VI. sin 5 = |f, tan 5= f$, sec 5= ft, cos 5= If, cot 5= ff, esc 5= ff. VII. sin 5= !, tan 5=^-, sec5=-V, cos5= \, cot 5= 3 %, csc 5=|. VIII. sin 5= if a, tan 5 -fff, sec 5= if \ ;, cos 5= 1 f, cot 5= i^f, csc 5= iff. 10. (1) (2) 1. (3) 1. (4) (5) 1. (6) 1. (7) 0. (8) 1 ANSWERS 22. AD = 218.4. sin A = p* + q 2 28. 30. p + q 9 Cl> = 358.7. 23. .854. cos A = DB = 181.3. 24. 56.75. 5 = 283.86. p + q smA = \ m ^ secA=^ + n l tan A = 2 mn m 2 -TO 2< 31. 32. 33. 34. 35. sin *==-5, w + cos 5 = TO + 71 sin .B=fff , tan 5= ^, sec B=-\% sin -<4=f \/5, tan ^4=2, sec^l=V5, =-, tan ^1= W TO =|5, cot -A=, cscJB=||f. CSC 4=^? 2 ' 36. 6 37. 38. 41. 1. 2. 3. 6 7 7 sin ^4 = ||, tan A = -^ , sec A = -^ , cos A = T \, cot vl = & , esc ^4 = if. 1.62. 42. f, f. Exercise 9 cos 30. 4. tan 34 24'. 7. cos 1'. sin 75. 5. sec 68 35' 30". 8. sin 88 42'. cot 24 36'. 6. esc 5 44'. 9. V3. Exercise 10 sin A = if, sec A = J^, sin J.=i*, 10. 4. 11. * y 12. p. esc A = Vm 2 + 1. 6. sin A = 7. tan A 8. sin A 9. sin A 10. tan .4 11. sin A 12. tan x = ^ tan A 5 : 0, sec A 1, tan A 0, sec A co , sec A tan^l V5 IT 1 cosA = 1, sec A = oo , cos -4 = 1, cos A = 0, = 0, 1 cos A 2V5 cot A co , cot .4 = 0, cot .4 = co, cot A = 0, cot A = 0, cot A = 2. csc .4 = oo. csc .4=1. csc A = co. csc -4=1. csc A = 1 . cot x = 13. sin A 14. sin A 15. sin .4 16. sin A = 17. tan A 18. sin^l 19. sin A = 2 ' "" 2 ' 21. sin x = 0, tan x = 0, sec x = 1 , cot x = co , csc x = oo . tan A = 4^, sec ^1 = ^-, cos A = ? 9 T , cot A = ? 9 ,j, ni 2 + ri 2 csc J. = J*,. cos .4 = ^-^, cot^ = ?>i 2 + n 2 2 mn 2 mn sec J. = tan A = V2 1, = V4 + 2V2. cos ^4 = 0, cot .4 = 0, csc .4=1. 26. sin 22 1 = V2 - V2, cos _ V2 + V2 ANSWERS 7 29. sin A = Vl - 7f 2 , , , , tan 15 = 2 -V3, cos 15 = 30 sec 15 = 2 V2 - V3, esc 15 = 2 V2 + V3. 31. cos ^1 = Vl - sin 2 .4, tan ^ = 32. sin A = l- cos 2 ^4, tan ^ = Vl - cos 2 A sec A = -- , esc A = cos A Vl - cos 2 A 33. sin A=- ^__, cos.4= -,cotA = Vl -|- tan 2 A Vl + tan 2 A , 2 ^, esc A = tan^l 34. tan^4 = , esc A = Vl + cot 2 ^4, sin A = Vl + cot 2 A cot -A i . . , tan A = Vsec 2 A 1, cot^l= Vsec 2 A - 1 Vsec 2 A 36. . esc A Vcsc 2 ^i - 1 sec A = - gg^ , cot ^1 = Vcsc 2 ^- 1. Vcsc 2 ^1 1 37. cos A = 1 vers A, sec .4 = 1-versJL' 1 vers A 1 versJ. v 2 vers .4 vers 2 A sin ^4 = V2vers A vers 2 ^4, esc A = V2 vers J. - vers 2 A 38 - 1 - A- 39. rf^ViSTS. 43. . 40. W3. 44. 4V42. 48. 2 sin 2 x + sin x = 1. 41. 3 8 ? V39. 45. 1- cos 2 ,4 + cos A 49. tan 2 x 2tanx = 1. ANSWERS Exercise 12 13. 2 i- 17. -1-V2. 22. iV6. 36. 150; 259.8. 14. iV3(& + c). 18. -i- 23 . 5. 38. 961. 3+. 15. 2+V2. 20. (v"2 1). 35 . 86.6. 39. 165. 16. 1-2V3. 21, *- Exercise 13 1. 60. 4. 60. 7. 45*. 10. 60. 13. 60. 16. 30. 19. 60 2. 60. 5. 0. 8. 457/ 11. 45. 14. 30. 17. 45. 20. 90. 3. 30. 6. 45. 9. 3Q?/ 12. 30, 90. 15. 45. 18. 45. 21. 0. 22. 27 13' 12". 26 . 22. 28. 18. 33. 30. 23. 15. 27 90 2 9. 45. 34. 60. 24. 10. _1_ 1 30. 38 50'. 35. 30. 25. 60. Exercise 14 1. 9.64647 - 10. 9. 8.95017 - 10. 19. 6.1493. 26. 9.9523 - 10. 2. 9.98997 - 10. 10. 9.97991 - 10. 20. 14.991. 27. 0.3076. 3. 9.86603 - 10. 11. 0.11532. 21. 9.4214 10. 28. 0.6489. 4. 9.38699- 10. 12. 9.99194 - 10. 22. 9.8297 - 10. 29. 9.8832 - 10. 5. 0.15908. 13. 1.24820. 23. 0. 175t). 30. 0.2522. 6. 9.43707 - 10. 14. 8.91931 - 10. 24. 0.7033. 31. 0.6413. 7. 8.73767 - 10. 15. 9.84324 - 10. 25. 9.6622 - 10. 32. 15.24. 8. 9.86126 - 10. 16. 9.74610 - 10. Exercise 15 1. 23 15'. 8. 85 5' 15". 15. 28.7. 21. 61.07. 2. 28 40'. 9. 65 10' 20". 16. '18.5. 22. 0.541. 3. 35 43'. 10. 5 20' 29". 17. 56.26. 23. 88.465. 4. 40 23'. 11. 40'47". 18. 70.14. 24. 65.67. 5. 66 15' 24". 12. 85 59' 13". 19. 64.43. 25. 78.14. 6. 70 16' 21". 13. 26.5. 20. 46.11. 26. 14.47. 7. 70 0' 26". 14. 50.2. Exercise 16 1. 8.21421 - 10. 14. 4' 31". 27. 8.1238 - 10. 40. 4.662. 2. 8.34812 - 10. 15. 2' 39". 28. 8.1070 - 10. 41. 84.35. 3. 8.49128 - 10. 16. 89 45' 6". 29. 8.2701 - 10. 42. 8.3638 10. 4. 1.72220. 17. 42 5' 26". 30. 1.6657. 43. 1.6362. 5. 1.64078. 18. 8252'1". 31. 1.8744. 44. 89.266. 6. 8.18538- 10. 19. 83 24' 25". 32. 8.3446 - 10. 45. .613. 7. 8.28456 - 10. 20. 017'7.3". 33. 7.9686 - 10. 46. 89.285. 8. 8.47866- 10. 21. 17' 7.1". 34. 89.266. <47. .624. 9. 26' 10". 22. 89 54' 15". 35. 1.036. 48. 1.6375. 10. 88 53' 6". 23. 8.245. 36. 89.216. 49. 2.792. 11. 42' 53". 24. .1504. 37. .634. 50. 112.82. 12. 89 32' 27". 25. 1.6687. 38. 89.553. 51. .7348. 13. 89 57 '. 26. 8.3353 - 10. 39. .507. 52. .026694. ANSWERS 9 Exercise 17 1. Sine 4 = T 8 r . Cosine 4 = ^f . Cotangent 4 = J / Secant 4 = {|. Cosecant 4 = ^. 6 = 30. c = 34. 2. 5 T 6^. 8. cot 37 > tan 37. 22. 1. 5. sin 49 > cos 49. 19. x = 45. 23. fVS \/2 f. 6. 4<45. 20. ic = 60. . 11-3V3 7. 4>60. 21. x = 45. 2 25. cot 4 = ^, esc 4 = ^ 26. *?. 27. .3056. 28. 300. 29. 270.12 r - Exercise 18 4. 5 = 62. 7. B = 61 43'. 10. 5 = 51 43' 36". a = 6.3804. a =11. 448. a = 2.2478. c = 13.591. 6 = 21.276. b = 2.849. . 5. 5 = 12. 8. 4 = 35 17'. 11. 4 = 17 43' 18". a = 26:15. a = 648.67. 6 = 70.985. b = 5.5585. 6 = 916.7. c = 74.5217. 6. 5 = 43 42'. 9. 4 = 52 41'. 12. .23661. a = 50.78. a = 385.436. 13. .282726. c = 70.24. c = 484.644. 14. B = 26 31' 20". 15. 4 = 2 43' 30". 16. 5 = 38 50' 54". b = 127.976. a = 13.85129. a = .153254. c = 286. 5875. b = .674616. b = .12343. 17. B = 63 41 '24". 18. .96565. 6 = 256.406. 19. 164.93. c = 286.033. 20. 1416.13. 21. 1614.26 yd. = depth of canon. 5521.125 yd. = distance of river. 24. 5=57.4. 30. 5 = 68.68. 39 '. 352.1. a = 11.5125. b = 41.65. 41 . 5=60. c = 21.37. c = 44.71. a = V3 = 4.0425. 25. 5 = 34. 31. 4 = 23.73. c = i/ V3 = 8.083. a = 2.22. a = .003824. 4g . a = b = 6V2 = 8.484. b = 1.4976. c = .009504. 26. 4 = 51.8. 32. .3907. c ^\/3 = 28.86. a = .604. 33. .11388. 44 b = iAJLQ-v/3 = 577 4. b = .4753. 34. 50.933. 27. 4 = 7.5. 35. 5 = 1.83. 45 & 1.2^0^3-1154 8* b = 95.42. a - 13.125. c = i.^LQ-v/3 = 2309.5. c = 96.225. b = .4194. 46 . a = 600 V3 = 1039.25. 28. 5 = 62.33. , 36. 4 = 47.84. 5 _ goo. a = 77.43. & = 52.33. ? 29 4 = 13.75. b = .4757. 4- c = .7086. 37. 129.15. . a =200. c = 200V2 = 282.8. 6 = 3.7845. H . L J-lJ Ct c = 3.89583. 38. 1.081. & = 10dV3= 17.32 d. 10 ANSWERS 49. Same as the respective answers for numbers 6 and 7 in this exercise. 51. Z>5 = 50. BC = 25. DC = \ 5 -VS = 21.65 x. Exercise 19 1. A = 35 33' 27". 16. B = 17 56' 5". 31. 50.43. 6 = 14.969. b = 8.6188. 32. A = 18.96. 2. A = 33 18' 3". 17. 13 7' 18". a- 50.91. 6 = 31.147. 18. Z = 67 22' 48", 33. B = 7.812. 3. A = 42 24' 43". .-. 7' 12" too small. b = 117.166. b = 29.2557. 21. A = 41.49. 34. 57.26. 4. A = 39 48' 20". b = 17.755. 35. 26.77. c = 7.81016. 22. ^ = 45.17. 37. ^4 = # = 45. 5. A = 49 44' 5". a = .39855. c = 13V2 = 18.384. b = .579587. 23. A = 50.66. 38. ,4 = 30. 6. .4 = 49. c = 43.04. b = 9V3 = 15.888. a = 16.3608. 24. A = 32.02. 39. JB = 30. 7. A = 52 12' 25". c = 9.432". a=100V3 = 173.2. c = .079471. 25. A = 46.31. 40. J5 = 30. 8. .4 = 43 52'. a = 7.015. c = 2. b = .184875. 26. ^ = 48.43. 41. ^ = 60. 9. 53 7' 48". c = .19107. 6 = 3. 10. 21 53' 58". 27. ^1 = 40.67. 42. ^4 = 45. 11. 42 24' 39". a = 86.64. 6 = 1. 12. c = 8.48. 28. A = 40.95. 43. .4=60. 13. 25 48' 40". 6 = .0839. 6 = 50. 14. B = 16 11' 7". 29. A = 52.33. 44. ^1 = 30. b = 32.702. c = 2987.33. a = 6. 15. A = 8 31' 31". 30. A = 43.44. c = 12. a = 53.666. Exercise 20 1. Leg = 120. 8. Base Z = 46 16' 41". Vertex Z = 60. Vertex Z = 87 26' 38". 2. Base = 353.87. Leg = 6690. 16. 3. Base = 9.6837. 9. r = 8.2583. Vertex Z = 67 24'. R = 10.208. 4. Leg = 50.699. Perimeter = 60. Base = 79.578. Area = 247.75. Vertex Z = 103 24' 20". 10. r= 1.5388. 5. Vertex Z = 69 23' 12". R= 1.618. Leg = 927.84. Perimeter = 10. Base = 1056.225. Area = 7.694. 6. Leg = 8.8204. 11. Side = 8.282. Base Z = 62 10'. r = 15.455. Vertex Z = 55 40". Area = 768. 7. Base Z = 33 21' 30". 12. Side = 9. 112. Leg = .075978. r = 17. Area = 929.24. ANSWERS 11 13. 14. 15. 16. Side = 8.6524. r = 5.9546. Perimeter = 43.262. Area = 128.8. Perimeter = 4.70498. Area = 1.6417. h = IsmD. ra = 2 1 cos D. C = 180 - 2 D. 17. 18. 19. sin D = " . cos - C = - 2 I m = C = 180 - 2 D. h = i m tan Z>. Z = -i D = 90 - A C. fc = - 2 2 V */ I esc C. m 2.. 2 tan ^ ( ' ~2~7T 20. Base = 3.889. BaseZ = 42 15' 34". Vertex Z = 95 28' 52". 21. 12.7001. 34. Base = .0588. 41. Side 9.318. 22. 95.94. Leg = . 12027. r 5= 17.387. 23. 15.1848. 35. BaseZ = 54.275. Area = 972. 26. 8.1183. Leg = 26.77. 42. 22.025. 27. 48.2055. 36. Base = .8462. 43. 111.4. 28. Base = 61.86. BaseZ = 14.15. 44. Altitude 2 /-\/3 Vertex Z = 114.8. 37. r = 16.9. 14.435. 29. Leg = 2081.5. Area = 946.5. BaseZ E= 30. Vertex Z = 45.2. 38. Perimeter = 143.166. 45. BaseZ 3= 30. 30. Leg = 34.47. 39. Side = 1.0878. Base = 50 V3 Base = 59.026. r = 1.6737. = 86.6. BaseZ = 31.14. 40. Side = 20.22. 46. 2 T- ^3 11.547 31. BaseZ = 52.86. r = 21. 3= leg = base. Leg = .61014. E = 23.3. 47. BaseZ = 45. 32. BaseZ = 61.1. Area = 1486.34. Vertex Z = 90. Base = 124.4. Perimeter = 141.54. Altitude 6. 33. Base = 114.2. 48. 120. Vertex Z = 114.54. 49. 7.07. Exercise 21 1. 12560.57. 2. 5911.7. 9. b = 3.416. c = 4.2881. A = 36 11' 53". 10. a = 2.67815. b = 5.41875. c = 6.0445. 11. a = 13.1945. b = 8.4405. A = 57 23' 36". 12. 42.847. 3. 172.756. 5. 3122. 7. .19936. 4. 545.44. 6. 21519.5. 8. 202281.818. 13. .088996. 19. I = 7.1773. 14. .0287326. c = 12.299. 15. 244.79. h = 3.7011. 16. 300.61. 20. .7723. 17. h = 5.2496. 21. 9.58675. I = 6.1403. 22. 1.5458. A = 58 45' 17". 23. .8874. 18. 1 = 1.5086. 24. fi = 3.22046. c = 2.6811. c = 2.2029. h = .69175. r = 3.0263. 12 ANSWERS 25. Perimeter = 21.265. 42. 151.4. 54. 72 = 18.34. 26. p = 23.181. 43. 80.8. c = 10.3332. R = 3.9448. 44. .2084. r = 17.6. 28. 938. 45. h = 8.828. 55. 22 = 4.031. 29. 47577. A = 22.03. c = 2.7575. 30. 882. I = 23.54. r = 3.788. 31. .01618. 46. I = 1.2351. 56. 101.36. 32. 31.47. ft = .7478. 57. 2886.8 = ift-jp V3. 33. 137.33. c = 1.9656. 58. 180000 \/3 = 301760. 34. 6000000. 47. I = 54.51. 59. 298.78. 35. .00003529. c = 91.06. 60. 4050 v/3 = 7014.6. 36. a = 8.283. h = 30.04. 61. 3200 V3 =5542.4. A = 52.44. 48. c = .8598. 62. 800. c = 10.45. h = .2384. 63. 2000000^/3 = 3464000. 37. c = 77.22. .4=29. 64. 7200. a = 68.9. 49. 58.75. 65. 2500 V3 = 4330. 6 = 34.84. 50. .8308. 38. Impossible. 51. 36950. 66. mono v/3 = 5773.3. 39. .13833. 52. 15.172. 67. 400 V3 = 692^. 40. 149.07. 53. H = 2.262. 68. 80,000. 41. 4813.3. c = 1.9624. r = 2.038. Exercise 22 In this exercise, where two answers are given to an example, the first is the result obtained by use of five-place log tables, and the second answer is the result obtained by use of four-place tables. 1. 389.7 = Ht. 9. 695.414. 19. 23.013. 2. 474.788. 695.35. 23.012. 474.8. 10. 17 31' 7". 20. 5246.25. 3. 114.1. 17.52. 5246.6. 4. 10 33' 25". 11. 82.056. 21. 43.3 = ht. of tree. 10.56. 82.06. 25 = width of river. 5. 491.511. 12. 287.25. 22. KR = 12. 491.44. 287.47. HP = 6 V3~= 10.392. 6. Base = 76. 79. 13. 231.7. It $ =6^/6 = 14.694. Base = 76.8. 231.68. /ST= 12V3 = 20.784. Alt. =49.6955. 14. 1534.96. SF = 24. Alt. =49.7. 1535. TF = 12. Area = 1908.5. 16. Ht. of hill 1673.038. 23. 13.071. Area = 1908.08. Ht. of hill 1673.67. 13.053. 7. 37 58' 46". Dis. of ship 621 5. 143. 24. 71.264. 37.975. Dis. of ship 62 15. 7. 71.28. 8. Distance of ladder 17. KR = 12 V3 = 20.784 . 25. 616.771. from house = 12.588. KA = 24. 616.5. 12.58. KT = 6 V3 = 10.392. 26. 450'37". Z. ladder makes with HT = 18. 45. house = 30 14' 8" FT = 18 V3 =31.176 50.6375. = 30.22. 2tF = 36. 50.62. ANSWERS 13 27. AB = sin y. OB = cos y. BO sin x cos y. OC = cos x cos y. 29de = 26.0.9 A = 108 14' 40" 108.26. 71 45' 20". 71.74. Exercise 23 1. 2. 3. 3. 5. 4. 7. 4. 9. 3. 11. 1. 13. 2. 2. 4. 4. 6. 1. 8. 3. 10. 3. 12. 2. 14. 16. (1) Same as the signs of the functions in the second quadrant. (3) Same as the signs of the functions in the third quadrant. (5) Same as the signs of the functions in the fourth quadrant. 15. 4. 17. 21. 27. 34. 385. 18. 745. - 335. - 695. 65. 22. 60. Second. Thirf 8.052 (by use of five-place tables). 55.73. 330. 19. 460. 690. . 820. - 390. - 260. - 750. - 620. 23. 60. 24. 155. 29. Second. 30. Third. 20. 260. 620. - 460. - 820. 25. 40. 31. Fourth. 32. Second. 8.06 (by use of four-place tables). 26. 53 C 1. 2. 2. oo. Exercise 24 3. 0. 5. 4. 4. c 2 - a 2 -f 4 ac. 6. 2 a. 7. 0. 8. 3m. 1, sin 390 =1. cos 390 = tan 390 = sec 390 = f V3. 2. cos 780 = . tan 780 = V3. sin 780 = cot 780 = 4. 5. 6. sm =: cos = |. tan = V3. cot = -V3. sin = i. cos = tan = cot = V3. sin = cos = tan = 1. cot = 1. 7. 10. 11. Exercise 25 sin = i. cos = \ V3. tan = \/3. cot = VS. sm = cos = tan = cot = sin cos = tan cot = sinx = tanx = cotx = secx = CSC X = sin x = cos x = cotx = v. 1V3. JV2. iV2. 1. 1. 1- Tf 1- if. T ,V- 12. 13. 14. secx cscx = COSX =: tan x secx = cotx = cscx sin x = _ 13 x/5 5 COS X = tan x = \. cot x = 2. 5 sec x = v - -A/5. sm x = Vm 2 -l cos x = m 14 ANSWERS cscx = sec x = VIO. Vio 3 V35 6 CSC X = sin x = 18. sin y = ^V5. csc y = | VS. 19. sin x = -i. V3 COS X 2 Vw 2 -l 15. sin x = 10 cosx = VlO 10 = -. cot x = 4. 1. -J. 2. J. 3. 10. --^+- 5 . 11. - 12. sin 38. 13. -tan 17. 14. sin 40. 15. sec 5. 16. tan 5. 34. a cos x -f b sin x cos x = I . tan x = V35. sec x = 6. 6V|5 35 tan x = CSC X = cotx = V35 cot x = VS. V3 3 secx = 2^. csc x = 2. 21 - VS. 6. - 1. 10. sin : COS : tan: COt : sec: csc 11. sin cos tan cot csc sec 12. sin; cos tan cot sec csc 2. VS. 3. : COS 29. : - sin 29. : COt 29. : tan 29. : - CSC 29. i -sec 29. i - cos 9. i sin 9. : COt 9. = - tan 9. = sec 9.' = csc 9. - sin 15. = cos 15. = - tan 15. = - cot 15. -. sec 15. = csc 15. Exercise 26 4. -VS. 5. -V2. 17. -tan 45. 18. - sin 20. 19. -sin 27. 20. -cot 25. 21. sec 30. 22. -sin 27. 23. cot 22. 24. -cos 10 16'. c tan x. 36 (a + &) cos x (a 6) Exercise 27 _i. 4. _V3. 5. -VS. 6. 0. 13. sin = sin 15. cos = cos 15. tan= tan 15. cot = cot 15. sec = sec 15. csc = csc 15. 14. sin = cos 17. cos = sin 17. tan = - cot 17. cot = - tan 17. sec = csc 17. csc = sec 17. 15. sin = cos 10. cos = sin 10. tan = cot 10. cot = tan 10. sec = csc 10. csc = sec 10. -1. M- V/8. 8. -f. 9. -J. 25. - cot 30 17'. 26. - sec 25. 27. sin 8. 28. - tan 20. 29. - cot 30. 32. 9|. 33. 11 cosx. 35. p sin x cos x. sinx. 7. -2. 8. i V. 9. -iV2. 16. sin = sin 0. cos = cos 0. tan = tan 0. cot = cot 0. sec = sec 0. csc = csc 0. 17. sin = sin 36 43'. cos = - cos 36 43'. tan =- tan 36 43'. cot = cot 36 43'. sec = - sec 36 43'. csc = csc 36 43'. 18. sin = cos 37. 24. cos = sin 37.24. tan = cot 37.24. cot = tan 37.24. sec = csc 37.24. csc = sec 37. 24. ANSWERS 15 21. 22. 28. 29. COS X. COS X. a cos x + ?>l COS J[ - 23. sinx. 25. 24. t&nx. 26. b sin x c tan x. 30. -# cot .4 g cot A. 31. Exercise sec x. sec x. sin 2 x cos x. COS X. 28 27. 3 cos x. 1. 30, 150. 5. 30, 150. 9. 45, 225. 2. 30, 150, 210, 330. 6. 60 J , 300, 180 . 10. 60, 240. 3. 45, 135, 225, 315. 7. 30, 150. 11. 45 C , 225. 4. 30, 150, 210, 330 D . 8. 45, 225. 12. 45, 135, 225, 315. 13. 30, 150, 45, 225 D , 315. 15. 30, 150, 210, 330, 14. 60, 120, 240, 300, 60, 120, 240, 300. 45, 135, 225, 315. 16. 30, 150, 210, 330. 17. 30, 150. 18. 30, 150. Where two answers are given, the first answer is found by the five-place tables, the second answer is found by the four-place tables. 19. 66.35 mi. east. 66.34 mi. east. 27.14 mi. north. 20. 39 10' 25". 39.18. 21. 760.316. 760.33. 22. Distance of the spring from the house = 217.39. 217.4. Distance of the spring from the barn =229.12. 229.16. 1. 2. 3. 8. 14. sin(x + 2/) = f| sin (x - y) = f cos(x-*/)=. sin (x + 45) = cos (30 x) Exercise 29 ;. 4. (x + y) = co . |. 5. cot (x y) = 0. 1- c V6-V2 ' 90 = 1. 90 =0. 1 4 , (sin x + cos x). 7. 2 + V3. sin (x - 60) = V6 4- V2 2 slnx cosxV3 2 ,/Q in V6-V2 tt V2-V6 12. sin 4 tan ("45 -4- v^ 4 4 cos 1+tany 15 cot / 60 o - N \/3cot 2 y-4coty '1-tany 3cot 2 y-l 1 + tany C t(3 ' y) cot 2 2/-3 ' Exercise 30 1. cos 60 = $. 2. tan 60 = V3. 3. 9. 3 sin x 4 sin 3 x. 10. 4 cos 3 x - 3 cos x. ii 3 tan x- tan 3 a; t l 3 tan2 * 13 - - 14. -/ ? . 21. cos4x + cos 2 a; -f f. 16 2. sin 15 = i V2 V3 = .2588. tan 15 = 2 A/3 = .2679. cos 15 = i A/2 + V3 = .9659. 3. cot 221 = V2 + 1 = 2.4142. cos 221 = i V2_+V2 = .9239. sin 221 - i V2 - \/2 = .3827. 4. sin45 = cos45=:i-\/2 = .7071. tan 45 = cot 45 = 1. sec 45 = esc 45 = v/2 = 1.4142. ANSWERS Exercise 31 6. cos - = _ V2 + 2 a. 2 2i cot^ = - 2 1-a tan i = 1_ 2 l+o 12. cos.i=A/ 1+COB2 ^. \ 9 /I cos 2 ^4 1 cos 2 ^1 13. 14. - 16. 13. A = 79 36' 40". A = 79.726. 3 \/5 + 25 21 17. 2 44' 40". 2.744. Exercise 32 14. sin (60 + 30) = 1. sin J. ^ = _ VT5-\/3 sin 60 + sin 30 = v d + f , 2 8 15. - sin 29.5 cos 7.5 C sin 27 sin 11 16. -. cos 6 A 17. sin (.4 + 5) sin (.4 -J5). 18. 3.44. .2136. cos 2 5 = 1. esc e = cot 9 = 4-. Exercise 34 5. tan (180 -0) = - cos 15 = i A/2 + \/3. Csc 15 = 2 A/2 + V3. tan 15 = 2 V3~. the sign depending on whether \x is taken in the first or fourth quadrants. In like manner : 6. (a) (0 10 10 10 ANSWERS 17 = -V3. 00 sin (?r 0) = sin 6. COS (TT 0) = COS 0. tan (TT - 0) = - tan 0. cot (TT 0) = cot 0. _i\= w 11 oiu i jt/ i \^\jojiy. (0 25 V3 - 48 (c} cos i x ~ I sin x 39 en V5 v v / tan (x j = cot x. 7. (a) 2 = 1- cot (x J = tan x. ( c ) o sin (TT + x) = sin x. (*) -I VS. / ^7\ COS (TT + x) = COS x. sin I x \ = cosx. (c?) tan (TT + x) = tan x. cot (IT + x)= cotx. 8. (a) cos { x j = sin x. V w / tan [ x -^ = cotx. cot 1 x j = tan x. 34. -. 35. -|. 36. f. 37. -|V3- 38 - ~ 2& - 39. tan = = |. 41. _i|. K0 3 4 cos 4 x + cos 8 x sin : f 128 54. r (35 - 64 cos 2 x + 32 sin 2 2 x cos 2 x + 28 cos 4 x + cos 8 x). Exercise 35 3. a = c cos ^. 7. (I) * tan ( A 4*V^ nr d a right triangle Lclll I _/l ^O ) dl (II) a + &=(a-6)(2 + A/3) an isosceles triangle with the angles 30, 30, 120. 9. smB = -< a smA = ^ Exercise 36 1. c = 9.1226. 4. A = 109 19'. 7. A = 99 29' 12. C = 417'. a = 4899.56. 6 = 1.0943. b = 13.288. b = 4106. c = . 488667. 2. A = 134 20'. 5. C = 69 57' 36". 8. B = 43 18' 36". 6 = 74.9916 a = .85442. 6 = 1.3487. c = 242.755 6 = .81196. c = 1.8286. 3. .4 = 57 52'. 6. A = 29 1' 2'. 9. C = 68 15' 30' . a = 1116.98. a = 56.541. a = .182095. c = 1260.26. 6 = 90.164. 6 = .188745. 18 ANSWERS 10. 16. 11. 12. 13. 14. 15. b = 5.267 V2. = 7.4486. c= 2.6335 (V6+V2). = 11.175. 17. C = 105. C = 75. a = 500(3V2 -V6).18. = 896.55. 6 = 600(2 V3- 2). c = 38.52. 6 = 57.412. .4 = 79.9. a = 13283.34. c = 13346.67. A = 80 46'. a = 600.4. 6 = 602. C = .75. = 732.1. 19. c = 7.295. 4.0954. 11.697. 6 = 14.83. b = 17.08. A = 117.67. c = 15.097. 20. b = .2592. (7=56.73. a = .2181. a = 634.3. (7=55.87. 6=632.89. 21. a = 186.25. ^ = 81.32. c = 32.5. c = 1.022. A = 101.96. ! a = 1.4815. 22. c = 4377. B = 25.57. 6 = 5641.43. A = 55.69. 30. Distance of balloon from first point = 2033 yd. Distance of balloon from second point = 2363 yd. Height of balloon = 1739 yd. 23. a = 20.343. c = 28.66. 5 = 27.77. 24. a = 838.83. 6 = 595.1. C = 56.6. 25. 6 = c = a = 100. B = C = A = 60. 26. (7 = 30. a = 200 V3 = 346.42. 6 = c = 200. 27. (7 = 45^ 6 = 250(3 V2- V6) =448.3. c = 250(2 V3- 2) = 365.7. 28. 5 = 30. c = 200V2 = 282.,8. a = 100( V6 -j- V2) = 386.4. 29. 925.8. Exercise 37 1. 2. 3. 4. 5. 6. c = 26.8675. B = 39 45' 17". A = 72 14' 43". a = 385.43. B = 74 38' 19". C = 37 3' 41". O= 110 22' 10". 5 = 39 25' 30". a = .1912. A = 48 42' 12". C = 67 42' 18". b - .0748566. C = 34 6' 36". B = 22 36' 54". a = 4.70177. a = 336.446. B = 99 55' 36". C = 27 58' 24". 7. 8.185 5= C. 13. 7? = 141.99. 8. C 109 36' 5". A 25.89. B 38 5' 25". c = 3.972. a = 14.962. 14. A 79.82. 9. C = 6 49' 41". C 21.56. b = 317.8. b 1712.3. A 4 51' 41". 15. a = 7.93. 10. A 49.06. 16. B 6.23. c = 208.1. C = 4.97. B 79.117. a = 5.906. 11. a = .9418. 17. c = 102.425. B 117.99. A 65.83. C 33.85. B 45.93. 12. A 32.24. 18. A = 33.84. C 35.58. B = 102.98. b .6566. c = 1474.67. 19. b = 10.7. Where two answers place tables, and the tables. are given, the first answer is obtained by using the five- second answer is obtained by the use of the four-place ANSWERS 19 20. Distance = 234.34 ft. Distance = 234.32 ft. 21. 4.36 mi. 22. Kesultant = 14.989. Resultant = 15.08. Z with OA = 77 11' 20". Z with OA = 77.23. Zwith 0.8 = 43 31' 40". Zwith 05 = 43.49. 23. 3.59. 152.268. 152.22. 238.31. 238.22. Exercise 38 1. A = 78 5' 36". 78.1. B 58 23' 28". 58.38. C = 43 30' 58". 43.52. 2. A 44 32' 4". 44.53. B = 86 25'. 86, 41. C = 49 2' 58". 49.05. 3. A 26 19' 54". 26.33. B = 98 18' 54". 98.32. C = 55 21' 14". 55.36. 4. A = 45 11' 50". 45.19. B 101 22' 18" . 101.38 C = 33 25' 58". 33.43. 5. A 43 53' 14". 43.88. B =#0 3' 36". 60.06. C = 76 3' 18". 76.06. 6. A = 61 53' 38". 61.88. B = 72 46' 4". 72.78. C = 45 20' 20". 45.34. 7. A = 91 48'. 91.80. B 47 36' 56". 47.61. C = 40 35' 10". 40.59. 16. .53224. .5323. 8. A 37 50' 40". 37.84. B 127 3'. 127.05. C = 15 6' 22". 15.11. 9. A = 40 38' 22". 40.64. B 49 21' 56". 49.36. C = 89 59' 46". 90. 10. A = 52 20' 30". 52.34. B = 107 19' 12", , 107.32. C = 20 20' 26''. 20.34. 11. A 13 12' 8". 13.2. B = 30 2' 46". 30.04. C = 136 45' 6". 136.76. 12. A - 46 19' 52". 46.33. B 31 17' 50". 31.3. C = 102 22 ' 18" . 102.37. 13. A 107 55 ' 12. 107.92. B 35 15' 34". 35.26. C = 36 49' 18". 36.82. 14. 104 28' 42" . 104.48. 15. 16 o 44, 6 // 16.736. 21. 17. .1188. 18. 14.8586. 14.86. Q is 53 7' 48" (53.14) north of west from P. Q is 38 52' 48" (38.88) north of west from R. P is due west of 11. P is 36 52' 12" (36.86) east of south from Q. R is due east of P. R is 38 52' 48" (38.88) south of east from Q. When R is northeast from P : Q is 8 7' 48" (8.14) north of west from P. Q is 6 7' 12" (6.12) south of west from R. R is 6 7' 12" (6.12) north of east from Q. P is southwest from R. P is 8 7' 48" (8.14) south of east from Q. 28 57' 17". 28.96. 20 ANSWERS Exercise 39 1. 2. 3. 4. 5. 6. 7. v 8. 9. 10. One solution. 15. A = 32 55' 57". A' = 147 4' 3". Two solutions. ^ _ 131 o 33; 51/ ; One solution. C' = 17 25' 45". c = 1643.96. No solution. c t - 661.15. No solution. jg. A = 43 38'. One solution. 5 = 58 3' 42". 6 = .32868. One solution, a right A. 17. A = 90 . No solution. c = 25.64. Two solutions. 18. B = 28 16' 25". B 32 36' 33" C = 20 25' 11". b = .56045. 22. .4 = 25,22. (7=49.51. a = 135.46. 23. .4 = 20.79. B = 132.99. b = 136.733. 24. A = 16.25. 4' = 163.75. C = 149.45. C" = 1.95. c = 36.63. c' = 2.4518. 25. 5 = 122.81. B' = 12.45. C = 109 5' 27". C = 34.81. c = 211.48. 19. A = 103 50' 22". C' = 145.19. 11. B = 40 40'. B' = 16 44'. A 1 = 13 7' 8" = A. a = 15.354. a' =3.589. b = 441.7. 6' = 113.2. 0=78*2'. B = 44 38' 23". 26. A = 70.78. C" = 101 58', B' = 135 21' 37". C = 45.91. b = 15.787. a = 10.08. b' = 6.9753 20. A = 35.91. A' = 144.09. 27. 4 = 72.16. 12. B = 42 44' 23". C= 111.72. A' = 9.22. A = 33 1' 49". C" = 3.54. B = 58.53. a = 92.942. c = 219.7. 5' = 121.47. 13. ,4 = 18 19' 43". c' = 14.6. a = .19685. C = 139 17' 59". 21. 5=55. a' = .03313. c = 1.3952. B 1 = 10.26. 14. 5 = 70 47'. C = 67.63. . B' = 14 35'. C" = 112.37. > C = 61 54'. & =20.118. C' = 118 6'. 6' = 4.372. b = 128.465. b' = 34.2515. * 2 8. f 129.1. Other side = mi25< 29. 1010.58. 1010.2. Other diagonal = { 173 15' 8". Larger angle of parallelogram = 1 17 g 26 Smaller angle of parallelogram _ f 6 44' 52". ~ [6.74. ANSWERS Exercise 40 1. 106.79. 106.8. 4. 14290.6. 14290. 2. .30733. .30726. 5. 38983.64. 38983.33. 3. 125.229. 125.225. 11. Area of parallelogram 13. 600 V3= 1039.2. 6. 113.55. 7. .054776. .0547875. = cd sin A. 14. 106.798. 106.8. Exercise 41 8. 9. 10. 1056.66. 1056.25. 1283.5. 42150. 42130.77. In this exercise when two answers are given' to an example, the first answer is found by the use of five-place tables, and the second answer is found by four-place tables. 4. 69.372. 5. 72.268. 69.37. 72.27. 6. 8968.5 ft. above the Colorado plain. 8958 ft. above the Colorado plain. 14144.5 ft. above sea level. 14134 ft. above sea level. 7. 373.3. 11. Height = 97.083. 8. 69.98. Height = 97.08. 9. 136.9. Distance = 71.787. Distance = 71.78. 10. 1016.6. 1016.8. 10.274. 6.61. 13. 16.83. 14. Other side = 43.43. 15. Height = 42.93. Height = 42.92 ft. Distance = 104.63. Distance = 104. 675 ft. 16. 11.36. 18. 4.2818. 5.573. 4.283. 17. .1189. 19. 1496.517. 1496.66. 20. First answer = 4.4867 mi., 4.488 mi. Second answer = 9.16 mi. 21. Other diagonal = I 58.342. { 58.346. 24. 996.94. 997.6. 401.52. 401.54. 443.54. 443.5. 974.145. 973.9. 25. 220.7. 26. 16.58. 30. 146 52' 47". 146.88. 33 7' 13". 33. 12. Difference of latitude = difference of departure = 247.5 mi. New latitude =-34 23' North. New longitude = 48 9' W. 152.69ft. 31. 114.5ft. 152.7 ft. 85.854 ft. 85.89 ft. 38.566 ft. = distance of first observer from the rock. 2008 = resultant. 72 16' 1 [ = angle the resultant makes with OX. 4 2.27 I 27. 6739m. 6740 in. 28. 9 6'. = distance between observers. 22 ANSWERS 35. 298. 39. 367.89 ft. j =gide opposite tower . 37. 161.8ft. 90.04 ft. and | the other two sides 38. 97 2' 32 379.125ft. respectively. 97.06 and 379>1 ft . 14 57' 28" 14.94 respectively. 40. 48 ft. and 108 ft. respectively. 41. 40 0' 16" 1 _ le the gl ma kes with the embankment. 40 J 29.45 ft. = width of base. 42. 161.3. 43. 22 49' 46". 44. 85.27 mi. 22.83. Exercise 42 ofto T 2 T_3o 3. 1 = . 01745 radian. = 6 ' e - 16" = .00007757 radian. 2' 16" = .0006545 radian. 135 = | = 45. 5 o 14 , = .0913374 radian. 4. 2 radians = 114 35' 30". 3= 3.2 radians = 183 20' 48". .003 radian = 10' 18.8". 90 = - v = 120 23 5. Arc 21 in. long = f radian. 7 _ 4 v Arc 7 in. long = \ radian. 210 = "-as 144. 6 5 6. # = 28. 270= - P-Z^iQS . 7. Radians = 1.118. Angle = 64 3' 22.5". 225 = *. 1 -~ = 252. 8 . Angles = 85 ; 25. = 1.47325 radians; .43625 72 = . -^ = 96. radian. 5 15 315 = ^. 9. Complement of ' = - , 60 ; supplement = ^ , 150. Complement of - = , 30 ; supplement = ^ , 120. 3 6 o Complement of - , 45 = - , 45 ; supplement = ^ , 135. 4 4 Complement of = 7 ~ , 70 ; supplement = - T , 160. 9 18 " Complement of |f = ^ , 40 ; supplement = 1^ , 130. ANSWERS 23 10. sin = ^. cos = -V3. s i n ^ = iV2. cos = -^V2. 62 2 42 2 tan = i V3. cot = V3. tan cot = 1. sec = | V3. esc = 2. sec = V2. esc = VI. sin^ = iV3. cos = -- S m^=-i C os=-lV3. 3 2_ 2 6 2 tan = V3. cot = V3. tan = J- V3. cot = V3. sec = 2. esc = | V3. sec = | \/3. esc = 2. sin = cos * = 1 V2. 8inZ r = - 5 V ^ COS = 5 v/1 442 2 ten^ooC? = I. tan=cot = -l. 4 4 sec = V2. esc = V2. sec 7 ' = csc^ = V2. 11. H radians = 68 45' 18". 4 4 sin^ = l. cot-=0. 13. B = 4. ^ = 143 14' 22.5". cos- = 0. sec -=oo. 14. a = 12.5. A = 14 19' 261". tan- = 00. csc-= 1. 15. p = 8. 22 A = 458 22'. 16. p = .26175. 17. p = .64565. 18. 4' 35". 20. 4' 20". a = 10.9935. It = 154.89. 19. 69.102ft. 21. 1117mi. 22. 437320 mi. 23. 35374500 mi. 24. f V2 - 6. Exercise 44 2. 7T 2ir 4*- STT^ 5. 7T 57T 3' 2 8 3 * 6' 6 ' 3. 7T 37T STT 77T 6. 7T STT 4' 4 ' 4 4, ' 3' 3 ' 4 IT 27T 4ir STT 7. 7T S^r T^r ll^r 3' 3 ' 3 3 ' 6' o o Q 7T 37T 57T 77T 7T 2lT 47T STT . 4' 4 ' 4 4 ' 3 ' 3 ' 3 ' 3 ' 9. 7T 3_7T 7T 5jr 16. 7T 7?r 2' 2 6' 6 6' 6 ' 10. ?r , iz. 17. * P-Z, Lz, HE, o, z. 6' 6 6' Q O u o 11. 0, IT. 18. 7T 3fl- 7T 57T 4' 4 ' 3' 3 12. 37T 4 7 7T 4 7T ' 2 3jr 2 19. o, 7T 7T 5 7T 3' 6' IT' 13. i- T 1 f' T' T' T' 20. o, TT TT 5?r ^ 3?r 2 ' 6 ' 6 ' T ' 2 , , 7T. 21 7T - * ' Q ' Q' Q' O O o o o - ^ 7T 4 7T 5 7T go ^T 7T 7T 2 TT 4 7T 5 7T oooo o ^ o o o 3 4 ANSWERS Exercise 45 1. = 30, 210. 4. sc=50. 7. rK = 36.87. x =100, -100. 2/ = 40. 2/ = 22.62. 2. = 36.5, 216.5. 5. a; = 1000. 8. z = 1000, z = 200, -200. 2/ = 2000. = 72.5. 3. = 58.51, 301.49. 6. x = 60. 9. x = acos^ + 6sinA x = 500, -500. y = 45. y = b cos J. a sin .4. Exercise 46 4 2. cos (cot- 1 f) = f . -lA/S 60 * 3. tan (sin" 1 T 5 j ) = r 5 ^. ' 3* 4. sec(tan- 1 T ^) = H. sin- 1 430 T . 5 sin (cot" 1 rt^ ' 6 sec-i-y/2 45 *" 6 cot(co- lfl5 )- aV ^- :2 ' 4 csc- 12 V3-60 -. 7. tan (2 sin- 1 1) = V3. 3 8. sin(2tan-iA) = Hft. ' 6 " 9. cos (2 sec- 1 V) = - Mi- JQ gjjj / 1 C Qg-l 1 N lV3 COB-4=W,J. ^2 SJ 3 3 11. cot ( J tan- 1 Y) = f sec- 1 2 = 60, -. 12. sin (3 sin- 1 i) = 1. 3 13 -in (-in-" CO-- 1 *) 2 ~ V ^, sin- 1 |V3 = 60, |. 6 14. tan (tan - 1 2 + cof 1 3) = 7. /~ T 3 30. "2nir, 31. - 2 WTT, 6 6 tan" 1 ^ V3 = 30, ^2r. 7 ^2^. 3, fS ~, 35. -2w7r, 38. 7r 2mr, T 2n *' 6 " 4 33. 2 nr, 36. ^_j-2n7r, 39. 2 mr, 3 2 6 4 7T ^2 M ,r. 1T 2 *- 34. ^ 2 W 7r, 37. -2wir, 42. x = ^- ANSWERS 25 43. 30 = sin- 1 \ = cos" 1 \/3 = tan- 1 $ V3 = cor 1 \/3. 60 = sin- 1 * V3 = cos- 1 ^ = tan" 1 V3 = cof 1 ^ >/3. 90 = sin- 1 1 = cos- 1 = tan- 1 oo = cotr 1 0. 45 = sin" 1 1 V2 = cos" 1 1\/2 = tan" 1 1 = cotr 1 1. = sin' 1 = cos- 1 1 = tan' 1 = cor 1 oo . n 180 = sin- 1 = tan- 1 0. n 90 = cos' 1 = cot' 1 0. ANSWEKS Exercise 50 1. A 36 12' 14". 10. A = 132 43'. 16. A = 74 22' 18". B 85 52' 19". B 80 6' 46". B = 29 0' 22". c = 84 20' 30". b = 76 29' 10". c = 59 41' 57". 2. a 26 19' 48". 11. a = 113 53' 56'. 17. a = 69 6' 12". B = 74 4' 42". b = 156 33' 10". b = 106 19' 45". b = 51 47' 41". c = 68 10' 51". c = 95 45' 20". 3. a = 44 36". 12. A _ 19 55' 51". 18. A 68 22' 26". b = 28 48' 51". A' = 160 4' 9". B = 26 50' 36". c = 51 22' 13". a = 13 44' 24". a 35 17' 40". 4. A _ 76 20' 45". a' = 166 15' 36". a 71 34' 7". c 44 9' 51". 19. B = 118 6' 14". b 46 47 '50". c' = 135 50' 9". b = 127 44' 46". a 43 36' 55". 5. A 51 2' 30". 13. B = 24 3' 27". b = 17 26' 29". b _ 15 2' 18". 20. A = 35 55' 46". c 63 27'. = 39 31' 49". a = 25 30' 58". 6. B a b = 70 5' 13". 52 18' 30". 65 24' 9". B' = C' = 155 56' 33". 164 57' 42". 140 28' 11". c A' = a' = c' = 47 13' 55". 144 4' 14". 154 29' 2". 132 46' 5". 7. Ji = 65 9' 27". 14. B = 23 21' 14". a _ 118 6' 23". B' = 156 38' 46". 21. A = 7 54'. c 102 38' 49". b' _ 158 54' 42". b = 70 46' 52". 8. A 131 27' 18". b = 21 5' 18". c = 70 58' 11". B _ 80 55' 27". c 65 10' 50". 22. B = 40 28' 56". c = 98 6' 42". c' = 114 49' 10". 139 31' 4". 9. A 70 23' 52". 15. A = 95 10' 9". 23. c = 89 59' 5". a 54 40' 14". b = 54 56' 29". b = 149 50' 25". c = 93 37' 4". 24. b = 8 40' 30". 26. A = 36.2. 29. A = 76.35 o. 32. B = 65.16. 35. A = 132.71 o B 85.87. a -71.57. a = 118.11. 5 = 80.11 c = 84.34. b = 46.8, c = 102.65. b = 76.49 . 27. A 26.33. 30. ^1 = 51.03 o 33. A = 131.46. 36. a = 113.9 . B 74.019. b = 17.44 o B = 80.92. b = 156.53. b = 51.8. c = 63.44 o c 98.11. c = 68.18 28. a _ 44.6. 31. B = 70.09. 34. A = 70.4. 37. A = 19.93. b = 28.77. b = 65.41 a = 54.68. A'= 160.07 c = 51.37. a = 52.3. b = 149.84. a = 13.74 ANSWERS 38. a' c = 166.26. = 44.19. b' c = 158.94. = 65.15. 43. A B = 68.37. = 26.84. 46. a = 7.9. 6 = 70.78. c' = 135.81. c' = 114.85. a = 35.32. c = 70.97. B = 24.06. 40. A = 95.17. 44. B = 118.11. 47. B = 40.49. h = 15.04. b = 54.94. b = 127.75. B' = 139.51. c, = 39.53. c = 93.62. c = 43.625. = 155.94. 41. A = 74.37. 45. A = 35.92. 48. c = 89.985. b' = 164.96, B = 29. A' = 144.08. 49. b = 8.8. c' = 140.47. c = 59.7. a = 25.61. B = 28.35. 42. a = 69.1. a' = 154.49. B' = 156.65. b = 106.36. c = 47.24. b = 21.06. c = 95.76. c' = 132.76. Exercise 51 1. B - 145 26'. 7. A = 38 4' 46". c = 103.05. a 98 35' 33''. c = 91 27 '50". A 1 = 38.94. C 102 14' 1". 8. A = 7 29' 34". a' = 40.18. 2. A _ 166 37' 20". A' = 172 30' 26". c' = 76.95. B _ 139 10' 16". a = 80 50' 30". 14. A = 37.84. C _ 137 24' 22". a' = 99 9' 30". B = 37.84. 3. a 110 10' 11". 10. B = 145.44. C = 133. b _ 172 35' 46". a = 98.59. 15. C = 159.2. C = 106 25' 11". C = 102.23. c = 143.78. 4. a _ 139 49' 16". 11. A = 166.65. 16. A = 38.08. A _ 141 3' 46 '. B = 139.17. B = 38.08. c _ 103 4'. C = 137.41. c = 91.47. A 1 38 56' 12". A' = 13.35. 17. A = 7.49. a' 40 10' 44". B' = 40.83. d = 80.9. c' = 76 56'. C' = 42.59. A' = 172.51. 5. A 37 50' 18". 12. a = 110.17. a' = 99.1. C = 133 3'. b = 172.55. 1} = 37 50' 18". C = 106.42. 6. .C = 159 12' 12". 13. A = 141.06. c = 143 46' 39". a = 139.82. 19. Tetrahedron: 70 31' 46" = 70.55. Octahedron : 109 28' 27" = 109.48. Dodecahedron : 116 33' 45" = 116.6. Icosahedron : 138 11' 36" = 138.27. 22. Dihedral angle between two adjacent faces = 109 28' 13" = 109.47. Dihedral angle between each face and the base = 54 44' 6" = 54.74. 20. Surface = 2064.57 [2064.29]. Volume = 7662.8 [7668.3J. 21. Cot i M - VCos m. 28 ANSWERS 23. Face angle at base of frustum equals 81 6' 32" = 81.11. Dihedral angle between two adjacent faces = 91.24 = 91.4. 24. 62 53' 14" = 62.89. 25. 157 31' 20" = 157.4. C = 53 30' 3". a = 48 30' 20". 2. A= 61 28' 30". C= 42 J 12'54". b = 94 41' 17". 3. A = 129 40' 45". 5=61 38' 9". c = 65 54'. 4. B = 44 8' 29". C = 133 51' 51". a = 70 47 7". 5. B = 125 40' 7". O = 34 9' 47". a = 73 34' 40". Exercise 53 6. B = 162 38' 21". C= 147 52' 21". a = 77 8'. .7. A = 132 12' 37". C = 64 49' 57". b = 69 59' 47". 8. B = 92.04. C = 53.5. a = 48.55. 9. A = 61.47. (7 = 42.21. 5 = 94.76. 10. A = 129.68. B = 61.64. c = 65.92. 11. B = 44.15. C = 133.86. a = 70.77. 12. B = 125.67. = 34.17. a = 73.53. 13. 5=162.64. C = 147.87. a = 77.13. 14. A = 132.21. C = 64.83. B = 70. Exercise 54 1. A = 64 36' 45". 6. .5 = 49 13'. 11. (7 = 83.76. b = 49 0' 15". a = 67 49' 54". a = 123.74. c = 36 41' 37". c = 117 26' 34". b = 134.91. 2. C = 157 52' 54". 7. C = 137 40' 54". 12. A = 123.3. a = 114 42' 41". a=8337'28". b = 70.26. b = 39 3'. b = 45 12' 20". c = 145.95. 3. B = 48 49' 12". 8. .4 = 64.6. 13. B = 49.25. a = 124 42' 42". b = 49.01. a = 67.82. c = 103 23' 42". c = 36.7. c = 117.44. 4. C = 83 39' 16". 9. (7=157.88. 14. C = 137.68. a = 123 43' 44" a = 114.47. a = 83.61. b = 134 55' 16". 6 = 39.07. b = 45.21. 5. A = 123 18' 15". 10. JS = 48.82. b = 70 15' 24". a = 124.71. c = 145 56' 38". c = 103.4. Exercise 55 1. A = 85 35' 14". 2. A = 143 3' 48''. 3. A-- = 34 15' 4". B = 49 35' 34". B = 79 54' 4". B = 42 15' 16". C = 59 38' 40", C = 55 3' 4", (7 = 121 36' 20", ANSWERS 29 4. 4 = 113 39' 17". B = 123 40' 19". C= 159 43' 22". 5. A = 110 51' 20". = 38 26 '46". C = 48 56' 8". 6. B = 151 44' 47' . 7. C =128 53' 9". 8. .4 = 85.6. B = 49.59. (7=59.66. 9. A = 143.07. B = 79.91. C = 55.06. 10. 4 = 34.24. B = 42.25. 0=121.61. 11. A = 113.66. B = 123.67. O= 159.72. 12. 4 = 110.88. B = 38.43. O = 48.93. 13. B = 151.74. 14. O = 128.88. 1. a = 147 23' 29". b = 122 16' 32". c = 60 41' 31". 2. c = 126 58' 19". 3. a = 44 11' 33". 6 = 113 9' 29". c = 113 9' 29". 4. a = 73 57' 28". 5. a = 71 8' 55". b = 75 12'. c = 59 58' 4". Exercise 56 6. b = 102 46' 10". 7. a =27 31' 12". 5 = 86 14' 34". c = 83 31' 42". 8. c = 146 37' 16". 10. a = 147.39. b = 122.29. c = 60.69. 11. c = 126.97. 12. a = 44.2. b = 113.16. c= 113.16. 13. a = 73.96. 14. a = 71.14. b = 75.2. c = 59.98. 15. b = 102.79. 16. a = 27.52. b = 86.24. c = 83.53. 17. c = 146.61. 1. B = 39 48' 30". = 50 30' 44". c = 40 30' 3". 2. .B = 55 52' 40". O = 20 9' 48". c = 20 16' 30". 3. 4 = 115 57' 58". B = 57 34' 53". a = 95 18' 14". A' = 25 44' 34". B' = 122 25' 8". a' = 28 45' 6". 4. 5 = 56 29' 13". O= 136 31' 8". c = 126 1' 22". S' = 123 30' 47". O' = 14 34'. c' = 17 11' 42", Exercise 57 5. a = 67 3' 48". 6. O = 40 24' 30". 0' = 139 35' 30". 7. B = 74 7'. 4 = 80 24' 46". a = 117 37 '23". B' = 105 53'. 4' = 37 31' 29". ' = 146 48' 37". 8. = 39.8. = 50.5. c = 40.5. 9. 5 = 55.88. O = 20.17. c = 20.28. 10. 4=115.98. B = 57.56, a = 95.33. A' = 25.69. B' = 122.44. a' = 28.7. 11. O= 83.76. a = 113.74. b = 134.91. 12. 4 = 67.1. 13. O = 40.39. C' = 139.61. 14. 4 = 80.37. B = 74.15. a = 117.66. 4' = 37.59. .B' = 105.85, a' = 146.8. 30 1. 2. B - 57 37' 36' . b = 34 34' 56". c = 35 35' 56". C = 73 24' 50". b = 33 27' 6". c = 42 14' 44". 3. 4 = 136 51' 12". a = 128 19' 56". b = 143 32' 40". 4. B = 143 40' 5". 6 - 157 33' 6". c = 39 24'. B' = 59 12' 46". &' = 3337'14". c' = 140 36'. 5. B = 115 58' 30". a = 54 21' 7". 6. 8. 10. 11. ANSWERS Exercise 58 ^' = 162 48' 34". a' = 125 38' 53". b' =161 16' 50''. C = 33 36' 24". b = 21 13' 16". c = 13 54' 18". a = 66 59' 50''. a' = 113 0' 10". 5 = 57.59. 6 = 34.57. c = 35.6. C = 73.42. b = 33.45. c = 42.25. A = 136.85. a = 128.34. 6 = 143.54. 12. 13. 14. 15. B = 143.63. b = 157.53. c = 39.41. B 1 = 59.21. b' = 33.63. c' = 140.59. B = 115.94. a = 54.34. b = 77.68. B' = 162.82. a' = 125.66. b' = 161.27. C = 33.62. b = 21.22. c = 13.91. a = 67. a' = 113. 7. 19.505 sq. in. 19.503 sq. in. Exercise 59 In this exercise, where two answers are given to an example, the first answer is obtained by use of five-place tables and the second answer by use of four-place tables. 1. 2.513 sq. ft. 4. 254.82 sq. in. 2. 13.548 ft. 5. 43,793 sq. mi. 13.547 ft. 43,780 sq. mi. 3. 17.279 sq. mi. 6. 4,379,300 sq. mi. 17.265 sq. mi. 4,379,000 sq. mi. 8. Each angle = 62 23' 22" = 62.39. Perimeter = 6273.42 statute miles = 6277.14 statute miles. 9. 137.439 sq. ft. 11. 827.96 sq. m. 13. 8008 sq. m. 10. 195.36 sq. in. 12. 18.767 sq. in. 14. 2547.53 sq. ft. 15. 4867.33 sq. in. = area of 1st triangle. 1135 sq. in. = area of 2d triangle. 16. 137.4 sq. ft. 18. 827.8 sq. m. 20. 8006 sq. m. 17, 195.3 sq. in. 19. 18.76 sq. in. 21. 2547.65 sq. ft. 22. 4858.9 sq. in. = area 1st triangle. 1134.2 sq. in. = area 2d triangle. ANSWERS 31 Exercise 60 In this exercise, where two answers are given to an example, the first answer is obtained by use of five-place tables and the second answer by use of four-place tables. 1. Distance = 6485.5 nautical miles. 6484.8 nautical miles. N. 47 48' 37" W. = bearing of Halifax from Cape Town. S. 59 48' 34" E. = bearing of Cape Town from Halifax. 2. Lat. 42 9' 36" N. = 42. 1 N. 4. Lat. = 53 25' 47" N. Lon. 70 8' 38" W. = 70. 14 W. 53.43 N. 3. Lon. = 13 11' 37" W. Distance = 3 f 2 ' 7 statute mi ^ 13 2 W 3301.5 statute miles. Distance = 3113.64 statute miles. 3112.86 statute miles. 5. Distance = 2080.5 nautical miles. 2082 nautical miles. N. 53 38' 44" E. = bearing of San Francisco from Honolulu. N. 53.65 E. = bearing of San Francisco from Honolulu. S. 71 44' 14" W. bearing of Honolulu from San Francisco. S. 71.75 W. = bearing of Honolulu from San Francisco. 6. Lat. = 33 40' 21 "N. = 33.67 N. 7. Length of arc* of great circle = 1688.1 nautical miles = 1888.4 nautical miles. Length of parallel of latitude 1700.6 nautical miles = 1701 nautical miles. 8. Distance = 5769.43 nautical miles = 5769 nautical miles. N. 60 24' 50" W. = bearing of Manila from Seattle. N. 60.42 W. = bearing of Manila from Seattle. 9. Lat. = 53 52' 47" N. 15. (1) 2641.2 statute miles. = 53.88 N. 2641.25 statute miles. Long. = 152 38' W. (2) 2950.6 statute miles. = 152.63 W. 2950.67 statute miles. 10. 7 hours 41 minutes 43 seconds A.M. (3) 2948 statute miles. 7 hours 41 minutes 24 seconds A.M. ( 4 ) 18539.17 statute miles. 18535 statute miles. 11. 4 hours 41 minutes A.M. 16. 2 hours 7 minutes P.M. 12. 5 hours 27 minutes P.M. 17. 14 hours 51 minutes 10 seconds. 14 hours 51 minutes 10 seconds. 121.87. 14. 6 hours 14 minutes A.M. 32 ANSWERS 18. Angle TOP = 37 17' 14" * = 37.29. OP makes with the plane XO T an angle = 30. OP makes with the plane XOZ an angle = 52 42' 46". OP makes with the plane ZO Y an angle = 20. 19. Length of perpendicular from Pto OX = 17.173 = 17.175. Length of perpendicular from Pto OT= 11.072. Length of perpendiculer from Pto OZ = 15.827. Length of projection of OP on OX 6.254 = 6.251. Length of projection of OP on OY= 14.54. Length of projection of OP on OZ = 9.138. 20. Length of perpendicular from P to plane XOY = 9.138. Length of perpendicular from Pto plane XOZ = 14.54. Length of perpendicular from P to plane YOZ = 6.25. Length of projection of OP on plane XOY = 15.827. Length of projection of OP on plane XOZ = 11.072. Length of. projection of OP on plane YOZ- 17.175. 21. OP makes with the plane XO Y an angle = 22 52' 42" = 22.88. Cutting plane makes with OX an angle = 33. Cutting plane makes with O Y an angle = 48. Cutting plane makes with OZ an angle = 22 52' 42" = 22.88. Cutting plane makes with plane XOY&n angle = 67 7' *18" = 67.12 Cutting plane makes with plane XOZ an angle = 42. Cutting plane makes with plane YOZ an angle = 57. 25. If D represents the diagonal of the parallelepiped, then D = a? + 6 2 + c 2 + 2 ab cos 7 + 2 be cos a + 2 ac cos 0. w A/ tyL ( ' */ UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. : 25'cents on first day overdue 50 cents on foufcth day overdue One dollar on seventh day overdue. 13 1948 MAR 1948 14Dec'49SS LD 21-100m-12,'46(A2012 19Nov52HD MOV t OCT201955LU REC'D LD Mi* 4 1957 : REC'D LD 241957 I30ct'59FK RECTO LD OCT 9 YC 2228C UNIVERSITY OF CALIFORNIA LIBRARY