AT LOS ANGELES GIFT OF Mrs. M. H. Wood A ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO ELEMENTS OF BY DAVID A. ROTHROCK, PH.D. PROFESSOR OF MATHEMATICS, INDIANA UNIVERSITY BLOOMINGTON, INDIANA THE MACMILLAN COMPANY 1911 All rights reserved COPYRIGHT, 1910, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published September, 1910. Reprinted January, 1911. NortoooO J. 8. Gushing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. Ens'inesnng & Mathematical Sciences Library KM PREFACE IN this work the author has endeavored to prepare a text which would serve as a basis for a fifty- or sixty-hour course in Plane and Spherical Trigonometry as ordinarily presented in advanced secondary and elementary college courses. Emphasis is placed upon drill work in the trigonometric identities, upon the applications of trigonometry to practical problems, and upon approximate calculations by means of natural functions. The more accurate results obtained by logarithmic calculations are emphasized in the solutions of oblique triangles ; a uniform style of tabulating logarithmic calculations is suggested. For the benefit of those who may wish to pursue advanced courses in mathematics, a brief discussion of analytic trigo- nometry is presented in Chapter IX. In Part II the elements of spherical trigonometry are developed in so far as to include the ordinary formulse necessary in the solution of right and oblique spherical triangles. DAVID A. ROTH ROCK. BLOOMINGTON, INDIANA, December, 1909. 374041 CONTENTS PART I PLANE TRIGONOMETRY CHAPTER I Trigonometric Functions of Acute Angles ART. PAGB 1. Trigonometry 1 2. Functions of Acute Angles . .2 3. Functions of Complementary Angles . 3 Exercises .... ....'... 5 4. Fundamental Relations among the Trigonometric Functions . . 6 5. Fundamental Identities ......... 7 6. Variation of the Trigonometric Functions . .... 7 7. Transformation of Identities . 7 Exercises 8 Exercises 10 8. Functions of Particular Angles : 0, 30, 45, 60, 90 . ... 13 Exercises ........... 14 9. Table of Trigonometric Functions 15 CHAPTER II Solution of Right Triangles 10. Fundamental Formulas 16 11. Projections 17 12. Components 18 13.' Resultant 18 14. Projected Areas 21 15. Areas of Right Triangles 21 16. The Isosceles Triangle 22 17. Notations of Direction 23 18. To solve Right Triangles 25 Exercises 26 CHAPTER III Trigonometric Functions of Any Angle 19. Axes. Quadrants . -, : 31 20. Coordinates, Abscissa, Ordinate 31 21. Definitions of the Functions 32 viii CONTENTS ART. PAGE 22. Laws of Signs 33 Exercises 34 23. Functions of Negative Angles 34 Exercises 35 24. Functions of 90 and 90 + 36 25. Functions of 180 and 180 + 37 Exercises .38 26. Line Values of the Functions 39 Exercises 41 27. Graphs of the Trigonometric Functions 42 CHAPTER IV Measurement of Angles 28. Units of Measure 44 29. Relations "between Degree and Radian Measure 44 Exercises 45 30. The Length of any Arc 46 31. Segment and Sector Areas ......... 47 Exercises 47 CHAPTER V Functions of Two Angles. Multiple Angles 32. To develop sin (a + /3) and cos (a + /3) 50 33. To develop tan (a + /3) 51 34. Important Formulas 52 Exercises 52 Functions of Multiple and Sub-Multiple Angles 35. Functions of 2 a 53 36. Functions of 3 a 64 37. Half-angle Formulas . . . . ' 54 Exercises 55 Sum and Difference Formulas 38. Converting to Products .56 39. Converting to Sum or Difference 67 Exercises , 57 CHAPTER VI Logarithms 40. The Index Laws 60 41. Definition of Logarithms 61 CONTEXTS ix ART. PAGE 42. Systems of Logarithms 62 43. Laws governing the Use of Logarithms 62 44. Characteristic and Mantissa 63 45. Use of Tables 65 46. Conversion of Common to Napierian Logarithms ..... 67 Exercises in Use of Logarithms ....... 67 CHAPTER VII Solution of Triangles in General 47. The Theorem of Sines 69 Applications of the Theorem of Sines 71 48. The Theorem of Tangents 73 Applications of the Theorem of Tangents 74 49. The Theorem of Cosines 75 Exercises ........... 77 50. The Half-angle Theorems 77 Application of the Half-angle Theorems 79 Areas of Triangles 51. Area in Terms of Sides and Angles 80 52. Area in Terms of r 81 53. Expressions for Area 82 Exercises . 82 CHAPTER VIII Inverse Functions. Trigonometric Equations 54. Inverse Notation ........... 87 55. Inverse Identities ........... 88 Exercises 89 56. Definitions 90 57. Solutions 91 58. Simple Equations .91 Exercises . 92 59. Equations of the Form r cos = a, r sin = b ..... 93 60. Equations of the Form r sin 6 cos = a, r sin 6 sin $ = &, r cos 6 = c . 94 61. To solve a- sinz-f b- cosx = c 95 62. To solve sin (x -f 0) = a sin a; .95 63. To solve tan (x + 0) = a tan x 96 64. To solve x = + sin x 97 CHAPTER IX Complex Numbers. DeMoivre's Theorem. Trigonometric Series. Exponential and Hyperbolic Functions 65. Roots of Quadratic Equations 99 Exercises . 100 x CONTENTS AKT. PAGB 66. Complex Numbers expressed Trigonometrically 100 67. DeMoivre's Theorem .101 68. Raising to Powers and extracting Roots 102 69. Value of sin a;, cos x in Terms of x 104 70. Summation of Series 108 Examples 109 71. The Exponential Series 110 72. Euler's Formulas . Ill Exercises 112 73. The Hyperbolic Functions 112 74. The Gudermannian 116 PART II SPHERICAL TRIGONOMETRY CHAPTER X General Definitions. The Right Triangle 75. Definitions and Geometric Properties ....... 117 76. The Polar Triangle 118 The Right Spherical Triangle 77. Definitions 119 78. Trigonometric Relations 119 79. Important Formulas . . . 120 80. Napier's Rules of Circular Parts 121 81. Relative Dimensions of Sides and Angles 122 82. The Isosceles and Quadrantal Triangles ...... 122 83. Solution of Right Spherical Triangles 123 Exercises 124 CHAPTER XI The Oblique Spherical Triangle 84. The Theorem of Sines 126 85. The Theorem of Cosines. Side and One Angle 127 Modified Formula . . . 128 86. The Theorem of Cosines. Angles and One Side 128 87. The Half-angle Formulae 129 1) The Half-angles in Terms of the Sides 129 2) The Half-sides in Terms of the Angles 130 88. Napier's Analogies 131 89. The Area of a Spherical Triangle . . . . . . . .133 90. Solution of Oblique Spherical Triangles 133 Exercises . . . 135 CONTENTS xi Applications of Spherical Trigonometry ART. PAOB 91. The Earth as a Sphere. Definitions and Notation .... 137 92. The Terrestrial Triangle 138 93. The Celestial Sphere 140 94. The Celestial Triangle 141 Exercises . 142 CHAPTER XII FORMULAS . ... 143 TRIGONOMETRY PART I PLANE TRIGONOMETRY CHAPTER I TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 1. Trigonometry. The word Trigonometry is derived from two Greek words, triangle (rpiycovov*) and measurement (perpov), which would suggest that the subject has to do with the meas- urement of the triangle. At the present time the subject of trigonometry, though treating of the measurement of the tri- angle, has a much wider scope, and includes all manner of investigations depending upon certain functions of angles called Trigonometric Functions. These functions are called sine, cosine, tangent, cotangent, secant, and cosecant. For any angle 6 * these words are abbreviated into sin 0, cos 0, tan 0, cot 0, sec 0, esc 0, respectively. In the next section definitions are given for the trigonometric functions of an acute angle, but it should be borne in mind that similar definitions are applicable for any angle whatever, and later will be extended to angles varying from to 360. * In trigonometric notation, angles are frequently denoted by Greek letters. The Greek alphabet is here inserted. Letters Names Letters Names Letters Names A a Alpha I t Iota P p Rho B /3 Beta K K Kappa 2 a s Sigma T 7 Gamma A X Lambda T T Tau A 5 Delta M fj. Mu T v Upsilon E e Epsilon N v Nu * Phi Z Zeta H Xi X x Chi H TI Eta O o Omicron * $ Psi 6 d Theta II * o Pi fi w Omega B 1 2 PLANE TRIGONOMETRY [ 2 2. The Trigonometric Functions of an Acute Angle. (1) Definitions. For an acute angle constructed in a right triangle, Fig. 1, with #, y, r, as the base, altitude, and hypote- nuse, respectively, we define . ft _ opposite side _ y n ~ hypotenuse ~r^ fl adjacent side _ x >s ~ hypotenuse r ' * _ opposite side _ y ~ adjacent side ~ x ' Fig. 1. a _ hypotenuse 1 _r 16 ~ opposite side ~ gin 6 ~~ V ' Q _ hypotenuse 1 _r ~ adjacent side ~ cos 6 ~ & ' fl _ adjacent side _ 1 _x * ~ opposite side ~~ tan 6 ~" V ' To these six functions are sometimes added : versed sine 6 = 1 cos 9, written rers 8, coTersed sine 6 =1 sin 8, written covers 6. These six functions of the angle are called trigonometric functions (trigonometric ratios) of that angle. The symbols sin #, cos 0, tan 0, cot #, sec 0, esc #, are usually read sine 0, cosine 0, tangent 0, cotangent 0, secant 0, cosecant 0, respectively. (2) Elementary relations among the functions. The six func- tions sin 0, cos 0, tan 0, cot 0, sec 0, esc 0, are not independent. By comparing the definitions we find : (1) tan6 = -^-r, or tan 6 -cot 6 = 1, cot 6 (2) cos 8 = -~ . or cos 6 sec 8 = 1, sec 8 (3) sin 8=-^-, or sin 8 -esc 8 = 1. esc 8 2] TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES (3) Exponents. When the trigonometric functions are to be affected by exponents, the following notation is usually em- ployed : (1) if the exponent be positive, the index is placed thus, sin 2 #, tan 3 #, sec"#, and read sine square 6, etc. ; (2) if the exponent be negative, the index is usually attached to a bracket as in bracketed expressions in algebra, thus (sin ^)~ 1 = -^ , (cot BY 2 = -^, (cos0)-"=?:. These are read sin e cot 2 6 cos" 6 sin exponent minus one, cot 6 exponent minus two, cos 6 expo- nent minus n. (4) Functions are constant for any given angle. The trigo- nometric functions of an acute angle are ratios of lines in a right triangle. They are constant for any fixed angle, and do not change value for different lengths of the sides of the triangle. Thus, Fig. 2, C" B' C Fig. 2. C' BC B'G' B"O' = OB = OW = OO = PC' OB OB' BO B'O' tana = 0(rw OB"' 00" OB"' B"C" 00" ' Tables of these ratios have been constructed, showing their numerical values for all angles from to 90 (see Table p. 15). 3. Functions of Complementary Angles. In Fig. 3, and < are complementary angles : 6 + < = 90. PLANE TRIGONOMETRY' [3 sin 6 = ^ = cos , jr. cos 6 = - = sin <|>, tan 6 = ^ = cot , JO csc 9 = = sec , T cot 6 = tan tail a sma (B) sec 2 a - tan 2 a = 1, (C) esc 2 a -cot 2 a = 1, ( F ) sin a i (D) tana- -- , (G) esca= . cos a sin a These identities should be memorized. 6. Variation of the Trigonometric Functions. It should be noticed that as the angle increases from to 90, the functions vary as follows : (1) sine increases from to 1, (2) cosine decreases from 1 to 0, (3) tangent increases from t o oo , (4) cotangent decreases from oo to 0, (5) secant increases from 1 to oo , (6) cosecant decreases from ob to 1. These facts may be observed from the above identities, as well as from the definitions of the functions in 2. 7. Transformation of Identities. By means of the funda- mental identities, 5, any trigonometric function may be changed into various forms. For example, , /i - s r H 1 v sec 2 1 tan <6 sm< = VI cos 2 < = A/l - = = - -s etc. sec 2 sec sec 9 These forms for the value of the sine of an angle are ob- tained by using identities (-4), (-^), C#)^ and algebraic manipu- lation. Again, sin a sec cot a = - - - cot a = tan a cot a = 1. cos a 8 PLANE TRIGONOMETRY [ 7 In verifying an identity, we may proceed, (a) by transforming the left member of the equation into the right, (6) by transforming the right into the left, or (cT) by reducing each side to the same form. For example, show cos 4 a: sin 4 a: = 2 cos 2 a: 1. () Transform the left member into the right. cos 4 a: sin 4 or = (cos 2 a: + sin 2 a:) (cos 2 a; sin 2 a:) = cos 2 a: sin 2 a; by (.4) = cos 2 a: (1 cos 2 a:) by (A) = 2 cos 2 a; - 1. (6) Transform the right member to the form of the left. 2 cos 2 a: 1 = cos 2 a; (1 cos 2 a?) = cos 2 a; sin 2 a: by (.4) = (cos 2 a: sin 2 x) (cos 2 x + sin 2 a;), multiplying by 1, = cos 4 x sin 4 x. (c) Transform each side to a common form. cos 4 x sin 4 x = 2 cos 2 x 1 = cos 2 x sin 2 x. Remove the factor cos 2 a; + sin 2 a; = 1 from the left member, and we have cos 2 a; sin 2 a: = cos 2 a; siu 2 ar. The student should attain a good degree of skill in manipu- lating trigonometric functions ; to this end is now inserted a list of the more common forms of identities involving a single angle. EXERCISES Verify the following identities : 2. (sec 2 ^-- = . 3. tan A + cot A = sec A x esc A. 4. sin 2 A (esc 2 A 1) = cos 2 A. 5. cot 2 A cos 2 A = cos 2 A cot 2 A. cos A 6. - = tan A. sin A cot 2 A 7. sin A cos A (tan A + cot A) = 1. 7] TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 9 8. (tan A + cot A~) 2 = sec 2 A + esc 2 A. 9. cos A esc A tan A = 1 . 10. cos A (cot A + tan A) = esc A. 11. cot A + Sm A = esc ^1. 1 + cos A sec A tan ^ cos A cot .4 13. sec 2 A sec 2 ^4. sin 2 .4 = 1. 14. (esc A - cot A) 2 = l - cosA . I + cos A 15. sec A cos A = sin .4 tan A. sin .A _,_ cos A _ 1 J ~" 7 csc .4 sec A tan A \ 1 cot .4 18. tan J. + 1 1 + cot A sec A csc A tan .4 sec .4 + csc A tan .4 + 1 19. cos 4 A sin 4 A= cos 2 A sin 2 .4= 1 - 2 sin 2 J.= 2 cos 2 A 1. 1 + cot 2 A , 9 n 20. - = COt 2 A. 1 + tan 2 A 21. (sin ^4. + cos ./I) 2 + (sin ^4 cos A) 2 = 2. 22. sec A + tan A = (sec .4. tan A)~ l . 23. sin 4 J. + cos 4 A = 1 2 sin 2 ^4. cos 2 A. 24. sin 3 ^1 + cos 3 A = (sin ^. + cos A)(\ sin A cos ^i). 25. sec 4 A - tan 4 ^1 = sec 2 A + tan 2 A = l+2 tan 2 ^4. 26. * Show rr 2 + y 2 = r 2 , when x = r cos , y = r sin 0. / ~ //"" 27. Show + f- = 1, when a; = a cos ^>, y = b sin <. a^ tr x 2 v 2 28. Show f- = 1, when x = a sec ^, y = 5 tan <. a j o^ 29. Show x* + y 2 + z* = r*, when a: = r sin cos <>, y = r sin # sin <, z = r cos 0. * The algebraic equations in Exercises 26-29 are, respectively, the equations of a circle, ellipse, hyperbola, and sphere. The equations giving the values of z, y, z are the parametric equations of the same curves and sphere, respectively. 10 PLANE TRIGONOMETRY [7 30. Show ic 2 + y 2 + 2 2 = r 2 , when x = r(cos 6 cos + sin $ sin cos -^r), y = r(cos sin cos T/T sin d cos 0), z = r sin sin -v/r. 31. Show 2;* + y* = T , when x = a cos 3 0, ?/ = a sin 3 0. EXERCISES Simplify the following expressions, construct the angle in each case, and read the values of the remaining functions. NOTE. When square roots are to be taken, use only the + sign. l. sin sec 0-f 1 = 5. In this exercise, we have sin sec = 4, or, Hence, V17 , cos d> = = , V17 I tan < = 4, cot d> = - , sec = V 17, t In the following the values of a single function should be found, then the angle may be constructed. 2. tan = 2 sin sec 0. 3. tan cos = | . 4. sec cot = f 5. sec = 4 cos 0. 6.- cot = 3 tan 0. 7. tan = 4 cot 0. 9. cos -*- sin = |^. 10. cot cos tan = - Ans. cos = - b b 11. sin 2 sec cos esc = ^-. Ans. sin = y 3 ^. 12. 1 cos 2 = -|. ^Iws. sin = 4. tan = !. sin = f . Ans. sin <^> = | . sec 0=2. , Ans. tan = 2. . COS = |. cot = 4- 7] TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 11 13. sec 2 4> 1 = i 2 4\- Ans. tan < = T 5 ^. 14. 2 3 sin d> = i. Ans. sin = i . ' ^ ' J 15. (1 + sin )(! sin)= -^. Ans. cos< = f. 16. tan 2 = 2. 17. tan $ = 2 2 sec -f- esc . Ans. tan < = | . 18. tan (f> cos 2 = sin 2 < + 5. Ans. tan<=6. 19. sin < = ^g- cos sec < esc <. J.TJS. sin = f . 5 tan d> x cos d> x esc rf> -< . , 4 20. -J-ael. ^.WS. Sinrf> = -- 4 cot x sec 5 1x7-1, < ^ ^ i f si n a ' sec a 21. When tan = 4, find value of - cos 2 a n . 5 T , P cos 2 a + sin 2 a 22. When sm = -, find value of 6 tana UIT-I 1 j i f si n a tan a 23. When cos a = -, nnd value 01 - 2 sec 2 a + 1 24. When sec a = 5, find value of cos a esc a > 25. When cot a = -, find value of t f ng , . 8 esc 2 a 1 V yj Tr 5 T T T sin 2 a + cos 2 26. It sin = , what value has - : 18 l + tan 2 27. If tan a = ^ V3, what value has 3 sin a 4 sin 3 a? 28. sin 30 = 1, find the value of 4 cos 3 30 - 3 cos 30. 29. sec 45 = V2, find the value of ^-^ 1- tan 2 45 30. sin 30 = I, cos 45 = | V2, find the value of tan 45 x cot 30 -=- (sec 30 x sin 45). Change the following functions so that only sin a appears : o cot a , tan a 31. sin a cos"* : 1 cos a sin a cos a sin a 32. esc a sin + cos a tan a cos 2 a 33. tan 3 a 4- 1 esc 2 + 2 sin a cos a, 34. sec 4 a tan 2 a + cos 2 a cot 2 a. 12 PLANE TRIGONOMETRY [ 7 Change the following so that only tan a appears : 35. cot a + sec a esc a. 36. (1 + cos a)(l cos a) + tan a (cot 2 a 1). 37. sin a + cos a + cos a tan a + cot a sin a. 38. (1 sin a cos a) -5- (1 + sin a cos a) cot a sin a. Change the following so that only cos a appears : sin 2 a . tan a 1 , tan a 39. | 40. sec a | cos a cot a cos a sin a 41. cot 2 a + tan 2 a sin 2 a cos 2 a. 42. sin a cos a tan a cot a. 43. Express sin x in terms of each of the other functions. Thus, (1) sin x = VI cos 2 #, from identity (A), 5. (2) sin x tan x H- Vl + tan 2 x, from (D), (F) and (B). (3) sinx 1 from (G) and (C). Vl + cot 2 x (4) sin x = A/1 -- V = A/ 860 ^" 1 , from (A) and (F). 2 sec a; sec a; (5) sina; = , from (G). esc x 44. Express cos# in terms of the other functions. 45. Express tan x in terms of the other functions. 46. Express each function in terms of sin x. [NOTE. Construct the angle x with hypotenuse unity, the opposite side sin x, and the base Vl sin 2 a;.] 47. Express each function in terms of (1) cos #, (2) tan #, (3) cot a?. 48. Prove that sec J.(sin A cos -4) + esc A(s'mA + cos .4) = sec A esc A. 49. Show sin a + cos a > 1, ^ a <; 90. . 50. Show tan a + cot a ^> 2. 51. Show sin a + cos a < V2. 52. Show sin a cos < 4-. _d 8] TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 13 Fig. 6. 8. Functions of Particular Angles. The trigonometric func- tions of certain angles may be found from geometrical draw- ings. (1) Functions of 45. From the drawing, Fig. 6, we have sin 45 = I V2, esc 45 = V2, cos 45 = I V2, sec 45 = V2, tan 45 = 1, cot 45 = 1. (2) Functions of 30 and 60. From Fig. 7, sin 30 = i = cos 60, cos 30 = ~ = sin 60, tan 30 = ~ = cot 60, o cot 30 = V3 = tan 60, sec 30 = f V3 = esc 60, esc 30 = 2 = sec 60. VT Fig. 7. (3) Functions of and 90. The trigonometric functions of and 90 are limiting values which may be seen from the drawing, Fig. 8. As the side y approaches 0, the opposite angle approaches and the base x approaches r. 00 cos no lim tanO= lim - = 1 x=r f : sin 90, = cot 90, : tan 90, sec = h l n - = 1 = esc 90, X 1 x csc = lim n - = oc = sec 90. =O 14 PLANE TRIGONOMETRY C9 The angles 0, 30, 45, 60, 90, occur so frequently that it will be found convenient to keep in mind the numerical values of the trigonometric functions of each. Tabulate the values of these functions as shown in the follow- ing table : = COS 6 tan sec 6 30 60 !>0 .:_:_._ EXERCISES Find the values of the following by inserting numerical values of the trigonometric functions and reducing to simplest form. 1. 5 cos 60 - 3 sin 30 + tan 45. Ans. 2. 2. 5 cos 30 + 4 cos 60 - 5 sin 60. Ans. 2. 3. 8 tan 30 -4 cot 45 + cos 90 -8 cot 60. Ans. -4. 4. 10 cos 30 + 16 cos 60 - 5 V3. Ans. 8. 5. (4 tan 45 -11 tan 60) (4 cot 45 + 11 cot 30). Ans. -347. 6. (16 tan + 10 cot 90 + 6 sin 30) cos 90. Ans. 0. 7. sin 30 x cos x tan 45 x sec 60. 8. (a; + y) cos - <> - y) tan 45 - 2 y sin 30. 9. O + yf sin 60 - O - #) 2 cos 30 - 2 ^ tan 60 C 10. (a + 6) sec 60 + (a - 6)cos 90 + a tan 90. 11. a sin 30 - (a - 6) tan 45 -b sin 30. 1. 1. Ans. y. Ans. 0. Ans. co. Ans. ^ (6 a). 12. (a sin 60 -6 cos 30) x esc 60. . a b. 9. Tables. For convenience in the numerical calculations which follow on p. 26 a condensed table of trigonometric functions, true to three decimals, will now be inserted. By use of this table approximate results may be obtained for nu- merical problems. 9] TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 15 NATURAL TRIGONOMETRIC FUNCTIONS ANC;I.K SIN Csc TAX COT- SEC Cos 0.000 00 0.000 00 1.000 1.000 90 1 1 0.017 57.30 0.017 57.29 1.000 1.000 89 2 0.035 28.65 0.035 28.64 1.001 0.999 88 3 0.052 19.11 0.052 19.08 1.001 0.999 87 4 0.070 14.34 0.070 14.30 1.002 0.998 86 5 0.087 11.47 0.087 11.43 1.004 0.996 85 6 0.105 9.567 0.105 9.514 1.006 0.995 84 7 0.122 8.206 0.123 8.144 1.008 0.993 83 8 0.139 7.185 0.141 7.115 1.010 0.990 82 9 0.156 6.392 0.158 6.314 1.012 0.988 81 10 0.174 5.759 0.176 5.671 1.015 0.985 80 11 0.191 5.241 0.194 5.145 1.019 0.982 79 12 0.208 4.810 0.213 4.705 1.022 0.978 78 13 0.225 4.445 0.231 4.331 1.026 0.974 77 14 0.242 4.134 0.249 4.011 1.031 0.970 76 15 0.259 3.864 0.268 3.732 1.035 0.966 75 16 0.276 3.628 0.287 3.487 1.040 0.961 74 17 0.292 3.420 0.306 3.271 1.046 0.956 73 18 0.309 3.236 0.325 3.078 1.051 0.951 72 19 0.326 3.072 0.344 2.904 1.058 0.946 71 20 0.342 2.924 0.364 2.747 1.064 0.940 70 21 0.358 2.790 0.384 2.605 1.071 0.934 69 22 0.375 2.669 0.404 2.475 1.079 0.927 68 23 0.391 2.559 0.424 2.356 1.086 0.921 67 24 0.407 2.459 0.445 2.246 1.095 0.914 66 25 0.423 2.366 0.466 2.145 1.103 0.906 65 26 0.438 2.281 0.488 2.050 1.113 0.899 64 27 0.454 2.203 0.510 1.963 1.122 0.891 63 28 0.469 2.130 0.532 1.881 1.133 0.883 62 29 0.485 2.063 0.554 1.804 1.143 0.875 61 30 3 0.500- 2.000 0.577 1.732 1.155 0.866 60 31 0.515 1.942 0.601 1.664 1.167 0.857 59 32 0.530 1.887 0.625 1.600 1.179 0.848 58 33 0.545 1.836 0.649 1.540 1.192 0.839 57 34 0.559 1.788 0.675 1.483 1.206 0.829 56 35 0.574 1.743 0.700 1.428 1.221 0.819 55 36 0.588 1.701 0.727 1.376 1.236 0.809 54 37 0.602 1.662 0.754 1.327 1.252 0.799 53 38 0.616 1.624 0.781 1.280 1.269 0.788 52 39 0.629 1.589 0.810 1.235 1.287 0.777 51 40 0.643 1.556 0.839 1.192 1.305 0.766 50 41 0.656 1.524 0.869 1.160 1.325 0.755 49 42 0.669 1.494 0.900 1.111 1.346 0.743 48 43 0.682 1.466 0.933 1.072 1.367 0.731 47 44 0.695 1.440 0.966 1.036 1.390 0.719 46 45 0.707 1.414 1.000 1.000 1.414 0.707 45 3 Cos SEO COT TA.V OK SIN ANGLE CHAPTER II SOLUTION OF RIGHT TRIANGLES A right triangle is known when a side and any other part are known. The use of trigonometric functions enables us to compute the unknown parts. 10. Fundamental Formulas. We have, Fig. 9, the following 8 relations : (1) a 2 + 6 2 = c 2 , (2) sin ^4 = - = cosB, c (3) cos ^4 =- = sinB, c (4) tan^4=^ = cotB, b Fig. 9. (5) A + B = 90. The first of these formulas is a statement of the Pythagorean Theorem ; (2), (3), (4) are results from the definition of sine, cosine, tangent. With any two dimensions given (not A and .#) the other three dimensions may be ,..-.. ^ found. EXAMPLES. 1. Given A 37, c = 12 in. ; find a and ft. Solution. From formula (2), a = c sin A = 12 x sin 37. From the approximate table, p. 15, sin 37 = .602. Therefore, a = 12 x .602 = 7.2 in. From (3), ft = c cos 37 = 12 x .799 = 9.59 in. (a) 16 -37 11] SOLUTION OF RIGHT TRIANGLES 17 2. Given a = 20 ft., c = 35 ft., find angle A. a Solution, sin A = - = ~ = .571. c 35 From table, p. 15, the angle whose sine is .571 is 34 50' (approximately). Hence .1 = 34 50'. 11. Projections. Let perpendiculars fall from two points, A and B (Fig. 10) upon any line Z, intercepting PQ. PQ is called the projection of the line AB upon L. This manner of projection is called orthogonal. s P Q Fig. 10y (1) Horizontal and vertical projections. Any line AB may be projected orthogonally in any direction ; the usual projec- tions, however, are upon the horizontal PQ, or upon the verti- cal RS. (2) Laivs of orthogonal projection. The orthogonal projec- tion of any line upon another line involves the base and alti- tude of a right triangle, B with the hypotenuse given. Hence, (a) The horizontal pro- jection, x, = r . cos 8, (5) The vertical projec- tion, r~~ Q Fig. 11. y = r . sin 8. THEOREM. The horizontal projection of any line segment equals the length of the segment multiplied by the cosine of the 18 PLANE TRIGONOMETRY [ 12-13 angle of inclination; the vertical projection equals the length of the segment multiplied by the sine of the angle of inclination. 12. Components. Forces, velocities, and accelerations may be represented by directed lines. A line of given length and fixed direction is sometimes called a vector. Thus, in Fig. 12, P'is a vector, also V x , V y . Fig. 12. Suppose a body is moving with a velocity V in the path AB, Fig. 12, making an angle a with the horizontal OX', we may require the velocity in the horizontal A (the horizontal com- ponent of the velocity, V x ~), also the vertical velocity CB (the vertical component of the velocity, V y }. These components are projections of AB upon the horizontal and vertical, respectively. ., or Vx - Fcosa; CB = AB sin a, or Vy = Fsin a. Squaring and adding, . F* 2 + V y * = F 2 (cos 2 a + sin 2 a) = F 2 . THEOREM. The sum of the squares of the horizontal and ver- tical components of a velocity equals the square of the velocity. 13. Resultant. If two forces act upon a particle at A, the one, AC, acting horizontally, the other, AD, acting vertically, the particle at A will be moved along the diagonal of the parallelogram to B, Fig. 13. The line AB is called the re- 13] SOLUTION OF RIGHT TRIANGLES 19 sultant of the forces represented by the two lines A O and AD. The forces A C and AD need not act at right angles, but what- ever their angle, the resultant will be the diagonal of the par- allelogram constructed upon AC and AD. Fig. 13. The relations between resultant and components which act at right angles to each other involve the simple trigonometric properties of right triangles. EXAMPLES. 1. A train of cars is moving northeast with a velocity V of 40 mi. per hour. Find its rates of travel east and north. Solution. Let the vector V represent the given velocity. Then V x , V v are the velocities east and north, respectively. See Fig. 12. V x = V cos 45 = 40 x J V2 = 20v/2, V,j = V sin 45 = 40 x \ V2 = 20 V2. Hence, the velocities east and north are equal, 20 V2 mi. per hour. 2. A' point is moving with a velocity of 25 ft. per second along a line making an angle of 38 with the horizontal. Find the horizontal component. Ans. 19.7 ft. 3. Find the velocity of a point moving at an angle of 64 with the horizontal, if the vertical component of the velocity be 100 ft. per second. Ans. 111.3 ft. 4. A point describes a circle of radius 20 inches at a uniform rate of two revolutions per minute. Find the distance from the centre of the circle to the projection of the point upon a 20 PLANE TRIGONOMETRY [13 horizontal diameter 5 seconds after passing the extremity of that diameter. Ans. 10 in. 5. A force F of 100 Ib. is applied to a block at the point 0, see Fig. (>). If the force makes an angle of 60 with the hori- zontal, what force tends to draw the block horizontally (0(7) ? What force tends to lift the block upwards Solution. Here the solutions called for are the horizontal component F x and the vertical component F y of the force F. F x = F x cos a = 100 cos 60 = 50 Ib., F y = F x sin = 100 sin 60 = 86.6 Ib. 6. A balloon rising vertically at a uniform velocity of 704 ft. per minute encounters a wind blowing horizontally at the rate of 24 mi. per hour. Find the angle at which the balloon will rise and its velocity after meeting the wind. Solution. The angle required is represented by in Fig. c, the velocity of wind and vertical velocity of the balloon by OB and OA, respectively. 04 _1 3' tan = 2112 2112 a = 18 26'. V = 2112 x sec = 2112 x 1.054 = 2226 ft. 7. A force of 400 Ib. acting in a direc- tion inclined 40 from the horizontal is ap- plied to a heavy block. Find the force which tends to move the block (1) horizontally, (2) the force which tends to lift the block vertically. Ans. (1) 306.4 Ib., (2) 257.2 Ib. 8. The horizontal and vertical components of a force acting upon a heavy block are 160 Ib. and 120 Ib. respectively. Find the force and its direction of action. Ana. 200 Ib., 36 52'. 5 14, 15] SOLUTION OF RIGHT TRIANGLES 21 14. Projected Areas. Any plane area ABCD inclined to a Fig. U. plane LM at an angle 6 may be projected orthogonally upon LM into the area ABFE. ABFE = ABCD x cos 9 Law of projection. The projection of a given plane area upon a plane equals the given area multiplied by the cosine of the angle of inclination of the two planes. EXAMPLES. 1. How much horizontal area will be covered by 60 sq. yd. of roofing, the roofing making an angle of 45 with the horizontal ? Ans. 60 cos 45 = 42.42 sq. yd. 2. Find the roofing required to cover a horizontal space 15 ft. by 24 ft., the roofing rising at an angle of 40. Ans. 15 x 24 x sec 40 = 469.8 sq. ft. 15. Area of the Right Triangle. The area of a right triangle may be expressed in various ways. Let K= area ; then we have, (1) K = i a x It \ ac cos A = \ be sin A. (2) K = I b 2 tan A = la? tan B (3) K = I c 2 sin A cos A = \ c 2 sin B cos B. 22 PLANE TRIGONOMETRY [10 In (1) a cos A = b sin A = p ; and in (2) b tan A = a, a tan B = b. A* Fig. 15. EXAMPLES. Find the area of each of the following right triangles, the lettering being shown in Fig. 15 : 1. a = 20, ^4 = 35. 2. c = 65, .6 = 28. 3. a = 100, .4 = 80 4. 6=42, #=67. 5. is the angle of elevation of P: the angle Q OH ^r is called the angle of depression of Q. A depression angle could also be considered as a negative elevation angle. The direction of a line OP with respect to a north and south line NS, Fig. 18, is read North a de- grees East, N. E. ; the same direction with regard to the east and west line ^TTcould be read East ft degrees North, E. /3 N. 24 PLANE TRIGONOMETRY [ 17 Similar readings for directions may be applied to south and N I 7 \ ' s Fig. 18. west lines. A direction of a line given as above is called the bearing of the line when read from JV". or S. EXAMPLES, i. If the angle of elevation of the sun be 35, how high is a pole whose shadow upon the ground is 40 ft. ? Solution. (1) Construct drawing showing data. (2) Select formula (4), p. 16, Ji 40' h = 40 x tan 35 = 40 x 0.7 = 28 ft. tan 35 = 2. Find the radius of a circle in which a chord 30 in. long subtends a 20 arc. Also find the area of the triangle formed by the chord and the radii to its extremities. Solution. (1) Construct figure ; draw radius perpendicular to middle of chord. Let B be the middle point of the chord. Angle AOB = 10, AB = 15 in. AB or, sin AOB = , R = AB -s- sin AOB, R R = ABx esc AOB. SOLUTION OF RIGHT TRIANGLES 25 (3) Substitute the values AB = 15, AOB = 10, and we have R = 15 x esc 1U = 15 x 5.759 = 86.38 in. (4) The area is given by K - \ base x altitude. OB = 15 x cot 10 = 85.06. Then K = 15 x OB = 1275.9 sq. in. 18. To solve Right Triangles. In the solution of right tri- angles the beginner should provide himself with (1) a gradu- ated ruler, (2) squared paper, and (3) a protractor graduated to degrees. With this equipment a triangle approximating very closely to any given data may be constructed. An approximate construction will enable the student to detect any considerable error in calculation. In solving a right triangle the following directions are sug- gested : (1) Draw, on squared paper, a figure as accurately as possible from the given data, and estimate approximate values for the unknown parts. (2) Select formulas, 10, each of which contains one unknown. (3) Substitute the given data in the proper formula, using approximate values of the trigonometric functions, p. 15, and solve for the unknoivns. (4) Check results by using some formula not employed in the calculation. In many ordinary measurements approximate results only are desired. The following list of exercises involving the right triangle is inserted to give practice in approximating results. The trigonometric functions found on p. 15 give approximations to three decimals, and when employed in solu- tion will give results true to one or two decimals. When greater accuracy is desired, the more complete tables of the natural functions should be used. For still more accurate results, the logarithmic tables, explained in Chapter VI, should be employed. 26 PLANE TRIGONOMETRY [18 EXERCISES Solve the following right triangles, giving results to nearest tenth and nearest minute. Use Approximate Tables of trigo- nometric functions, p. 15, for solutions of Ex. 1 to 20, approxi- mating angles to minutes. For areas employ formulas of 15. No. DATA ANSWERS AREA = A' 1 a = IS ft = 40 ^4 = 18 B = 12 c = 42 260 2 a =20 ft =8 ,4 = 68 12' B = 21 48' c = 21.5 80 3 a =80 ft =30 A = 69 27' B = 20 33' c = 85.4 1200 4 a = 16.16 ft =25.28 A = 32 35' B = 57 25' c = 30 204.3 5 a = 10 c = 50 A = 11 32' ^ = 78 28' ft = 49 245 6 a = 71 c = 78 A = 65' 32' 7? = 24 28' 6 = 32.3 1146.6 7 a = 84.9 c = 93.5 A = 65 14' J3 = 24 46' ft = 39.17 1662.8 8 c =42 A = 81 30' B = 8 30' a =41.5 ft =6.2 128.9 9 c = 100 A = 36 B = 54 a = 58.8 ft = 80.9 2378.5 10 c =67.7 A = 23 30' B = 66 30' a = 27 ft = 62.08 838.1 11 c =250 7J = 47 A =43 a = 170.5 6 =182.8 15583.7 12 c =400 ft =240 A = 53 8' B = 36 52' a = 320 38400 13 A^=55.42 .4 = 30 a =8 ft =8\/3 c =16 14 1T=28.93 ^=.26 45' a = 5.4 ft = 10.7 c = 12 15 K = 145.8 6 = 18 A =42 c =24.2 a = 16.2 16. A monument 283 ft. high casts a shadow 100 ft. long upon the ground. Find the angle of elevation of the sun at that instant. Ans. 70 32'. 17. A ladder 30 ft. long resting upon the ground reaches a point 24 ft. high upon a vertical wall. Find the angle of ele- vation of the ladder. Ans. 53 6'. 18. Gable rafters 20 ft. long project 2 ft. beyond the walls of a house and are set with a pitch (angle of elevation) of 40. Find the height h of the ridge-pole and the width of the house. Ans. h = 11.57 ft., w= 27.57 ft. 19. Find the bearing of a road which leads to a point 5 mi. east and 8 mi. north. Ans. N. 32 E. 20. A rectangle is 98 by 148. Find the angle made by a diagonal with the longer side. Ans. 33 30'. 21. The sides of an isosceles triangle are 30, 45, 45, respec- tively. Find the angles. [Note. Use four-place tables.] Arts. 70 32', 38 56'. 18] SOLUTION OF RIGHT TRIANGLES 27 22. Find the radius of a circle in which a 100 ft. chord sub- tends an angle of 18 at the centre. Ans. 319.69 ft. 23. A chord 50 in. long subtends an angle of 36 at the centre. Find the radius R of the circle and the area A of the inscribed square. Ans. R= 80.9 in., A= 13,089.6 sq. in. 24. Find the height of a tree which casts a horizontal shadow 60 ft. long when the sun's elevation is 65. Ans. 128.67 ft. 25. Find the angle of inclination of the faces of a wedge /^* whose base is 3 in. and whose face is 14 in. long. Ans. 12 18'. 26. The exterior angle between two 500 ft. tangents is 72. Find the radius of the circle. Ans. 688.2 ft. 27. The length of a kite-string is 200 m. Find the height of the kite when its angle of elevation is 34. J.ws.111.84 m. 28. The radius of a circle is 5 cm. and the length of a chord is 4 cm. Find the angle subtended by the chord at the centre. Ans. 47 9' 20". 29. Find the radius of a circle inscribed in an equilateral triangle whose perimeter is 42 cm. Ans. 4.04 cm. 30. What is the height of a tower if a 16 ft. flagpole upon the top of the tower subtends an angle of 4 at a point on the ground and the angle of elevation of the bottom of the pole is 40 ? Ans. 106 ft. 31. At a certain point the angle of elevation of a mountain peak is 27 ; at a distance of 2 mi. farther away in the same direction its elevation angle is 25. Find the horizontal distance from the first point of observation to the peak. Ans. 21.58 mi. 32. Two consecutive milestones on a level road in the same vertical plane as a tower have depression angles of 42, 3, respectively, from the top of the tower. Find the height of the top of the tower and its horizontal distance from the nearer milestone. Ans. h = 293.8 ft. or 261.4 ft,, d = 326.3 ft. or 290.3 ft. 33. A tower stands upon the same plane as a house whose height is 60 ft. The elevation and depression of the top and bottom of the tower from the top of the house are 41 and 35, respectively. Find the height of the tower. Ans. 134.49 ft. L> 28 PLANE TRIGONOMETRY [ 18 34. From a point 10 ft. above the water, the angle of eleva- tion of the top of a tree standing at the edge of the water is 46, while the depression angle of its image in the water is 50. Find the height of the tree, and its horizontal distance from the point of observation. Ans. A = 142. 5 ft., # = 128 ft. ^ 35. In measuring the width of a river a line AB is measured 240 ft. along one bank. A perpendicular to AB at A is erected which locates a point O upon the opposite bank, and the angle ABC is found to be 65. Find the width AC of the stream. Ans. 514.68 ft. 36. Two towers upon the same horizontal plane are of such heights that their elevation angles from a point midway be- tween them are 40, 60, respectively. Find the ratio of their heights. Ans. 8391-5-17,321. 37. From each of two stations a mile apart upon a north and south road, the angle of elevation of a balloon is observed to be 30, and its bearings are, respectively, N.E. and S.E. Find the height of the balloon. Ans. 2155.5 ft. 38. A balloon is exactly over the middle point between two cities. The balloon is a mile high and the distance between the cities subtends an angle of 136 at the balloon. Find the dis- tance between the cities and the distance of the balloon from either of them. Ans. 4.95 mi., 2.67 mi. 39. Find the height of a tree if the angle of elevation of its top changed from 36 to 42 on walking toward it 75 ft. in a horizontal line through its base. Ans. 282.16ft. 40. A hill rises uniformly 4 ft. in a horizontal distance of 85 ft. What is the difference in elevation of two points 3500 ft. apart, the distance being measured along the ground ? Ans. 164.1 ft. 41. What is the angle of incline of a railroad track if it rises 30 ft. in a horizontal distance of a mile ? Ans. 19' 32". 42. What is the bearing of a road which leads to a point 14 mi. east and 8 mi. north? Ans. N. 60 15' 20" E. 43. If the radius of the earth (3956 mi.) subtends 57' at the moon, what is the moon's distance from the earth ? Ans. 238,300 mi. 18] SOLUTION OF RIGHT TRIANGLES 29 44. Find the radius of one's horizon if he be located 1320 ft. above the earth. How large when located 2 mi. above the earth? Am. 44.48 mi., 125.8 mi. 45. How high above the earth must one be in order to see a point located on the surface 50 mi. away? Ans. 1668.2 ft. i 46. The radius of a circle is 240 ft. Find the perimeter of a regular inscribed pentagon. Ans. 1410.72 ft. . 47. The radius of a circle is 85 ft. What is the area of the regular inscribed decagon? Ans. 21,234.27 sq. ft. 48. Find the perimeter of a regular dodecagon inscribed in a circle whose radius is 25 in. Ans. 155.25 in. v 49. What is the radius of a circle inscribed in an equilateral triangle whose perimeter is 99 ft.? Ans. 9.52 ft. so. The area of a regular pentagon inscribed in a circle is 475.53 sq. cm. Find the area of a regular decagon inscribed in the same circle. Ans. 587.8 sq. cm. 51. The area of a regular pentagon is 441 sq. ft. Find the apothem ; also find the radius of the circumscribed circle. Ans. 11 ft.; 13.6 ft. 52. What is the length of a diagonal joining the first and fourth vertices of a regular polygon of 12 sides inscribed in a circle whose radius is 30 ft. ? Ans. 42.426 ft. 53. If R = radius of a circle, show that the area of a regular -& 180 180 inscribed polygon ot n sides is A = nR sin cos n n 54. From the result in Ex. 53 construct a table showing, in terms of R, areas of the regular inscribed polygons of 3, 4, 5, 9, 10, 12, sides. 55. Show that the area of a regular circumscribed polygon of n sides is given by A = nR 2 tan 56. A train moving at a uniform speed of 40 mi. per hour on a straight track passes a station A at noon ; at 2:15 o'clock p. M. it has arrived at a station B, 56 mi. farther north. How far east of A is jB, and what is the bearing of the road ? Ans. 70.46 mi.; N. 51 31' 30" E. 30 PLANE TRIGONOMETRY [ 18 57. A boat running at a rate of 10 mi. per hour starts to steam directly across a river one mile in width. If the water be flowing uniformly at a rate of 4 mi. per hour, find the point a,t which the boat will land, and its velocity in the water. Ans. 0.4 mi. downstream, 10.77 mi. velocity. 58. Find the projection of the altitude of an equilateral tri- angle upon a side. Let a = side. Ans. | a. 59. A line extends N. 20 E. 125 rd. from a point A. Find its projection upon a line extending N. 60 E. from A. Ans. 95.75 rd. 60. Find the projection of a line 450 ft. long running N. 47 E. upon a line running E. 15 S. Ans. 238.45 ft. 61. Two forces of 160, 120 Ib. act upon a heavy body, the first at an angle of 30 with the horizontal, the second at an angle of 75. Find the total forces which tend to move the body (1) horizontally, (2) vertically. Ans. (1) 169.6 Ib., (2) 195.9 Ib. 62. Find the number of square yards in a conical tent with circular base, the vertical angle being 60, and the centre pole 12 ft. high. [Take 7r = \ 2 -.] Ans. 33.5 sq. yd. 63. Find the area in acres of the following tract of land: starting from a point A, the boundary line runs N. 24 E. 80 rd. to B, thence N. 65 E. 120 rd. to C, thence S. 180 rd. to D, thence back to A. Find also the length and bearing of DA. Ans. 99.157 acres ; DA = 152.07 rd.; N. 68 18' 23" W. 64. Find the area of the following described tract of land : starting from a point J., the boundary line runs N. 10 E. 100 rd., thence N. 47 E. 150 rd., thence E. 40 rd., thence S. 10 W. 100 rd., thence W. 40 rd., thence to A the place of beginning. Ans. 81.04 acres. CHAPTER III ii TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. GRAPHS In 2 the trigonometric functions have been defined for positive acute angles only. We shall now extend these defi- nitions to include angles of any size whatever. 19. Axes. Quadrants. To locate an angle which may be either acute or obtuse it is convenient to employ Coordinate Axes as shown in Fig. 19. The horizontal line is called the Jf-axis, the vertical is called the Y-axis. These axes divide the plane into four quadrants marked I, II, III, IV. A positive trigonometric angle is described when a vector OP is rotated about counter-clockwise from the initial position OX into a terminal position OP ; XOP = a. If the rotation be clockwise about 0, the angle described is nega- tive ; XOP l = -j3. See Fig. 19. 20. Coordinates. Abscissa, Ordinate. The position of a terminal line OP (Fig. 20 a, b, c, d) is determined by two measurements x, y, called coordinates of the point P. The x-measurement is called the abscissa of the point P, and is the projection of OP on the X-axis. If the projection of OP falls to the right of (Fig. 20 a, J), the abscissa is positive, if to the left (Fig. 20 5, K / \ , / , 1 \ Y Y - ' 3 S f 1 ) ^ s r II 3 / V S u ,x 1 -^ S / > 5 ^ ^ s, / f 3 - . > , ' I Y \l I \ 2 c. I \ ^ 2 ) 21. Definitions of the Functions. From the above figures, (a), ^), ( distance y _ ordinate sec a = = distance = -r > a? abscissa a? abscissa = ' 22] TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 33 In this notation the signs of the abscissa and ordinate deter- mine the algebraic signs of the trigonometric functions for angles terminating in the respective quadrants. 22. Laws of Signs. (1) For all angles terminating in the first quadrant, the functions are positive. (2) For angles terminating in the second quadrant, the six functions are negative except sine and cosecant. (3) For angles terminating in the third quadrant, the tangent and cotangent are positive, all others are negative. (4) For angles terminating in the fourth quadrant, the cosine and secant are positive, all others are negative. These laws are shown in the following diagram : Six Cos TAN COT SEC Csc I + + + + + + II + - - - - + III - - + + - - IV -f - - + - It should be noticed that these general definitions apply to angles larger than 360. For example, sin 400 is the sine of an angle terminating 40 above the initial line ; hence sin 400 = sin 40. tan 500 = tan (360 + 140) is the tangent of an angle terminating in the second quadrant, tan 500 = tan 140 = - tan 40. sin (n 360 + ) = sin . The fundamental identities, 5, hold for any angle. sin' 2 + cos 2 = 1, sec' 2 tan- a = 1, esc' 2 cot' 2 a = 1, sin a esc a = 1, cos a sec = 1, tan cot= 1. EXAMPLES. 1. What values have the functions when wnf-f? Solution. The angle < may be constructed in either of two positions, A OP^jn the first quadrant, or the supplement A OP 2 in the second quad- rant. Let A OP l = $ r A OP-, - < 2 . In the first case, sin < t = 4, cos ^ = |, tan <>! = f, etc. In the second case, sin . 2 = f, cos . 2 = $, tan < 2 = f, etc. 34 PLANE TRIGONOMETRY [22-23 2. Find all the trigonometric functions when cos = ^. Solution. Locate angle < as a positive angle in the first quadrant, and as an equal negative angle in the fourth quadrant. The angle whose cosine is - is either < or . Then, sin = , cos < = -, tan < = Vtf, cot < = , etc. V3 3. Locate the positive angle , when P has the following coordinates: (a) (4, 3); (6) (-4, 5); (c) (5, -3); ( terminates when : 1. sin = |. 6. tan = 5. 2. tan< = 4. 7. csc<= 2. 3. cos = f 8. sin < = f , tan < < 0. 4. siii(= ^. 9. cos = |^, cot < 0. 5. cot = 0. 10. tan = 3, sin < < 0. Express the following as functions of acute angles : 11. sin (440) = sin (360 + 80) = sin 80. 12. sin 370. 13. tan 430. 14. cos (2 7r+20). . tan[7rH ). 16. cotfmrH ). 17. sin(2/wrH ) V 3/ \ QJ \ 4/ 18. sec 300. 19. tan 700. 20. sin 500. 15 23. Functions of Negative Angles. To express the trigo- nometric functions of a negative angle in terms of an equal positive angle construct the negative angle and an equal posi- tive angle, Figs. 21 and 22. In either drawing (6 acute, or 6 obtuse) the triangles A OP and A OP l are equal, 23] TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 35 sin(-e)= * = =-sine, rj r QC* Uu cos ( 6) = - = - - cos 0, r i r csc (-0) --esc 6, sec ( 0) = sec 0, tan (- 0) = -J = - r =- tan 0, cot (- 0) =- cot 0. THE LAWS GOVERNING FUNCTIONS OF NEGATIVE ANGLES ARE : (1) Any trigonometric function of a negative angle equals the same function of an equal positive angle, (2) the sign being changed in all cases except for the cosine and secant. EXERCISES 1. Write equivalents of the following functions with the signs of the angles changed : (1) cos (-48) ; (4) cot (- 87) ; (7) sin 47 ; (2) tan (- 65) ; (5) sec (- 75) ; (8) tan (-); (3) sin (- 50) ; (6) cot (- 100) ; (9) sin (0 - <). 2. Write numerical values of the following : (1) sin (- 30) ; (3) cot (- 60) ; (5) cos (- 60) ; (2) tan (- 45) ; (4) sec (-45) ; (6) tan (- 90). 3. Reduce to numerical values : (1) sin 90 x sin (- 90) -=- tan (- 45). (2) tan (- 60) x sin (- 30) x csc 60. (3) sin 2 (- 45) x cos (- 60) x csc (- 45). (4) sin 2 (- 45) + cos 2 (- 45) + tan (- 45). (5) sec 2 (120) - tan 2 (120) + sin (-90). (6) cos (- 80) x sin (- 20) x sec 80 x csc (- 20). 36 PLANE TRIGONOMETRY [ 23-24 24. Functions of 90 - 8 and 90 + 8. (1) Complemented angles. The functions of 90 with acute, Fig. 23 a, or obtuse, Fig 23 b ? obey the same laws as have been developed in 3. 4Y .D i* iY *X A 1 , i Y' Fig. 23 a. In the drawings, Figs. 23 a, 5, the triangles OP A and are equal, and x i=y* y\= x i r i= r - Defining the functions, we have : sin (90 - 0) = & = * = cos 0, esc (90 - 0) = sec 0, #j / cos (90 - 0) = ^1 = y~ = sin 0, sec (90 - 0) = esc 0, r \ r tan (90 -0) = 1 = = cot 0, cot (90 -0) = tan 0. LAW FOR COMPLEMENTAL ANGLES. The functions of the complement of any angle equal the corresponding co-functions of the angle. (2) Functions of 90 + 0. The functions of 90 + may be derived from the above functions of 90 by changing into and noting the results of 23. sin (90 + 6) = cos ( - 6) = cos 6, esc (90 + 6) = sec 6, cos (90 + ) = sin ( - 8) = - sin 6, sec (90 + 9) = - esc 6, tan (90 + 9) = cot (-6) = - cot 0, cot (90 + 6) = - tan 8. These results show that the same law holds for functions of 90 +0 as for 90 0, except the algebraic sign is changed in all cases except that of the sine and cosecant. 25] TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 37 25. Functions of 180 -9 and 180 + 9. Supplemental Angles. The functions of the supplement of an angle 6 may be expressed in terms of the functions of 0. Construct acute, Fig. 24 a ; also construct 6 obtuse, Fig. 24 b. In either figure the triangle OAP equals the triangle OA-^P^ and x =-x, y = y, r =r. Hence, sin (180 - 9) = ^ = | = sin 9, esc (180 - 9) = esc 9, r i cos (180 - 9) = *-! = =~ = -cos 9, sec (180 - 9) =- sec 9, tan (180 - 9) = ^ = ^ = -tan 9, tan (180 - 9) = - tan 9. LAWS FOR SUPPLEMENTAL ANGLES. Any trigonometric function of an angle equals the same function of its supplement, the algebraic sign being changed in all cases except the sine and cosecant. The functions of 180 + may be obtained from those of 180 6 by changing 6 to 6 in the above formulas. We find : sin (180 + 9) = sin (- 9) = - sin 9, esc (180 + 9) = - esc 9, cos (180 + 9) = - cos ( - 9) = - cos 9, sec (180 + 9) = - sec 9, tan (180 + 9) = - tan (- 9) = tan 9, cot (180 + 9) = cot 9. LAW. Any trigonometric function of 180 equals the same function of the angle 0, regard being had for the algebraic sign. 374041 38 PLANE TRIGONOMETRY [25 EXERCISES 1. Fill the blanks with the proper function of the supplement of each angle : (1) sin 150 = sin 30 ' (6) esc 100 20' = (2) tan 97 20'= (7) tan (90 + <) = (3) cos 160 40' = (8) sin (90 -)= X" (4) cot 175 10' = (9) cos (45 - <) = ' (5) sec 120 10' = (10) cot (60 + <) = 2. By taking supplements of 0, 30, 45, and 60 find the trigonometric functions of 180, 150, 135, and 120. Fill the blanks in the following table : 30 45 60 90 120 135 150 180 sin (f> .1*1 P r \ cos / 0- ',- - tan 1 \w ', cot < oO / . "tfl sec (f> n^ c*e>. ~" f CSC (f> V us? r 3. Find numerical values of the following : (1) sin 120 x sin 60. (2) tan 45 x cot 12 x cos 90. (3) 5 x cos 135 x sin 90 x cos 180. (4) 'tan 135 x cot 130 -4- sin 60. (5) tan 150 x cos 150 -5- sin 30. (6) 2 sin 120 + cot 150. (7) tan 135 + cot 45 - cos 180. (8) cos 30 + cos 150 + tan 60 + tan 120. (9) (tan 120 - tan 135) x (tan 120 + tan 135). (10) sec (180 - 0) x cos 6 X tan (180 - a) x cot a. 4. From proper drawings obtain values of the functions of 180 -f 6 in terms of the functions of 6. 26] TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 39 5. Verify sin 210 = - | ; cos 225 = - l V2 ; tan 225 = 1 ; tan 240 = Vo. [NOTE. Construct drawings.] 6. By adding 180 to the angle in the functions of 90 derive the functions of 270 0. Verify the results by con- structing proper drawings. IT,- T ,i r cos (90 + a) . tan ( a) 7. i ind the value of ^ ^ + ^ } - sin(-) tan (180 + ) . -, ,, f sin (180 - ) tan (180 + 0} B. I 1 ind the value of ^ ^ X - - cos (90 + ) cot (90 + 0) 9. What sign has sin x + cos x for the following values of x : (1) x = 90 ; (2) x = 120 ; (3) x = 135 ; (4) x =210; (5) x =300? 10. Find all angles less than 360 which satisfy : (1) tan0 = -l; (2) sin0 = V3; (3) cos0 = -i. 26. Line-values of the Functions. (1) Acute angles. Let the angle be constructed at the cen- tre of a circle whose radius is unity, Fig. 25. Then arc AB = 6. COT e 00= cos 0, OT = sec0, OA = vers 0, ED = covrs 0. These lines represent graphically the values of the trigonometric func- tions when the radius of the circle is taken as unity. It should be noticed that trigonometric func- tions considered as line-values may be described as follows : (1) The sine of an angle is the length of the perpendicular let fall from the terminal end of the arc (.B) upon the diameter through the initial end (yl). (2) The cosine of an angle is the Fig. 25. 40 PLANE TRIGONOMETRY [26 part of the radius (0(7) cut off by the foot of the perpendicular (5(7). (3) The tangent of an angle is the geometric tangent erected at the initial extremity (^1) of the arc and terminated by the diameter produced through the terminal extremity (.5) of the arc. (4) The secant is the line from to the extremity of the tan- gent (OT). (5) The cotangent is the geometric tangent erected at the 90 point (D) of the circle, and terminated by the terminal radius produced to T', (D2 7 '). (6) The cosecant is the length of the line from the centre of the circle to the extremity of the cotangent (02 7 '). Convention of signs. In the above definitions the sines and tangents are verticals, and are taken positive when drawn up- ward, negative when drawn downward. The cosines and cotan- gents are horizontals, and are positive when drawn to the right, negative when drawn to the left. The secants and cosecants are positive when drawn from along the terminal boundary OB, negative when drawn from backward, as OT in Fig. 26 (a). The older text-books defined the trigonometric functions from this line-value standpoint, but at the present time the definitions are usually given, as in 2 and 21, from the ratio standpoint. It is often convenient to use the line-values. (2) Angles larger than 90. Drawings are here inserted to show the line-values of the functions for angles larger than 90, Figs. 26 (a), (5), (c). In each case OT= sec 6, OT' = esc 6. V COT Oi D D COT Bg T' Fig. 26 (a), (6), (c). These drawings show clearly that the fundamental relations, 5, hold whatever the angle may be. The following table shows the signs of the functions of 6, r 2 , 3 , shown in Figs. 25, 26 (a), (6), (c), respectively. 26] TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 41 SIN Cos COT SBC Csc It should be remarked that the functions of an angle between 270 and 360 are the same as those of a negative acute angle, and that the functions of an angle between 180 and 270 are the same as those of a negative obtuse angle. EXERCISES Draw figures and show : 1. sin 70 = cos 20 = sin 110. 2. sin 120 = cos 30 = -sin (-60). 3. sin 117 = cos ( - 27) = cos 27. 4. cos 300 = cos (-60)= sin 30. 5. sin 220 = sin (- 40) = - sin 40. 6. cos 195= -cos (-15)= -cos 15. 7. sin (- 10) = sin 190 = - sin 10. 8. sin 60 = - cos 150 = cos 30. 9. tan 120 = - tan 60 = tan ( - 60). 10. cot 165 = - cot 15 = - tan 75. If vl + + 2. Change from degrees to radians : 18 ; 124 ; 290 ; 30 ; 500 45'. 3. Change from radian measure to degrees: 1.5; |; 1.65; 2.75; -0.4; 51. 46 PLANE TRIGONOMETRY [ 29-430 4. Find the number of degrees in : (a) 1 + ; (6) 1 r ; 00 140 - 11; 00 30 + 2 TT; () 40- - ; (/) 8 - | TT. -Lv o 5. What does each of the following symbols mean? NOTE, n is any integer. 00 TT; (6)mr; OO'Swir; (rf) wj; (e) (2w + 1)|; 6. Name the quadrant in which each of the following angles terminates : 7T + 48 ; 0) 240 + 2 ; (t a cot P 1 (0 cot(ap)=- <- . -, cot a cot P "^ EXERCISES 1. Show sin 75 = sin (45 + 30) = J ( V6 + V2). SOLUTION, sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30 _ V2 _ V| V2 _ 1 2 ' 22 '2 Hence, sin 75 = cos 15 = \( Vf> + V2). 2. Show sin 15 = i( VB - V2) = cos 75. 3. Show tan 75 = 2 + V3 = cot 15. 4. Show tan 15 = 2 - V3 = cot 75. V2 5. Show sin (45 + a) = (cos a -f sin a). A NOTE. Apply formula (1). 6. Prove cos (30 a) = \ ( V3 cos a + sin a). 7. Prove tan (45 + x) = * + tap x = cot (45 - x). 1 tan x 8. Prove - cos x - sin x = sin (60 x). 9. Prove -COSZ + sin a; = cos (60 x~). u A r> sar\o ^ V3 tan X 10. Prove tan(60 x) = 1 + V3 tan x 34-35] FUNCTIONS OF TWO ANGLES 53 11. Derive the formula sin (oc + ft + 7) = sin a cos (3 cos 7 + cos a sin p* cos 7 + cos a cos /3 sin 7 sin a sin /3 sin 7. SUGGESTION. Write sin (a + /3 + y) = sin (a + [/? + y]), and apply (1). 12. Derive cos (cc + ft + 7) = cos cos /3 cos 7 cos sin ft sin 7 cos /3 sin sin 7 cos 7 sin a sin /3. is. Derive tan (x + y + 2) _ tan a; + tan y + tan 2 tan a: tan y tan 2 1 tan x tan y tan y tan 2 tan 2 tan 2; 14. If sin x = 0.5 and cos y = 0.6, find the value of sin (# + ?/), and cos (x+ y*). SUGGESTION. Construct angles x, y, and expand sin (x -f y), then substi- tute. Likewise with cos (# + ?/). sin (x + y) = sin x cos y + cos a: sin ?/ = 0.5 x 0.6 + 0.5 V3 x 0.8 = ( V3 + 4). cos (x + y~) = cos x cos y sin x sin r/. = 0.5V3 x 0.6 - 0.5 x 0.8 = (V3 - 4). 10 V 15. If sin # = 0.8, tan^=l, what value has sin(# y)l tan (z + y)? 16. When tan x = V3, cos ^ = 0.5, find tan (x y}\ also cos #/. 17. When cos #=1, tan ?/ = !, find sin (#+/); also cot ( 18. Show cos (# + y) = 0, when tan # = 0.5 and cot y = 0.5. 19. Show sin (x + y) = 1, when sin # = cos y. 20. Show tan (# + y) = cc , when cos # = sin /, or tan # = cot y. FUNCTIONS OF MULTIPLE AND SUB-MULTIPLE ANGLES The formulas enumerated in 34 give rise to new and im- portant results by making angle /3 equal to angle a. 35. Functions of 2 a. In the formulas, (1) sin ( + /3) = sin a cos /3 + cos a sin /3, (3) cos (a + /?)= cos a cos /3 sin a sin /3, , o-. tan a + tan $ .rrx (5) 1 tan a tan 54 PLANE TRIGONOMETRY [35-37 let /3 = a, and we derive : (7) sin 2 a = 2 sin a cos a, (8) cos 2 a = cos 2 a - sin 2 a = 2 cos 2 a- 1 = 1 - 2 sin 2 a, x \ 2 tan a (9) tan 2 a = 1 tan 2 a 36. Functions of 3 a. Let /3 = 2 a in (1), (3), (5) above, and employ results of (7), (8), (9), and we obtain: (10) sin 3 a = 3 sin a 4 sin 3 a, (11) cos 3 a = 4 cos 3 a 3 cos a, 3 tan a tan 3 a (12) tan 3 a = 3tan 2 a 37. Half-angle Formulas. In (7), (8), (9) 35, let = |, and we find : (13) sin x = 2 sin -cos-, (14) cos x = cos 2 ? - sin 2 - = 2 cos 2 - - 1 = 1 - 2 sin 2 ?, it ' A A 2* 2 tan | (15) tan x = 1-tan 2 - Solving each of the latter two values of cos x for sin - and a cos ?, we have the important formulas : M (16) (17) and dividing sin- by cos-, 2i /] g\ ^jj a? _ ll cos x _ _sin^ 1 - cosag. 2 M + coso; 1+cosic sin a? (16), (17), (18) are known as the half-angle formulas. 37] FUNCTIONS OF TWO ANGLES 55 EXERCISES 1. If sin x = ^, find sin 2 x ; also tan 2 x. 2. If cos x = ^, find tan 2 a? ; also cos 2 a;. 3. If esc x = 2, find sin 2 a; ; also cos 2 a;. 4. Find sin a;, if cos-= 0.3. x 5 5. Find tan a:, if sin - = . I > 6. Find sin a;, if cos 2 a; = ^- *7. Find tan a;, if sin 2 x = ^. 1 8. Find tan 60, knowing tan 30 = V3. 9. Find tan 53 8', if tan 26 34' = 0.5. 10. Find cos 2 a;, if sec x = %. 11. If sin a = 0.4, find sin 3 a. 12. When tan a = 0.1, find tan 3 . A- 13. Find sin 22 and cos 22 from the functions of 45. /-14. Find tan 15 from the functions of 30. 15. Find the sin 67| from the. functions of 22|. Verify the following identities : 16. (sin x cos a;) 2 = 1 sin 2 x. 17. cos 4 x sin 4 x = cos 2 x. 2 tan- 1-tan 2 ? ""19. 18. sin a; = -- ""19. cos a; = 1 + tan 2 n . . 2 sin 3 x 20. 2 sin a; + sin 2 x = 1 cos x 21. tan a; + cot # = 2 esc 2 a;. 22. tan [ h - ] = sec x -f tan a:. V4 2J 23. cot a; tan x = 2 cot 2 z. 56 PLANE TRIGONOMETRY [37-38 cot a; 25. =1 + 2 cos 2 x. 27. tan - = sin x 21 + sin x + cos x 28. sin 40 = 4 sin cos (cos 2 sin 2 0) = 8 cos 3 sin 4 cos sin 0. 29. cos 4 = 8 cos 4 0-8 cos 2 + 1. 30. When sin 3 sin 6 = 0, show cos 3 = . 31. sin 50=16 sin 5 0-20 sin 3 + 5 sin 0. 32. cos50 = 16cos 5 0-20cos 3 + 5cos0. - 33. sin 2 A sin 2 B = sin (J. + B) sin (A B). 34. cos 2 .A cos 2 B = sin {A + B) sin (.B A). 35. cos 2 J. sin 2 B cos (J. + B} cos(J. #). 36. sin - + cos - = vT^f- sin 0. 37. sin cos = Vl sin 0. (Explain use of double sign in 36, 37.) SUM AND DIFFERENCE FORMULAS 38. Converting to Products. Taking the results (1), (2), (3), (4), 34, sin (a + /3) = sin a cos /3 + cos a sin /3, sin (a /3) = sin a cos /3 cos a sin & cos (a + /3) = cos a cos /3 sin a sin $, cos (a /3) = cos a cos /3 + sin a sin /3, we have on adding and subtracting : sin (a + /3) + sin (a /9) = 2 sin a cos /3, sin (a + /3) sin (a /3) = 2 cos a sin /3, cos (a + /3) + cos (a /3) = 2 cos a cos /3, cos .( + /3) cos (a /3) = 2 sin a sin /S... Let a + /3 = JT, a-/3=P; then jr+r JT-F 38-39] FUNCTIONS OF TWO ANGLES 57 Making these substitutions, we have : (19) sin X + sin Y = 2 sin x + Y cos X ~ Y , (20) sin X - sin Y = 2 cos J sin A ~ 1 , (21) cos A' + cos Y = 2 cos X + r cos X ~ r , A L (22) cosJi:-cosI r =-2sm X ^ ^sm X ~ Y ' '- These formulas should be remembered as the sum of two sines equals twice the sine of the half sum, times the cosine of the half difference, and so on for the remaining formulas. Formulas (19)-(22) are the so-called Sum and Difference Formulas; when read forward they convert a sum or difference of two sines or cosines into a product ; when read backwards they con- vert a product of two sines or cosines into a sum or difference. 39. Converting to Sum or Difference. The formulas (19), (20), (21), (22) should be recognized when read conversely. Writ- ing formulas (I), 38, conversely, and replacing a by A, /3 by B, we have : (19') 2 sin A cos B = sin (A + B} + sin (A - B), (20') 2 cos A sin B = sin (A + B} - sin (A - B), (21') 2 cos A cos B = cos (A + B) + cos (A - B), (22') 2 sin A sin B = cos (A - B) - cos (A + B>. 9 a set of important relations which should be recognized. EXERCISES Read the following exercises, applying the sum and difference formulas, reducing answers : 1. sin 10 + sin 40 = 8. sin (- 10) + sin 40 = 2. sin 80 + sin 30 = 9- cos 80 - cos (- 20) = 3. cos 60 + cos 40 = 4. sin 70 - sin 40 = 10. cos + cos ^ = b o 5. cos 28 + cos 42 = 6. cos 28 -cos 42= u sin40 + sin30 = 7. sin 35 + cos 25 = 12. sin 3 6 sin 6 = 58 PLANE TRIGONOMETRY 13. cos (n + 1) 6 cos (n 1) = 14. sin (n 4- 1) sin (n 1) = 4 4 Express as a sum or difference : 15. 2 sin 40 cos 20 = 18. 2 cos 60 cos 10 = [39 19. 2 sin ^ cos ~ = 4 4 16. 2 cos 50 sin 40 = 17. 2 sin 20 sin 40 = 20. 2 sin (n0) cos (ro - 1) = "'Prove the following identities : sin 2 a; + sin 2 y _ tan (a: + y) sin 2 a: sin 2 y tan (a; y) cos 3 2; cos 5 x 22. 23. 24. 25. sin 3 x + sin 5 x sin x 4- sin y _ cos x + cos y sin 9 x 4- sin = tan = tan 5 x. cos 9 x + cos x tan sin x + sin y _ 2 sn x sn cos a; + cos y 27. sinrcH tan tan - 2 2 snrc -- = sn a;. 28. sin (30 + A) 4- sin (30 - J.) = cos A. fir \ fir \ 29. cos f + A j 4- cos f A } = cos A. 30. sin (n 2} z + sin (waO cos (n 2) # cos (nx) = cot x. 31. 2 sin f x + - j sin f x - j = sin 2 x cos 2 x. 32. cos a; +"cos 3 x + cos 5 x -f- cos 7 a; = 4 cos a; cos 2 a; cos 4 a:. 33. sin x + sin 3 x + sin 5 x + sin 7 a; = 4 cos x cos 2 a; sin 4 a;. 39] FUNCTIONS OF TWO ANGLES 59 f) fh f) -L ft\ 34. sin 6 + sin < + sin (0 + ) = 4 cos - cos -2 sin ft rt\ ft -L- rt\ 35. sin 6 -f sin < sin (6 -f- <) = 4 sin - sin 2 s i n _JP. sin $ + sin $ + sin i/r sin (9 + < + -^r) COS + COS < + COS i/r -f COS (6 + T/r + -{- c, or if csinA = a, Fig. (1), (2); two solutions (triangles ABC, ABC') will occur when a c sin A, Fig. (3). If a < c sin A, no solution will exist, Fig. (4). If A be obtuse, one solution will exist provided a > c, otherwise the construction is impossible. EXAMPLE. Given a = 556, 5 = 678.4, A = 31 10' 30", to find B, C, c. In this example the angle A is acute and the side a < 6, hence -B. two solutions will be found. In one solution the angle B l will be acute, in the other solution B 2 = 180 B r Solution. (1) Make an approximate construction of the data. (2) Formulas and tabulation of the calculations. Data and Results Given a I A 556 . 678.4 31 10' 30" B\ c, C 2 39 10' 12" 140 49' 48" 109 39' 18" 7 59' 42" 1011.5 149.39 Ji c ; ~, sin A Logarithms log a log sin A log b 2.74507 9.71404 2.83149 log sin B 9.80046 log sin Ci log sin C 2 9.97393 9.14329 logc t Iogc 2 3.00496 2.17432 47-48] SOLUTIONS OF TRIANGLES IN GENERAL 73 EXERCISES > 1. Given a = 140.6, A = 48 30' 10", B = 76 24'; find 6, <^ -3. Given a = 148.6, b = 121.78, ^ = 69 20' 10"; find B, (7, e. *. Given a = 2311, b = 1600.7, B = 34 42' 29"; find A, C, c. 5. Given a = 1906, 5 = 224.8, A = 61 24' 18"; find B, C, c. 6. Given b = 1009, c= 796.4, <7== 85; find B, ^4, a. 7. Given A = 67 54', B = 34 52', 6 = 4356.7; find a, c. 48. The Theorem of Tangents. From the theorem of sines sin A sin B sin C 1 a c 'IR we have a= 2 R sin ^4, b = 2 R sin .B. Adding and subtracting, and reducing by 38, (1) + 5 = 2 12(sin A + sin 5)= 4 R sm AS (2) a-b = 2 ^(sin ^1 - sin B} = 4 Dividing (1) by (2), B + b 2 We may derive in a similar manner, or write by symmetry, b + c 2 c + THEOREM. Jw aw^ triangle the sum of two sides is to their difference as the tangent of the half sum of the opposite angles is to the tangent of their half difference. The theorem of tangents may be used in the solution of a tri- angle when two sides and their included angle are given. For example, let a, >, (7, be given. Then we know A + #+(7=180. A + B = 180 - C = 90 o _ C 2 2 "2" 74 [48 Hence, tan' and the first formula above becomes a + b cot- a b tan A-B A B This formula enables us to find the unknown tan - , and A P A I P then the angle >- . The angle - - being known, we find at once the values .of A and B : The application of the theorem of sines now determines side c. APPLICATIONS OF THEOREM OF TANGENTS EXAMPLE. Given 5=1436.7, c= 1141.2, ^1=42 14' 35"; find B, C, a. Solution. (1) Make an approxi- mate construction. C (2) Select formulas and tabulate the calculations. b= 1436.7 Fig. 40. b-c tan -(B + C), B+ C = 180 - A, a = b x sin .1 sin B Data and Results Logarithms Given b c A 1436.7 1141.2 42 14' 35" b-c b + c 295.5 2577.9 B + C 68 52' 42" 16 31' 40" 2 B - C 2 B C 85 24' 22" 52 21' 2" a 969.0 log (b - c) lop-tan" 8 + C J.47056 0.41308 3.41126 log (ft + c) 9.47238 log b log sin A log sin B 3.15737 9.82755 9.99860 log a 2.98632 48-49] SOLUTIONS OF TRIANGLES IN GENERAL 75 EXERCISES Solve for the unknown parts of the following triangles : 1. a = 281, c = 153, B = 34 42' 29". 2. ft = 296, c = 178, A = 78 21' 40". >3. ft = 199.37, c = 642.75, A = 130 9' 24". 4. a = 101. 47, ft = 9936.7, C= 47 48' 12". . ft = 1134.7, c = 2277.9, A = 19 34' 24". 6. a = 1434.2, b = 9767.2, (7= 109 19' 36". 7. ft = .538, c= 1.245, J.= 62 14' 40". 8. ft = 234. 7, c = 185.4, ^1 = 84 36'. 9. a = 1896.9, ft = 3463.7, C= 124 10 . 10. c= 9. 876, a = 4. 921, B =76 20.4'. 49. The Theorem of Cosines. (1) First derivation. Draw the perpendicular p from B to ft, Fig 41. Then, from the right triangle BXC we have But we have from the right triangle AXE p* + AX 2 = c 2 , and AX ex cos A. Making these substitutions, we find and by symmetry, b X Fig. 41. a- & 2 + c 2 2 be cos ^1, 52 _ c a _|_ CT 2 _ 2 c cos B, e* = 2 + & 2 - 2 ab cos C. If the point X should fall upon the base produced, it may readily be shown that the above results still hold. 76 PLANE TRIGONOMETRY [49 THEOREM. In any triangle the square of any side equals the sum of the squares of the other two sides diminished by twice their product into the cosine of their included angle. Fig. 42. (2) Second derivation. From Fig. 42, we have AX = c x cos A, CX= a x cos (7, and therefore, Similarly, b = c x cos A + a x cos C. c = a x cos -B+ b x cos A, a = b x cos C + c x cos B. (-5) (-0 00 Multiply these equations, as indicated, by ( 6), ( 2 (* - a) The form of this radical may be changed so as to make it sym- metrical in a, J, c. = 1 /(g-a * ^ C The radical here employed will be the same for tan , tan '2> 2 If we set = AT f r tan = , tan = , 2 s-a 2 s-b tan- = (7 r 2 -c Then, we have as results the following half-angle formulas. (1) The sines of the half-angles of a triangle. gin = 2 50] SOLUTIONS OF TRIANGLES IN GENERAL (2) The cosines of the half -angles of a triangle. cosf = be p ftc ,# _ A /O- &) cos 2 ~ -\ ~ ' ca 79 * (3) 27ze tangents of the half -angles of a triangle. tan tan /(*-)(*- c} _ r s a B _ 1 /O-CT)Q-&)O-c) _ r 8 -6' s s-b C _ 1 /( 2 s-c> s c In calculating the angles of a triangle, the tangents of the half-angles should be used, as the complete calculation of A, S, O may be performed by taking only four logarithms from the tables, viz. log s, log(s a), log(s 6), log(s B > C. (2) Select formulas and tabulate the calculations : , 2 s-a - r tan = - r o ~ j.' 1 ' 2 s 6 2 * c 80 PLANE TRIGONOMETRY [ 50-51 Data and Results Logarithms Given a (55.43 b 58.26 c 49.35 2s 173.04 s 86.52 s a 21.0!) s -I 28.26 X C 37.17 A 37 11' 18" 2 B 29 31' 10" 2 C o 23 17' 31" A 74 22' 36" B 59 2' 20" C 46 35' 2" log(s - a) 1.32408 log(.s - b) 1.45117 log(s - c) 1.57019 logs 1.93712 log r 2 2.40832 logr 1.20416 A log tan 9.88008 log tan - 9.75299 log tan 9.63397 As a check, A + B + C = 180. EXERCISES Find the angles in each of the following triangles : 1. a = 98.76, 5 = 104.97, c= 140.76. 2. a = 57,896, 5 = 49,784, c = 35,891. 3. a = 5.769, 5 = 9.8764, c= 11.675. 4. a = 0.0587, 5 = 0.09765, c = 0.1067. 5. a = 94.28, 5 = 112.68, e = 180.47. AREAS OF TRIANGLES Many expressions may be obtained for the area of a triangle. Some of these will be enumerated. 51. Area in Terms of Sides and Angles. Let the area be denoted by K, and let the triangle be lettered as shown in the drawing. Then, from plane geometry, (1) K=lbxh. Making use of trigo- nometric relations h = c sin A, (2) K= % be sin A. 51-52] SOLUTIONS OF TRIANGLES IN GENERAL 81 THEOREM. The area of any triangle equals one half the prod- uct of two sides by the sine of the included angle. xox T^ lex. sin B (*>} Since sin and . sln 2 sin C 1 c 2 sin A x sin B 2 ' sin G - - = z sin cos , A 9 A 50 we may express K directly in terms of the three sides. K=-bc sin JL = sn cos b ) ( a c ) be V 1 ^ where 2s = a + b + c. 52. Area in Terms of r. The area of a triangle may be Fig. 45. expressed in terms of the radius of the inscribed circle. Let a circle be inscribed in the triangle ABC. See Fig. 45. Then, AAOC=br, Adding, we have A COB = ar. A ABC= K= = ar, 82 PLANE TRIGONOMETRY [53 53. Expressions for the Area of a Triangle. 1. K = } be x sin A = I ca x sin B = \ ab x sin C. - c- sin A sin B 2. K = sinC 3. K = 4. X = s x r, r = radius of inscribed circle. EXERCISES Solve the following triangles for the unknown parts : ^i. a = 7950, 4 =79 59', .5 = 44 41'. 2. a = 80.86, 4 = 19 29', =33!'. 3. a = 62.65, 6=89.81, C= 55 5'. 4. a = 2071, 6 = 1887, C=55 12'3". 5. a = 0.2034, 6 = 0.1123, (7= 72 15' 19". 6. a = 48.5, 6=84, ^=21 31'. 7. a =838.56, 6 = 841.53, B = 68 10' 24". 8. a = 49, 6 = 45, j9=1741'9". 9. 6 = 117.4, c = 726.3, .6 = 80 10'. 10. c= 1047.51, a = 943.27, A =63 17' 18". 11. a = 5, 6 = 7, = m, cos = Vl m\ tan < = 1 - m 2 These equations may be expressed inversely, i.e. solved for <, thus: = arc sin m = arc cos VI m 2 = arc tan = Vl-w 2 = sin" 1 m = cos" 1 Vl m 2 = tan Y From Fig. 46 6, with Z tan -^r = L sin -dr = = >/r, we have cos ijr = 1=1= i cot ilr = - , or = arc tan t = arc sin = tan" 1 1 = sin" 1 vrr = arc cos VI = arc cot - ; t = cos" Vl + 1 2 t The symbols sin- 1 w, cos" 1 Vl m 2 , etc., are sometimes called anti-sine, anti-cosine, etc.. but a better reading is to call these ar<> .n'w m, arc cosine Vl ?n 2 , etc. From the above drawings it is to be noted that each inverse function has two initial or primary values. Thus sin < = ra, or = arc sin m 87 88 PLAXE TRIGONOMETRY [54-55 is satisfied by the angle or ?r $ = <'. The equation tan i/r = t, or ty = arc tan , is satisfied by the angle ty or TT + i/r = -^r'. In addition to the primary solutions, any number of solutions to a trigonometric equation may be obtained by adding any integral multiple of 2 TT = 360. EXAMPLES, i. sin x=\. Solution. x = arc sin = 30, 150, primary solutions. x = 2 mr + - , 2 nir + - TT, multiple solutions, 6 6 where n = any integer. 2. COS = ^V2. Solution. = cos-^Vi} = 45, -45, = 2fi7r + J, 2mr j, n = any integer. 3. tan (f) = VS. Solution. = tan- 1 V3 = 60, 240. = MTT + - , n = any integer. 3 55. Inverse Identities. A number of important inverse iden- tities will be introduced. I. sin- 1 a? + sin- 1 !/ = sin- 1 (a? VI y' 2 + y Vl - x 2 ). This identity may be established as follows : (1) construct an angle < whose sine is a;, also an angle i/r whose sine is y. (2) The left member of the identity is + i/r. Take sin ( 4- \/r) = sin cos i/r + cos sin i/r = x Vl Hence, = sin" 1 (a; Vl y^ y Vl In a similar manner sin" 1 re II. cos- 1 :*; cos This identity may be established by constructing two angles, ^ = tos" 1 x, i/r = cos" 1 /, taking the cosine of the sum and differ- 55] INVERSE FUNCTIONS 89 ence, substituting as above, arid then passing to inverse no- tation. Prove the following : III. tan- 1 IV. sin- 1 x = \ Kin l (2 x VI -a?) = - Hin- 1 (- V. cos- 1 a? = i cos- 1 (2^- 1). EXERCISES Construct the acute angles indicated in the inverse notation, and find values of the following : 1. cos (sin" 1 1). SUGGESTION. Construct < nin~*(%), then cos< = jV3. 2. tan (sin" 1 f ). 9. cos (90 cos" 1 a). 3. tan (arc cot -MO- * 10. sin (tan" 1 V3) - cos - J . V3\ 6 4. cot arc sin 11. sin(2cot-V3). 5. sin (arc tan 1). 12. cof(8fijr*!). 6. cos (arc cot 0). 7. tan(arccotl). 13 ' tan (90- sec' W2). 8. sin (90 -sin' 1 !). 14. cos (90- sin' 1 ^). Verify the following : 15. sin' 1 i + cos' 1 \ = 90. 16. arc tan 1 + arc cos = 90. V'2 17. arc sin 1 arc tan 1 = 45. 18. arc cos 1 + arc tan oc arc cot 1 = 45. 19. arc vers 1 arc sec V2 = 45. 20. sin (90 - tan- 1 VH)+ tan(90 - sec' 1 V2) = f . 21. arc sin ^ + arc cos '- = ^- 23. arc tan - -f- arc tan - = ^ . 5 52 b 74 22. arc sin x + arc cos x = . 24. sin ( 2 arc sin - ) = - V3. 2i \ 2i) L 90 PLANE TRIGONOMETRY [ 55-56 11 7T 25. arc tan h arc tan - = arc tan 1 = 23 4 2 2 ~ 26. arc tan = 2 arc tan x = arc sin 27. sin~ 1 a;=^ cos 1 a: = -cos 28. cos~ 1 :r = -- sin -1 a;= 2tan 2 29. sin" 1 (3 a; 4z 3 ) = 3siu~ 1 2;. 30. cos" 1 (4 a^ 3 x) = 3 cos" 1 x. 31. tan- 1 f ^1=3 tan- 1 a;. VI 3 32. ~ 1 \/ -- w l + a; _ i 33. 2 sin" 1 x = sec" 1 1 ,/1 i 2 34. tan" 1 = COS" 1 X* = - sill" 1 iE 2 . TRIGONOMETRIC EQUATIONS 56. Definitions. An equation containing one or more trigo- nometric functions of an unknown angle is called a trigonometric equation. Thus, (a) sin x = , (5) tan a: -f- sin x = 5, J sina;+ cos^ = |, x A I 3 sin x cos y = 1, (c) sm a; + cos - = 0, are trigonometric equations. Equations (c?) above are simultane- ous trigonometric equations. The operations of ordinary algebra clearing of fractions, transposing, multiplying by constants are applicable to trigo- nometric equations. In addition to these operations, the transformations of trigo- nometric identities may also be brought into use. For exam- ple, the equation sin 2 x = cos x 57-58] TRIGONOMETRIC EQUATIONS 91 may be changed trigonometrically into 2 sin x cos x = cos x. then transpose and factor, cos #(2 sin x 1) = 0. Equating to zero each factor, cosx=0, x =90, or 270 3 . 2 sin a: = 1, x =30, or 150. 57. Solutions. Trigonometric equations differ from algebraic equations in one important particular, viz. they have a multi- tude of solutions, whereas algebraic equations have a finite num- ber of solutions. As illustrations notice the following examples. (1) sin x = 1 has x = 30, 150, or x = 2 mr + -, or 2 b (2n+l>-, O = 0, 1, 2, ...) b (2) tan# = V3 has for solutions x = 60, 240, or nir + , o when n = any integer. We shall enumerate and illustrate some of the more impor- tant types of trigonometric equations. 58. Simple Equations. Under this heading may be included any equation which reduces readily to one of the forms : 1 i where -ff"is not greater numerically than 1. ( ) (_/(_) o */ f\ ^ I (3) tan x = K, K= any number. EXAMPLES. 1. Solve 4 sin x = esc x for the angle x. Solution. Multiply by sin ar, and divide by 4, giving sin' 2 x = \, or sin x = \. Hence, x = sm~\ ) = 30, - 30 ; 150, - 150 ; or, in general notation, x = nir , n = 0, 1, 2, 3, . b 92 PLANE TRIGONOMETRY [58 2. Find x from tan 2 a; + 4 = 2 sec 2 a:. Solution. Replace sec 2 x by 1 + tan 2 x, transpose, collect, and change signs, tan 2 x = 2, or tan x = \/2. Then, x = tan-!( V2)= tan- ( 1.4142)= 54 44', - (54 44'), or x = mr (54 44'). NOTE. Take the angle tan-^v^) from the table of natural functions. 3. Find #, when 6 cot x + 5 = tan x. Solution. Multiply by tan x f transpose, and change signs, tan 2 x - 5 tan x - 6 = 0, a quadratic equation with tan x as the variable, which factors into (tan x + l)(tan x - 6) = 0, giving j tan x = 1, and 1 tan x 6. Hence, x 1 = 135, - 45, or x 2 = tan- 1 6 = 80 32', 260 32'. The general values of x are Xl = mr - y , a: 2 = mr + 80 32'. 4 EXERCISES Solve the following trigonometric equations, giving only the solutions which are between 180 and + 180, inclusive. 1. 3 sin x = 2. Am. sin- 1 f = 41 48' 35", or 138 11' 25". 2. sin 2 x cos x = 0. [sin 2 x = 2 sin x cos a:.] Ans. 90, 30, 150. 3. cos 2 x + sin x = 1. Ans. 0, , , TT. 6 6 4. cos 2 x = sin x. Ans. 30, - 90, 150. 5. tan 2 x -f- 2 sin z = 0. Ans. 0, 60, 180. 6. cos 3 x sin 2 a; = 0. SUGGESTION. Change cos 3 a; to 4 cos 8 a; 3 cos x, sin 2 x = 2 sin x cos z, giving 4 cos 8 a: 3 cos x = 2 sin a; cos x ; factor and solve. Or, cos 3 x = sin(3 x + 90) ; then cos 3 x - sin 2 x = sin(3 x + 90) - sin 2 a? c + 45 J sinff + 45J=0. 58-59] TRIGONOMETRIC EQUATIONS 93 Hence, cosf^z + 45) = 0, and Bin(| + 45) = 0, \" / \* / giving | x + 45 = 90, 270, 460, and * + 45 = 0, 180. Then, and x = 18, 90, 162, x = - 90. 7. cos 3 x + sin 2 x -cos x = 0. Am. 0, 30, 90, 150, 180. 8. cos(> + 60) - sin(> + 30) = |V3. Ant. - 30, - 150. 9. sinO+60 )-smO-60 ) = -iV3. Ant. 120. 10. sin 4 x = 2 sin 2 x. Ans. 0, 90, 180. 59. Equations of the Form r cos (> = , r sin (> = b. Here is a set of two simultaneous equations in two unknowns, (1) To find <, divide the second equation by the first, ,_b ,_ -if&\ a' \c Or, d> = HTT + tan" 1 ;- ) W (2) To find r, square and add, recalling the identity sin 2 < r = Va 2 + i 2 . The proper sign of r must be chosen so that r cos < = a, r sin = b. EXAMPLES. 1. Find r, < from {r cos < = 3, r sin < = 4. We have tan< = f, < = tan^Cf) = 53 8'. Squaring and adding, r 2 = 4 2 + 3 2 = 25, 94 PLANE TRIGONOMETRY [ 59-60 Find r, in the following : rcos0 = 12, r sin = 5. r cos = 6, 2. 3. rsin = 12. 4. 5. r sin 5, rcos0 = 5V3. r sin = 4.876, r cos = 2. 396. 60. Equations in the Form (/ sin 6 cos 4> < / , r sin 6 sin (j> = b, r cos 6 = c, where r, 0, are variables. Divide the second equation by the first, b ,b tan 9 = , = tan - a a Squaring all three equations and adding, we have r z (sin 2 [cos 2 4- sin 2 0] + cos 2 0) = a 2 + b z + c 2 , or r 2 = a 2 -f J 2 + c 2 , From the third equation, /i c c COS0=- = r . r -f <9 = cos- 1 EXERCISES Find r, 0, in the following : r sin cos = 2, r sin sin 0=2, r cos = 1. 1. f r sin cos = 6, 2. < r sin sin = 3, rcos0 = 2. 3. r sin cos = 2, r sin sin = 6, r cos 0=9. [Vsin 0cos0 = 12, 4. j rsin 0sin0 = 12, rcos0 = 1. {r sin cos = 1, r sin sin 0=4, r cos = 8. r sin cos 0=5, 6. ( r sin sin = 2, rcos0= 0. 01-6-2] TRIGONOMETRIC EQUATIONS 95 61. To solve a sin x + 6 cos x = c. Divide this equation by ~~ ,-.x a sin x b cos x Now let ^ ==== -_ = sin , ===== = cos , and hence, - = tan $, (f> = tan J - o 6 Then equation (1) becomes (2) sin sin x + cos < cos a; = , or cos (as ) = = x (j) = cos J rr = d> cos" 1 ( = == } Wa a + 6V EXAMPLES, i. Solve 5 sin x + 12 cos x = 6.5. Divide both members by 13, (1) j 6 ^ sin x + i|^ cos re = 0.5. Write ^ 5 g = sin <, 1| = cos ^>, tan < = ^. Hence, = tan" 1 ^) = 22 37'. Now equation (1) becomes (2) sin (f> sin # + cos cos re = 0.5, or cos (re ) = 0.5, .r = cos-^O-S) = 22 37' 60 = 82 37', - 37 23'. 2. Solve 3 cos x + 5 sin x 4. 3. Solve 12 sin x + 5 cos x = 3.9. 4. Solve 8 cos x + 15 sin x = 5.1. 5. Solve 3 cos x 2 sin re = f V13. 6. Solve 5 sin re 6 cos re = f V61. 62 . To solve sin (x + cf) = a s * n * sin (re -f- ) a XT- i e have T sin x 1 96 PLANE TRIGONOMETRY [62-63 Take by composition and division, sin (x + 0) + sin x _ a + 1 sin (x + <) sin x a 1 ' / \ 2 sin he + ^ cos ^ . V 2/ 2 a+ 1 2 cos ( x + ) sin V Zy Z The right member is now known, and the solution may be written out. EXAMPLES, i. Solve sin (re + 37 14') = 2 sin x. Substituting in the above result, and retaining the smallest angle, / Q71J.'\ Q71J.' tan (re + - 1 ) = Stan --^- = 3(0.3369), \ Zi / A 2. Solve sin (x + 65 21') = 3 sin x. 3. Solve sin (x - 28 40' ) = f sin x. 4. Solve sin (x + 56 24') = 5 sin (x - 10 20'). 5. Solve sin (x + 94 10') = 4 sin #. 6. Solve sin (x - 124) = f sin x. 63. To solve tan (x + <}>) = a tan jr. Divide by tan #, and take by composition and division, tan (x + (fe") _ v * ^ ri tt, tan x tan (rr + ^>) + tan x a + 1 tan (a; + ) tan x a 1 Simplifying, *)1 sin < a 1 sin (2 a; + ) = ^1 "" 63-64] TRIGONOMETRIC EQUATIONS 97 EXAMPLES. i. Solve tan (x + 20) = 5 tan x. Comparing with the above result, ^ = 20, a = 5, sin (2 x + (/>) = | sin 20 = f(0.342) = 0.513, 2 a + = sin" 1 (0.513) = 30 52', 2 x = 30 52' -20 = 10 52', a: = 5 26'. 2. Solve tan (x + 30)= 6 tan x. 3. Solve tan (x + 47 20') = 7 tan x. 4. Solve tan (x + 25 10') = 10 tan x. 5. Solve tan (x + 60) =13 tan x. 64. To solve x = a + p sin jr. In this equation a, /3 are usually given as angles (degrees or radians) ; /3 expressed in radian measure is smaller than unity. Two plans of solution will be sketched. (1) Trial solution. Let a, (B be expressed in degrees. Then an upper limit to x will be shown by a + /3, since the multiplier ft is smaller than unity. Take a trial solution, substitute in the equation, note the error, make another approximation, and so continue until the required degree of accuracy is attained. (2) G-raphical solution. Let a, ft be expressed in radian measure : a = a radians, /3 = b radians. Then we have to determine x so that x a = b sin x. Let y l = x a, y^ = b sin x. Construct upon rectangular axes a curve representing each of these equations. The first is a straight line through the point a on the X-axis ; the second is a modified sine curve. The x of the point of section of these two graphs is the required solution. As an illustration of the graphical solution let us solve x = 57 17' 44" + 45 x sin x. Here, 57 17' 44" = 1 radian, 45 = 0. 785 radians. Then we are to solve x = \-\- 0.785 x sin x. Let y v = x 1, 3/2 = 0.785 sin x. 98 PLANE TRIGONOMETRY [64 Now construct the straight line y l = x\, and the modified sine curve y z = 0.785 sin x. The abscissa of the point of intersection of the straight line and curve is approximately x = 1.769 radian = 101 27'. Y Fig. 47. EXAMPLES, i. Solve x = 24 + 30 x sin x. 2. Solve x = 64+ 28 x sin x. 3. Solve x = 30+ 50 x sin x. 4. Solve x = 10 20' + 40 x sin x. 5. Solve x = 45 + (37 30') x cos x. 6. Show sin x 1^x, O^x^^- 7T 2 CHAPTER IX COMPLEX NUMBERS. DEMOIVRE'S THEOREM. TRIGONO- METRIC SERIES. EXPONENTIAL AND HYPERBOLIC FUNCTIONS 65. Roots of Quadratic Equations. In ordinary algebra we have such equations as a?-6a; + 250, whose roots, z = 3 +4V~^1, are called complex numbers. These numbers contain a real unit, 1, and a so-called imaginary unit, V 1. (1) Properties of V 1. The imaginary unit is usually represented by i. We may easily show that when V 1 = i, i 2 = 1, z 3 = i, i* = 1, z 5 = i , and generally, (2) Graphical representation of x + yi. To any complex number as 3 + 4 i, 3 4 i, x + ?/i corresponds a point in a plane. If we multiply a real number a by i, and this product again by z, the result is a. Thus, multiplying twice by i changes a number to its negative. Multiplying a number by i may be in- terpreted as turning its direction through 90. To locate 3 +4 i upon a plane, lay off 3 units, OQ, along the real axis (horizon- tal in Fig. 48), then at Q erect a perpendicular 4 units long ; the point P l represents the complex number 3 + 4 i. Lay off - 4 perpendicular to OQ at Q, and we locate 3 4 i, at P v Any complex number x 4- yi may be represented upon a plane as shown in Fig. 49. The point P may be anywhere in the plane. (3) Modulus, arc (x + yi). The line OP =r is a vector equal in length to the modulus of x + yi, or the absolute value of 99 100 PLANE TRIGONOMETRY [ 65-66 x + yi. The angle XOP = is called the arc of x -\- yi, or amplitude, or argument of x + yz. As abbreviations r = mod (# + yi), = amp (x + yi). Y -4 Fig. 48. -Y Fig. 49. The following notation should be recognized : Modulus of a? 4- yi = r = Va? + y 2 = \& + yi\. Arc of x + yi = = tan- 1 ^ = amplitude (a? + yi). Xs EXERCISES l. Locate the following complex numbers : (1) 3 + 2*; (2) 2 + 6*; (3) -3 + 3; (4) -4 + z; (5) _3-4z; (6) -6; (7) 4-5f; (8) (1+ ^,' 5. 2. Find the modulus and arc of each of the numbers in Ex. 1. 3. Solve the following equations and locate the roots as complex numbers : (1) z 2 -4z + 13 = 0; (2) z 2 + 62 + 13 = 0; (3) z 2 + z + l = 0; (4) z 3 -l=0; (5) z 3 +l = 0. 4. Locate the following products : (1) i x (2 + 4 z) ; (2) i'x(-3 + 2i); (3) x(5-3i); (4) ix*xt'(2-0. 66. Complex Numbers expressed Trigonometrically. From Fig. 49, we have a? = rcos 4>, j/ = r sin = tan- 1 ^ 66-67] DE MOIVRE'S THEOREM 101 Hence, oc + yi = r(cos (j) + i sin ) = Vac? 2 + j/ 2 (cos + i sin (j>). THEOREM. ^4. complex number equals its modulus multiplied by the expression cos (f> + i sin -f 2 i sin < cos <), (1) z 2 = r 2 (cos 2 + i sin 2 <), 35, (7), (8). Now, multiply (1) by z = r(cos < + i sin <). z 3 = r 3 [cos 2 cos < sin 2 sin + z'(sin 2 cos + cos 2 < sin )]. (2) z 3 = r 3 (cos 3 4* + i sin 3 <^>), 34. 102 PLANE TRIGONOMETRY [ 07-68 The law of exponents shown in (1), (2) would indicate that for n = any positive integer, (3) z n r"(cos n<$> + i sin w). Assuming law (3) to hold for any integral w, let us see if it holds when n is replaced by n + 1. Multiply (3) by z = r(cos (f> + i sin <), 2 n+l _ r n+l |- cog n ^ CQS ^ _ g j n w< g j n ^ _j_ ^(gin n fy cos < + cos nsin <)]. (4) z n+l = r n+1 [cos (n + !)< + i sin (n + 1)0] . Hence, the law assumed in (3) for the integer n holds for n + 1 . We see this law holds for n = 2 and n = 3, hence it holds for n = 4, 5, , n= any positive integer. Hence, we have DEMOIVRE'S THEOREM: c DeMoivre's Theorem holds when n is an integer, a fraction, or a negative number. (cos (j> + i sin ) = cos + i sin . n n P ft n (cos <|> + i sin <(>) q = cos + i sin 4>, (cos + i sin ) + i sin( m({)) = cos mj> i sin w.(|>. 68. Raising to Powers and Extracting Roots. DeMoivre's Theorem enables us to raise z = x + iy = r(cos <> + *' sin 0) to any power, or to extract any root of 2. Thus, z 2 = r 2 (cos 2 < + i sin 2 0). Hence, to square a complex number, square its modulus and double its amplitude. To cube a complex number, cube its modulus and multiply its amplitude by three. 68] DEMOIVRE'S THEOREM 103 EXAMPLES, l. Raise z = 3 -f 4 i to the 2d power ; to the 3d power. z 3 + 4 i = 5(cos + i sin <), < = tan- 1 * = 53 8', nearly. 2 2 = (3 + 4 ) = 2o(cos 2 < + i sin 2 <). 2 3 = (3 + 4 f)s _ I25(cos 3 + i sin 3 <). 2. Raise z = - + i to 2d, 3d, rth powers, and locate these ^J ^ respective numbers on a diagram. See Fig. 49. 3. Find z 2 , z 4 , g 6 , when z = |- -i. *. Zi 4. Find z 5 , z 10 , when z = 1 i. The extraction of roots may be performed by use of DeMoivre's Theorem : 1 If , . . \ zn = r> i { cos + i sin r ). \ n nj This formula seems to give but one of the n wth roots of a, but we may obtain n different roots by writing z = r[cos( + 2 &TT) + i si i If + 2&7T , .d> + = r cos-* h * ^ - L A: = 1, 2, d, , w 1. L' * then will i If + 2&7T , .d> + 2&7 ^ - j EXAMPLES l. Extract the square root of *. Here, z = i = 1 [ cos- + i sin ^ J ? + 2 y 1 a /+27T\ / + 2far\ Then, 2- = i 5 = cosf ^ ^ - J + t sinf J. Let zj, 2 2 be the two roots, then 2, = cos? + f sin^ = J- + JL j f fc = o, 4 4 V2 V V2 V2 = --^-4^ V2 V2 104 PLANE TRIGONOMETRY [68-69 2. Extract the cube root of 8, and locate the roots on a diagram. In this case z = 8 = 8 (cos TT + i sin TT) = 8[cos(7r + 2 kir) + i sin(ir + 2 /br)]. Extract the cube root, Giving fc values 0, 1, 2, we find the three roots, z r z y z 3 , 2j = 2( cos- + i sin - ) = 1 + V3 i, k = 0, \ 3 3/ 2 2 = 2 (cos ?r + i sin TT) = 2, _ Q/ 5 TT . . 5 ir\ i * V ~3~ ~3~J~ 3. Find the five 5th roots of 32 ; of i. 4. Find the six 6th roots of 1 ; the cube roots of - ~ m 69. Value of sin x, cos x in Terms of x. (1) Value of when $ = 0. When an angle is small, sin approaches arc . From the tables of natural functions and radian measure we have : sin = 0.00000 = 0.00000 radian sin 10' = 0.00291 10' = 0.00291 radian sin 40' = 0.01164 40' = 0.01164 radian sin 1= 0.01745 1 = 0.01745 radian sin 2 = 0.03490 2 = 0.03491 radian which show that sin = < for values of (f> from to near 2, true to five decimals. D To show this property generally, we have from geometry ^ BC < 90, or, if OA = 1, Fig. 50, sin < < < tan . Divide this inequality by sin <, 1 <$<^. sin 4> cos 69] DE MOIVRE'S THEOREM 105 Now let (f> approach zero, (f> = 0, and cos 0=1; hence, or, limit f ^ _ -i (2) Value of sin n(f>, cos n(f> in terms of sin <, cos . In algebra it is shown that if x + yi = a + fo', then x = a, y = b- THEOREM. If two complex numbers are equal, the real parts are equal and the imaginary parts are equal. By DeMoivre's Theorem, (A) (cos + i sin <)" = cos w< + i sin w<. But by the Binomial Theorem, (J?) (cos + i sin 0)" = cos" + n cos"" 1 (i sin 0) , n(n 1^ o j. /- jx9 . n(n l")(w 2} _ , ,^. H ^- J- cos n ~ 2 (t sin 0) 2 + - ^ - t -^ cos"" 3 (i sin 1 1 o n (n ,,_ 4 , ,. , N , L - - * cos n 4 (i sm 0) -f the series terminating if n is a positive integer, and becoming infinite if n be a fraction or negative. Now i 2 = 1, & = i, i*=l, ; hence, the first, third, fifth, -, terms of the right member of () are free of i, and the even terms contain i. The right members of (A) and (J9) are equal. Equating the real and imaginary parts, respectively, we have ( (7) cos w0 = cos" (f> n Y ~~n cos n ~ 2 sin 2 (f> cos 106 PLANE TRIGONOMETRY [69 (D) sin n(f> = n sin"" 1 cos -- ^ - Q ' ^- cos n ~ 3 (f> sin 3 (f> L a ' O n ( - !)( - 2)(. -8)Q - 4) If in (#), (-#) we give w the values 2, 3, 4, , we may obtain the ordinary expressions for sin 2 <, cos 2 < ; sin 3 $, cos3<; etc., in terms of sin <, cos . See 35, 36. EXAMPLES, l. Show from (<7) cos 2 = cos 2 sin 2 , 2. Show from (D) sin 2 = 2 sin cos 0. 3. Show cos 3 = 4 cos 3 3 cos 0. 4. Show sin 3 < = 3 sin < 4 sin 3 0. (3) Trigonometric series. In formulas (O'), (D) above, let 99 us substitute n = x, or n = , < and we have ((7') cos a; = cos** - Y Q cos"' 2 sin 2 (/> + , x(x d>} = cos n <#> - -5 ^ cos" o , /sn "- 2 - \ 9 "" 1 sin a; = T cos" > sn > ^r - cos" (p L "Z ' 6 21 N / * J \ Q

Y " * \r?) ~ i'-V-8 cos " 8 * r# -) + "" Now let n=oo in such manner that nx, then - ^ = 1, see (1) above, and (0"), (-Z)') become the infinite trigonometric series 69] DEMOIVRE'S THEOREM Kfl where [2=1-2, [3=1-2.3, [4=1-2-3.4, etc., these symbols being read, factorial two, factorial three, factorial four, etc. Dividing sin a? by cos x and cos # by sin x, we find III. tan,^ + + + + .. IV. 2x 3 45 945 The series for cos x and sin x are convergent for any value of x. In the expansion for any trigonometric function the radian measure of x must be used in the series. Tables of the numerical values of the trigonometric functions for any set of angles may be computed by means of the series I, II, III. EXAMPLES, l. Compute sin 10 correct to four decimals. (, = ^ = 0.17453). 2. Compute sin 12. 3. Compute cos 10, tan 10. 4. Compute sin 20 from Ex. 1 and 3. 5. Find sin 80, cos 80, cos 78, cos 70. (4) Trigonometric products, sin x when X= 0, X= 7T, = 2 7T, = 3 7T, . This fact suggests that sin x can be expressed as a product of factors, and indeed an infinite number of factors. Likewise, cos x = 0, when 7T 3 7T , 5 7T x = ^ x = -Y' x= -^~- Arranging the factors properly, 108 PLANE TRIGONOMETRY [ 70 70. Summation of Series. In analysis it sometimes becomes necessary to sum the following series : (1) S l = sin 6 + sin (6 + a) + sin (<9 + 2 a) + - (2) $j = cos + cos (0+ a)+cos(0 + 2a) + + cos[0-f (n !)]. S l is a series of n sines in which the angles are in arithmeti- cal progression, the common difference being a. S z is a similar series of cosines. To find the sum S v multiply both members by 2 sin - + 2 sin | sin [6 + (n - 1)] = COS 0-= - + cos cos^ + pa, 39 (22') Dividing by 2 sin - , we have . a sm- 70] TRIGONOMETRIC SERIES 109 In a similar manner, by multiplying S t by 2 sin- and sepa- rating the double products as in S v we may reduce the value of $ 2 to $2= cos#+ cos(0 + a) + cos(# + 2a)+ + c< . a sin- EXAMPLES Verify the following : \ o/ \ 2. cosz+cosz + cosa; -+ +cos \ o/ \ o Q \ -1 -)+ +sin x+(n 1) o / ^ I 2. I , s -i N 7r~\ . mr sin x + (n 1) sm r SUGGESTION. Common difference = ^ , compare S r x (n 1)^ o J [SUGGESTION. The common difference is ; apply 5 2 with a = ^. o o J X . -4 ~\ X liX sm(w + l)-sin y 2 2 3. sinz + sm2:r + sm32;+ +smwa;= . x sm- , ^ ^ x vix cos (n + 1) - sin sin^ Sill 5. sin a; + sin 3 x + sin 5 z + -f sin(2 l)a: = sinx 110 PLANE TRIGONOMETRY [70-71 6. cos a; + cos 3 a; + cos 5 a; -f- -f-cos(2 n V)x = cos nx si n nx sin x sin 2 nx 2 sin a; o A -a n sin ("w 4-1")^ sin nx 7. sin 2 x +sin 4 a; 4- sin 6 x 4- + sm2wa;= v ' -- sin a; 8. cos 2 x + cos 4 a; -f- cos 6 x + + cos 2 wa: _ cos (n + 1) x sin wa; sin a; 9. s _ n _ cos [2 x + (11 1)] sin net '2 2 sin a [SUGGESTION. Multiply this series by 2 and separate each term thus : 2 sin 2 a; = 1 - cos 2 x, 2 sin 2 (x + a) = 1 - cos(2 x + 2 a), ... 2 sin 2 [x + (n - l)a] = 1 - cos [2 x + (n - 1) 2 a] ; add and sum the cosines as in Ex. 8.] 10. cos 2 x + cos 2 (x + a) + cos 2 (x + 2 a) + or , / i\ n n , cos [2 a; +(n 1)1 sinwa + cos^[a; + (7i l)a] = -H . 2 2 sin a 71. The Exponential Series. If we take the Binomial Series, l and substitute we find m * , l + w 14- m Now, divide the numerator and denominator of the respective fractions by m, w 2 , ra 3 , , and finally let m = x ; then we have 71-72] EXPONENTIAL AND HYPERBOLIC FUNCTIONS 111 The numerical value of this series, when x = 1, is denoted by e. (A) e = I + 1 + I + -| + 1 + = 2.7182818 (.B) e x = 1 + x + ^ + ^ + ^ + ... =(2.7182818 )* [2_ [. LL Series (jB) is called the exponential series; e is the base of the Napierian Logarithmic system. 72. Euler's Formulas. Series (5) readily identifies the ex- ponential functions with the trigonometric functions. In (.B) substitute x = iQ, where i = V 1, i z = 1, , 65 1 li = cos 6 + i sin 0, 8 69, I, II. o Changing i to i, e~ ifl = cos 6 i sin 9. Subtracting and dividing by 2 i, and adding and dividing by 2, respectively, we have Euler's Formulas for sin & and cos 0. tan = sin -=- cos 0. V. sin9 = e * ~f~* . _vfl , ;ft VI. VII. The reciprocals of these fractions define the esc 0, sec 0, cot 0, respectively. These analytic definitions of sin 0, cos 0, tan 0, may be em- ployed instead of the ratio definitions given in 2. 112 PLANE TRIGONOMETRY [ 72-73 EXERCISES Prove the following identities by use of Euler's definitions, V, VI, VII. 1. sin 2 = 2 sin 6 cos 0. 2. cos 2 B= cos 2 6- sin 2 0. 3. sin (a + #)= sin a; cos?/ + cos x sin #. o . x-\- 11 x 11 4. sin x + sin y = 2 sin & cos ^ 5. sin 2 a; + cos 2 # = 1. 6. sec 2 rc tan 2 # = 1. 7. Write trigonometric values for each of the following : (1) e>-, (2) ei (3) . 8. Express in exponential notation : (1) cos 30 + i sin 30; ,,,v (b) 9. Prove the following: (1) e t(a+2n) = e ia ; (2) e 2+ " r = -e 2 ; (3) / +t 2 = t2; (4) e 2+2 ^=e 2 . 10. Find approximately the value of Ve by substituting x = l in OB), 71. 73. The Hyperbolic Functions. In V, VI, VII, 72, let 6 = iar, sin 12: = e~ x e x . e x e~ x tan ix = - - =i -- -- 73] EXPONENTIAL AND HYPERBOLIC FUNCTIONS 113 The fractions , , are taken as defi- 2 2 ' e*+ e - x nitions of the hyperbolic sine of x, hyperbolic cosine of x, and hyperbolic tangent of x, respectively. These functions are written : VIII. IX. X. tank x = . cosh x &* + e~ x (1) Relations between trigonometric and hyperbolic functions, From the above definitions it is seen that sin ix = i sinh #, cot ix = i coth x, cosix = cosh a;, sec ix = sech rr, tan ix = i tanh a?, esc ix = i csch a;. This table of relations enables us to determine the identities existing among the hyperbolic functions, corresponding to the trigonometric identities. (2) Identities. Let the student verify the following identi- ties among the hyperbolic functions : (1) cosh 2 x- sinh 2 x = 1. ^ tanh2a . = 2 tanh a; (2) sech 2 z + tanh 2 x = 1. 1 + tanh 2 a: (3) coth 2 a; - csch 2 z = 1 . (7) sinh ( - ar) = - sinh x. (4) sinh 2 z = 2 sinh x cosh a;. (8) cosh ( - x) = cosh x. (5) cosh 2 x = cosh 2 a; + sinh 2 x. (9) sinh (a; + y) = sinh a; cosh y cosh a: sinh y. (10) cosh (xy)= cosh a; cosh ?/ sinh x sinh y. , 11N , tanh x tanh y (11) tanh(zy)=- ^- 1 tanh x tanh y (12) e* = cosh x + sinh #. fT* I 7/ i^* *--" *?/ (13) sinh x + sinh y = 2 sinh ^ " cosh ^ (14) cosh x cosh y = 2 sinh ^"j" ^ sinh --^-^' 114 PLANE TRIGONOMETRY [73 (3) Infinite series forms for sink x, cosh x. Take the series for sin a;, cos a;, tana;, II, I, III, 69, sin, -_ + --..., X 2 , X* cosx= __+__..., X 3 9 3? tana; =x + +^+ , and replace x by ix, giving 3 lo - 11 It- cosh^ l tank x = x - 3 15 (4) Graphs of the hyper- bolic functions. The numeri- cal values of the hyperbolic functions for various values of the variable x are col- lected in Tables of Hyper- bolic Functions. The ap- proximate values of sinh #, cosh a;, tanh x are shown upon the following graphs, Fig. 51, where the ordinates parallel to the !F-axis are the values of these functions for the corresponding values of x. The graphs of the csch a;, sech a;, and coth x are not shown, but these may be easily constructed upon the same diagram by taking re- ciprocals of the ordinates of the sinh x, cosh x, and tanh #, respectively. 7:]] EXPONENTIAL AND HYPERBOLIC FUNCTIONS 115 (5) The inverse hyperbolic functions. If we write we may express x in terms of y by an inverse notation similar to that employed in inverse trigonometric notation. y = sinh a-, x = sinh" 1 y. By means of the exponential value of sinh #, we may obtain another expression for x in terms of y. Thus, _ ?/ />.r _ 1 '/ ii) 3 y- y e ' or e 2 *' 2 i/r r 1 = 0, a quadratic in e x . Solving, e* = y V/ 2 + 1, the negative sign being excluded since e^ is positive. Take logg of each member, a- = log e (j/ + V/ + 1) = sinh' 1 y. EXERCISES Prove the following : 1. cosh" 1 .r = log (> + ^v x 2 1). 2. tanh- 1 * = lo 1-1 i i 1 i I+A l-.r2 3. seen" 1 .r = cosh" 1 - = log, - x x 4. sinh" 1 x = cosh" 1 Vl + x 2 = tanlr 1 Vl+z 2 5. tanlr 1 x+ tanh"- 1 y = tanh" 1 /- - 1 + xy 6. cosh = 1, cosli ~ = 0, sinh iri = 0. 7. sinh = /. cosh TTI = 1. tanh = 0. 8. sinh 2 NTH = 0, cosh 2 ?t7r = 1. tanh mri = 0. 116 PLANE TRIGONOMETRY [ 74 74. The Gudermannian. If an angle is related to a num- ber #, so that sec 6 cosh #, then 6 is denned by 6 = gudermannian of #, or briefly, = gd x = sec~ 1 (cosh #). Inversely, x = inverse gudermannian of #, or x = gd" 1 6 = cosh -1 (sec 0) . When sec 6 = cosh x, the following relations are true : cos 6 = sech x, sin = tanh x, tan = sinh x, esc = coth x, cot = csch x, r x tan - = tanh - Any one of these relations defines 6 as the gudermannian of x. PART II SPHERICAL TRIGONOMETRY CHAPTER X GENERAL DEFINITIONS. THE RIGHT TRIANGLE 75. Definitions and Geometric Properties. From spherical geometry we recall the following facts : (1) The intersection of the surface of a sphere by a plane through its centre is a great circle. (2) A great circle divides the surface of a sphere into two equal parts called hemispheres. (3) Two great circles upon the same sphere divide its sur- face into four parts each of which is called a Lune. In Fig. 52, BCB'AB is a lime, bounded by the semi-circumferences BOB' and BAB'. (4) The angles of a lime are equal, and are measured by the diedral angle between the planes of the great circles forming the sides of the lune. In the drawing, angle B is meas- ured by the diedral angle CBB'A. Since tangents to arcs BC, BA at B are per- pendicular to the edge of the diedral angle, the spherical angle B is measured by the plane angle between the tan- gents t, Bt' to the arcs BC and BA at B. (5) Any third great cir- cle w r ill divide a lune into two parts called spherical triangles. Thus, arc AC, Fig. 52, divides the lune into the spherical triangles ABO and AB'C. 117 118 SPHERICAL TRIGONOMETRY [ 75-76 (6) is the apex of a spherical pyramid whose base is the spherical triangle ABC. The face angles of this pyramid are equal to the corresponding sides (measured in degrees) of the spherical triangle ; in Fig. 52, a = Z B C, b = Z C A, c = Z A OB. The diedral angles of the pyramid are equal to the correspond- ing angles A, B, C, of the spherical triangle. (7) The length of a side a of a spherical triangle expressed in linear units is given by arc a = radius x radian measure of a. The length of the radius of the sphere is usually not considered in the trigonometric discussion of a spherical triangle. (8) A side of a spherical triangle lies between and 180; likewise an angle lies between and 180. The following limitations should be recognized (see notation, Fig. 52): (1) < a + b + c < 360, (2) 180 < A + B + C < 540. (9) The angles and sides of a spherical triangle are in the same order of magnitude ; if a > b > c, then will A > B> C. 76. The Polar Triangle. To any spherical triangle ABC there corresponds another called its polar which may be con- structed as follows : take each vertex A, B, (7, as a pole and describe arcs (90 away) B'C', C'A', A'B', forming the spherical triangle A'B' C 1 , Fig. 53. The triangle A' B'C' is the polar of the triangle ABC. In geometry it is shown that the sides and angles of a spherical triangle ABC and the angles and sides of its polar A' B'C' are related by the following equations : (1) A = 180 - a', B = 180- &', C - 180 - c', (2). a = 180 - A', b = 180 - B', c = ISO - C'. Fig. 53. "6-78? THE RIGHT TRIANGLE 119 THEOREM. The angles of a spherical triangle are the supple- ments of the corresponding sides of its polar ; the sides of the given triangle are the supplements of the corresponding angles of the polar triangle. Polar triangles are so related that the vertices A, B, C, of one are poles of the corresponding sides a', 6', c' of the other, and the vertices A', ', C' are poles of a, 6, c. THE RIGHT SPHERICAL TRIANGLE 77. Definitions. If one angle (7, Fig. 54, be 90, the triangle ABO is called a right spherical triangle. The side c opposite the right angle is called the hypotenuse. (1) The angles A, B, may both be acute, both obtuse, or one acute and the other obtuse. (2) The sides a, 5, lie in the same quadrant as the opposite angles. If A > 90, a>90; if _>90, >90; if J.<90, a<90; if <90, 6<90. (3) The hypotenuse c is smaller than 90 if A and B are both smaller or both larger than 90 ; c is larger than 90 if A is smaller than 90 and B larger than 90, or inversely. See 81. 78. Trigonometric Relations. In the spherical triangle AB C, Fig. 54, let C = 90, and let A, B, be acute. To measure the angle A draw at any point A' in the edge of the diedral angle CAOB, a plane per- o< pendicular to OA ; the traces of this plane upon planes OA C and OAB are the lines A'C', A'B\ and the angle B'A'C' equals the angle A. Draw Fi g . 54. B'C'. The following are plane right-angled triangles : A'C'B', OA'C', OAB', 00' B'. 120 SPHERICAL TRIGONOMETRY [78-79 Defining sin JL, cos A, and tan A from the drawing, we have _B'C' _B'C> OB'_sina ~^A r B i ~A r B' X ~OB'- A'C' A'C' OA 1 A'B' A'B' OA' tanc A = = A'C' A'C' OC' sin b Also, from triangle B' OA', cos B'OA' = cos c = = . x = cos a cog ft> By drawing perpendiculars to the edge O.B and constructing a measure of angle B, we could derive similar results for the functions of jB, namely : T> sin b rt tan a , ^ tan sm.Z?= sin c tan c sin a Taking the reciprocals of tan A and tan B, and multiplying their values together, , T> sin b sin a r cot A cot jt? = -- x - - = cos a cos b. tan a tan b Hence, cos c = cos a cos 5 = cot A cot 5. Again, tan b sin 5 cos c sin 6 cos a cos 5 sin J cos A = - = -- - x = - - x - = - - x cos a tan c coso sine cos 6 sine sine = sin B cos a. Similarly, cos B sin A cos b. 79. Important Formulas. Collecting the results of 78, we have the following formulas relating to the right spherical triangle : (1) sin^=, (5) sin c sine (2) cos^ = , (6) , tan c tan c (3) te n^ = , (7) sin 6 sin a (4) cos A = sin U cos a, (8) cos J5 = sin .4 cos &, (9) cos c = cos a cos 6 = cot A cot -B. 79-80] NAPIER'S RULES 121 These formulas are true for any right spherical triangle, whether the angles and sides be acute or obtuse. It should be noticed that the values of sine, cosine, and tangent, formulas (1), (2), (3), (5), (6), (7), are very similar to the definitions of those functions in a plane triangle. This similarity enables one to remember the trigonometric relations of the angles and sides of the right spherical triangle. A right spherical triangle may be completely solved for all its parts when any two parts, other than the right angle, are known. 80. Napier's Rules of Circular Parts. Another method of remembering the formulas (1) to (9) of 79 is embraced in what are called Napier's Rules of Circular co, B Parts. Let a right spherical triangle be given with the usual notation, Fig. 54, the right angle being C. Then write #, b, comple- ment of A, complement of c, complement of B, Fig. 55. The notation shown on the drawing consists of five parts, known as Napier's Circular Parts. If any part as co. c be taken as a middle part, the next two parts to the right and left, co. A, co. B are called the adjacent parts, and the remaining two parts, a, b, are called the opposite parts. NAPIER'S RULES. (1) The sine of the middle part is equal to the product of the tangents of the adjacent parts. sin(mid. pt.^) = tan(adj. pt.*) x tan(adj. pt.^). (2) The sine of the middle part is equal to the product of the cosines of the opposite parts. sin(mid. pt.~) = cos(op. pt.~) x cos(op. pt.~). As illustration of these rules, identify each of the following with some one of the formulas (1) to (9), 79 : 1. sin (co. c)= cose = 2. sin(a) = 3. sin( 90, b < 90. Formula (9), 79, ( ~ ) + A cos c = cos a cos o, shows cos c negative ; hence, c > 90, the supplement of the tabular angle determined by formula (9). Formulas (2), (6), 79, + - tan b r> tana tan c tan c show A > 90, 5 < 90. (3) Given a > 90, b > 90. Formula (9), (+) " i cos c cos a cos 0, determines cos c positive ; hence c < 90. A , (-> tan a ( ~ } D tan Also, cos A = - , cos -D = tan c tan c show that cos .A, cos B are negative ; hence, A > 90, 5 > 90. THEOREM. In any rigid spherical triangle the hypotenuse c is acute if a and b lie in the same quadrant; cis obtuse if a and b lie in opposite quadrants. 82. The Isosceles and Quadrantal Triangles. If two sides of a spherical triangle be equal, the opposite angles are equal, and 82-83] 123 A=c, the triangle is isosceles. An isosceles spherical triangle may be divided into two symmetrical right triangles, ADB, CDS, by drawing the arc of a great circle from , Fig. 56, perpendicular upon side b. This perpendicular p bisects the angle B and the opposite side b. The solution of an isosceles triangle ABO depends upon the solution of the two right triangles ADB, CDB. If a side be 90, the triangle is called a quadrantal spherical triangle. The solution of a quadrantal triangle may be made to depend upon the solution of a right spherical triangle by tak- ing the polar triangle, one of whose Fi - 5(3 - angles will be 90, the supplement of the given side. See 76. To solve a quadrantal triangle, solve its polar and take the sup- plement of each part of the polar. 83. Solution of Right Spherical Triangles. Six cases arise in the solution of right spherical triangles. CASE I. Given two sides, a, b. CASE II. Given one side and the hypotenuse, a, c. CASE III. Given two angles, A, B. CASE IV. Given one angle and the adjacent side, A, b. CASE V. Given one angle and the opposite side, A, a. Double solution. CASE VI. Given one angle and the hypotenuse, A, c. That Case V has a double solution may be seen from the drawing, Fig. 57. B Fis. 57. 124 SPHERICAL TRIGONOMETRY [ 83 In calculating the unknown parts of a triangle falling under any one of the above Cases, (1) either select a proper set of formulas, 79, or employ Napier's Rules to obtain the required trigonometric relations ; (2) use logarithms in performing the operations of multiplication and division. EXERCISES Solve the following right spherical triangles for each un- known part: 1. a = 64 20', b = 70 24'. 2. a = 49 28', 90. Then we have tan f 90 _ i ^ _^/sin [(90 - (S- A')] sin [90 - ( S- "^~ sin (270 -S) sin [90 -(^-^') or, - \~_costf cos (S-') with similar expressions for cot ^ c', and cot ^ a'. Taking the reciprocal, and omitting accents, we have (11) ten I ft = V - c g os >Sf <** ^ ~ g ^-- = X cos (S - ^cos (S - A) cos (/S - C) 87-88] THE OBLIQUE SPHERICAL TRIANGLE 131 (12) tanr.c = A/ cos & cos (.S < 'cos (S - A} cos (8 - -B) cos (S - C) NOTE. It may be shown that k, used in (8), (9), (10), is equal to the tangent of the radius of the small circle inscribed in the spherical triangle ABC, and that K, used in (11), (12), (13), is equal to the tangent of the radius of the small circle circumscribing the triangle ABC. 88. Napier's Analogies. Dividing tan | B by tan | (7, for- mulas (8), (9), 87, we have tan I B _ sin (s c) tan | (7 sin(s 6)* Taking this equality by composition and division, tan * B + tan | C _ sin (s c) + sin (s 5) tan ^ B tan | C sin (s c) sin (s &) which reduces to sin * (B + (7) _ sin | ( 2 8 b c) cos \ (5 e) sin | (.6 6' ) cos \ - (2 s 5 c) sin | (i c) ' and finally, .1 i (U) Hi g-c = tan^- c ' Similarly, fiin ( ~*. -4- A \ i*i _ 7 ar x oiii j \ T.V T ^t ^ 5) _^ : (16) tan?,(c-) tanc (0-6)" Another formula may be obtained by multiplying tan \ B by i TI . i n si n C 8 ~ ^ tan 1 5 tan %C= - ^ - L . tan l (7. sins 132 SPHERICAL TRIGONOMETRY [ 88 Write tan ^ B and tan ^ C in terms of sines and cosines, and take the equation by division and composition, cos | B cos \ C sin ^ B sin \G _ sin s sin (s a) cos | jB cos C + sin ^ jB sin (7 sin + sin ( a)' and reduce, giving (17) cos|(J5 + C) tan^a cos^JB-C) Similarly, COS^(C + .d) tail .', /> cos I (A + B) tan \ c cos \ (A - B) tan ^ (a + 6) The formulas (14) to (19) are known as Napier's Analogies. Numbers (14) and (17) enable us to compute the sides 6, c, when the opposite angle B, (7, and the included side" a are known. The Theorem of Sines, 84, then determines the remaining angle A. Napier's Analogies appear also in another form which may be derived by means of the polar triangle. By substituting a = 180 - A', A = 180 - a', and so on, let the student deduce from (14) to (19) the following : sin I (6 + c) cot \ A sin I (c + a) cot I B sin 3 (c a) tan^CC - & & sin \ (a + b) cot \ C (22) and cos \ (b + c) cot \ A (23) = = j-^ , cos a (&-) tan 5 (B + C) Z cos (c + a) cot .^ B (24) | = T- , cos ( c - a) 88-90] THE OBLIQUE SPHERICAL TRIANGLE 133 cos I (a + 6) cot ~ C 89. The Area of a Spherical Triangle. The sum of the three angles of a spherical triangle is greater than 180. If A, B, C be the angles, A + g + p_ 180 o _ E determines the spherical excess of the triangle. In spherical geometry it is shown that the areas of spherical triangles are to each other as their spherical excesses. A tri- rectangular triangle has an area \ TrR? and its excess is 90. Hence, if A denote the area of any spherical triangle whose spherical excess is E, and R be the radius of the sphere, we have or 180 The spherical excess E is readily found when the angles of a spherical triangle are known. The value of E may be determined directly when the three sides are known. The resulting formula is known as ISHuiliers Theorem, and is given by E tan = Vtan ^ s tan |(s a) tan |(s b) tan |(s c). The derivation of this formula will not be given here. 90. Solution of Oblique Spherical Triangles. When any three parts of a spherical triangle are given, the remaining parts may be found by use of the Theorem of Sines, the Half -angle Formulas, or by Napier's Analogies. Six cases occur. CASE I. G-iven the three sides, a, 5, c. To determine the angles A, B, C, we use the half -angle formulas - -^ -, tanijB=- -, tan 1(7 = . * sin a) sin (s 5) sin (s cy , 7 /sin(s a) sin (s 6) sin (* tan i ( C+ A) = - 2A -- Z C ot 1 B, i / /> / tani((7 J.)=- ^coH.B. sml(c + a) In the first of these formulas cot | B is positive, cos |( 90. After the angles (7 and A have been found, the side b may be determined by means of the Theorem of Sines, sin a sin B sin b = sin A In using the Theorem of Sines care must be exercised in determining whether b lies in the first or second quadrant. CASE IV. Criven two angles and the included side, B, C, a. Formulas (17) and (14) of Napier's Analogies determine b and c. The first of these formulas shows that if ^B + (7) > 90, then will ^ (b + c*) > 90, and inversely. The angle A may be determined by the Theorem of Sines. 90] THE OBLIQUE SPHERICAL TRIANGLE 135 CASE V. Given two sides and an angle opposite one of them, b, c, B. Here, the angle C is given by . n sin c sin B sm C = : . sin b (1) If sin c sin B > sin b, no solution exists. (2) If sin c sin B sin b, the angle C = 90. (3) If sin c sin B < sin b, either one or two solutions will occur; see Fig. 60. After (7 has been found, angle A and side a may be obtained from cos A (6 + C)* cot I A = - * tan *(B + C\ cos (6 - i COsi(.B + (7 s ) , -i ST. , N ton * a -co g i(J-C)* n * ( * + ' ) - CASE VI. Given two angles and a side opposite one of them, B, (7, b. The side c is given by sin b sin (7 sm c = - - sin B This case may also have no solution, one solution, or two solutions. After c has been determined the remaining parts, a, A, may be derived from Napier's Analogies, as in Case V. EXERCISES Solve the following spherical triangles for the unknown parts: 1. A = WO, 5=75, C=65. 2. = 2740', 5=48, c = 5040'. 3. a = 65 48', b = 120 21', e=8421'. 4. JL = 9650', =75 10', C= 96 50'. 5. B = 72 30', ^4 = 41 27', c = 4917'. 6. b = 118 48', 90 and 7l = Z 8BF '= 180 - 7. When the bearing, or bearing and distance, of two points whose latitude and longitude are given, is required, Napier's Analogies, (20), (23), 88, may be used to find the angles NBF and NFB. The side BF may then be found by use of the Theorem of Sines, (1), 84. 140 SPHERICAL TRIGONOMETRY [ 92-93 EXERCISES 1. Find the distance in knots between Boston, lat. 42 21' N., long. 71 4' W., and Liverpool, lat. 53 24' N., long. 3 4' W. Find also the bearing of each port from the other. 2. Find the distance in statute miles along the arc of a great circle from New York, lat. 40 43' N., long. 74 W., to San Francisco, lat. 37 48' N., long. 122 28' W. 3. Find the distance in nautical miles from New York, lat. 40 43' N., long. 74 W., to Greenwich, lat. 51 29' N., and the bearing of each point from the other. 4. If a vessel sails directly east from Sandy Hook, lat. 40 28' N., long. 74 1' W., along the arc of a great circle at the uniform rate of 16 knots per hour, find its latitude and longitude (1) at the end of 48 hours' sailing, (2) at the end of five days' sailing. 5. Find the distance in statute miles from San Francisco, lat. 37 48' N., long. 122 28' W., to Calcutta, lat. 22 33' N., long. 88 19' E., and the bearing of each point from the other. 93. The Celestial Sphere. Astronomical problems furnish many applications of spherical trigonometry. One class of these problems will be noticed here. The daily rotation of the earth upon its axis from west to east causes the stars to seem to rotate from east to west upon the surface of an immense sphere named the celestial sphere. To a person located at any point upon the earth one-half of this sphere is visible. The celestial sphere is represented in Fig. 62, the earth being a mere point at the centre. (1) The Horizon of any point upon the earth is the intersec- tion of the horizontal plane through the point with the celes- tial sphere. In Fig. 62, the horizon is the great circle HLH 1 ' . (2) The Zenith of any point is the intersection of the perpen- dicular erected to the plane of the horizon at the point. The point on the celestial sphere diametrically opposite the zenith is the Nadir. In the figure Z is the zenith, Z' is the nadir. (3) The Celestial Poles are the intersections of the line of the earth's axis with the celestial sphere. The point N is the 93-94] THE CELESTIAL SPHERE 141 north pole of the celestial sphere, S is its south pole. The celestial sphere rotates (apparently) about the axis NS once in 24 hours. (4) The Celestial Equator is the intersection of the plane of the earth's equator with the celestial sphere. In the figure, EME' is the celestial equator, N and S are its poles. (5) The Celestial Meridians are the great circles through N and 8. The celestial meridian through any star is called the hour circle of that star. Thus, NPMis the hour circle of P. 94. The Celestial Triangle. The position P of any star is known if we know its distance PL above the horizon, and its distance PM north or south of the celestial equator. (1) The Altitude PL of a star at P is the arc of a zenith circle ZL intercepted between the point P and the horizon. See Fig. 62. (2) The Declination of a star P is the arc PM of the celes- tial meridian NPM between the point and the celestial equator. (3) The Latitude of the Zenith of any point on the earth is the arc EZ and is equal to the latitude of the observer. (4) The Hour Angle of any star is the angle between the 142 SPHERICAL TRIGONOMETRY [94 meridian through the star and the zenith meridian. In the figure PNZ is the hour angle. Since the celestial sphere apparently rotates through 360 in 24 hr., or 15 in one hour of time, we may express the hour angle in hours, minutes, and seconds, and thus determine the time required for a star to rotate into the meridian of the zenith. The points P, N, Z determine the celestial triangle PNZ, sometimes called the astronomical triangle. This triangle is determined when the latitude of the observer EZ, the altitude of the star PL, and the declination of the star PM are known. We have : NZ = 90 latitude of the observer, ZP = 90 - altitude of the star, PN= 90- declination of the star. If an observation of the sun's altitude be made in the fore- noon, let us sky, and the declination of the sun upon that day of the year be known, then we may compute the hour angle provided the latitude of the observer be known. Having the hour angle, we may find the corresponding equivalent in time, and thus determine the hour of day at which the observation was made. The sun's declination varies from near 23 30' south to near 23 30' north declination. At the vernal equinox and at the autumnal equinox the sun's declination is zero. EXERCISES 1. At San Francisco, lat. 37 48' N., a forenoon observation shows the sun's altitude as 40 21'. If the sun's declination be 10 41' N., what is the time of observation ? 2. In latitude 40 21' N., the sun's altitude in the afternoon was found to be 35 40'. What was the time of observation, if the sun's declination is 8 4' S.? 3. Find the time of sunrise at Boston, lat. 42 21' N., the sun's declination being 20 31' N. 4. Find the time of sunrise (approximately) at a point whose latitude is 39 10' N. twenty days after the vernal equinox. CHAPTER XII FORMULAS PLANE TRIGONOMETRY Fundamental Identities. 1. sin x csc x = cos x sec x = tan x cot x = 1. 2. sin 2 x -f- cos 2 z = 1, sec 2 x tan 2 x = 1, esc 2 # cot 2 x = 1. sin a: 1 sec a? Vl cos 2 x \ s\ Tl 'J* "" ~~~ r~^ - cos a; cot a; cscx cos a; 1 tana: 4. sin x = cos x tan x = 5. cos x = sin x cot a; = Vl -f cot 2 x Vl + tan 2 a; 1 cot a; Vl + cot 2 z Sum and Difference Formulas. 6. sin (x y} = sin x cos y cos x sin y. 7. cos (x y) = cos x cos y T sin x sin ^. x tan x tan ^ 8. tan (x y) = - - *- 1 T tan a; tan y cot z cot y T 1 9. cot(z?/) = - ^T -- cot y cot a: 10. sin x + sin y = 2 sin ^ (a; + ?/) cos ^ (x y). 11. sin x sin y = 2 cos |(a: + #) sin -| (a; y). 12. cos a; + cos # = 2 cos ^ (x + y) cos |(a; y) . 13. cos a; - cos # = 2sini(z + #) sinl(a;-T/). sin (a; y) 14. tan g tany = c08a . cosy . sin (a: v) 15. cot a; cot y = 16. sin x sin y sin x + sin y , cos x + cos y 143 144 FORMULAS sin x + sin y _ tan \(x-\- y) sin x sin y tan ^ (xy) 18. sin 2 a; sin 2 y = sin (x + y) sin (# #). 19. cos 2 a; cos 2 1/ = sin (a: + y)sin(a; ?/ 20. cos 2 a? sin 2 y = cos (a? + y} cos (x y). Half Angle and Multiple Angle Formulas. 2 tan A a; 21. sin x = z sin TT a; cos * a; = A & l+tan 2 | 22. cos x = cos 2 \x sin 2 1- x = 2 cos 2 \x 1 = 1 2 sin 2 1 tan 2 1 x ~ 1 + tan 2 ^- a; 2 tan -i x 2 cot 1 a; 2 1 tan 2 1- a; cot 2 ^ a; 1 cot ^ a; tan |- a; 23. tan a; = COt 3T - - 1 24. cot a; = ^ 2 . = iYcoti a; tani-af). V nrf. I i- " ^ * IS 25. sin^ a: = V|(l cos a;). 26. cos| a;= V|(l + cosa;). i^ cos a; sin a; 1 cos a; 27. tania; = \/- =- = : j- cos x 1 + cosa; sin a; 28. sin 2 x 2 sin x cos a;, cos 2 a; = cos 2 x sin 2 x. 29. sin 3 a; = 3 sin x 4 sin 3 a;, cos 3 a; = 4 cos 3 a; 3 cos x. 30. sin 4 x = sin a; (8 cos 3 x 4 cos a;), cos 4 a; = 8 cos 4 x 8 cos 2 a; + 1. n 2 tan a; 3 tan x tan 3 a; Plane Triangles. rrn ,; sin A sin B sin C 1 32. 1 heorems of sines : - - = ; = = a b c 2R 33. Theorem of cosines : a 2 = V 2 + c 2 2 be cos A. ** ' i 34. Theorem of tangents : a + b _ sin A + sin B _ tan ^ ( JL + j?) a b sin J. sin B tan i (^L B) FORMULAS 145 35. The half angles : sin = A , ON (2) (3) tan = s s wVlPVP Q fl "^ VY 1 1 v I O M V * ~ w z^ i i 19 sin .4 sin J5 36. Area : K = k be sin A = tit 03 01 284 326 368 410 452 494 536 578 620 662 i 4-4 4-3 4.2 2 8.8 8.6 8.4 04 703 745 787 828 870 912 953 995 *036 *078 3 13-2 12.9 12.6 05 02 119 160 202 243 284 325 366 407 449 490 4 n.6 17.2 16.8 06 531 572 612 653 694 735 776 816 857 898 5 22.0 21.5 2I.O 6 26.4 25-8 2S.2 07 938 979 *019 *060 *100 *141 *181 *222 *262 *302 7 30.8 30.1 29-4 08 03 342 383 423 463 503 543 583 623 663 703 8 35-2 34-4 33-6 09 743 782 822 862 902 941 981 *021 *060 -100 9 39-6 38.7 37-8 110 04 139 179 218 258 297 336 376 415 454 493 11 532 571 610 650 689 727 766 805 844 883 41 40 39 12 922 961 999 *038 *077 *115 *154 *192 *23] *269 13 05 308 346 385 423 461 500 538 576 614 652 i 4.1 4.0 3-9 2 8.2 8.0 7.8 14 690 729 767 805 843 881 918 956 994 *032 3 12.3 12.0 11.7 IS 06 070 108 145 183 221 258 296 333 371 408 4 16.4 16.0 iS-6 16 446 483 521 558 595 633 670 707 744 781 S 20.5 20.0 19-5 6 24.6 24.0 23-4 17 819 856 893 930 967 *004 *041 *078 *115 *151 7 28.7 28.0 27-3 18 07 188 225 262 298 335 372 408 445 482 518 8 32.8 32.0 31-2 19 555. 591 628 664 700 737 773 809 846 882 9 36-9 36.0 3S-I 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 21 08 279 314 350 386 422 458 493 529 565 600 38 87 36 22 636 672 707 743 778 814 849 884 920 955 23 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 I 3-8 3-7 3-6 2 7-6 7-4 7-2 24 09 342 377 412 447 482 517 552 587 621 656 3 11.4 n. i 10.8 25 691 726 760 795 830 864 899 934 968 *003 4 15-2 14.8 14.4 26 10 037 072 106 140 175 209 243 278 312 346 5 19.0 18.5 18.0 6 22.8 22.2 21.6 27 380 415 449 483 517 551 585 619 653 687 7 26.6 25-9 25.2 28 721 755 789 823 857 890 924 958 992 *025 8 30.4 29.6 28.8 29 11 059 093 126 160 193 227 261 294 327 361 9 34-2 33-3 32-4 130 394 428 461 494 528 561 594 628 661 694 31 727 760 793 826 860 893 926 959 992 *024 35 34 33 32 12 057 090 123 156 189 222 254 287 320 352 33 385 418 450 483 516 548 581 613 646 678 i 2 3-5 7-o 3-4 6.8 3-3 6.6 34 710 743 775 808 840 872 905 937 969 *001 3 10.5 IO.2 9-9 35 13 033 066 098 130 162 194 226 258 290 322 4 14.0 13-6 13-2 36 354 386 418 450 481 513 545 577 609 640 S I7-S 17.0 16.5 6 2I.O 20.4 19.8 37 672 704 735 767 799 830 862 893 925 956 7 24-5 23-8 23-1 38 988 *019 *051 *082 *114 *145 *176 *208 *239 *270 8 28.O 27.2 26.4 39 14 301 333 364 395 426 457 489 520 551 582 9 3I-S 30.6 29-7 140 613 644 675 706 737 768 799 829 860 891 41 922 953 983 *014 *045 *076 *106 *137 *168 *198" 32 31 30 42 15 229 259 290 320 351 381 412 442 473 503 I 3-2 3-1 3*O 43 534 564 594 625 655 685 715 746 776 806 2 6.4 6.2 6.0 44 836 866 897 927 957 987 *017 *047 *077 *107 3 4 9.6 12.8 9-3 12.4 9.0 I2.O 45 16 137 167 197 227 256 286 316 346 376 406 ^ 16.0 *5-5 T C O 46 435 465 495 524 554 584 613 643 673 702 6 19.2 18.6 i j.w 18.0 47 48 732 17 026 761 056 791 085 820 114 850 143 879 173 909 202 938 231 967 260 997 289 7 8 22.4 25.6 28.8 21.7 24.8 27 O 21.0 24.0 ->7 r> 49 319 348 377 406 435 464 493 522 551 580 /*y -/ 150 609 638 667 696 725 754 782 811 840 869 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 15O-200 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 150 17609 638 667 696 725 754 782 811 840 869 51 898 926 955 984 *013 *041 *070 *099 *127 *156 29 28 52 18 184 213 241 270 298 327 355 384 412 441 53 469 498 526 554 583 611 639 667 696 724 i 2.9 2.8- 2 5-8 5-6 54 752 780 808 837 86^ 893 921 949 977 *005 3 8.7 8.4 55 19033 061 089 117 145 173 201 229 257 285 4 n.6 II. 2 56 312 340 368 396 424 451 479 507 535 562 5 I4-S I4.O 6 17.4 16.8 57 590 618 645 673 700 728 756 783 811 838 7 20.3 19.6 58 866 893 921 948 976 *003 *030 *058 *085 *112 8 23.2 22.4 59 20 140 167 194 222 249 276 303 330 358 385 9 26.1 25.2 160 412 439 466 493 520 548 575 602 629 656 61 683 710 737 763 790 817 844 871 898 925 27 26 62 952 978 *005 *032 *059 *085 112 *139 *165 *192 63 21 219 245 272 299 325 352 378 405 431 458 i 2.7 2.6 2 5-4 5-2 64 484 511 537 564 590 617 643 669 696 722 3 8.1 7.8 65 748 775 801 827 854 880 906 932 958 985 4 10.8 10.4 66 22011 037 063 089 115 141 167 194 220 246 S 13-5 13-0 6 16.2 15.6 67 272 298 324 350 376 401 427 453 479 505 7 18.9 18.2 68 531 557 583 608 634 660 686 712 737 763 8 21.6 20.8 69 789 814 840 866 891 917 943 968 994 *019 9 24-3 23.4 170 23045 070 096 121 147 172 198 223 249 274 71 300 325 350 376 401 426 452 477 502 528 25 72 553 578 603 629 654 679 704 729 754 779 73 805 830 855 880 905 930 955 980 *005 *030 I 2.S 2 5.0 74 24055 080 105 130 155 180 204 229 254 279 3 7-5 75 304 329 353 378 403 428 452 477 502 527 4 10. o 76 551 576 601 625 650 674 699 724 748 773 5 12.5 6 15.0 77 797 822 846 871 895 920 944 969 993 *018 7 17-5 78 25042 066 091 115 139 164 188 212 237 261 8 20.O 79 285 i 310 334 358 382 406 431 455 479 503 9 22-5 180 527 551 575 600 624 648 672 696 720 744 81 768 ' 792 816 840 864 888 912 935 959 983 24 23 82 26007 031 055 079 102 126 150 174 198 221 83 245 269 293 316 340 364 387 411 435 458 I 2.4 2.3 2 4.8 4.6 84 482 505 529 553 576 600 623 647 670 694 3 7-2 6.9 85 717 741 764 788 811 834 858 881 905 928 4 9.6 9-2 86 951 975 998 *021 *045 *068 *091 *114 *138 *161 5 I2.O "5 6 14.4 13-8 87 27 184 207 231 254 277 300 323 346 370 393 7 16.8 16.1 88 416 439 462 485' 508 531 554 577 600 623 1 8 19.2 18.4 89 646 669 692 715 738 761 784 807 830 852 9 21.6 20.7 190 875 898 921 944 967 989 *012 *035 *058 *081 91 28 103 126 149 171 194 217 240 262 285 307 22 21 92 330 353 375 398 421 443 466 488 511 533 93 556 578 601 623 646 668 691 713 735 758 i 2.2 2.1 ' a 4.4 4-2 94 780 803 825 847 870 892 914 937 959 981 3 6.6 6-3 95 29003 026 048 070 092 115 137 159 181 203 4 8.8 8.4 96 226 248 270 292 314 336 358 380 403 425 S 6 II. 13-2 10.5 12.6 97 447 469 491 513 535 557 -579 601 623 645 1 IS-4 14.7 98 667 688 710 732 754 776 798 820 842 863 8 17.6 16.8 99 885 907 929 951 973 994 *016 *038 *060 *081 9 19.8 18.9 200 30 103 125 146 168 190 211 233 255 276 298 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 2OO-25O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 200 30 103 125. 146 168 190 211 233 255 276 298 01 320 341 363 384 406 428 449 471 492 514 22 21 02 535 557 578 600 621 643 664 685 707 728 03 750 771 792 814 835 856 878 899 920 942 I 2.2 2.1 2 4.4 4.2 04 963 984 *006 *027 *048 *069 *091 *112 *133 *154 3 6.6 6.3 05 31 175 197 218 239 260 281 302 323 345 366 4 8.8 8.4 06 387 408 429 450 471 492 513 534 555 576 S n.o 10.5 6 13.2 12.6 07 597 618 639 660 681 702 723 744 765 785 7 iS-4 14-7 08 806 827 848 869 890 911 931 952 973 994 8 17.6 16.8 09 32015 035 056 077 098 118 139 160 181 201 g 19.8 18.9 210 222 243 263 284 305. 325 346 366 387 408 11 428 449 469 490 510 531 552 572 593 613 20 12 634 654 675 695 715 736 756 777 797 818 13 838 858 879 899 919 940 960 980 *001 *021 i 2.O 2 4.0 14 33041 062 082 102 122 143 163 183 203 224 3 6.0 15 244 264 284 304 325 345 365 385 405 425 4 8.0 16 445 465 486 506 526 546 566 586 606 626 S IO.O 6 12.0 17 646 666 686 706 726 746 766 786 806 826 7 14.0 18 846 866 885 905 925 945 965 985 *005 *025 8 16.0 19 34044 064 084 104 124 143 163 183 203 223 9 18.0 220 242 262 282 301 321 341 361 380 400~ 420 21 439 459 479 498 518 537 557 577 596 616 19 22 635 655 674 694 713 733 753 772 792 811 23 830 850 869 889 908 928 947 967 986 *OOS I 1.9 2 3-8 24 35025 044 064 083 102 122 141 160 180 199 3 5-7 25 218 238 257 276 295 315 334 353 372 392 4 7.6 26 411 430 449 468 488 507 526 545 564 583 S 9-5 6 11.4 27 603 622 641 660 679 698 717 736 755 774 7 13-3 28 793 813 832 851 870 889 908 927 946 965 8 15-2 29 984 *003 *021 *040 *059 *078 *097 *116 *135 154 17.1 230 36173 192 211 229 248 267 286 305 324 342 31 361 380 399 418 436 455 474 493 511 530 18 32 549 568 586 605 624 642 661 680 698 717 33 736 754 773 791 810 829 847 866 884 903 I 1.8 2 3-6 34 922 940 959 977 996 *014 *033 *051 *070 *088 3 5-4 35 37 107 125 144 162 181 199 218 236 254 273 4 7.2 36 291 310 328 346 365 383 401 420 438 457 5 9.0 6 10.8 37 475 493 511 530 548 566 585 603 621 639 7 12.6 38 658 676 694 712 731 749 767 785 803 822 8 14.4 39 840 858 876 894 912 931 949 967 985 *003 9 16.2 240 38021 039 057 075 093 112 130 148 166 184 41 202 220 238 256 274 292 310 328 346 364 17 42 382 399 417 435 453 471 489 507 525 543 43 561 578 596 614 632 650 668 686 703 721 i 2 ' 1.7 3-4 44 739 757 775 792 810 828 846 863 881 899 3 5 i f> a 45 917 934 952 970 987 *005 *023 *041 *058 *076 4 u.o 46 39094 111 129 146 164 182 199 217 235 252 5 6 8.5 IO.2 47 270 287 305 322 340 358 375 393 410 428 7 11.9 48 445 463 480 498 515 533 550 568 585 602 8 13-6 49 620 637 655 672 690 707 724 742 759 777 9 15-3 250 794 811 829 846 863 881 898 915 933 950 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 25O-30O R. L. i 1 2 3 ; 4 56789 Prop. Pts. 250 39 794 811 829 846 863 881 ; S9S 915 933 950 51 967 985 *002 *019 *037 *054 *071 *08S *106 *123 18 52 40 140 157 175 192 209 226 243 261 278 295 53 312 329 346 364 381 398 415 432 449 466 i 1.8 2 3-6 54 483 500 i 518 535 552 569 586 603 620 637 3 5-4 55 654 671 688 705 i 722 739 756 773 790 807 4 7-2 56 824 841 858 875 ; 892 909 926 943 960 976 5 9.0 6 10.8 57 993 *010 *027 *044 *061 *078 *095 *111 *128 *145 7 12.6 58 41 162 179 196 212 229 246 1 263 280 296 313 8 14.4 59 330 347 363 380 397 414 430 447 464 481 9 16.2 260 497 514 531 547 564 581 597 614 631 647 61 664 68L 697 714 731 747 764 780 797 814 17 62 830 847 863 880 896 913 929 946 963 979 63 9% *012 *029 *045 *062 *078 *095 111 *127 *144 I 1-7 2 3-4 64 42 160 177 193 210 226 243 259 275 292 308 3 5-1 65 325 341 357 374 390 406 423 439 455 472 4 6.8 66 488 504 521 537 553 570 586 602 619 635. 5 8.5 6 10.2 67 651 667 684 700 716 732 749 765 781 797 7 II.9 68 813 830 846 862 878 894 911 927 943 959 8 13.6 69 975 991 *OOS *024 *040 *056 *072 *OSS *104 *120 9 15-3 270 43 136 i 152 169 ; 185 ] 201 217 233 249 265 281 71 297 313 i 329 345 361 377 393 409 425 1 441 16 72 457 473 489 505 521 537 553 569 584 600 73 616 632 648 664 680 696 712 727 743 759 i 1.6 2 3-2 74 775 791 807 823 838 854 870 886 902 917 3 4.8 75 933 949 965 981 996 *012 *028 *044 059 *075 4 6.4 76 44091 107 122 138 154 170 185 201 217 232 5 8.0 6 9.6 77 248 264 279 295 311 326 342 358 373 389 7 II. 2 78 404 420 436 451 467 483 498 514 529 545 8 12.8 79 560 576 592 607 623 638 654 669 685 700 9 14.4 280 716 731 747 762 778 793 809 824 840 855 81 871 886 902 917 932 948 963 i 979 994 *010 15 82 45025 040 056 071 t 086 102 117 | 133 148 163 83 179 194 209 225 240 255 271 286 301 317 I 1.5 2 3-0 84 332 347 362 378 393 408 423 439 454 469 3 4-5 85 484 500 515 530 ' 545 561 576 591 606 621 4 6.0 86 637 652 667 682 697 712 728 743 758 773 S 7-5 6 9.0 87 788 803 818 834 849 864 879 894 909 924 7 10.5 88 939 954 969 984 ,*000 *015 *030 *045 *075 8 I2.O 89 46090 105 120 135 i 150 165 180 195 210 225 9 13-5 290 240 255 270 285 300 315 330 345 359 374 91 389 404 419 434 449 464 479 494 509 523 14 92 538 553 568 583 ! 598 613 627 642 657 672 93 687 702 716 731 746 761 776 790 805 820 I 2 1.4 2.8 94 835 850 864 879 894 909 923 938 953 967 3 4.2 95 982 997 *012 *026 *041 *056 *070 *085 *100 114 4 5-6 96 47 129 144 159 173 188 202 217 232 246 261 5 6 7.0 8.4 97 276 I 290 305 319 334 349 363 378 392 407 7 9.8 98 422 i 436 451 465 480 494 509 524 538 553 8 II. 2 99 567 582 596 611 625 640 654 669 683 698 9 12.6 300 712 >7 I 741 756 770 784 799 ; 813 828 842 N. L. 1 2 3 4 5 a 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 3OO-35O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 300 47 712 727 741 756 770 784 799 813 828 842 01 857 871 885 900 914 929 943 958 972 986 02 48001 015 029 044 058 073 087 101 116 130 03 144 159 173 187 202 216 230 244 259 273 15 04 287 302 316 330 344 359 373 387 401 416 i i-S 05 430 444 458 473 487 501 515 530 544 558 2 3-0 06 572 586 601 615 629 643 657 671 686 700 3 4-5 4 6.0 07 714 728 742 756 770 785 799 813 827 841 5 7-S 08 855 869 883 897 911 926 940 954 968 982 6 9.0 09 996 *010 *024 *038 *052 *066 *080 *094 *108 *122 7 IO-S 310 49136 150 164 178 192 206 220 234 248 262 8 I2.O 11 276 290 304 318 332 346 360 374 388 402 9 13-5 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 14 16 969 982 996 *010 *024 *037 *OS1 *065 *079 *092 I 1.4 17 50106 120 133 147 161 174 188 202 215 229 2 2.8 18 243 256 270 284 297 311 325 338 352 365 3 4.2 19 379 393 406 420 433 447 461 474 488 501 4 5 "5-6 7.O 320 515 529 542 556 569 583 596 610 623 637 6 8.4 21 651 664 678 691 705 718 732 745 759 772 7 9.8 22 786 799 813 826 840 853 866 880 893 907 8 II. 2 23 920 934 947 p 961 974 987" *001 *014 *028 *041 9 12.6 24 51055 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 13 28 587 601 614 627 640 654 667 680 693 706 I 1-3 29 720 733 746 759 772 786 799 812 825 838 2 2.6 330 851 865 878 891 904 917 930 943 957 970 3 3.9 31 983 996 *009 *022 *035 *048 *061 *075 *088 *101 4 5-2 ft n 32 52 114 127 140 153 166 179 192 205 218 231 S 0.5 33 244 257 270 284 297 310 323 336 349 362 6 7.8 7 9.1 34 375 388 401 414 427 440 453 466 479 492 8 10.4 35 504 517 530 543 556 569 582 595 608 621 9 11.7 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 53020 033 046 058 071 084 097 110 122 135 12 340 148 161 173 186 199 212 224 237 250 263 i 1.2 41 275 288 301 314 326 339 352 364 377 390 2 2.4 42 403 415 428 441 453 466 479 491 504 517. 3 3-6 43 529 542 555 567 580 593 605 618 631 643 4 4.8 5 6.0 44 656' 668 681 694 706 719 732 744 757 769 6 7-2 45 782 794 807 820 832 845 857 870 882 895 7 8.4 46 908 920 933 945 958 970 983 995 *OOS *020 8 9.6 9 10.8 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. 307 320 332 345 357 370 382 394 350 407 419 432 444 456 469 481 494 506 518 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 35O-4OO I. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 53 111 790 802 814 827 839 851 864 876 888 54 900 913 925 937 949 962 974 986 998 *011 55 55 023 035 047 060 072 084 096 108 121 133 13 56 145 157 169 182 194 206 218 230 242 255 T 1.3 2 2.6 57 267 279 291 303 315 328 340 352 364 376 3 3-9 58 388 400 413 425 437 449 461 473 485 497 4 5-2 6 f 59 509 522 534 546 558 570 582 594 606 618 6 v . j 7.8 360 630 642 654 666 678 691 703 715 727 739 8 O.I 10.4 61 751 763 775 787 799 811 823 835 847 859 9 ii. 7 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 12 67 467 478 490 502 514 526 538 549 561 573 68 585 597 608 620 632 644 656 667 679 691 i 2 1.2 2.4 69 703 714 726 738 750 761 773 785 797 808 3 3-6 370 820 832 844 855 867 879 891 902 914 926 4 S 4.8 6.0 71 937 949 961 972 984 996 *008 *019 *031 *043 6 7-2 72 57 054 066 078 089 101 113 124 136 148 159 8 8.4 0.6 73 171 183 194 206 217 229 241 252 264 276 9 10.8 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 11 79 864 875 887 898 910 921 933 944 955 967 380 978 990 *001 *013 *024 *035 *047 *058 *070 *081 a 2.2 81 58 092 104 115 127 138 149 161 172 184 195 3 3-3 82 206 218 229 240 252 263 274 286 297 309 4 5 4.4 5-5 83. 320 331 343 354 365 377 388 399 410 422 6 6.6 7 7-7 84 433 444 456 467 478 490 501 512 524 535 8 9 8.8 0.0 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 10 91 218 229 240 251 262 273 284 295 306 318 f I.O 92 329 340 351 362 373 384 395 406 417 428 3 2.0 93 439 450 461 472 483 494 506 517 528 539 3 3-o 4 4.0 94 550 561 572 583 594 605 616 627 638 649 6 6.'o 95 660 671 682 693 704 715 726 737 748 759 7 7.0 96 770 780 791 802 813 824 835 846 857 868 8 9 8.0 9.0 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 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 400-450 N. L. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 11 08 61 066 077 087 098 109 119 130 140 151 162 09 172 183 194 20 1 215 225 236 247 257 268 i 2 i.i 2.2 410 278 289 300 310 321 331 342 352 363 374 3 3-3 11 384 395 405 416 426 437 448 458 469 479 4 S 4*4 S-S 12 490 500 511 521 532 542 553 563 574 584 6 6.6 13 595 606 616 627 637 648 658 669 679 690 8 7-7 8.8 g 0.9 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 10 23 634 644 655 665 675 685 696 706 716 726 i I.O 24 737 747 757 767 778 788 798 808 818 829 2 3 2.O 3.0 25 839 849 859 870 880 890 900 910 921 931 4 4.0 26 941 951 961 972 982 992 *002 *012 *022 *033 5 5- 6 6.0 27 63 043 053 063 073 083 094 104 114 124 134 8 7.0 8.0 28 144 155 165 175 185 195 205 215 225 236 9 9.0 29 246 256 266 276 286 296 306 317 327 337 430 347 357 367 377 387 397 407 417 428 438 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 9 37 64 048 058 068 078 088 098 108 118 128 137 38 147 157 167 177 187 197 207 217 227 237 I 2 o.g 1.8 39 246 256 266 276 286 296 306 316 326 335 3 2.7 f\ 440 345 355 365 375 385. 395 404 414 424 434 4 S 3-0 4-5 41 444 454 464 473 483 493 503 513 523 532 6 5-4 42 542 552 562 572 582 591 601 611 621 631 8 6.3 7.2 43 640 650 660 670 680 689 699 709 719 729 9 8.1 44 738 748 758 768. 777 787 797 807 816 826 45 836 846 856 865 875 885 895 904 914 924 46 933 943 953 963 972 982 992 *002 *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 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 450-500 H. L. 1 2 3 4 * o 6 7 8 9 Prop. Pts. 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 6J9 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 10 58 66 087 096 106 115 124 134 143 153 162 172 j 59 181 191 200 210 219 229 238 247 257 266 2 2.O 460 276 285 295 304 314 323 332 342 351 361 3 4 3-0 4.0 61 370 380 389 398 408 417 427 436 445 455 5 S.o 62 464 474 483 492 502 511 521 530 539 549 6 6.0 63 558 567 577 586 596 605 614 624 633 642 I 7.0 8.0 9 0.0 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 I 403 413 422 431 440 449 459 468 477 9 73 486 495 504 514 523 532 541 550 560 569 i 0.9 2 1.8 74 578 587 596 605 614 624 633 642 651 660 3 2.7 75 669 679 6SS 697 706 715 724 733 742 752 4 3-6 76 761 770 779 788 797 806 815 825 834 843 6 4-S 5-4 7 6.3 77 852 861 870 879 888 897 906 916 925 934 8 7.2 78 943 952 961 970 979 988 997 *006 *015 *024 g 8.1 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 8 87 753 762 771 780 789 797. 806 815 824 833 I 0.8 88 842 851 860 869 878 886 895 904 913 922 2 1.6 89 931 940 949 958 966 975 984 993 *002 *011 3 2.4 32 490 69 020 028 037 046 055 064 073 082 090 099 4 5 4- 91 108 117 126 135. 144 152 161 170 179 188 6 7 4.8 <>.6 92 197 205 214 223 232 241 249 258 267 276 8 6.4 93 285 294 302 311 320 329 338 346 355 364 9 7.2 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 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. LOGARITHMS OF NUMBERS 500-55O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 9 08 586 595 603 612 621 629 638 646 655 663 09 672 680 689 697 706 714 723 731 740 749 i 2 o.o 1.8 510 757 766 774 783 791 800 808 817 825 834 3 2.7 _ jc 11 842 851 859 868 876 885 893 902 910 919 4 s 3.0 4-5 12 927 935 944 952 961 969 978 986 995 *003 6 5-4 13 71 012 020 029 037 046 054 063 071 079 088 8 6.3 7.2 g 8.1 14 0% 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 6.54 642 650 659 667 675 21 684 692 700 709 717 725 : 734 742 750 759 22 767 775 784 792 800 809 817 825 834 842 8 23 850 858 867 875 883 892 900 908 917 925 I 0.8 24 933 941 950 958 966 975 983 991 999 *008 2 3 1.6 2.4 25 72 016 024 032 041 049 057 066 074 082 090 4 3-3 26 099 107 115 123 132 140 148 156 165 173 6 4.0 4.8 27 181 189 198 206 214 222 230 239 247 255 8 5-6 6.4 28 263 272 280 288 296 304 313 321 329 337 9 7>2 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 7 37 997 *006 *014 *022 *030 *038 *046 *054 *062 *070 i 0.7 38 73 078 086 094 102 111 119 127 135 143 151 2 1.4 39 159 167 175 183 191 199 207 215 223 231 3 4 2.1 2.8 540 239 247 255 263 272 280 288 296 304 312 S 3-5 41 320 328 336 344 352 360 368 376 384 392 6 7 4.2 4-9 42 400 408 416 424 432 440 448 456 464 472 8 5.6 43 480 488 496 504 512 520 528 536 544 552 9 6.3 44 560 568 576, 584 592 600 608 616 624 632 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. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 10 LOGARITHMS OF NUMBERS 550-600 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 8 i 0.8 64 128 136 143 151 159 166 174 182 189 197 2 1.6 65 205 213 220 228 236 243 251 259 266 274 3 M 66 282 289 297 305 312 320 328 335 343 351 ' 4 3- 2 S 4- 64-8 67 358 366 374 381 389 397 404 412 420 427 75-6 68 435 442 450 458 465 473 481 488 496 504 8 6.4 9 7.2 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 7 83 567 574 582 589 597 604 612 619 626 634 i 0.7 2 1.4 84 641 649 656 664 671 678 686 693 701 708 3 2.1 85 716 723 730 738 745 753 760 768 775 782 4 2.8 S 3-5 86 790 797 805 812 819 827 834 842 849 856 6 4.2 7 4-0 87 864 871 879 886 893 901 908 916 923 930 85.6 9 6.3 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 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 11 LOGAKITHMS OF NUMBERS 6OO-65O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 600 77815 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 78032 039 046 053 061 068 075 082 089 097 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 8 08 390 398 405 412 419 426 433 440 447 455 j 08 09 462 469 476 483 490 497 504 512 519 526 2 i!6 610 533 540 547 554 561 569 576 583 590 597 3 4 2.4 3.2 11 604 611 618 625 633 640 647 654 661 668 S 4.0 12 675 682 689 696 704 711 718 725 732 739 6 7 4.8 r 6 13 746 753 760 767 774 781 789 796 803 810 8 3 w 6.4 9 7-2 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 79029 036 043 050 057 064 071 078 085 092 18 099 106 113 120 127 134 141 148 155 162 19 169 176 183 190 197 204 211 218 225 232 620 239 246 253 260 267 274 281 288 295 302 21 309 316 323 330 337 344 351 358 365 372 22 379 386 393 400 407 414 421 428 435 442 7 23 449 456 463 470 477 484 491 498 505 511 . 1 0.7 24 518 525 532 539 546 553 560 567 574 581 2 3 1.4 2.1 25 588 595 602 609 616 623 630 637 644 650 4 2.8 26 657 664 671 678 685 692 699 706 713 720 6 3-S 4.2 7 4-9 27 727 734 741 748 754 761 768 775 782 789 8 5-6 28 796 803 810 817 824 831 837 844 851 858 9|6. 3 29 865 872 879 886 893 900 906 913 920 927 630 934 941 948 955 962 969 975 982 989 996 31 80003 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 6 37 414 421 428 434 441 448 455 462 468 475 I 0.6 38 482 489 496 502 509 516 523 530 536 543 2 1.2 39 550 557 564 570 577 584 591 598 604 611 3 1.8 640 618 625, 632 638 645 652 659 665 672 679 4 | 2.4 3.0 41 686 693 699 706 713 720 726 733 740 747 6 3.6 42 754 760 767 774 781 787 794 801 808 814 1 8 4.2 4.8 43 821 828 835 841 848 855 862 868 875 882 9 5-4 44 889 895 902 909 916 922 929 936 943 949 45 956 963 969 976 983 990 996 *003 *010 *017 46 81023 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. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 12 LOGARITHMS OF NUMBERS 65O-7OO R. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 650 SI 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 82020 027 033 040 W6 053 060 066 073 079 62 086 092 099 105 112 119 125 132 138 145 7 63 151 158 164 171 178 184 19J 197 204 210 i 0.7 64 217 223 230 236 243 249 256 263 269 276 2 1.4 312.1 65 282 289 295 302 308 315 321 328 334 341 42.8 66 347 354 360 367 373 380 387 393 400 406 53-5 64.2 7 4-9 67 413 419 426 432 439 445 452 458 465 471 85.6 68 478 484 491 497 504 510 517 523 530 536 916.3 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 83059 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 6 83 442 448 455 461 467 474 480 487 493 499 i 0.6 2 1.2 84 506 512 518 525 531 537 544 550 556 563 31-8 85 569 575 582 588 594 601 607 613 620 626 4 2-4 86 632 639 645 651 658 664 670 677 683 689 5 3-0 63-6 74-2 87 696 702 708 715 721 727 734 740 746 753 84.8 88 759 765 771 778 784 790 797 803 809 816 95-4 89 822 828 835 841 847 853 860 866 872 879 690 885 891 897 904 910 916 923 929 935 942 91 948 954 960 967 973 979 985 992 998 *004 92 84011 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 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 13 LOGARITHMS OF NUMBERS 7OO-75O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 700 84510 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 7 08 85003 009 016 022 028 034 040 046 052 058 09 065 071 077 083 089 095 101 107 114 120 i 2 0.7 1.4 710 126 132 138 144 150 156 163 169 175 181 3 2.1 2.8 11 187 193 199 205 211 217 224 230 236 242 4 5 3-5 12 248 254 260 266 272 278 285 291 297 303 6 4-2 13 309 315 321 327 333 339 345 352 358 364 8 4.9 5.6 9 6.3 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 6 23 914 920 926 932 938 944 950 956 962 968 I 0.6 24 974 980 986 992 998 *004 *010 *016 *022 *028 2 3 1.2 1.8 25 86034 040 046 052 058 064 070 076 082 088 4 2-4 26 094 100 106 112 118 124 130 136 141 147 s 3-o 6 3-6 27 153 159 165 171 177 183 189 195 201 207 7 8 4.2 4.8 28 213 219 225 231 237 243 249 255 261 267 9 5-4 29 273 279 285 291 297 303 308 314 320 326 730 332 338 344 350 356 362 368 374 380 386 31 392 398 404 410 415 421 427 433 439 445 32 451 457 463 469 475 481 487 493 499 504 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 5 37 747 753 759 764 770 776 782 788 794 800 38 806 812 817 823 829 835 841 847 853 859 i 2 -5 I.O 39 864 870 876 882 888 894 900 906 911 917 3 i-S 740 923 929 935 941 947 953 958 964 970 976 4 5 2.O 2.5 41 982 988 994 999 *005 *011 *017 *023 *029 *035 6 3-0 42 87040 046 052 058 064 070 075 081 087 093 8 3.5 4.0 43 099 105 111 116 122 128 134 140 146 151 9 4-S 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 F. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 14 LOGARITHMS OF NUMBERS 75O-8OO Jf. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 88024 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 6 i 0.6 64 309 315 321 326 332 338 343 349 355 360 2 1.2 65 366 372 377 383 389 395 400 406 412 417 3 1.8 66 423 429 434 440 446 451 457 463 468 474 4 2 -4 5 3-0 63.6 67 480 485 491 497 502 508 513 519 525 530 7 4-2 8 4.8 68 536 542 547 553 559 564 570 576 581 587 9 5-4 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 74 874 880 885 891 897 902 908 913 919 925 75 930 936 941 947 953 958 964 969 975 981 76 986 992 997 *003 *009 *014 *020 *025 *031 *037 77 89042 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 81 265 271 276 282 287 293 298 304 310 315 82 321 326 332 337 343 348 354 360 365 371 5 83 376 382 387 393 398 404 409 415 421 426 i 0.5 2 1.0 84 432 437 443 448 454 459 465 470 476 481 3I-S 4 2.0 85. 487 492 498 504 509 515 520 526 531 537 5 2.5 86 542 548 553 559 564 570 575 581 586 592 6 3.0 7 3-5 87 597 603 609 614 620 625 631 636 642 647 84.0 94-5 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 9] 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 90037 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. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 15 LOGARITHMS OF NUMBERS 8OO-85O s. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 fi i 0.6 14 062 068 073 078 084 089 094 100 105 110 2 1.2 15 116 121 126 132 137 142 148 153 158 164 3 1.8 16 169 174 180 185 190 196 201 206 212 217 4 2.4 5 3-0 6 3.6 17 222 228 233 238 243 249 254 259 265 270 7 4-2 8 4.8 18 275 281 286 291 297 302 307 312 318 323 9 5-4 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 5 33 065 070 075 080 085 091 096 101 106 111 i 0.5 2 1.0 34 117 122 127 132 137 143 148 153 158 163 3 i-5 35 169 174 179 184 189 195 200 205 210 215 4 2.O S 2.5 36 221 226 231 236 241 247 252 257 262 267 6 3.0 7 3-S 37 273 278 283 288 293 298 304 309 314 319 8 4.0 9 4-5 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. L. O 1 2 3 4 5 6 7 8 9 Prop. Pts. 16 LOGARITHMS OF NUMBERS 850-90O N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 850 92942 947 952 957 962 967 973 978 983 988 51 993 998 :*003 *008 *013 *018 *024 *029 *034 *039 52 93044 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 6 58 349 354 359 364 369 374 379 384 389 394 59 399 404 409 414 420 425 430 435 440 445 i 2 0.6 1.2 860 450 455 460 465 470 475 480 485 490 495 3 4 1.8 2.4 61 500 505 510 515 520 526 531 536 541 546 5 3-0 62 551 556 561 566 571 576 581 586 591 596 6 3.6 63 601 606 611 616 621 626 631 636 641 646 8 4.2 4.8 9 5-4 64 651 656 661 666 671 676 682 687 692 697 65 702 707 712 717 722 727 732 737 742 747 66 752 757 762 767 772 111 782 787 792 797 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 94002 007 012 017 022 027 032 037 042 047 72 052 057 062 067 072 077 082 086 091 096 5 73 101 106 111 116 121 126 131 136 141 146 I o.S 2 I.O 74 151 156 161 166 171 176 18-1 186 191 196 3 i.S 75 201 206 211 216 221 226 231 236 240 245 4 2.0 76 250 255 260 265 270 275 280 285 290 295 6 2.5 3-o 7 3.5 77 300 305 310 315 320 325 330 335 340 345 8 4.0 78 349 354 359 364 369 374 379 384 389 394 9 4-5 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 4 87 792 797 802 807 812 817 822 827 832 836 Z 0.4 88 841 846 851 856 861 866 871 876 880 885 ' 9 0.8 89 890 895 900 905 910 915 919 924 929 934 3 4 1.2 1.6 890 939 944 949 954 959 963 968 973 978 983 I 2.O 91 988 993 998 *002 *007 *012 *017 *022 *027 *032 t 7 2.4 2.8 92 95 036 041 046 051 056 061 066 071 075 080 8 3-2 93 085 090 095 100 105 109 114 119 124 129 9 3-6 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 99 376 381 386 390 395 400 405 410 415 419 900 424 429 434 439 444 448 458 463 468 I. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 17 LOGARITHMS OF NUMBERS 900-950 N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 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 V 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 *02S *033 *03S *042 13 96 047 052 057 061 066 071 076 080 085 090 5 i 0.5 14 095 099 104 109 114 118 123 128 133 137 2 I.O 15 142 147 152 156 161 166 171 175 180 185 3 i.S 16 190 194 199 204 209 213 218 223 227 232 4 2.O 5 2-5 6 3.0 17 237' 242 246 251 256 261 265 270 275 280 73.5 8 4.0 18 284 289 294 298 303 308 313 317 322 327 94-5 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 4 33 988 993 997 *002 *007 *011 *016 *021 *025 *030 i 0.4 2 0.8 34 97 035 039 044 049 053 058 063 067 072 077 3 1.2 4 1.6 35 081 086 090 095 100 104 109 114 118 123 5 2.0 36 128 132 137 142 146 151 155 160 165 169 6 2.4 72.8 37 174 179 183 188 192 197 02 206 211 216 83.2 3.6 38 220 225 230 234 239 243 248 253 257 262 39 267 271 276 280 285 290 294 299 304 308 940 313 317 322 327 331 336 340 345 350 354 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 If. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 18 LOGARITHMS OF NUMBERS 95O-10OO N. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 950 97 772 777 782 786 791 795 800 804 809 813 51 818 823 827 832 836 841 ~84T 850 855 i 859 52 864 868 873 877 882 886 891 896 900 905 53 909 914 918 923 928 932 937 941 946 950 54 955 959 964 968 973 978 982 987 991 9% 55 98000 005 009 014 019 023 028 032 037 041 56 046 050 055 059 064 068 073 078 082 087 57 091 0% 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 63 363 367 372 376 381 385 390 394 399 403 5 i o-S 64 408 412 417 421 426 430 435 439 444 448 2 I.O 65 453 457 462 466 471 475 480 484 489 493 3 i.S 66 498 502 507 511 516 520 525 529 534 538 4 2.O 5 2-5 6 3.0 67 543 547 552 556 561 565 570 574 579 583 73-5 68 588 592 597 601 605 610 614 619 623 628 8 4.0 9 4-5 69 632 637 641 646 650 655 659 664 668 673 970 677 682 686 691 695 700 70 * 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 8% 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 99034 038 043 047 052 056 061 065 069 074 79 078 083 087 ; 092 0% 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 4 83 255 260 264 269 273 277 282 286 291 295 i 0.4 2 0.8 84 300 304 308 313 317 322 326 330 335 339 3 1.2 85 344 348 352 357 361 366 370 374 379 383 4 1.6 86 388 392 3% 401 405 410 414 419 423 427 52.0 6 2.4 72.8 87 432 436 441 445 449 454 458 463 467 471 83-2 88 476 480 484 489 493 498 502 506 511 515 9 3o 89 520 524 528 533 537 542 546 550 555 559 990 564 568 572 577 ! 581 585 590 594 599 603 91 607 612 616 621 615 629 634 638 642 647 92 651 656 660 66 f 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 00000 004 009 013 017 022 026 030 035 039 H. L. 1 2 3 4 5 6 7 8 9 Prop. Pts. 19 TABLE II FIVE -PLACE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 91 TABLE II contains the common logarithms of the trigonometric functions for angles from to 90 as follows : (1) log sin a = log tan a, for every second to 3', and log cos /3 = log cot /3, for every second, 89 57' to 90, page 22. (2) logarithms of sine, tangent, arid cosine for every 10" from to 2 ; also the logarithms of sine, cosine, and cotangent for every 10" from 88 to 90, pages 23-28. (3) logarithms of each function for every 1' from to 90, pages 29-73. log sin = log tan // O' 1' 2' n n 0' 1' 2' n 00 6.46 373 6.76 476 60 30 6.16 270 6.63 982 6.86 167 30 1 4.68 557 6.47 090 6.76 836 59 31 6.17 694 6.64 462 6.86 455 29 2 4.98 660 6.47 797 6.77 193 58 32 6.19 072 6.64 936 6.86 742 28 3 5.16 270 6.48 492 6.77 548 57 33 6.20 409 6.65 406 6.87 027 27 4 5.28 763 6.49 175 6.77 900 56 34 6.21 705 6.65 870 6.87 310 26 5 5.38 454 6.49 849 6.78 248 55 35 6.22 964 6.66 330 6.87 591 25 6 5.46 373 6.50 512 6.78 595 54 36 6.24 188 6.66 785 6.87 870 24 7 5.53 067 6.51 165 6.78 938 53 37 6.25 378 6.67 235_ 6.88 147 23 8 5.58 866 6.51 808 6.79 278 52 38 6.26 536 6.67 680 6.88 423 22 9 5.63 982 6.52 442 6.79 616 51 39 6.27 664 6.68 121 6.88 697 21 10 5.68 557 6.53 067 6.79 952 50 40 6.28 763 6.68 557 6.88 969 20 11 5.72 697 6.53 683 6.80 285 49 41 6.29 836 6.68 990 6.89 240 19 12 5.76 476 6.54 291 6.80 615 48 42 6.30 882 6.69 418 6.89 509 18 13 5.79 952 6.54 890 6.80 943 47 43 6.31 904 6.69 841 6.89 776 17 14 5.83 170 6.55 481 6.81 268 46 44 6.32 903 6.70 261 6.90 042 16 15 5.86 167 6.56 064 6.81 591 45 45 6.33 879 6.70 676 6.90 306 15 16 5.88 969 6.56 639 6.81 911 44 46 6.34 833 6.71 088 6.90 568 14 17 5.91 602 6.57 207 6.82 230 43 47 6.35 767 6.71 496 6.90 829 13 18 5.94 085 6.57 767 6.82 545 42 48 6.36 682 6.71 900 6.91 088 12 19 5.96 433 6.58 320 6.82 859 41 49 6.37 577 6.72 300 6.91 346 11 20 5.98 660 6.58 866 6.83 170 40 50 6.38 454 6.72 697 6.91 602 10 21 6.00 779 6.59 406 6.83 479 39 51 6.39 315 6.73 090 6.91 857 9 22 6.02 800 6.59 939 6.83 786 38 52 6.40 158 6.73 479 6.92 110 8 23 6.04 730 6.60 465 6.84 091 37 53 6.40 985 6.73 865 6.92 362 7 24 6.06 579 6.60 985 6.84 394 36 54 6.41 797 6.74 248 6.92 612 6 25 6.08 351 6.61 499 6.84 694 35 55 6.42 594 6.74 627 6.92 861 5 26 6.10 055 6.62 007 6.84 993 34 56 6.43 376 6.75 003 6.93 109 4 27 6.11 694 6.62 509 6.85 289 33 57 6.44 145 6.75 376 6.93 355 3 28 6.13 273 6.63 006 6.85 584 32 58 6.44 900 6.75 746 6.93 599 2 29 6.14 797 6.63 496 6.85 876 31 59 6.45 643 6.76 112 6.93 843 1 30 6.16 270 6.63 982 6.86 167 30 60 6.46 373 6.76 476 6.94 085 // 59' 58' 57' n n 59' 58' 57' n 89 log cos = log cot 22 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS / // log sin log tan log cos " ' i a log sin log tan log cos // / 10 20 30 40 50 oo 5.68 557 5.98 660 6.16 270 6.28 763 6.38 454 oo 5.68 557 5.98 660 6.16 270 6.28 763 6 38 454 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 060 50 40 30 20 10 100 10 20 30 40 50 7.46 373 7.47 090 7.47 797 7.48 491 7.49 175 7.49 849 7.46 373 7.47 091 7.47 797 7.48 492 7.49 176 7.49 849 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 050 50 40 30 20 10 1 10 20 30 40 50 6.46 373 6.53 067 6.58 866 6.63 982 6.68 557 6.72 697 6.46 373 6.53 067 6.58 866 6.63 982 6.6S 557 6.72 697 10JOOOOO 10.00000 10.00000 10.00000 10.00000 10.00000 059 50 40 30 20 10 110 10 20 30 40 50 7.50 512 7.51 165 7.51 808 7.52 442 7.53 067 7.53 683 7.50 512 7.51 165 7.51 809 7.52 443 7.53 067 7.53 683 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 049 50 40 30 20 10 2 10 20 30 40 50 6.76 476 6.79 952 6.83 170 6.86 167 6.88 969 6.91 602 6.76 476 6.79 952 6.83 170 6.86 167 6.88 969 6.91 602 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 058 50 40 30 20 10 120 10 20 30 40 50 7.54 291 7.54 890 7.55 481 7.56 064 7.56 639 7.57 206 7.54 291 7.54 890 7.55 481 7.56 064 7.56 639 7.57 207 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 048 50 40 30 20 10 3 10 20 30 40 50 6.94 085 6.96 433 6.98 660 7.00 779 7.02 800 7,04 730 6.94 085 6.96 433 6.98 661 7.00 779 7.02 800 7.04 730 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 057 50 40 30 20 10 130 10 20 30 40 50 7.57 767 7.58 320 7.58 866 7.59 406 7.59 939 7.60 465 7.57 767 7.58 320 7.58 867 7.59 406 7.59 939 7.60 466 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 047 50 40 30 20 10 4 10 20 30 40 50 7.06 579 7.0S 351 7.10 055 7.11 694 7.13 273 7.14 797 7.06 579 7.08 352 7.10 055 7.11 694 7.13 273 7.14 797 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 056 50 40 30 20 10 140 10 20 30 40 50 7.60 985 7.61 499 7.62 007 7.62 509 7.63 006 7.63 496 7.60 986 7.61 500 7.62 008 7.62 510 7.63 006 7.63 497 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 046 50 40 30 20 10 5 10 20 30 40 50 7.16 270 7.17 694 7.19 072 7.20 409 7.21 705 7.22 964 7.16 270 7.17 694 7.19 073 7.20 409 7.21 705 7.22 964 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 055 50 40 30 20 10 150 10 20 30 40 50 7.63 982 7.64 461 7.64 936 7.65 406 7.65 870 7.66 330 7.63 982 7.64 462 7.64 937 7.65 406 7.65 871 7.66 330 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 045 50 40 30 20 10 6 10 20 30 40 50 7.24 188 7.25 378 7.26 536 7.27 664 7.28 763 7.29 836 7.24 188 7.25 378 7.26 536 7.27 664 7.28 764 7.29 836 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 054 50 40 30 20 10 160 10 20 30 40 50 7.66 784 7.67 235 7.67 680 7.68 121 7.68 557 7.68 989 7.66 785 7.67 235 7.67 680 7.68 121 7.68 558 7.68 990 10.00000 10.00000 10.00000 10.00000 9.99999 9.99999 044 50 40 30 20 10 7 10 20 30 40 50 7.30 882 7.31 904 7.32 903 7.33 879 7.34 833 7.35 767 7.30 882 7.31 904 7.32 903 7.33 879 7.34 833 7.35 767 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 053 50 40 30 20 10 170 10 20 30 40 50 7.69 417 7.69 841 7.70 261 7.70 676 7.71 088 7.71 496 7.69 418 7.69 842 7.70 261 7.70 677 7.71 088 7.71 496 9.99999 9.99999 9.99999 9.99999 9.99999 9.99999 043 50 40 30 20 10 8 10 20 30 40 50 7.36 682 7.37 577 7.38 454 7.39 314 7.40 158 7.40 985 7.36 682 7.37 577 7.38 455 7.39 315 7.40 158 7.40 985 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 052 50 40 30 20 10 180 10 20 30 40 50 7.71 900 7.72 300 7.72 697 7.73 090 7.73 479 7.73 865 7.71 900 7.72 301 7.72 697 7.73 090 7.73 480 7.73 866 9.99999 9.99999 9.99999 9.99999 9.99999 9.99999 042 50 40 30 20 10 9 10 20 30 40 50 7.41 797 7.42 594 7.43 376 7.44 145 7.44 900 7.45 643 7.41 797 7.42 594 7.43 376 7.44 145 7.44 900 7.45 643 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 051 50 40 30 20 10 190 10 20 30 40 50 7.74 248 7.74 627 7.75 003 7.75 376 7.75 745 7.76 112 7.74 248 7.74 628 7.75 004 7.75 377 7.75 746 7.76 113 9.99999 9.99999 9.99999 9.99999 9.99999 9.99999 041 50 40 30 20 10 10 7.46 373 7.46 373 10.00000 050 200 7.76 475 7.76 476 9.99999 040 / n log cos log cot log sin a f / n log cos log cot log sin // / 89 23 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS / // log sin log tan log cos // / i n log sill log tan log cos // / 200 10 20 30 40 50 7.76 475 7.76 836 7.77 193 7.77 548 7.77 899 7.78 248 7.76 476 7.76 837 7.77 194 7.77 549 7.77 900 7.78 249 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 040 50 40 30 20 10 300 10 20 30 40 50 7.94 084 7.94 325 7.94 564 7.94 802 7.95 039 7.95 274 7.94 086 7.94 326 7.94 566 7.94 804 7.95 040 7.95 276 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 030 50 40 30 20 10 210 10 20 30 40 50 7.78 594 7.78 938 7.79 278 7.79 616 7.79 952 7.80 284 7.78 595 7.78 938 7.79 279 7.79 617 7.79 952 7.80 285 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 039 50 40 30 20 10 310 10 20 30 40 50 7.95 508 7.95 741 7.95 973 7.96 203 7.96 432 7.96 660 7.95 510 7.95 743 7.95 974 7.96 205 7.96 434 7.96 662 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 029 50 40 30 20 10 220 10 20 30 40 50 7.80 615 7.80 942 7.81 268 7.81 591 7.81 911 7.82 229 7.80 615 7.80 943 7.81 269 7.81 591 7.81 912 7.82 230 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 038 50 40 30 20 10 320 10 20 30 40 50 7.96 887 7.97 113 7.97 337 7.97 560 7.97 782 7.98 003 7.96 889 7.97 114 7.97 339 7.97 562 7.97 784 7.98 005 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 028 50 40 30 20 10 230 10 20 30 40 50 7.82 545 7.82 859 7.83 170 7.83 479 7.83 786 7.84 091 7.82 546 7.82 860 7.83 171 7.83 480 7.83 787 7.84 092 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 037 50 40 30 20 10 330 10 20 30 40 50 7.98 223 7.98 442 7.98 660 7.98 876 7.99 092 7.99 306 7.98 225 7.98 444 7.98 662 7.98 878 7.99 094 7.99 308 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 027 50 40 30 20 10 240 10 20 30 40 50 7.84 393 7.84 694 7.84 992 7.85 289 7.85 583 7.85 876 7.84 394 7.84 695 7.84 993 7.85 290 7.85 584 7.85 877 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 036 50 40 30 20 10 340 10 20 30 40 50 7.99 520 7.99 732 7.99 943 8.00 154 8.00 363 8.00 571 7.99 522 7.99 734 7.99 946 8.00 156 8.00 365 8.00 574 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 026 50 40 30 20 10 250 10 20 30 40 50 7.86 166 7.86 455 7.86 741 7.87 026 7.87 309 7.87 590 7.86 167 7.86 456 7.86 743 7.87 027 7.87 310 7.87 591 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 035 50 40 30 20 10 350 10 20 30 40 50 8.00 779 8.00 985 8.01 190 8.01 395 8.01 598 8.01 801 8.00 781 8.00 987 8.01 193 8.01 397 8.01 600 8.01 803 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 025 50 40 30 20 10 260 10 20 30 40 50 7.87 870 7.88 147 7.88 423 7.88 697 7.88 969 7.89 240 7.87 871 7.88 148 7.88 424 7.88 698 7.88 970 7.89 241 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 034 50 40 30 20 10 360 10 20 30 40 50 8.02 002 8.02 203 8.02 402 8.02 601 8.02 799 8.02 996 8.02 004 8.02 205 8.02 405 8.02 604 8.02 801 8.02 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 024 50 40 30 20 10 270 10 20 30 40 50 7.89 509 7.89 776 7.90 041 7.90 305 7.90 568 7.90 829 7.89 510 7.89 777 7.90 043 7.90 307 7.90 569 7.90 830 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 033 50 40 30 20 10 370 10 20 30 40 50 8.03 192 8.03 387 8.03 581 8.03 775 8.03 967 8.04 159 8.03 194 8.03 390 8.03 584 8.03 777 8.03 970 8.04 162 9.99 997 9.99 997 9.99 997 9-99 997 9.99 997 9.99 997 023 50 40 30 20 10 280 10 20 30 40 50 7.91 088 7.91 346 7.91 602 7.91 857 7.92 110 7.92 362 7.91 089 7.91 347 7.91 603 7.91 858 7.92 111 7.92 363 9.99 999 9.99 999 9.99 999 9.99 999 9.99 998 9.99 998 032 50 40 30 20 10 380 10 20 30 40 50 8.04 350 8.04 540 8.04 729 8.04 918 8.05 105 8.05 292 8.04 353 8.04 543 8.04 732 8.04 921 8.05 108 8.05 295 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 022 50 40 30 20 10 290 10 20 30 40 50 7.92 612 7.92 861 7.93 108 7.93 354 7.93 599 7.93 842 7.92 613 7.92 862 7.93 110 7.93 356 7.93 601 7.93 844 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 9.99 998 031 50 40 30 20 10 390 10 20 30 40 50 8.05 478 8.05 663 8.05 848 8.06 031 8.06 214 8.06 396 8.05 481 8.05 666 8.05 851 806 034 8.06 217 8.06 399 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 021 50 40 30 20 10 300 7.94 084 7.94 086 9.99 998 030 400 8.06 578 8.06 581 9.99 997 020 / n log cos log cot log sin // / / // log cos log cot log sin // / 89 24 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS c / /' log sin log tan log cos // / / // log sin log tan log cos // / 400 10 20 30 40 50 8.06 578 8.06 758 8.06 938 8.07 117 8.07 295 8.07 473 8.06 581 8.06 761 8.06 941 8.07 120 8.07 299 8.07 476 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 020 50 40 30 20 10 500 10 20 30 40 50 8.16 26S 8.16 413 8.16 557 8.16 700 8.16 843 8.16 986 8.16 273 8.16 417 8.16 561 S.I 6 705 8.16 S4S 8.16 991 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 010 50 40 30 20 10 410 10 20 30 40 50 8.07 650 8.07 826 8.08 002 8.08 176 8.08 350 8.08 524 8.07 653 8.07 829 8.08 005 8.08 ISO 8.08 354 8.08 527 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 019 50 40 30 20 10 510 10 20 30 40 50 8.17 128 8.17 270 8.17 411 8.17 552 8.17 692 8.17 832 8.17 133 8.17 275 8.17 416 8.17 557 8.17 697 8.17 837 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9 50 40 30 20 10 420 10 20 30 40 50 8.08 696 8.08 868 8.09 040 8.09 210 8.09 380 8.09 55_0 8.08 700 8.08 872 8.09 043 8.09 214 8.09 384 8.09 553 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 018 50 40 30 20 10 520 10 20 30 40 50 8.17 971 8.18 110 8.18 249 8.18 387 8.18 524 8.18 662 8.17 976 8.18 115 8.18 254 8.18 392 8.18 530 8.18 667 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 8 50 40 30 20 10 430 10 20 30 40 50 8.09 718 8.09 886 8.10 054 8.10 220 8 10 386 8.10 552 8.09 722 8.09 890 8.10 057 8.10 224 8.10 390 8.10 555 9.99 997 9.99 997 9.99 997 9.99 997 9.99 997 9.99 996 017 50 40 30 20 10 530 10 20 30 40 50 8.18 798 8.18 935 8.19 071 8.19 206 8.19 341 8.19 476 8.18 804 8.18 940 8.19 076 8.19 212 8.19 347 8.19 481 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 7 50 40 30 20 10 440 10 20 30 40 50 8.10 717 8.10 881 8.11 044 8.11 207 8.11 370 8.11 531 8.10 720 8.10 884 8.11 048 8.11 211 8.11 373 8.11 535 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 016 50 40 30 20 10 540 10 20 30 40 50 8.19 610 8.19 744 8.19 877 8.20 010 8.20 143 8.20 275 8.19 616 8.19 749 8.19 883 8.20 016 8.20 149 8.20 281 9.99 995 9.99 995 9.99 995 9.99 995 9.99 995 9.99 994 6 50 40 30 20 10 450 10 20 30 40 50 8.11 693 8.11 853 8.12 013 8.12 172 8.12 331 8.12 489 8.11 696 8.11 857 8.12 017 8.12 176 8.12 335 8.12 493 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 015 50 40 30 20 10 550 10 20 30 40 50 8.20 407 8.20 538 8.20 669 8.20 800 8.20 930 8.21 060 8.20 413 8.20 544 8.20 675 S.20 806 8.20 936 8.21 066 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 5 50 40 30 20 10 460 10 20 30 40 50 8.12 647 8.12 804 8.12 961 8.13 117 8.13 272 8.13 427 8.12 651 8.12 SOS 8.12 965 8.13 121 8.13 276 8.13 431 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 014 50 40 30 20 10 560 10 20 30 40 50 8.21 189 8.21 319 8.21 447 8.21 576 8.21 703 S.21 831 8.21 195 8.21 324 8.21 453 8.21 581 8.21 709 8.21 837 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 4 50 40 30 20 10 470 10 20 30 40 50 8.13 581 8.13 735 8.13 888 8.14 041 8.14 193 8.14 344 8.13 585 8.13 739 8.13 892 8.14 045 8.14 197 8.14 348 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 013 50 40 30 20 10 570 10 20 30 40 50 S.21 958 8.22 085 8.22 211 8.22 337 8.22 463 8.22 588 8.21 964 8.22 091 8.22 217 8.22 343 8.22 469 8.22 595 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 3 50 40 30 20 10 480 10 20 30 40 50 8.14 495 8.14 646 8.14 796 8.14 945 8.15 094 8.15 243 8.14 500 8.14 650 8.14 800 S.14 950 8.15 099 8.15 247 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 012 50 40 30 20 10 580 10 20 30 40 50 8.22 713 8.22 838 8.22 962 8.23 086 8.23 210 8.23 333 S.22 720 8.22 844 8.22 968 8.23 092 8.23 216 8.23 339 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 2 50 40 30 20 10 490 10 20 30 40 50 8.15 391 8.15 538 8.15 685 8.15 832 8.15 978 8.16 123 8.15 395 8.15 543 8.15 690 8.15 836 8.15 982 8.16 128 9.99 996 9.99 996 9.99 996 9.99 996 9.99 995 9.99 995 Oil 50 40 30 20 10 590 10 20 30 40 50 8.23 456 8.23 578 8.23 700 8.23 822 8.23 944 8.24 065 8.23 462 8.23 585 8.23 707 8.23 829 8.23 950 8.24 071 9.99 994 9.99 994 9.99 994 9.99 993 9.99 993 9.99 993 1 50 40 30 20 10 500 8.16 268 8.16 273 9.99 995 010 600 8.24 186 8.24 192 9.99 993 / // log cos log cot log sin tr I / // log cos log cot log sin // / 8 9 25 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS ] 1 / // log sin log tan log cos // i / // log sin log tan log cos // / 10 20 30 40 50 8.24 186 8.24 306 8.24 426 8.24 546 8.24 665 8.24 785 8.24 192 8.24 313 8.24 433 8.24 553 8.24 672 8.24 791 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 060 50 40 30 20 10 100 10 20 30 40 50 8.30 879 8.30 983 8.31 086 8.31 188 8.31 291 8.31 393 8.30 888 8.30 992 8.31 095 8.31 198 8.31 300 8.31 403 9.99 991 9.99 991 9.99 991 9.9) 991 9.99 991 9.99 991 050 50 40 30 20 10 1 10 20 30 40 50 8.24 903 8.25 022 8.25 140 8.25 258 8.25 375 8.25 495 8.24 910 8.25 029 8.25 147 8.25 265 8.25 382 8.25 500 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 059 50 40 30 20 10 110 10 20 30 40 50 8.31 495 8.31 597 8.31 699 8.31 800 8.31 901 8.32 002 8.31 505 8.31 606 8.31 708 8.31 809 8.31 911 8.32 012 9.99 991 9.99 991 9.99 991 9.99 991 9.99 991 9.99 991 049 50 40 30 20 10 2 10 20 30 40 50 8.25 609 8.25 726 8.25 842 8.25 958 8.26 074 8.26 189 8.25 616 8.25 733 8.25 849 8.25 965 8.26 081 8.26 196 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 058 50 40 30 20 10 120 10 20 30 40 50 8.32 103 8.32 203 8.32 303 8.32 403 8.32 503 8.32 602 8.32 112 8.32 213 8.32 313 8.32 413 8.32 513 8.32 612 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 048 50 40 30 20 10 3 10 20 30 40 50 8.26 304 8.26 419 8.26 533 8.26 648 8.26 761 8.26 875 8.26 312 8.26 426 8.26 541 8.26 655 8.26 769 8.26 882 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 9.99 993 057 50 40 30 20 10 130 10 20 30 40 50 8.32 702 8.32 801 8.32 899- 8.32 998 8.33 096 8.33 195 8.32 711 8.32 811 8.32 909 8.33 008 8.33 106 8.33 205 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 047 50 40 30 20 10 4 10 20 30 40 50 8.26 988 8.27 101 8.27 214 8.27 326 8.27 438 8.27 550 8.26 996 8.27 109 8.27 221 8.27 334 8.27 446 8.27 558 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 056 50 40 30 20 10 140 10 20 30 40 50 8.33 292 8.33 390 8.33 488 8.33 585 8.33 682 8.33 779 8.33 302 8.33 400 8.33 498 8.33 595 8.33 692 8.33 789 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 9.99 990 046 50 40 30 20 10 5 10 20 30 40 50 8.27 661 8.27 773 8.27 883 8.27 994 8.28 104 8.28 215 8.27 669 8.27 780 8.27 891 8.28 002 8.28 112 8.28 223 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 055 50 40 30 20 10 150 10 20 30 40 50 8.33 875 8.33 972 8.34 068 8.34 164 8.34 260 8.34 355 8.33 886 8.33 982 8.34 078 8.34 174 8.34 270 8.34 366 9.99 990 9.99 990 9.99 990 9.99 990 9.99 989 9.99 989 045 50 40 30 20 10 6 10 20 30 40 50 8.28 324 8.28 434 8.28 543 8.28 652 8.28 761 8.28 869 8.28 332 8.28 442 8.28 551 8.28 660 8.28 769 8.28 877 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 054 50 40 30 20 10 160 10 20 30 40 50 8.34 450 8.34 546 8.34 640 8.34 735 8.34 830 8.34 924 8.34 461 8.34 556 8.34 651 8.34 746 8.34 840 8.34 935 9.99 989 9.99 989 9.99 989 9.99 989 9.99 989 9.99 989 044 50 40 30 20 10 7 10 20 30 40 50 8.28 977 8.29 085 8.29 193 8.29 300 8.29 407 S.29 514 8.28 986 8.29 094 8.29 201 8.29 309 8.29 416 8.29 523 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 9.99 992 053 50 40 30 20 10 170 10 20 30 40 50 8.35 018 8.35 112 8.35 206 8.35 299 8.35 392 8.35 485 8.35 029 8.35 123 8.35 217 8.35 310 8.35 403 8.35 497 9.99 989 9.99 989 9.99 989 9.99 989 9.99 989 9-99 989 043 50 40 30 20 10 8 10 20 30 40 50 8.29 621 8.29 727 8.29 833 8.29 939 8.30 044 8.30 150 8.29 629 8.29 736 8.29 842 8.29 947 8.30 053 8.30 158 9.99 992 9.99 991 9.99 991 9.99 991 9.99 991 9.99 991 052 50 40 30 20 10 180 10 20 30 40 50 8.35 578 8.35 671 8.35 764 8.35 856 8.35 948 8.36 040 8.35 590 8.35 682 8.35 775 8.35 867 8.35 959 8.36 051 9.99 989 9.99 989 9.99 989 9.99 989 9.99 989 9.99 989 042 50 40 30 20 10 9 10 20 30 40 50 8.30 255 8.30 359 8.30 464 8.30 568 8.30 672 8.30 776 8.30 263 8.30 368 8.30 473 8.30 577 8.30 681 8.30 785 9.99 991 9.99 991 9.99 991 9.99 991 9.99 991 9.99 991 051 50 40 30 20 10 190 10 20 30 40 50 8.36 131 8.36 223 8.36 314 8.36 405 8.36 496 8.36 587 8.36 143 8.36 235 8.36 326 8.36 417 8.36 508 8.36 599 9.99 989 9.99 988 9.99 988 9.99 988 9.99 988 9.99 988 041 50 40 30 20 10 100 8.30 879 8.30 888 9.99 991 050 200 8.36 678 8.36 689 9.99 988 040 / // log cos log cot log sin // / / // log cos log cot log sin // / 8* * 26 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 1 / // log sin log tan log cos // / / // log sin log tan log cos // t 200 10 20 30 40 50 8.36 678 8.36 768 8.36 858 8.36 948 8.37 038 8.37 128 S.36 689 8.36 780 8.36 870 8.36 960 8.37 050 8.37 140 9.99 988 9.99 988 9.99 988 9.99 988 9.99 988 9.99 988 040 50 40 30 20 10 300 10 20 30 40 50 8.41 792 8.41 872 8.41 952 8.42 032 8.42 112 8.42 192 8.41 807 8.41 887 8.41 967 8.42 048 8.42 127 8.42 207 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 030 50 40 30 20 10 210 10 20 30 40 50 8.37 217 8.37 306 8.37 395 8.37 484 8.37 573 8.37 662 8.37 229 8.37 318 8.37 408 8.37 497 8.37 585 8.37 674 9.99 988 9.99 988 9.99 988 9.99 988 9.99 988 9.99 988 039 50 40 30 20 10 310 10 20 30 40 50 8.42 272 8.42 351 8.42 430 8.42 510 8.42 589 8.42 667 8.42 287 8.42 366 8.42 446 8.42 52^ 8.42 604 8.42 683 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 029 50 40 30 20 10 220 10 20 30 40 50 8.37 750 8.37 838 8.37 926 8.38 014 8.38 101 8.38 189 8.37 762 8.37 850 8.37 938 8 38 026 8.38 114 8.38 202 9.99 988 9.99 988 9.99 988 9.99 987 9.99 987 9.99 987 038 50 40 30 20 10 320 10 20 30 40 50 8.42 746 8.42 825 8.42 903 8.42 92 8.43 060 8.43 138 8.42 762 8.42 840 8.42 919 8.42 997 8.43 075 8.43 154 9.99 984 9.99 984 9.99 984 9.99 984 9.99.984 9.99 984 028 50 40 30 20 10 230 10 20 30 40 50 8.38 276 8.38 363 8.38 450 8.38 537 8.38 624 8.38 710 8.38 289 8.38 376 8.38 463 8.38 550 8.38 636 8.38 723 9.99 987 9.99 987 9.99 987 9.99 987 9.99 987 9.99 987 037 50 40 30 20 10 330 10 20 30 40 50 8.43 216 8.43 293 8.43 371 8.43 448 8.43 526 8.43 603 8.43 232 8.43 309 8.43 387 8.43 464 8.43 542 8.43 619 9.99 984 9.99 984 9.99 984 9.99 984 9.99 984 9.99 984 027 50 40 30 20 10 240 10 20 30 40 50 8.38 796 8.38 882 8.38 968 8.39 054 8.39 139 8.'39 225. 8.38 809 8.38 895 8.38 981 8.39 067 8.39 153 8.39 238 9.99 987 9.99 987 9.99 987 9.99 987 9.99 987 9.99 987 036 50 40 30 20 10 340 10 20 30 40 50 8.43 680 8.43 757 8.43 834 8.43 910 8.43 987 8.44 063 8.43 696 8.43 773 8.43 850 8.43 927 8.44 003 8.44 080 9.99 984 9.99 984 9.99 984 9.99 984 9.99 984 9.99 983 026. 50 40 30 20 10 250 10 20 30 40 50 8.39 310 8.39 395 8.39 480 8.39 565 8.39 649 8.39 734 8.39 323 8.39 408 8.39 493 8.39 578 8.39 663 8.39 747 9.99 987 9.99 987 9.99 987 9.99 987 9.99 987 9.99 986 035 50 40 30 20 10 350 10 20 30 40 50 8.44 139 8.44 216 8.44 292 8.44 367 8.44 443 8.44 519 8.44 156 8.44 232 8.44 308 8.44 384 8.44 460 8.44 536 9.99 983 9.99 983 9.99 983 9.99 983 9.99 983 9.99 983 025 50 40 30 20 10 260 10 20 30 40 50 8.39 818 8.39 902 8.39 986 8.40 070 8.40 153 8.40 237 8.39 832 8.39 916 8.40 000 8.40 083 8.40 167 8.40 251 9.99 986 9.99 986 9.99 986 9.99 986 9.99 986 9.99 986 034 50 40 30 20 10 360 10 20 30 40 50 8.44 594 8.44 669 8.44 745 8.44 820 8.44 895 8.44 969 8.44 611 8.44 686 8.44 762 8.44 837 8.44 912 8.44 987 9.99 983 9.99 983 9.99 983 9.99 983 9.99 983 9.99 983 024 50 40 30 20 10 270 10 20 30 40 50 8.40 320 8.40 403 8.40 486 8.40 569 8.40 651 8.40 734 8.40 334 8.40 417 8.40 500 8.40 583 8.40 665 8.40 748 9.99 986 9.99 986 9.99 986 9.99 986 9.99 986 9.99 986 033 50 40 30 20 10 370 10 20 30 40 50 8.45 044 8.45 119 8.45 193 8.45 267 8.45 341 8.45 415 8.45 061 8.45 136 8.45 210 8.45 285 8.45 359 8.45 433 9.99 983 9.99 983 9.99 983 9-99 983 9.99 982 9.99 982 023 50 40 30 20 10 280 10 20 30 40 50 8.40 816 8.40 898 8.40 980 8.41 062 8.41 144 8.41 225 8.40 830 8.40 913 8.40 995 8.41 077 8.41 158 8.41 240 9.99 986 '9.99 986 9.99 986 9.99 986 9.99 986 9.99 986 032 50 40 30 20 10 380 10 20 30 40 50 8.45 489 8.45 563 8.45 637 8.45 710 8.45 784 8.45 857 8.45 507 8.45 581 8.45 655 8.45 728 8.45 802 8.45 875 9.99 982 9.99 982 9.99 982 9.99 982 9.99 982 9.99 982 022 50 40 30 20 10 290 10 20 30 40 50 8.41 307 8.41 388 8.41 469 8.41 550 8.41 631 8.41 711 8.41 321 8.41 403 8.41 484 8.41 565 8.41 646 8.41 726 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 9.99 985 031 50 40 30 20 10 390 10 20 30 40 50 8.45 930 8.46 003 8.46 076 8.46 149 8.46 222 8.46 294 8.45 948 8.46 021 8.46 094 8.46 167 8.46 240 8.46 312 9.99 982 9.99 982 9.99 982 9.99 982 9.99 982 9.99 982 021 50 40 30 20 10 300 8.41 792 s.-n so; 9.99 985 030 400 8.46 366 8.46 385 9.99 982 020 r n log cos log cot log sin // / / // log cos log cot log sin // / 88 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 1 / // log sin log tan log cos // / 1 II log sin log tan log cos // / 400 10 20 30 40 50 8.46 366 8.46 439 8.46 511 8.46 583 8.46 655 8.46 727 8.46 385 8.46 457 8.46 529 8.46 602 8.46 674 8.46 745 9.99 982 9.99 982 9.99 982 9.99 981 9.99 981 9.99 981 020 50 40 30 20 10 500 10 20 30 40 50 8.50 504 8.50 570 8.50 636 8.50 701 8.50 767 8.50 832 8.50 527 8.50 593 8.50 658 8.50 724 8.50 789 8.50 855 9.99 978 9.99 978 9.99 978 9.99 978 9.99 977 9.99 977 010 50 40 30 20 10 410 10 20 30 40 50 8.46 799 8.46 870 8.46 942 8.47 013 8.47 084 8.47 155 8.46 817 8.46 889 8.46 960 8.47 032 8.47 103 8.47 174 9.99 981 9.99 981 9.99 981 9.99 981 9.99 981 9.99 981 019 50 40 30 20 10 510 10 20 30 40 50 8.50 897 8.50 963 8.51 028 8.51 092 8.51 157 8.51 222 8.50 920 8.50 985 8.51 050 8.51 115 8.51 180 8.51 245 9.99 977 9.99 977 9.99 977 9.99 977 9.99 977 9.99 977 9 50 40 30 20 10 420 10 20 30 40 50 8.47 226 8.47 297 8.47 368 8.47 439 8.47 509 8.47 580 8.47 245 8.47 316 8.47 387 8.47 458 8.47 528 8.47 599 9.99 981 9.99 981 9.99 981 9.99 981 9.99 981 9.99 981 018 50 40 30 20 10 520 10 20 30 40 50 8.51 287 8.51 351 8.51 416 8.51 480 8.51 544 8.51 609 8.51 310 8.51 374 8.51 439 8.51 503 8.51 568 8.51 632 9.99 977 9.99 977 9.99 977 9.99 977 9.99 977 9.99 977 8 50 40 30 20 10 430 10 20 30 40 50 8.47 650 8.47 720 8.47 790 8.47 860 8.47 930 8.48 000 8.47 669 8.47 740 8.47 810 8.47 880 8.47 950 8.48 020 9.99 981 9.99 980 9.99 980 9.99 980 9.99 980 9.99 980 017 50 40 30 20 10 530 10 20 30 40 50 8.51 673 8.51 737 8.51 801 8.51 864 8.51 928 8.51 992 8.51 696 8.51 760 8.51 824 8.51 888 8.51 952 8.52 015 9.99 977 9.99 976 9.99 976 9.99 976 9.99 976 9.99 976 7 50 40 30 20 10 440 10 20 30 40 50 8.48 069 8.48 139 8.48 208 8.48 278 8.48 347 8.48 416 8.48 090 8.48 159 8.48 228 8.48 298 8.48 367 8.48 436 9.99 980 9.99 980 9.99 980 9.99 980 9.99 980 9.99 980 016 50 40 30 20 10 540 10 20 30 40 50 8.52 055 8.52 119 8.52 182 8.52 245 8.52 308 8.52 371 8.52 079 8.52 143 8.52 206 8.52 269 8.52 332 8.52 396 9.99 976 9.99 976 9.99 976 9.99 976 9.99 976 9.99 976 6 50 40 30 20 10 450 10 20 30 40 50 8.48 485 8.48 554 8.48 622 8.48 691 8.48 760 8.48 828 8.48 505 8.48 574 8.48 643 8.48 711 8.48 780 8.48 849 9.99 980 9.99 980 9.99 980 9.99 980 9.99 979 9.99 979 015 50 40 30 20 10 550 10 20 30 40 50 8.52 434 8.52 497 8.52 560 8.52 623 8.52 685 8.52 748 8.52 459 8.52 522 8.52 584 8.52 647 8.52 710 8.52 772 9.99 976 9.99 976 9.99 976 9.99 975 9.99 975 9.99 975 5 50 40 30 20 10 460 10 20 30 40 50 8.48 896 8.48 965 8.49 033 8.49 101 8.49 169 8.49 236 8.48 917 8.48 985 8.49 053 8.49 121 8.49 189 8.49 257 9.99 979 9.99 979 9.99 979 9.99 979 9.99 979 9.99 979 014 50 40 30 20 10 560 10 20 30 40 50 8.52 810 8.52 872 8.52 935 8.52 997 8.53 059 8.53 121 8.52 835 8.52 897 8.52 960 8.53 022 8.53 084 8.53 146 9.99 975 9.99 975 9.99 975 9.99 975 9.99 975 9.99 975 4 50 40 30 20 10 470 10 20 30 40 50 8.49 304 8.49 372 8.49 439 8.49 506 8.49 574 8.49 641 8.49 325 8.49 393 8.49 460 8.49 528 8.49 595 8.49 662 9.99 979 9.99 979 9.99 979 9.99 979 9.99 979 9.99 979 013 50 40 30 20 10 570 10 20 30 40 50 8.53 183 8.53 245 8.53 306 8.53 368 8.53 429 8.53 491 8.53 208 8.53 270 8.53 332 8.53 393 8.53 455 8.53 516 9.99 975 9.99 975 9.99 975 9.99 975 9.99 975 9.99 974 3 50 40 30 20 10 480 10 20 30 40 50 8.49 708 8.49 775 8.49 842 8.49 908 8.49 975 8.50 042 8.49 729 8.49 796 8.49 863 8.49 930 8.49 997 8.50 063 9.99 979 9.99 979 9.99 978 9.99 978 9.99 978 9.99 978 012 50 40 30 20 10 580 10 20 30 40 50 8.53 552 8.53 614 8.53 675 8.53 736 8.53 797 8.53 858 8.53 578 8.53 639 8.53 700 8.53 762 8.53 823 8.53 884 9.99 974 9.99 974 9.99 974 9.99 974 9.99 974 9.99 974 2 50 40 30 20 10 490 10 20 30 40 50 8.50 108 8.50 174 8.50 241 8.50 307 8.50 373 8.50 439 8.50 130 8.50 196 8.50 263 8.50 329 8.50 395 8.50 461 9.99 978 9.99 978 9.99 978 9.99 978 9.99 978 9.99 978 Oil 50 40 30 20 10 590 10 20 30 40 50 8.53 919 8.53 979 8.54 040 8.54 101 8.54 161 8.54 222 8.53 945 8.54 005 8.54 066 8.54 127 8.54 187 8.54 248 9.99 974 9.99 974 9.99 974 9.99 974 9.99 974 9.99 974 1 50 40 30 20 10 500 8.50 504 8.50 527 9.99 978 010 600 8.54 282 8.54 308 9.99 974 / // log cos log cot log sin // / / n log cos log cot log sin // / 88 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS O *9Q 180 *2yo / log siu d. log tan | c. d. log cot log cos Prop. Pts. 1 2 3 4 6.46 373 6.76476 6.94 085 7.06 579 30103 17609 12494 9691 7918 6694 5800 5115 4576 4139 3779 3476 3218 2997 802 633 483 348 227 119 02 1 93 848 773 704 639 579 524 472 424 379 336 297 259 223 190 158 128 IOO 672 046 022 999 976 954 934 914 896 877 860 843 827 812 797 782 769 755 743 730 6.46 373 6.76476 6.94 085 7.06 579 30103 17609 12494 9691 7918 6694 5800 5115 4576 4139 3779 3476 3219 2996 2803 2633 2482 2348 2228 2119 2O2O 1931 1848 1773 1704 1639 1579 1524 1473 1424 1379 1336 1297 1259 1223 1190 ii59 1128 IIOO 1072 1047 IO22 998 976 955 934 915 895 878 860 843 828 812 797 782 769 756 742 730 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 d 30103 17609 12494 9691 79i8 6694 5800 5ii5 4576 4139 3779 3476 3219 3218 2997 2996 2803 2802 2633 2483 2482 2348 2228 2227 2119 202 1 2O2O 1931 1930 1848 1773 1704 1639 1579 1524 1473 1472 1424 1379 1336 1297 1259 1223 1190 "59 1158 1128 IIOO 1072 1047 1046 IO22 999 998 976 955 954 934 ppi" 501.72 293.48 208.23 161.52 131-97 ni-57 96.67 85-25 76.27 68.98 62.98 57-93 53-65 53-63 49-95 49-93 46.72 46.70 43-88 41-38 41-37 39-13 37-13 37-12 35-32 33-68 33-67 32.18 32.17 30.80 29-55 28.40 27.32 26.32 25.40 24-55 24-53 23-73 22.98 22.27 21.62 20.98 20.38 19.83 19.32 19-30 18.80 18.33 17-87 17-45 17-43 17-03 16.65 16.63 16.27 15.92 15.90 IS-57 d 915 914 896 895 878 877 860 843 828 827 812 797 782 769 756 755 743 742 730 ppl" 15-25 15-23 14-93 14.92 14.63 14.62 14-33 14-05 13-80 13-78 13-53 13-28 13-03 12.82 12.60 12.58 12.38 12.37 12.17 5 6 7 S 9 7.16270 7.24 188 7.30 882 7.36 682 7.41 797 7.16270 7.24 188 7.30 882 7.36 682 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 50 49 48 47 46 10 11 12 13 14 7.46 373 7.50512 7.54 291 7.57 767 7.60 985 7.46 373 7.50512 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 15 16 17 IS 19 20 21 22 23 24 7.63 982 7.66 784 7.69 417 7.71 900 7.74248 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 0.00000 0.00 000 9.99 999 9.99 999 9.99 999 45 44 43 42 41 lo 39 38 37 36 7.76475 7.78594 7.80615 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 2.19385 2.17454 2.15 606 9.99 999 9.99 999 9.99 999 9.99 999 9.99 999 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 2.10490 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 ~25~ 24 23 22 21 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 40 41 42 43 44 8.06 578 8.07 650 8.08 696 8.09 718 8.10717 8.06 581 8.07 653 8.08 700 8.09 722 8.10720 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 15 14 13 12 11 45 46 47 48 49 8.11 693 8.12647 8.13 581 8.14495 8.15 391 8.11 696 8.12651 8.13 585 8.14 500 8.15 395 1.88 304 1.87 349 1.86415 1.85 500 1.84 605 9.99 996 9.99 996 9.99 996 9.99 996 9.99 996 50 51 52 53 54 8.16268 8.17 128 8.17971 8.18 798 8.19610 8.16273 8.17 133 8.17976 8.18 804 8.19616 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 ~y 4 3 2 1 ~0 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 1.79587 1,78805 1.78 036 1.77 280 1.76 538 9.99 994 9.99 994 9.99 994 9.99 994 9.99 994 60 8.24 186 8.24 192 1.75 808 9.99 993 log cos d. log cot c. d. log tan log sin i Prop. Pts. *i 79 269 *359 89 29 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 1 *9i 181 *27i / log sin : (1. log tan j c. (1. log cot log cos Prop. Pts. 1 2 3 4 8.24 186 8.24 903 8.25 609 8.26 304 8.26 988 717 706 695 684 673 663 653 644 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 5U 508 502 496 491 485 480 474 470 464 459 455 45 445 441 436 433 427 424 419 416 411 408 404 400 396 393 390 386 382 379 376 373 369 367 363 8.24 192 8.24 910 8.25 616 8.26 312 8.26 996 718 706 696 684 673 663 654 643 634 625 617 607 599 591 584 575 568 56i 553 546 540 533 5?7 520 514 59 502 496 491 486 480 475 470 464 460 455 450 446 441 437 432 428 424 420 416 412 408 404 401 397 393 390 386 383 380 376 373 370 367 363 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 d 718 717 706 696 695 684 673 663 654 * 53 6 44 643 634 625 024 617 616 608 607 599 S9I 590 584 58 3 575 568 56i 560 553 547 546 540 539 533 527 526 520 5M 509 508 502 496 49i 486 ppi" 1.97 1.95 1.77 i. 60 1.58 1.40 1.22 I. OS O.90 0.88 0.73 0.72 0-57 0.42 0.40 0.28 0.27 0.13 0.12 9.98 9.85 9.83 9-73 9.72 9.58 9-47 9-35 9-33 9.22 9.12 Q.IO Q.OO 8.98 8.88 8.78 8-77 8.67 8.57 8.48 8.47 8.37 8.27 8.18 8.10 d 485 480 475 474 470 #64 460 459 455 450 446 445 441 437 436 433 432 428 427 424 42O 419 416 412 411 408 404 4oi 4oo 397 3g6 393 390 386 383 382 380 379 376 373 370 369 367 363 ppl" 8.08 8.00 7.92 7.00 7.83 7-73 7.67 7.6s 7-58 7-50 7-43 7.42 7-35 7.28 7.27 7.22 7.20 7.13 7.12 7.07 7.00 6.98 6-93 6.87 6.85 6.80 6.73 6.68 6.67 6.62 6.60 6.55 6.50 6.43 6.38 6.37 6.33 6.32 6.27 6.22 6.17 6.15 6.12 6.05 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 9.99 989 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 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 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 9.99 984 30 29 28 27 26 35 36 37 38 39 8.44 139 8.44 594 8.45 044 8.45 489 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 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 55 56 57 58 59 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 log cos z log cot c.d. log tan log sin / Prop. Pts. *I 7 8 268 * 35 8 88 30 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 2 *92 l82 *272 1 log sin d. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 8.54 282 8.54 642 8.54 999 8.55 354 8.55 705 360 357 355 351 349 346 343 341 337 336 332 330 328 325 8.54 308 8.54 669 8.55 027 8.55 382 8.55 734 361 358 355 352 349 346 344 341 338 336 333 330 328 326 323 321 319 316 314 3" 3io 307 305 303 301 299 297 295 292 291 289 287 285 284 281 280 278 276 274 273 271 269 268 266 264 263 261 260 258 257 255 254 252 251 249 248 246 245 244 243 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 d 361 360 358 357 355 352 35i 349 346 344 343 341 338 337 336 333 332 33 328 326 325 323 321 320 319 3i8 316 314 313 3" 310 309 307 305 303 302 301 299 298 297 296 295 294 293 292 ppl" 6.O2 6.00 5-97 5-95 5-92 5.87 5.8s 5-82 5-77 5-73 5-72 5.68 5.63 5.62 5.6o 5-55 5-53 5-50 5.47 5-43 5-42 5-38 5-35 5-33 5-32 5-30 5-27 5-23 5-22 5.18 5-17 5-15 5-12 5-o8 5-05 5-03 5.02 4.98 4-97 4-95 4-93 4.92 4.90 4.88 4.87 d 291 290 289 288 28.7 285 284 283 281 280 279 278 277 276 274 273 272 271 270 269 268 267 266 264 263 261 260 259 258 257 256 255 254 253 252 251 250 249 248 247 246 245 244 243 242 ppl" 4-85 4-83 4.82 4.80 4.78 4-75 4-73 4-72 4.68 4.67 4.6s 4.63 4.62 4.60 4.57 4.55 4.53 4-52 4.50 4.48 4.47 4-45 4.43 4.40 4.38 4.35 4.33 4.32 4.30 4.28 4.27 4.25 4.23 4.22 4. 2O 4.18 4.17 4.15 4.13 4.12 4.10 4.08 4.07 4-05 4-03 5 6 7 8 9 8.56 054 8.56 400 8.56 743 8.57 084 8.57 421 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 50~ 49 48 47 46 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 15 16 17 18 19 8.59 395 8.59 715 8.60 033 8.60 349 8.60 662 320 3i8 316 313 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 25 26 27 28 29 8.60 973 8.61 282 8.61 589 8.61 894 8.62 196 309 37 305 302 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 8.62 497 8.62 795 8.63 091 8.63 385 8.63 678 298 296 294 293 8.62 535 8.62 834 8.63 131 8.63 426 8.63 718 1.37 465 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 30 31 32 33 34 8.63 968 8.64 256 8.64 543 8.64 827 8.65 110 288 287 284 283 t%T 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 8.66 497 279 277 276 274 8.65 435 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 8.67 841 270 269 267 266 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 8.69 144 263 260 259 258 256 254 253 252 250 249 247 246 244 243 242 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 8.70 905 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 log cos d. log cot c.d. log tan log sin / Prop. Pts. *i77 267 357 87 31 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 3 *93 183 *2 73 5 / log sin (1. log tan c.d. log cot log cos Prop. Pts. 1 2 3 4 S.71 880 8.72 120 8.72 359 8.72 597 8.72 834 240 239 238 237 235 234 232 232 230 229 238 226 226 224 8.71 9-10 8.72 181 8.72 420 8.72 659 8.72 896 241 239 239 237 236 234 234 232 231 220 229 227 226 225 224 222 222 22O 219 219 217 216 215 214 213 211 211 2IO 2O9 208 2O6 206 205 2O4 203 202 2OI 201 199 198 198 196 196 195 194 193 192 192 190 190 189 1 88 1 88 186 186 IBs 184 184 182 182 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 I 241 240 239 238 237 236 235 234 233 232 231 230 229 228 227 226 225 224 223 222 221 220 2ig 2x8 217 216 215 214 213 212 211 2IO ppi" 4.02 4.00 3.98 3.97 3.95 3-93 3-92 3-9 3-88 3-87 3.8 S 3.83 3.82 3.80 3-78 3-77 3-75 3-73 3-72 3-7 3.68 3.6? 3.65 3.63 3.62 3.6o 3.58 3-57 3-55 3-53 3-52 3-5 d 209 208 207 206 205 204 203 202 20 1 200 199 198 197 196 195 194 193 192 191 190 189 188 187 186 185 184 183 182 181 ppl" 3.48 3-47 3-45 3-43 3.42 3-4 3.38 3-37 3-35 3-33 3.32 3-30 3.28 3-27 3.2S 3.23 3.22 3.20 3.18 3-17 3.15 3.13 3-12 3.io 3-08 3-07 3-OS 3-3 3.02 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 IS 19 8.75 353 8.75 575 8.75 795 8.76 015 8.76 234 222 220 22O 219 217 216 216 214 213 212 211 2IO 2OQ 208 208 2O6 205 204 203 2O2 2OI 2OI IQQ 199 197 197 196 195 194 193 192 192 190 190 189 1 88 187 187 186 185 184 183 183 181 181 8.75 423 8.75 645 8.75 867 8.76 087 8.76 306 1.24 577 1.24 355 1.24 133 1.23 913 1.23 .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 S.77 310 8.76 525 8.76 742 8.76 958 8.77 173 8.77 387 1.23 475 1.23 258 1.23 042 1.22 827 1.22 613 9.99 926 9.99 926 9.99 925 9.99 924 9.99 923 40 39 38 37 36 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 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 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 1.17 013 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 ..84 358 S.84 464 1.15 536 9.99 894 log cos (1. log cot c.d. log tan log sin / Prop. Pts. *I 7 6 266 * 35 6 86 32 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 4 t '94 184 *274 ' log sin d. log tan c. d. log cot log cos Prop . Pts. 1 2 3 4 8.84 358 8.84 539 8.84 718 8.84 897 8.85 075 181 179 179 178 8.84 464 8.84 646 8.84 826 8.85 006 8.85 185 182 180 180 179 178 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 8.85 955 177 176 175 175 8.85 363 8.85 540 8.85 717 8.85 893 8.86 069 177 177 176 176 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 8.86 816 173 173 171 171 8.86 243 8.86 417 8.86 591 8.86 763 8.86 935 174 1/4 172 172 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 IS 19 8.86 987 8.87 156 8.87 325 8.87 494 8.87 661 169 169 169 167 1 68 8.87 106 8.87 277 8.87 447 8.87 616 8.87 785 171 170 169 169 1 68 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 d 182 181 180 179 PP1" 3-03 3-02 3.0 2.98 20 21 22 23 24 8.87 829 8.87 995 8.88 161 8.88 326 8.88 490 166 166 163 164 8. 87 953 8.88 120 8.88 287 8.88 453 8.88 618 167 167 1 66 165 rfic 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 36 178 177 176 175 174 173 172 2.97 2.95 2.93 2.92 2.90 2.88 2.87 25 26 27 28 29 8.88 654 8.88 817 8.88 980 8.89 142 8.89 304 163 163 162 162 8.88 783 8.88 948 8.89 111 8.89 274 8.89 437 165 I6 3 I6 3 I6 3 161 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 31 171 170 169 168 167 166 165 2.85 2.83 2.82 2.80 2.78 2.77 2.7S 30 31 32 33 34 8.89 464 8.89 625 8.89 784 8.89 943 8.90 102 161 159 159 159 118 8.89 598 8.89 760 8.89 920 8.90 080 8.90 240 162 1 60 1 60 1 60 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 26 164 163 162 161 160 159 158 2.73 2.72 2.70 2.68 2.67 2.65 2.63 35 36 37 3S 39 8.90 260 8.90 417 8.90 574 8.90 730 8.90 885 157 157 156 155 8.90 399 8.90 557 8.90 715 8.90 872 8.91 029 158 158 157 157 Tcfi 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 IS7 156 155 154 153 152 151 2.62 2.60 2.58 2.57 2-SS 2-53 2.52 40 41 42 43 44 8.91 040 8.91 195 8.91 349 8.91 502 8.91 655 155 154 153 153 8.91 185 8.91 340 8.91 495 8.91 650 8.91 803 155 155 155 153 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 15 149 148 I4 I 146 145 2.50 2.48 2.47 2-4S 2.43 2.42 45 46 47 48 49 8.91 807 8.91 959 8.92 110 8.92 261 8.92 411 152 151 151 15 8.91 957 8.92 110 8.92 262 8.92 414 8.92 565 153 152 152 151 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 8.93 154 149 149 148 147 8.92 716 8.92 866 8.93 016 8.93 165 8.93 313 150 15 149 148 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 SS5_ 147 I4 6 146 U5 8.93 462 8.93 609 8.93 756 8.93 903 8.94 049 147 147 147 146 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 log cos d. log cot c.d. log tan log sin / Prop .Pts. *'75 265 *355 85 33 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 5 *95 185 *275 ' log sin d. log tan c. d. log cot log cos Prop . Pts. 1 2 3 4 8.94 030 8.94 174 8.94 317 8.94 461 8.94 603 144 143 144 142 143 8.94 195 8.94 340 8.94 485 8.94 630 8.94 773 145 145 us 143 144 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 8.95 310 141 142 ui 140 140 8.94 917 8.95 060 8.95 202 8.95 344 8.95 486 143 142 142 142 141 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.% 005 139 139 139 138 138 8.95 627 8.95 767 8.95 908 8.96 047 8.96 187 140 ui 139 140 138 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 IS 19 8.% 143 8.% 280 8.% 417 8.% 553 8.% 689 t37 137 136 136 136 8.% 325 8.96 464 8.96 602 8.% 739 8.96 877 139 138 137 138 136 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 20 21 22 23 24 8,96 825 8.96 960 8.97 095 8.97 229 8.97 363 135 135 134 134 133 8.97 013 8.97 150 8.97 285 8.97 421 8.97 556 137 135 136 135 135 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 d 145 144 143 142 ppi" 2 42 2 40 238 2 37 25 26 27 28 29 8.97 496 8.97 629 8.97 762 8.97 894 8.98 026 133 133 132 132 131 8.97 691 8.97 825 8.97 959 8.98 092 S.98 225 134 134 133 133 133 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 31 141 140 139 138 137 136 135 2 35 2 33 232 2 30 2 28 2 27 2 25 30 31 32 33 34 8.98 157 8.98 288 8.98 419 8.98 549 8.98 679 131 131 130 130 129 8.98 358 8.98 490 8.98 622 8.98 753 8.98 884 132 132 131 131 131 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 7% 9.99 795 30 29 28 27 26 134 133 132 131 130 129 128 2 22 2 20 218 2 17 2 IS 2 13 35 36 37 38 39 8.98 808 8.98 937 8.99 066 8.99 194 8.99 322 129 129 128 128 128 8.99 015 8.99 145 8.99 275 8.99 405 8.99 534 130 13 130 129 128 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 21 127 126 125 124 123 122 121 2 12 2 10 208 2 O7 205 203 202 40 41 42 43 44 8.99 450 8.99 577 8.99 704 8.99 830 8.99 956 127 127 126 126 126 8.99 662 8.99 791 8.99 919 9.00 046 9.00 174 129 128 127 128 127 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 1 2O 2.00 45 46 47 48 49 9.00 082 9.00 207 9.00 332 9.00 456 9.00 581 125 125 124 125 123 9.00 301 9.00 427 9.00 553 9.00 679 9.00 805 126 126 126 126 125 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 9.01 196 124 123 123 122 122 9.00 930 9.01 055 9.01 179 9.01 303 9.01 427 125 124 124 124 123 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 9.01 803 122 121 121 121 I2O 9.01 550 9.01 673 9.01 796 9.01 918 9.02 040 123 123 122 122 122 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 log cos d. log cot c.d. log tan log sin i Prop .Pts. *I74 264 *354 84 34 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS *96 1 86 *2y6 ' log sin d. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 9.01 923 9.02 043 9.02 163 9.02 283 9.02 402 1 20 1 20 120 IIQ 118 19 18 17 18 i? 117 116 116 116 116 "5 us 114 us "3 114 114 "3 112 "3 112 112 112 III III III no no no no 09 09 09 08 09 08 07 08 07 07 06 07 06 S 06 OS OS 05 05 04 04 04 03 03 03 9.02 162 9.02 283 9.02 404 9.02 525 9.02 645 121 121 121 1 2O 121 119 1 2O 119 118 119 118 118 H7 118 116 117 116 116 116 "5 "S us iiS "4 114 "3 114 113 112 "3 112 112 112 III III III no III IIO 109 IIO 09 09 08 09 08 08 07 08 07 06 07 06 06 06 06 05 OS 05 04 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 // 6 7 8 9 10 20 30 40 SO // 6 7 8 9 10 20 30 40 So // 6 121 12. 1 14.1 16.1 18.2 20. 2 40.3 60. S 80.7 100.8 118 11.8 13.8 iS-7 17.7 19-7 39-3 59-0 78.7 98.3 115 "5 13-4 iS-3 17.2 19.2 38.3 57-5 76.7 95-8 112 II. 2 I3.I 14-9 16.8 18.7 37-3 56.0 74-7 93-3 109 10.9 12.7 14.5 16.4 18.2 36.3 S4-S 72.7 90.8 106 10.6 12.4 14.1 15-9 17.7 3S-3 S3-o 70.7 88.3 120 I2.O I4.O 16.0 18.0 2O.O 4O.O 6O.O 80.0 IOO.O 117 ii. 7 13-6 IS.6 17.6 I9-S 39-0 58.5 78.0 97-5 114 11.4 13-3 IS-2 17.1 19.0 38.0 S7-o 76.0 95-0 111 11. i 13-0 14.8 16.6 18.5 37-0 55-5 74-0 92.5 108 10.8 12.6 14.4 16.2 18.0 36.0 S4-o 72.0 90.0 105 10.5 12.2 14.0 IS-* I7-S 3S- 52-5 70.0 87-S 119 11.9 13-9 IS-9 17.8 19.8 39-7 59- S 79-3 99.2 116 11. 6 13-5 IS-S 17.4 19-3 38.7 58.0 77-3 96.7 113 ii. 3 13-2 iS-i 17.0 18.8 37-7 56.5 75-3 94.2 110 n.o 12.8 14.7 16.5 18.3 36.7 55-0 73-3 91-7 107 10.7 12.5 14-3 1 6.0 17.8 3S-7 S3- 5 7L3 89.2 104 10.4 12. 1 13-9 15.6 17-3 34-7 52.0 69-3 86.7 5 6 7 8 9 9.02 520 9.02 639 9.02 757 9.02 874 9.02 992 9.02 766 9.02 885 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 9.01- 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 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 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 // 6 7 8 9 10 20 30 40 50 6 8 9 10 20 30 40 SO 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 9.99 693 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 log cos d. log cot c.d. log tan log sin i Prop. Pts. *i 73 263 *353 83 35 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 7 *97 187 *277 / log sin (1. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 9.08 589 9.08 692 9.08 795 9.08 897 9.08 999 03 03 02 02 02 OI 02 OI OI 00 OI 00 00 99 IOO 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 93 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 92 92 91 92 91 91 90 91 90 91 90 9.08 914 9.09 019 9.09 123 9.09 227 9.09 330 05 04 04 03 04 03 03 02 03 02 02 OI 02 OI OI OI OI oo oo oo IOO 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 91 92 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 6 8 9 10 20 3 40 50 6 7 8 9 IO 20 30 40 50 6 8 9 IO 20 30 40 SO // 6 8 9 IO 20 3 40 50 6 7 8 9 IO 20 30 40 So 6 7 8 9 10 20 30 40 50 105 10.5 12.3 14.0 15.8 17.5 3S-0 52.5 70.0 87.5 102 IO.2 1 1.9 13.6 15.3 17.0 34-0 Si.o 68.0 85.0 99 9.9 n.6 13-2 14.8 16.5 33-0 49-5 66.0 82.5 96 9.6 II. 2 12.8 14.4 16.0 32.0 48.0 64.0 80.0 93 9-3 10.9 12.4 14.0 15-5 31-0 46.5 62.0 77-5 90 9-0 10. 5 12.0 13-5 15-0 30.0 45-0 60.0 75-0 104 10.4 12. 1 13-9 IS.6 17-3 34-7 52.0 69.3 86.7 101 IO.I n.8 13-5 15-2 16.8 33-7 50.5 67.3 84.2 98 9.8 11.4 I3-I 14.7 16.3 32-7 49.0 65-3 81.7 95 9-5 ii. i 12.7 14.2 15-8 3L7 47-5 63.3 79-2 92 9-2 10.7 12.3 13-8 15-3 30-7 46.0 61.3 76.7 2 O.2 O.2 0-3 0.3 0.3 0.7 I.O 1-3 1.7 103 10.3 12.0 13.7 15-4 17.2 34-3 51-5 68.7 83.8 100 IO.O ii. 7 13-3 15-0 16.7 33-3 50.0 66.7 83-3 97 :?:3 7 12.9 14.6 16.2 32.3 48.S 64.7 80.8 94 9.4 II.O I2.S I4.I 15-7 31.3 47-0 62.7 78.3 91 9.1 10.6 12. 1 13.6 15.2 30.3 45-5 60.7 75-8 1 O.I O.I O.I 0.2 O.2 0.3 o.S 0.7 0.8 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 6Uo 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 0.89 850 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 45 "44 43 42 41 15 16 17 18 19 9.10 lOo 9.10 205 9.10 304 9.10 402 9.10 501 9.10 454 9.10 555 9.10 656 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 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 36 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 0.87 765 087 668 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 0.87 187 999 618 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 0.86 711 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 9S5 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 9.13 901 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 log cos d. log cot c.d. log tan log sin / Prop. Pts. *I 7 2 262 * 35 2 82 36 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 8 *98 1 88 *2y8 / log sin d. log tan c.d. log cot log cos Prop. Pts. 1 2 3 4 9.14 356 9.14 445 9.14 535 9.14 624 9.14 714 8 9 90 89 90 8 9 88 89 89 88 88 88 88 87 88 87 87 87 87 86 86 87 86 8S 86 85 86 85 8S 85 84 85 84 84 84 84 83 84 83 83 83 83 83 8 8 83 8 8 8 8 8 8 8 8 8 8 80 80 80 80 80 9.14 780 9.14 872 9.14 963 9.15 054 9.15 145 92 91 91 91 91 91 90 91 90 90 89 90 89 90 89 89 88 89 88 88 88 88 88 87 88 87 8? 87 86 87 86 86 86 86 86 85 86 85 85 8S 85 84 85 84 84 84 84 83 84 83 83 83 83 83 83 82 82 82 82 82 0.85 220 0.85 128 0.85 037 0.84 946 0.84 855 9.99 575 9.99 574 9.99 572 9.99 570 9.99 568 60 59 58 57 56 6 1 i IO 20 30 40 5 6 7 8 9 10 20 3 40 SO 6 8 9 10 20 30 40 5 6 8 9 IO 20 30 40 SO 6 7 8 9 IO 20 30 40 So 92 9.2 10.7 12.3 13-8 -iS-3 '30.7 46.0 61.3 76.7 89 8.9 10.4 11.9 13.4 14.8 29.7 44-5 59-3 74-2 86 8.6 IO.O 11.5 12.9 14-3 28.7 43.0 S7-3 71-7 83 8.3 9-7 ii. I 12.5 13.8 27.7 41-5 55.3 69.2 80 8.0 9-3 10.7 I2.O 13-3 26.7 4O.O 53-3 66.7 91 9.1 10.6 12. 1 13.6 JI5.2 30-3 45-5 60.7 75-8 88 8.8 10.3 ii. 7 13.2 14.7 29-3 44-0 58.7 73-3 85 8.5 9.9 II-3 12.8 14.2 28.3 42-S 56.7 70.8 82 8.2 9.6 10.9 12.3 13-7 27-3 41.0 54- 7 68.3 2 O.2 O.2 0.3 0.3 0.3 0.7 I.O 1-3 1-7 90 9.0 10. 5 I2.O 13.5 15-0 I30.0 ks-o 1 60.0 75.0 87 8.7 IO.2 11.6 13.1 14.5 29.0 43-5 58.0 72-S 84 8.4 9.8 II. 2 12.6 14.0 28.0 42.O S6.O 7O.O 81 8.1 9.5 lo.g 12.2 I3-S 27.0 40-S 54-0 67-S 1 O.I O.I O.I O.2 O.2 0.3 0.5 0.7 0.8 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 9.16 046 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 0.83 776 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 25 26 27 28 29 30 31 32 33 34 9.16 545 9.16 631 9.16 716 9.16 801 9.16 886 9.17 016 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 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 O.S2 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 0.82 035 081 949 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 9.99 474 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 - : ' r_ 9.19 971 0.80 029 9.99 462 log cos | d. log cot c. d. log tan log sin / Prop. Pts. *i7i 261 *35' 81 37 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 9 *99 189 *279 / log sin d. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 9.19 433 9.19 513 9.19 592 9.19 672 9.19 751 80 79 80 79 79 79 79 79 78 78 79 78 78 77 78 78 77 77 77 77 77 77 76 77 76 76 76 76 75 76 75 76 75 75 75 74 75 75 74 74 74 74 74 74 73 74 73 73 73 73 73 73 72 73 72 72 73 71 72 72 9.19 971 9.20 053 9.20 134 9.20 216 9.20 297 82 81 82 81 81 81 81 81 80 81 80 80 80 80 80 79 80 79 79 79 79 79 78 79 78 78 78 78 78 78 77 78 77 77 77 77 77 76 77 76 76 77 76 76 75 76 75 76 75 75 75 75 75 74 75 74 75 74 74 74 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 6 8 9 10 20 30 40 50 n 6 7 8 9 lo 20 30 40 So // 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 80 8.0 9-3 10.7 12.0 13-3 26.7 40.0 53-3 66.7 77 7-7 9-0 10.3 n.6 12.8 25.7 38.5 51.3 64.2 74 7-4 8.6 9.9 n. i 12.3 24-7 37-0 49-3 61.7 71 7-1 8-3 9-5 10.7 n.8 23-7 35-5 47-3 59-2 79 7-9 9 2 10.5 II.Q 13.2 26.3 39-5 52-7 65.8 76 7.6 8.9 IO.I 11.4 12.7 25.3 38.0 50.7 63-3 78 7-3 8.5 9-7 II.O 12.2 24-3 36.5 48.7 60.8 3 0.3 0.4 0.4 0.5 0.5 1.0 l-S 2.O 2-5 78 7.8 9.1 10.4 11.7 13.0 26.0 39-0 52.0 65.0 75 7-5 8.8 IO.O "3 12.5 25.0 37-5 50.0 62.5 72 7.2 8.4 9.6 10.8 12.0 24.0 36.0 48.0 60.0 2 0.2 O.2 0.3 0.3 0.3 0.7 I.O 1-3 1-7 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 0.78 898 9.99 442 9.99 440 9.99 438 9.99 436 9.99 434 50 49 48 47 46 15 16 17 IS 19 20 21 22 23 24 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 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 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 35 36 37 38 39 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 9.99 398 9.99 396 9.99 394 9.99 392 30 29 28 27 26 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 4-14 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 To~ 9 8 7 6 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 55 56 57 58 59 60 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 9.23 967 9.24 632 0.75 368 9.99 335 log cos d. log cot c.d. log tan log sin / Prop. Pts. *i 7 o 260' * 35 o 8O LOGARITHMS OF THE TRIGOXOMETKIC FUNCTIONS 1O *IOO D 190 *28o ' log sin d. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 9.23 967 9.24 039 9.24 110 9.24 181 9.24 253 72 ; 71 71 i 72 71 71 71 70 71 7O !" 7I 70 ! 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 6? 67 67 67 67 67 67 66 67 66 67 66 66 66 66 6s 66 66 65 63 66 65 65 9.24 632 9.24 706 9.24 779 9.24 853 9.24 926 74 73 74 73 74 73 73 73 73 73 72 73 72 73 72 72 72 72 72 71 72 71 72 71 71 71 71 70 71 71 7O 70 71 70 ?o TO 70 69 70 69 70 69 69 69 69 69 69 69 63 69 68 69 68 68 68 68 6? 68 68 67 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 6 7 8 9 10 20 30 40 50 6 8 9 10 20 30 40 50 6 8 9 10 20 30 40 50 6 8 9 10 20 30 50 74 7-4 8.6 9-9 II. I 12.3 --4.7 37-0 49-3 61.7 71 7-1 8-3 9-5 10.7 11.8 23-7 35-5 47-3 59-2 68 6.8 7-9 10.2 "3 22.7 34-o 45-3 56.7 6S 6-5 7-6 8.7 9.8 10.8 21.7 32-5 43-3 54-2 73 7-3 8.5 9-7 H.O 12.2 24-3 36.5 48-7 6o.S 70 7-o 8.2 9-3 10.5 11.7 23-3 35-0 67 6.7 7-8 8.9 10.0 II.2 22.3 33-5 44-7 55-8 S 0.3 0.4 0.4 o.i I.O 1-5 2.O a-5 72 7-2 8.4 9-6 10.8 12.0 24.O 36.0 48.0 6O.O 69 6.9 8.0 9-2 10.4 11.5 23.0 34-5 46.0 57-S 66 6.6 7.7 8.8 9-9 II.O 22.0 33-0 44-0 55-0 2 0.2 O.2 0-3 0.3 0.3 0.7 1.0 1-3 1-7 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 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 0.74 563 0.74 490 0.74 418 0.74 345 9.99 313 9.99 310 9.99 308 9.99 306 9.99 304 50 49 48 47 46 15 16 17 IS 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 85S 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 30 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 3S 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 4% 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 IS 17 16 45 46 47 4S 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 9.27 405 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 S 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 log cos d. log cot c. d. log tan log sin ' Prop. Pts. *i69 259 * 3 49 79 39 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 11 *ioi 191 *z8i / log sin (1. log tan C. (1. log cot log cos Prop. Pts. 1 2 3 4 9.28 060 9.28 125 9.28 190 9.28 254 9.28 319 6s 65 64 6S 65 64 64 65 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 62 63 62 63 62 62 63 62 62 6! 62 62 61 62 61 61 61 61 6r 61 60 61 60 61 60 61 60 60 60 59 60 60 59 60 9.28 865 9.28 933 9.29 000 9.29 067 9.29 134 68 67 67 67 67 67 67 6? 66 67 66 67 66 66 66 66 66 66 66 6s 66 65 6s 66 65 6s 6S 6S 65 64 6S 64 65 64 64 6S 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 63 62 63 62 62 62 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 n 6 8 9 10 20 30 40 5 // 6 8 9 10 20 30 40 SO 6 8 9 10 20 30 40 SO 6 8 9 10 20 30 40 50 68 6.8 7-9 9-1 10.2 .3 22.7 34-0 45-3 S6.7 65 6.S 7.6 8.7 9.8 10.8 21.7 32.5 43-3 54-2 62 6.2 7.2 8.3 9-3 10.3 20.7 31-0 41-3 Si-7 59 5-9 6.9 I 9 8.9 9.8 19.7 29-5 39-3 49-2 67 6.7 7.8 8.9 IO.O II. 2 22.3 33-5 44-7 55.8 64 6.4 7-i 8-5 9.6 10.7 21.3 32.0 42.7. S3-3 61 6.1 7-1 8.1 9-2 10.2 20.3 30.5 40.7 50.8 3 0.3 0.4 0.4 o.S 0.5 I.O 1-5 2.O 2-S 66 6.6 7-7 8.8 9.9 II.O 22.O 33-0 44-0 SS.o 63 6.3 7-4 8.4 9-4 10. 5 21.0 31-5 42.O 52.5 60 6.0 7.0 8.0 9.0 IO.O 2O.O 30.0 40.0 SO.O 2 0.2 O.2 0.3 0.3 0.3 0.7 I.O 1-3 i.7 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 IS 19 9.29 024 9.29 087 9.29 150 9.29 214 9.29 277 9.29 860 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 40 39 38 37 36 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 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 0.69 025 068 960 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 8S/ 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 78S 9.32 747 0.67 253 9.99 040 log cos d. log cot c.d. log tan log sin / Prop. Pts. *i68 258 *348 78 40 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 12 *I02 I92 *282 / log sin ' d. log tan c. d. log cot log cos Prop. Pts. 1 2 3 4 5 6 7 8 9 9.31 788 9.31 847 9.31 907 9.31 966 9.32 025 59 60 59 59 59 59 59 59 58 59 59 58 58 59 58 58 58 58 58 58 58 57 58 57 58 57 57 58 57 57 57 56 57 57 57 56 57 56 56 57 56 56 56 56 56 56 55 56 55 56 55 56 55 55 55 55 55 55 55 55 9.32 747 9.32 810 9.32 872 9.32 933 9.32 995 63 62 61 62 62 62 61 62 61 62 61 61 61 61 61 61 61 61 60 61 60 61 61 60 60 60 60 60 59 60 60 59 60 59 59 59 60 59 59 59 59 58 59 59 58 59 58 59 58 58 58 58 58 58 58 58 58 57 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 a 6 8 9 10 20 30 40 50 // 6 8 9 10 20 30 40 50 6 8 9 10 20 30 40 50 63 6.3 7-4 -8.4 9.4 io 21.0 31-5 42.O 52-5 60 6.0 7-o 8.0 9.0 IO.O 2O.O 30.0 4O.O 50.0 57 5-7 6.6 7.6 8.6 9-S 19.0 28.5 38.0 47-51 " I 60 TO 80 90 10 o 20 I 30 I 40 2 502 62 I 61 6.2 6.1 7.2 7.1 8.3! 8.1 9-3| 9-2 IO.3 'IO. 2 20. 7 '20.3 3I.OJ30.5 4I.3;40.7 5i-7iso.8 59 58 5-9 5-8 6.9 6.8 7-9 7-7 8.8 8.7 9-8 9-7 19.7 19.3 29.5 29.0 39-3'38.7 49-2 48.3 56 55 5-6 5-5 6.5 6.4 7-5 7-3 8.4 8.3 9-3 9-2 18.7 18.3 28.0 27.5 37-3 '36.7 46-7 45-8 ; 2 3o.a 40.2 40.3 50.3 50.3 00.7 5 1.0 01.3 5 i-7 9.32 084 9.32 143 9.32 202 9.32 261 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 IS 19 9.32 670 9.32 728 9.32 786 9.32 844 9.32 902 9.33 6/0 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 35 34 33 32 31 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 0.65 661 0.65 604 0.65 544 0.65 484 9.98 972 9.98 969 9.98 967 9.98 964 9.98 961 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 9*7 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 T5~ 14 13 12 11 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 50 51 52 53 54 ^sT 56 57 58 59 w 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 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 9.35 209 9.36 336 0.63 664 9.98 872 log cos (I. log cot c. d. log tan log sin t Prop. Pts. *i6 7 257 * 3 47 77 41 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 13 *'3 193 *2S 3 / log sin d. log tan c.d. log cot log cos Prop. Pts. 1 2 3 4 9.35 209 9.35 263 9.35 318 9.35 373 9.35 427 54 SS 55 54 54 55 54 54 54 54 54 54 54 54 54 53 54 53 54 S3 53 53 54 53 53 53 52 53 S3 53 9.36 336 9.36 394 9.36 452 9.36 509 9.36 566 58 58 57 57 S8 57 57 57 57 57 57 57 57 57 56 57 56 57 56 57 56 56 56 56 56 56 56 56 56 55 56 56 55 55 56 SS SS 56 SS SS SS 55 SS 54 55 55 >54 55 55 54 54 SS 54 54 54 54 54 54 54 54 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 // 6 7 8 9 10 20 30 40 50 // 6 8 9 10 20 30 40 5 // 6 7 8 9 10 20 30 4 5 58 5.8 6.8 7-7 8.7 9-7 19.3 29.0 38.7 48.3 55 5-5 6.4 7-3 8.3 9.2 18.3 27-S 36.7 45-8 62 5-2 6.1 6.9 7-8 8.7 17-3 26.0 34-7 43-3 " 6 o 7 o 8 o 9 o IO O 20 I 30 i 4-O 2 JO 2 57 5-7 6.6 7-6 8.6 9-5 19.0 28.5 38.0 47-5 54 5-4 6.3 7-2 8.1 9.0 18.0 27.0 36.0 45-0 51 S.i 6.0 6.8 7-7 8.5 17.0 25-S 34-0 42.S J ' 3 o 4 o 4 o 5. o S o S i O I S I 56 5-6 6.5 7-5 8.4 9-3 18.7 28.0 37-3 46.7 53 5-3 6.2 7-1 8.0 8.8 17.7 26.5 35-3 44*2 4 0.4 0.5 o-S 0.6 0.7 1-3 2.0 2.7 3.3 1 2 2 3 3 3 7 o 3 7 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 0.63 203 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 9.37 419 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 ^5" 26 27 28 29 30 31 32 33 34 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 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 9.36 819 9.36 871 9.36 924 9.36 976 9.37 028 52 S3 52- 52 S3 52 52 52 52 52 52 52 52 52 51 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 40 41 42 43 44 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 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 50 51 52 53 54 9.37 600 9.37 652 9.37 703 9.37 755 9.37 806 52 Si 52 Si 52 Si 5i Si Si Si Si Si 51 51 5i 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 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 log cos d. log cot c.d. log tan log sin / Prop. Pts. *i66 256 *346 76 42 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 1 1 o J.'i *IO4 194 *2O4 ' log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.38 368 9.38 418 9.38 469 9.38 519 9.38 570 SO Si 50 Si 50 50 Si SO SO 50 50 So SO SO So 49 SO So 49 50 49 49 SO 49 49 49 49 49 49 49 49 49 48 49 48 49 48 49 48 49 48 48 48 48 48 48 48 48 48 47 48 48 47 48 47 48 47 47 47 48 9.39 677 9.39 731 9.39 785 9.39 838 9.39 892 54 54 S3 54 S3 54 53 54 53 53 54 53 S3 53 53 53 53 52 53 S3 S3 52 53 52 S3 52 52 52 S3 52 52 52 52 52 52 52 51 52 52 51 52 51 S2 51 51 52 51 51 Si 51 51 51 51 51 Si 50 Si Si 5 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 3 3 3 3 3 4 3 3 3 3 3 4 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 4 60 59 58 57 56 " 5 6 S 7 6 8 7 9 8 lo 5 20 I? 30 27 40 36 S4S " 5 6 5 7 6 8 6 9 7 10 8 20 17 30 25 40 34 5042 // 6 7 8 9 10 20, 30 4 so! n 6 8 9 10 20 30 40 So I 5 4 5 3 6 2 7 I 8 o 8 .0 26 o 35 O A-\ 1 5 i S o S .8 6 7 7 S 8 o 16 S 25 o 33 5141 48 4.8 5-6 6.4 7-2 8.0 16.0 24.0 32.0 40.0 4 0.4 o.S 0.5 0.6 0.7 1-3 2.O 2.7 3-3 3 52 3 S-2 .2 6.1 .1 6.9 .0 7.8 .8 8.7 7 17-3 -5 26.0 3i34.7 2 43-3 9 49 .0 4.9 .8 5-7 7 6.5 5 7-4 .3 8.2 7 16.3 .0 24.5 3 32.7 .7 40.8 47 4-7 H 6.3 7-0 7-8 iS-7 23-S 39-2 3 0.3 0.4 0.4 o.S o-s I.O 2.O 2-5 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 60 49 48 47 46 15 16 17 IS 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 20 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 "35" 34 33 32 31 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 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 0.58 422 058 371 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 0.57 550 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 log cos d. log cot c. d. log tan log sin d. ' Prop. Pts. *i6 5 255 * 345 75 43 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 15 *io 5 195 * 285 - log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.41 300 9.41 347 9.41 394 9.41 441 9.41 488 47 47 47 47 47 47 46 47 47 46 47 46 47 46 47 46 46 47 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 46 45 45 46 45 45 45 45 45 45 45 44 45 45 45 44 45 44 45 44 45 44 44 44 45 44 44 9.42 805 9.42 856 9.42 906 9.42 957 9.43 007 51 50 51 50 5 51 5 So 50 5 50 5 50 5 50 49 50 50 49 5 49 5 49 50 49 49 49 50 49 49 49 49 49 49 49 48 49 49 48 49 49 48 49 48 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 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 3 3 4 3 4 3 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 4 3 4 3 4 4 3 4 60 59 58 57 56 55 54 53 52 51 " 6 6 . 7 < 8 ( 9 ' 10 i. 20 r 30 2. 403^ 504: " 4 6 i 7 . 8 ( 9 ' 10 i 20 i( 30 it. 403: 50 4C // 6 7 8 9 10 20 30 40 50 i> 6 7 8 9 10 20 30 40 50 i M ).O .8 .7 .3 .0 >-5 \.o 5 k8 .6 >-4 .2 .0 ).0 \-o .0 ).0 4 A 6 6 7 IS 22 30 37 4 0. 0. 0. 0. o. I. 2. 2. 3- 6 1 . ( 1 I< 2. 3; 4 ] 1 I 2. 3 3C i S 3 o 8 5 S 5 i 4 5 6 7 3 7 3 .0 .8 -7 5 3 >-7 .0 3 7 7 t-7 5 >-3 .0 .8 .7 5 3 .2 4 4 S S 6 7 14 22 29 36 1 0. o I. I. 2. 2. 40 4.9 5-7 6.5 7-4 8.2 16.3 24-5 32.7 40.8 46 4.6 5-4 6.1 6.9 7-7 15-3 23.0 30.7 38.3 1 4 .1 9 .6 3 7 .0 3 .7 ! 3 4 4 S. 5 S 5 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 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 9.42 186 943 558 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 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* 29 28 27 26 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 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 9.45 319 9.45 367 9.45 415 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 log cos d. log cot c. d. log tan log sin d. ; Prop, Pts. *i6 4 254 * 344 74 4.4. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 10 *io6 196 *286 ' log sin I d. log tan e. d. log cot log cos ; d. Prop. Pts. 1 2 3 4 ~5 6 7 8 9 9.44 034 9.44 078 9.44 122 9.44 166 9.44 210 44 44 44 44 43 44 44 44 43 44 44 43 43 44 43 44 43 43 43 43 9.45 750 9.45 797 9.45 845. 9.45 892 9.45 940 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 45 46 45 45 46 45 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 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 4 4 3 60 59 58 57 56 " 48 6 4.8 7 5-6 8 6.4 9 7-2 10 8.0 20 16.0 i 30 24.0 : 40 32.0 ; 50 40.0 : " 46 6 4-5 7 5-3 8 6.0 9 6.8 io 7-5 20 15.0 i 30 22.S 2 4O 3O.O 2 50 37-5 3 " 42 6 4.: 7 4-< 8 S-< 9 6., 10 7.c 20 I4.C 30 2I.C 40 28.C 50i35.c " 4 6 0.4 7 0.5 8 0.5 g 0.6 10 0.7 20 1.3 30 2.0 40 2.7 SO 3-3 17 4.7 5-5 6.3 7.0 7.8 5-7 .3-5 1.3 9.2 44 4-4 5-1 5-8 6.6 7-3 4-7 2.0 9-3 0.7 4 4 ) 4 > s 6 > 6 > 1.3 20 27 34 1 o. 0. 0. o. 0. I. I. 2. 2. 46 4.6 5-4 6.1 6.9 7-7 15-3 23.0 30.7 38.3 43 4-3 S-o 5-7 6.4 7.2 14-3 2i.S 28.7 35-8 I i 8 5 2 8 7 5 3 2 5 1 I i 5 a ; 3 5 9.44 253 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 Is" 16 17 IS 19 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 50 49 48 47 46 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 43 44 43 42 43 43 43 43 43 42 43 42 43 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 41 42 42 42 41 42 41 42 41 42 41 42 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 25 26 27 2S 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 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 0.52-011 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 v.M 9.48 534 0.51 466 9.98 OW log cos i d. log cot c. d. log tan log sin d. ' Prop. Pts. *i6 3 253' * 343 * 73 45 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 17 *i7 197 *287 / log sin d. log tan C. (1. log cot log cos d. Prop. Pts. 1 2 3 4 ~5~ 6 7 8 9 9.46 594 9.46 635 9.46 676 9.46 717 9.46 758 41 41 41 41 42 41 41 41 41 41 40 41 41 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 9.48 534 9.48 579 9.48 624 9.48 669 9.48 714 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 4S 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 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 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 S 4 4 4 4 4 60 59 58 57 56 " 4 6 A 7 i 8 e 9 10 7 20 IS 30 22 40 30 5037 // 6 8 9 10 20 30 40 50 // 6 8 9 10 20 30 40 5 // 6c 7< 8< 9< lo < 20 ] 30 : 40; SO/ 5 44 5 4- 3 5- -o 5- .8 6. 5 7. .0 14. 5 22. .O 2Q. .536. 42 4-2 4-9 5-6 6.3 7.0 14.0 21.0 28.0 35-0 , 40 4.0 4-7 5-3 6.0 6.7 13-3 i 2O.O ] 26.7 : 33-3 C 5 4 ).S 0.4 ).6 o.j >.7 o.< >.s o.e >.8 0.7 7 i-3 5 2.C 3 2.7 t.2 3-3 43 4 4-3 i 5-0 9 5-7 6 6.4 3 7.2 7 U-3 o 21. 5 328.7 735-8 41 4.1 4.8 fr* 6.2 6.8 3-7 o.S '7-3 14.2 39 3-9 4.6 5-2 5-9 6.S 3-0 9-5 6.0 2-S 3 0.3 0.4 0.4 0.5 o.S 1.0 i-S 2.O 2-S 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 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 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 40~ 41 42 43 44 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 9.97 914 9.97 910 9.97 906 25 24 23 22 21 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 10 9 8 7 6 50 51 52 53 54 9.48 607 9-48 647 9.48 686 9.48 725 9.48 764 40 39 39 39 39 39 39 39 39 39 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 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 4 4 4 4 4 5 4 3 2 1 60 9.48 998 9.51 178 0.48 822 9.97 821 log cos (1. log cot c. d. log tan log sin d. i Prop. Pts. *i62 252 *342 72 46 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 18 *io8 198 "288 i log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9. 48 998 9.49 037 9.49 076 9.49 115 9.49 153 39 39 39 38 39 39 38 39 39 38 39 38 38 39 38 38 39 38 38 38 38 28 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 37 37 37 37 37 38 37 37 37 36 37 37 37 37 36 37 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 43 43 42 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 42 43 _ 42 42 42 43 42 42 42 42 42 42 42 42 42 42 41 42 42 42 42 41 42 41 42 42 41 42 41 41 42 41 42 41 41 41 42 41 41 41 41 41 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 4 5 4 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 S 4 4 4 4 S 4 4 4 5 4 4 4 S 4 4 4 5 4 4 4 S 4 4 4 S 4 4 5 4 4 S 4 4 S 4 4 5 4 60 59 58 57 56 H 6 8 9 10 20 3 40 So 6 7 8 9 10 20 30 40 SO 6 7 8 9 10 20 3 40 SO 43 4-3 S-O 5-7 6.4 7-2 14-3 21.5 28.7 35-8 39 3-9 4.6 5-2 5-9 6.5 13-0 19-5 26.0 32.5 36 3-6 4-2 4.8 5-4 6.0 I2.O 18.0 24.0 30.0 42 4-2 4-9 5-6 6.3 7-0 14.0 21.0 28.0 35-0 38 3-8 4-4 S-i 5-7 6.3 12.7 IQ.O 25-3 31.7 5 0.5 0.6 0.7 0.8 0.8 i-7 2.5 3-3 4.2 41 4-1 4.8 5-5 6.2 6.8 13-7 20. S 27-3 34-2 37 3-7 4-3 4-9 5-6 6.2 12.3 18.5 24-7 30.8 4 0.4 o.S o.S 0.6 0.7 1-3 2.O 2.7 3-3 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 50 49 48 47 46 15 16 17 IS 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 25 26 27 2S 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 9.50 148 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 9.97 691 9.97 687 9.97 683 9.97 679 30 29 28 27 26 35 36 37 38 39 9.50 33o 9.50 374 9.50 411 9.50 449 9.50 486 9.52 661 9.52 703 9.52 745 9.52 787 9.52 829 0.47 339 0.47 297 047 255 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 4S 49 50 51 52 53 54 9-50 710 9.50 747 9.50 784 9.50 821 9.50 858 9.53 078 9.53 120 953 161 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 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i6i 251 *34i 71 47 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 19 *i09 199 *289 / log sin (1. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.51 264 9.51 301 9.51 338 9.51 374 9.51 411 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36' 35 36 35 36 35 35 36 35 35 35 35 35 35 35 36 34 35 35 35 35 35 35 34 35 9.53 697 9.53 738 9.53 779 9.53 820 9.53 861 41 41 41 41 41 41 41 41 40 41 41 40 41 41 40 41 40 41 40 41 40 41 40 40 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 40 39 40 39 40 39 39 40 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 4 5 4 4 5 4 5 4 4 5 4 4 5 4 5 4 5 4 4 5 4 5 4 5 4 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 60 59 58 57 56 // 41 4.1 4.8 5-5. 6.2 6.8 13-7 20.5 27.3 34-2 37 3-7 4-3 4-9 5-6 6.2 12.3 18.5 24.7 30.8 ' 84 5 3- J 4-< 5 4. J 5- 5 5- 5 II., 3 I7-< 3 22. 328. 40 4.0 4-7 5-3 6.0 6.7 13-3 20.0 26.7 33-3 3fi 3-6 4.2 4.8 5-4 6.0 [2.O 18.0 24.0 30.0 5 1 0.5 3 0.6 i 0-7 0.8 7 o.S 5 1-7 3 2-5 ' 3-3 i 4-2 39 3-9 4.6 5-2 5-9 6.5 13-0 19-5 26.0 32-5 35 3-5 4.1 4-7 5-3 5-8 ii. 7 17.5 23-3 29.2 4 0.4 jo.S o.S 0.6 0.7 i.3 2.O 2.7 3-3 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 6 7 8 9 10 20 30 40 50 // 6 8 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 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 9 10 20 30 40 50 / ( i< 2( 3< 4< 5< 35 36 37 38 39 9.52 527 9.52 563 9.52 598 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 0.44 805 044 765 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 9.5.5 554 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 23 L 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 5 4 5 5 4 3 2 1 60 9.53 405 9.56 107 0.43 893 9.97 299 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i6o 250 *340 7O 48 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS ^\/ *IIO" > 2OO *29O ' log sin d. log tan c. d log cot log cos d. Prop. Pts. 1 2 6 7 8 9 9.53 405 9.53 440 9.53 475 9.53 509 9.53 544 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 9.56 107 9.56 146 9.56 185 9.56 224 9.56 264 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 38 39 39 38 39 38 39 38 39 38 39 38 38 39 38 38 39 38 38 38 38 39 0.43 893 9.97 299 0.43 854 9.97 294 0.43 815 9.97 289 0.43 776 9.97 285 0.43 736 9.97 280 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 S 4 5 5 4 S 5 4 5 5 4 5 5 5 4 5 5 4 5 5 S 4 5 5 5 . 4 5 5 S 4 5 5 5 5 4 5 5 5 5 4 S 5 S 5 60 59 58 57 56 55 54 53 52 51 6 8 9 10 20 30 40 50 40 4.0 4.7 5-3 6.0 13-3 2O.O 26.7 33-3 38 3-8 4-4 5-7 6.3 12.7 19.0 25-3 3L7 35 3-5 4.1 4-7 5-3 5-8 ii. 7 17-5 23-3 29.2 3 5 .3 o. .80. .40. .0 0. -5 o. .01. 5 2. .03., 54- 39 3-9 4.6 5-2 5-9 6.5 13-0 19-5 26.0 32-5 37 3-7 4-3 4-9 5-6 6.2 12.3 18.5 24.7 30.8 34 3-4 4.0 4-5 5-7 "3 17.0 22.7 28.3 4 50.4 70.5 5 0.6 5o.7. 1 1.3 5 2.0 i 2.7 !l3-3 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 9.97 276 0.43 658 9.97 271 0.43 619 9.97 266 0.43 580 9.97 262 0.43 541 1 9.97 257 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 26 27 2S 29 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 6 8 9 IO 20 30 40 50 n 6 7 8 9 10 20 30 40 50 " 3 6 2 7 ; 8 A 9 i 10 ; 20 ii 30 it 4O 22 50 23 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 0.42 457 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 45 46 47 48 49 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 I5~ 14 13 12 11 9.54 936 9.54 969 9.55 003 9.55 036 9.55 069 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 9.57 849 9.57 887 9.57 925 9.57 963 9.58 001 38 38 38 38 38 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 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 38 38 38 38 38 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 38 37 38 38 38 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 log cos d. log cot c, d. log tan log sin d. ' Prop. Pts. *i 59 249' * 3 39 3 09 LOGARITHMS OF THE TRIGONOMETRIC FUNCTJONS 21 *IM 201 * 291 ' log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 5 6 7 8 9 9.55 433 9.55 466 9.55 499 9.55 532 9.55 564 33 33 33 32 33 33 33 32 33 33 32 33 32 33 32 33 32 33 32 32 33 32 32 33 32 32 32 32 32 33 32 32 32 32 32 31 32 32 32 32 32 31 32 32 32 31 32 31 32 32 31 32 31 31 32 31 32 31 31 32 9.58 418 9.58 455 9.58 493 9.58 531 9.58 569 37 38 38 38 37 38 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37' 36 37 37 36 37 36 37 36 37 36 37 36 37 36 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 5 5 4 S 5 5 S S S S 4 5 5 5 S S S 5 S 5 S 5 4 5 5 S S 5 5 5 S S 5 5 5 5 S S S S 5 S S 5 5 S S 5 6 5 5 S S S 5 S S 5 S S 60 59 58 57 56 6 8 9 10 20 30 40 50 // 6 8 9 10 20 3 40 5 ( ( i< 2( 3< 4< S< 88 3.8 4-4 S-i 5-7 6.3 12.7 iq.o 2S-3 31.7 .*> >> 3-3 3-9 4.4 5.o 5-5 n.o 16.5 22.0 27.5 ' 6 >0.6 7 0.7 .0.8 ) -9 3 1.0 5 2.0 > 3-0 3 4.0 55.0 37 3-7 4-3 4-9 5.6 6.2 12.3 18.5 24.7 30.8 3-2 3-2 3-7 4-3 4.8 5-3 10.7 1 6.0 *i.3 26.7 5 0.5 0.6 0.7 0.8 0.8 i.7 2-5 3-3 4-2 36 3-6 4.2 4.8 5-4 6.0 12.0 18.0 24.0 30.0 31 3.1 3-6 4 'A 4-6 5-2 10.3 15-5 20.7 25.8 4 0.4 o-S 0.5 0.6 0.7 1.3 2.O 2.7 3.3 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 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 60 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 20 19 18 17 16 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 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 ~y 4 3 2 1 55 56 57 58 59 9.57 201 9.57 232 9.57 264 9.57 295 9.57 326 9.60 459 9.60 495 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 60 9.57 358 9.60 641 0.39 359 9.96 717 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 5 8 248 * 33 8 68 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 22 * IJ 2 202 *292 ' log sin d. log tan :c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.57 358 9.57 389 9.57 420 9.57 451 9.57 482 31 31 31 31 32 31. 31 31 31 31 31 31 31 31 31 31 30 31 31 31 3 31 31 31 30 31 3 31 30 31 30 3i 30 31 30 31 30 30 30 31 30 30 30 31 30 30 30 30 30 30 3 30 30 30 30 30 29 30 30 30 9.60 641 9.60 677 9.60 714 9.60 750 9.60 786 36 37 36 36 37 36 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 36 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35 35 36 35 35 36 35 35 35 36 35 35 35 35 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 6 5 5 5 5 5 5 5 6 5 5 5 5 5 5 6 S 5 5 S 6 5 5 5 5 6 5 5 5 5 6 5 5 5 6 5 5 5 6 5 5 6 5 5 5 6 5 5 6 5 5 6 5 5 6 5 5 6 5 5 60 59 58 57 56 6 8 9 10 20 30 40 50 6 8 9 10 20 30 40 50 6 8 9 10 20 30 40 50 37 3-7 4-3 4-9 5-6 6.2 12-3 18.5 24.7 30.8 32 3-2 3-7 4-3 4.8 5-3 10.7 16.0 21.3 26.7 29 2.9 3-4 3-9 4-4 4-8 9-7 14-5 19-3 24.2 36 3-6 4.2 4.8 5-4 6.0 12.0 iS.o 24.0 30.0 31 3-i 3.6 4.1 4.6 5-2 10.3 I So 20.7 25.8 6 0.6 0.7 0.8 o-9 I.O 2.0 3-0 4.0 5-0 35 3-5 4-1 4-7 5-3 5-8 11.7 17-5 23-3 29.2 30 3.0 3-5 4-0 4-5 5-0 IO.O 15.0 2O.O 25.0 5 o.S 0.6 0.7 0.8 0.8 1.7 2-5 3-3 4.2 5 6 7 8 9 10 11 12 13 14 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 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 20 21 22 23 24 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 0.38 780 038 744 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 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 25 26 27 28 29 30 31 32 33 34 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 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 9.96 530 9.96 525 9.96 520 9.96 514 25 24 23 22 21 20" 19 18 17 16 40 41 42 43 44 45" 46 47 48 49 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 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 IT 56 57 58 59 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 5 4 3 2 1 ~0~ 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 25.0 9.96 429 9.96 424 9.96 419 9.96 413 9.96 408 60 9.59 1SS 9.62 785 0.37 215 9.96 403 log cos d. log cot c. d. log tan log sin d. i Prop. Pts. *i57 247 *337 67 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 23 *"3 203 *293 / log sin (1. log tan c. d. log cot log cos d. Prop. Pis. 1 2 3 4 9.59 188 9.59 218 9.59 247 9.59 277 9.59 307 30 29 3 30 29 30 30 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 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 6 5 5 6 5 6 5 5 6 5 6 5 5 6 5 6 5 6 5 6 5 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 5 6 6 5 6 5 6 5 6 6 5 6 5 6 60 59 58 57 56 " 86 6 3.6 7 4-2 8 4-8 9 5-4 10 6.0 2O I2.O 30 18.0 40 24.0 50 30.0 " 80 6 3.0 7 3-5 8 4.0 9 4-5 10 s-o 20 10.0 30 15-0 4O 2O.O 50 25.0 " 60. 70. 80. 9 o. IO I. 2O 2. 303. 404. sos- u 3.5 4.1 4-7 5-2 5.8 II. 7 17-5 23-3 2Q. 2 29 2-9 3-4 3-9 4-4 4.8 9-7 14-5 [Q-.S 24.2 5 6 o. 7 o.( 8 o. 9 o.> 3 O.J 3 I.' 3 2., 33-; D4-: 34 3-4 4.0 4-5 5-i 5-7 II-3 17.0 22.7 28.3 28 2.8 3-3 3-7 4-2 4-7 9-3 14.0 18.7 23-3 r ! ! 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 20 21 22 23 24 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 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 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 6 5 6 5 6 5 4 3 2 1 "o" 60 9.60 931 9.64 858 0.35 142 9.96 073 log cos d. log cot c. d. log tan log sin d. i Prop. Pts. *I 5 6 246 * 33 6 66 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 24 *"4 204 *2 94 / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 9.60 931 9.60 960 9.60 988 9.61 016 9.61 045 29 28 28 29 28 28 28 29 28 28 28 28 28 28 28 28 29 27 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 27 27 28 27 27 27 27 28 27 27 9.64 858 9.64 892 9.64 926 9.64 960 9.64 994 34 34 34 34 34 34 34 34 34 33 34 34 34 34 33 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 34 33 34 33 33 34 33 . 33 34 33 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 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 6 S 6 6 5 6 S 6 6 S 6 6 S 6 6 6 5 6 6 5 6 6 6 S 6 6 5 6 6 6 S 6 6 6 6 5 6 6 6 6 5 6 6 6 6 S 6 6 6 6 6 S 6 6 6 6 6 6 6 S 60 59 58 57 56 // 6 8 9 10 20 30 40 50 " 2 6 2 7 3 8 3 9 4 10 4 20 9 30 14 40 19 5024 // 6 7 8 9 10 20 30 40 So 34 3-' 4.c 4-. s. 5-' ii.. i-.c 22.' 28.. B 9 4 9 4 8 7 5 I 3 I 2 2 6 o.(. 0.7 0.? 0.5 I.C 2.C 3-c 4.c 5-c 33 i. 3.3 > 3.9 4.4 5-0 S-S II.O > 16.5 22.O 27-5 28 27 2.8 2.7 3-3 3-2 3-7 3-6 4.2 4.1 4-7 4-5 9-3 9-0 4-0 I3-S 8.7 18.0 3-3 22.5 5 0.5 0.6 0.7 0.8 0.8 1-7 2-5 3-3 4.2 9.61 073 9.61 101 9.61 129 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 50 49 48 47 46 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 15 16 17 IS 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 25 26 27 28 29 30 31 32 33 34 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 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 40 41 42 43 44 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 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i55 245 *335 65 53 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 25 *H5 D 25 295 i log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 26 27 26 27 26 27 26 27 26 27 26 27 26 26 27 26 26 27 26 26 26 27 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 9.66 867 9.66 900 9.66 933 9.66 966 9.66 999 33 33 33 33 33 33 33 33 32 33 33 33 33 3? 33 33 33 32 33 33 32 33 33 32 33 32. 33 33 32 33 32 33 32 33 32 32 33 32 33 32 32 33 32 32 33 32 32 32 33 32 32 32 32 33 32 32 32 32 32 32 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 6 6 6 6 6 6 6 6 6 6 60 59 58 57 56 55 54 53 52 51 " 33 6.3.. 7 3-< 8 4.^ 9 5..C 10 S-. 2O II.C 30 16.. 4OJ22.C S0;27., " 27 6 2.7 7 3-2 8 3 .t 9 4-1 10 4.5 20 Q.C 30 13.5 40 i8.c So 22.5 " 7 6 0.7 o 7 0.8 o 8JO.Q O 9 i.i IOjI.2 I 2O 2.3 2 30 3-5 3 4 4-7 4 5o;5-8 5 32 1 3-2 ) 3-7 \ 4-3 ) 4.8 S-3 > 10.7 16.0 > 21.3 26.7 26 2.6 3-0 3-5. 3-9 4-3 8.7 13-0 17-3 21.7 6 | 5 .60.5 7JO.6 .8:0.7 .9 0.8 .ojo.8 .0 1.7 0.2.5 o|3-3 .0 4.2 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 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 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 7 6 6 6 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 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 9.95 579 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 IS" 14 13 12 11 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 50 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 54 244 *334 64 54 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 20 *"6 206 *296 ' log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.64 184 9.64 210 9.64 236 9.64 262 9.64-288 26 26 26 26 25 26 26 26 26 25 26 26 25 26 26 23 26 23 26 23 26 25 26 23 26 23 26 23 23 26 23 23 26 23 23 23 26 23 25 25 23 23 26 23 25 23 25 25 23 23 23 25 23 25 24 23 25 23 25 23 9.68 818 9.68 850 9.68 882 9.68 914 9.68 946 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 32 32 31 32 32 32 31 32 32 31 32 32 31 32 31 32 32 31 32 31 32 3-t 32 31 32 31 32 31 31 32 31 32 31 31 32 31 31 32 31 31 31 32 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 6 6 6 7 6 6 6 6 7 6 6 6 6 7 6 6 6 7 6 6 6 7 6 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 7 6 6 7 6 7 6 6 7 6 6 7 6 7 6 6 7 6 7 6 6 7 60 59 58 57 56 n 6 7 8 9 10 20 30 40 50 " 2< 6 2 7 3 8 3 9 3 10 4 20 8 30 13 40 17 // 6 8 9 10 20 30 40 32 31 3-2 3-1 3-7 3-6 4-3 4-1 4.8 4.6 3-3 3-2 10.7 10.3 16.0 13.5 21.3 20.7 26.7 25.8 i 25 24 6 2.3 2.4 o 2.9 2.8 5 3-3 3-2 9 3.8 3-6 3 4-2 4.0 7 8.3 8.0 o 12.5 12.0 3 16.7:16.0 7 20.8 2O.O 7 6 0.7 0.6 0.8 0.7 0.9 0.8 i.o 0.9 1.2 I.O 2.3 2.0 3-3 3-0 4-7 4-0 . 5.8 5.0 5 6 7 S 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 IS 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 2S 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 30 29 28 27 26 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 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 "0" 60 9.65 705 9.70 717 0.29 283 9.94 9SS log cos d. log cot c. d. log tan log sin ! d. ' Prop. Pts. *'53 243 *333 63 55 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 27 *"7 207 *2 97 ' log sin d. log tan e. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.65 705 9.65 729 9.65 754 9.65 779 9.65 804 24 25 25 25 24 25 25 24 25 25 24 25 24 25 25 24 25 24 25 24 24 25 24 25 24 24 25 24 24 25 24 24 24 24 25 24 24 24 24 24 24 25 24 24 24 24 24 24 24 23 24 24 24 24 24 24 24 23 24 24 9.70 717 9.70 748 9.70 779 9.70 810 9.70 841 31 31 31 31 32 31 31 31 31 31 31 31 31 32 31 31 31 31 31 31 31 31 30 31 31 31 31 31 31 31 31 3 31 31 31 31 3 31 31 30 31 31 31 30 31 31 3 31 3 31 31 3 31 30 31 3 31 30 31 30 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 6 7 6 7 6 7 6 7 6 7 60 59 58 57 56 " 82 6 3.2 7 3-7 8 4-3 9 4-8 10 5-3 20 10.7 30 16.0 40 21.3 So 26.7 " 25 6 2.5 7 2- 8 3.3 9 3. 10 4.2 20 8.3 30 I2.J 40 16.7 50 20.J // 6( 7< 8 9 10 20 30 40 So 31 1:1 4.1 4-6 5-a 10.3 15.; 20.7 25.S 24 2.1 2.! 3- 3-< 4-< 8.< 12. 16. 20. 7 ( X7 o 3.80 3.g o I.I 1.2 I 2.3 2 3-5 3 4-7 4 5.85 30 3-0 3-5 4.0. 4-5 S-o IO.O 15.0 '2O.O !25.0 28 \ 2.3 5 2.7 ! 3-1 > 3-S > 3-8 > 7-7 3 II. S > iS-3 3 19.2 I .6 7 .8 9 .0 .0 .0 .0 .0 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 6 6 7 6 7 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 6 7 7 6 7 6 7 6 7 6 7 6 7 7 6 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 6 7 7 7 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 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 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 i 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 c i i 1 60 9.67 161 9.72 567 0.27 433 9.94 593 log cos d. log cot <-. d log tan log sin d. / Prop. Pts. *i 5 2 242 * 33 2 62 56 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 28 *"8 208 * 298" / log sili d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.67 161 9.67 185 9.67 208 9.67 232 9.67 256 i 24 23 24 24 24 23 24 11 24 23 24 23 24 23 24 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 23 24 23 23 23 23 23 23 23 23 23 24 23 23 22 23 23 23 23 23 23 23 23 23 22 23 9.72 567 9.72 598 9.72 628 9.72 659 9.72 689 31 30 31 3 31 30 30 31 30 31 3 30 31 30 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 31 30 30 30 30 30 3 30 30 30 30 30 30 30 30 30 30 30 30 3 30 30 29 30 30 30 30 30 29 30 0.27 433 0.27 402 0.27 372 0.27 341 0.27 311 9.94 593 9.94 587 9.94 580 9.94 573 9.94 567 6 7 7 7 7 7 6 7 7 7 6 7 7 7 7 6 7 7 7 7 6 7 7 7 7 7 6 7 7 7 7 7 7 7 6 7 7 7 7 60 59 58 57 56 " 81 6 3.1 7 3-6 8 4.1 9 4.6 10 5.2 20 10.3 30 IS-5 4O 2O.7 50 25.8 " 24 6 2.4 7 2.8 8 3.2 9 3-6 10 4.0 20 8.0 30 12.0 40 16.0 SO 2O.O // 6 8 9 IO 30 30 40 SO 30 3.0 3-5 4 .c 4-5 5-c IO.C 15.0 20. C 25.C 23 2-3 2.7 3.1 3-5 ,vS 7-7 ". 5 IS-3 19.2 7 0.7 c 0.8 c O.QC 1.0 C 1.2 2.3 : 3-5; 4-7- $i 29 2.9 3-4 3-9 4-4 4.8 9-7 I4-S 19-3 24.2 22 2.2 2.6 2-9 3-3 3.7 7-3 II.O 14.7 18.3 6 >.6 >.7 >.8 >-9 .0 .0 .0 t.o ;.o 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 IS 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 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 25 26 27 2S 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 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 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 1% 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 7 7 7 7 7 5 4 3 2 1 60 9.68 557 9.74 375 0.25 625 9.94 182 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 5 i 241 * 33 i 61 57 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 29 *"9 209 *2 99 / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.68 557 9.68 580 9.68 603 9.68 625 9.68 648 23 23 22 23 23 23 22 23 23 22 23 22 23 23 22 23 22 23 22 23 22 23 22 23 22 22 23 22 23 22 22 23 22 22 22 23 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 9.74 375 9.74 405 9.74 435 9.74 465 9.74 494 3 30 30 29 3 30 29 30 30 30 29 30 3 29 30 30 29 30 29 30 29 30 30 29 30 29 30 29 30 29 30 29 30 29 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 30 29 29 29 30 29 29 29 30 29 29 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 7 7 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 8 7 7 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 8 7 7 7 8 7 60 59 58 57 56 55 54 53 52 51 " { 6 7 ' 8 9 10 20 i 30 i 4O 2 502 [" 2 6 i 7 '' 8 .3 9 I 10 2 20 ', 30 I) 40 I. SO IS // 6 7 8 9 10 20 30 40 SO ,0 3.0 3-5 1-0 4-5 S.o 3.0 =5.0 3.0 5-0 3 3 7 .1 4 .8 7 S 3 .2 8 o.t o.g i.i i.: i-3 2-7 4.c 5-2 6.; 29 2.9 3-5 3-9 4-4 4.8 9-7. I4-S 19-3 24.2 22 2.2 2.6 2.9 3-3 .3-7 7-3 II.O 14.7 18.3 7 0.7 0.8 0.9 I.O 1.2 2.3 3-'S 4-7 5-8 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 10 11 12 13 14 968 784 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 23^ 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 20 19 18 17 16 40 41 42 43 44 9.69 456 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 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 log cos d. log cot c. d. log tan log sin d. i Prop. Pts. *i 5 o 240 * 33 o 6O 58 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 30 > *I20 210 *3oo 3 ' log sin (1. log tan C. (1. log cot log cos d. Prop. Pts. 1 2 3 4 9.69 897 9.69 919 9.69 941 9.69 963 9.69 984 22 22 22 21 22 9.76 144 9.76 173 9.76 202 9.76 231 9.76 261 29 29 29 30 29 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 8 7 7 60 59 58 57 56 5 6 7 S 9 9.70 006 9.70 028 9.70 050 9.70 072 9.70 093 22 22 22 21 22 9.76 290 9.76 319 9.76 348 9.76 377 9.76 406 29 29 29 29 29 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 8 7 8 7 55 54 53 52 51 " 80 29 28 10 11 12 13 14 9.70 115 9.70 137 9.70 159 9.70 ISO 9.70 202 22 22 21 22 22 9.76 435 9.76 464 9.76 493 9.76 522 9.76 551 29 29 29 29 29 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 7 8 7 8 7 50 49 48 47 46 6 3.0 2.9; 2.8 7 3-5 3-4 3-3 8 4-oj 3-9 3-7 9 4.5 4.4 4.2 10 s.oj 4.8 4.7 20 IO.OJ 9.7 9.3 15 16 17 IS 19 9.70 224 9.70 245 9.70 267 9.70 2SS 9.70 310 21 22 21 22 22 9.76 580 9.76 609 9.76 639 9.76 668 9.76 697 29 3 29 29 28 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 8 7 7 8 45 44 43 42 41 30\is.o 14.5 14.0 40120.0 19.3; 18.7 50125.0124.2:23.3 20 21 22 23 24 9.70 332 9.70 353 9.70 375 9.70 396 9.70 418 21 22 21 22 21 9.76 725 9.76 754 9.76 783 9.76 812 9.76 841 29 29 29 29 29 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 7 8 7 8 40 39 38 37 36 25 26 27 28 29 9.70 439 9.70 461 9.70 482 9.70 504 9.70 525 22 21 22 21 22 9.76 870 9.76 899 9.76 928 9.76 957 9.76 986 29 29 29 29 29 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 7 8 7 8 7 35 34 33 32 31 " 22 21 6 2.2J 2.1 7 2.6 2.4 8 2.9 2.8 30 31 32 33 34 9.70 547 9.70 568 9.70 590 9.70 611 9.70 633 21 22 21 22 21 9.77 015 9.77 044 9.77 073 9.77 101 9.77 130 29 29 28 29 29 0.22 985 0.22 956 0.22 927 0.22 899 0.22 870 9.93 532 9.93 525 9.93 517 9.93 510 9.93 502 7 8 7 8 7 30 29 28 27 26 9 3-3 3-2 1 3-7 3-5 20 7.3 7.0 3O;II.O IO.S 40 14.7 14.0 50 18.3,17.5 35 36 37 38 39 9.70 654 9.70 675 9.70 697 9.70 718 9.70 739 21 22 21 21 22 9.77 159 9.77 188 9.77 217 9.77 246 9.77 274 29 29 29 28 29 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 8 7 8 7 8 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 21 21 21 22 21 9.77 303 9.77 332 9.77 361 9.77 390 9.77 418 29 29 29 28 2 9 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 7 8 7 8 7 20 19 18 17 16 "87 45 46 47 48 49 9.70 867 9.70 888 9.70 909 9.70 931 9.70 952 21 21 22 21 21 9.77 447 9.77 476 9.77 505 9.77 533 9.77 562 29 29 28 29 29 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 8 7 8 7 8 15 14 13 12 11 6 0.8 0.7 7 0.9 0.8 8 i.i 0.9 9 1.2 I.O 10 1.3 1.2 20 2.7 2.3 30 4.0 3.5 50 51 52 53 54 9.70 973 9.70 994 9.71 015 9.71 036 9.71 058 21 21 21 22 21 9.77 591 9.77 619 9.77 648 9.77 677 9.77 706 28 29 29 29 28 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 7 8 7 8 8 10 9 8 7 6 40 5.3 4.7 5o!6.7 5.8 55 56 57 58 59 9.71 079 9.71 100 9.71 121 9.71 142 9.71 163 21 21 21 2! 21 9.77 734 9.77 763 9.77 791 9.77 820 9.77 849 29 28 29 29 28 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 7 8 7 8 5 4 3 2 1 60 9.71 184 9.77 877 0.22 123 9.93 307 log cos d. log cot c. d. log tan log sin d. / Pi-op. Pts. *I 4 9 239 *329 59 C i 59 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 31 *I2I 211 *30I / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.71 184 9.71 205 9.71 226 9.71 247 9.71 268 21 21 21 21 21 21 21 21 21 2O 21 21 21 21 21 21 20 21 21 21 2O 21 21 21 2O 21 21 20 21 21 2O 21 2O 21 20 9.77 877 9.77 906 9.77 935 9.77 963 9.77 992 29 29 28 29 28 29 28 29 29 28 29 28 29 28 29 28 29 28 28 29 28 29 28 29 28 28 29 28 29 28 28 29 28 28 29 28 28 29 28 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 29 28 28 28 0.22 123 0.22 094 0.22 065 0.22 037 0.22 008 9.93 307 9.93 299 9.93 291 9.93 284 9.93 276 8 8 7 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 8 8 7 8 8 8 7 8 8 8 8 7 8 8 8 7 8 8 8 8 8 7 8 8 8 8 8 8 7 8 8 8 8 60 59 58 57 56 n 6 8 9 IO 20 30 40 So // 6 8 9 IO 20 30 40 So it ( ; { ( 1C 2C 3< 4C 5< 29 2.9 3-5 3-9 4-4 4.8 9-7 14.5 iQ.3 24.2 21 2.1 2.4 2.8 3.2 3-5 7.0 10.5 14.0 I7-S 8 0.8 0.9 i.i 1.2 1.3 2.7 4.0 5-3 >6.7 28 2.8 3-3 3-7 4.2 4-7 93 14.0 18.7 23-3 20 2.0 2.3 2.7 3-0 3.3 6.7 IO.O 13.3 16.7 7 0.7 b.8 0-9 I.O 1.2 2-3 3.5 4-7 S-8 5 6 7 8 9 9.71 289 9.71 310 9.71 331 9.71 352 9.71 373 9.78 020 9.78 049 9.78 077 9.78 106 9.78 135 0.21 980 0.21 951 0.21 923 0.21 894 0.21 865 9.93 269 9.93 261 9.93 253 9.93 246 9.93 238 55 54 53 52 51 10 11 12 13 14 9.71 393 9.71 414 9.71 435 9.71 456 9.71 477 9.78 163 9.78 192 9.78 220 9.78 249 9.78 277 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 9.93 230 9.93 223 9.93 215 9.93 207 9.93 200 50 49 48 47 46 15 16 17 18 19 20 21 22 23 24 9.71 498 9.71 519 9.71 539 9.71 560 9.71 581 9.78 306 9.78 334 9.78 363 9.78 391 9.78 419 0.21 694 0.21 666 0.21 637 0.21 609 0.21 581 9.93 192 9.93 184 9.93 177 9.93 169 9.93 161 45 44 43 42 41 40" 39 38 37 36 9.71 602 9.71 622 9.71 643 9.71 664 9.71 685 9.78 448 9.78 476 9.78 505 9.78 533 9.78 562 0.21 552 0.21 524 0.21 495 0.21 467 0.21 438 9.93 154 9.93 146 9.93 138 9.93 131 9.93 123 25 26 27 28 29 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.78 590 9.78 618 9.78 647 9.78 675 9.78 704 0.21 410 0.21 382 0.21 353 0.21 325 0.21 296 9.93 115 9.93 108 9.93 100 9.93 092 9.93 084 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 40" 41 42 43 44 9.71 809 9.71 829 9.71 850 9.71 870 9.71 891 9.78 732 9.78 760 9.78 789 9.78 817 9.78 845 0.21 268 0.21 240 0.21 211 0.21 183 0.21 155 9.93 077 9.93 069 9.93 061 9.93 053 9.93 046 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 T5" 14 13 12 11 ^0 9 8 7 6 9.71 911 9.71 932 9.71 952 9.71 973 9.71 994 21 2O 21 21 20 2O 21 20 21 2O 21 2O 2O 21 2O 2O 21 2O 2O 21 2O 2O 21 2O 20 9.78 874 9.78 902 9.78 930 9.78 959 9.78 987 0.21 126 0.21 098 0.21 070 0.21 041 0.21 013 9.93 038 9.93 030 9.93 022 9.93 014 9.93 007 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 0.20 985 0.20 957 0.20 928 0.20 900 0.20 872 9.92 999 9.92 991 9.92 983 9.92 976 9.92 968 45 46 47 48 49 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 0.20 844 0.20 815 0.20 787 0.20 759 0.20 731 9.92 960 9.92 952 9.92 944 9.92 936 9.92 929 50 51 52 53 54 9.72 218 9.72 238 9.72 259 9.72 279 9.72 299 9.79 297 9.79 326 9.79 354 9.79 382 9.79 410 0.20 703 0.20 674 0.20 646 0.20 618 0.20 590 9.92 921 9.92 913 9.92 905 9.92 897 9.92 889 55 56 57 58 59 9.72 320 9.72 340 9.72 360 9.72 381 9.72 401 9.79 438 9.79 466 9.79 495 9.79 523 9.79 551 0.20 562 0.20 534 0.20 505 0.20 477 0.20 449 9.92 881 9.92 874 9.92 866 9.92 858 9.92 850 5 4 3 2 1 60 9.72 421 9.79 579 0.20 421 9.92 842 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 4 8 238 * 3 28 58 60 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 32 *I22 212 * 3 02 1 log sin (1. log tan c. d., log cot log cos d. Prop. Pts. 1 2 3 4 9.72 421 9.72 441 9.72 461 9.72 482 9.72 502 20 20 21 20 20 20 20 20 20 20 21 20 20 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 _2O 19 2O 20 2O 2O 19 2O 20 20 19 2O 2O 19 2O 2O 19 20 20 19 2O 19 20 19 20 19 2O 19 2O 19 2O 9.79 579 9.79 607 9.79 635 9.79 663 9.79 691 28 28 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 27 28 28 28 28 27 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 28 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 8 8 8 8 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 9 8 8 8 8 8 9 8 60 59 58 57 56 Ts~ 54 53 52 51 " 29 6 2.9 7 3-S 8 3-9 9 4-4 10 4.8 20 9.7 3 I4-S 40 19-3 50 24.2 " 21 6 2.1 7 2.4 8 2.8 9 3-2 10 3.5 20 7.0 30 10. 5 40 14.0 SO I7-S " 9 60.9 7 LI 81.2 9 1.4 10 1.5 203.0 304-S 40 6.0 5017.5 28 2.8 3-3 3-7 4.: 4-7 y.,- 14.0 18.7 *w 20 2.0 2.3 2.7 3-0 3-3 6.7 10.0 13.3 16.7 8 0.8 0.9 i.i 1.2 1-3 2-7 4.0 5-3 6.7 27 [ 2.7 3-2 3-6 4-1 4-5 9-0 13-5 18.0 22.5 19 1.9 2.2 2-5 2.9 3-2 6.3 9-5 12.7 15.8 7 0.7 0.8 0.9 i.i 1.2 2.3 3-5 4-7 5-8 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 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 IS 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 35 34 33 32 31 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 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 35 36 37 38 39 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 16 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 log cos (1. log cot c. d. log tan log sin d. i Prop. Pts. *I 47 237 * 3 2f 57 LOGARITHMS OF THE TRIGONOMETRIC FUXCTIOXS 33 I2 3 ' "3 *303 ' log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.73 611 9.73 630 9.73 650 9.73 669 9.73 689 19 20 19 20 19 19 20 19 19 20 19 19 20 19 19 20 19 19 19 19 20 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18 19 19 19 19 19 18 19 19 19 18 19 9.81 252 9.81 279 9.81 307 9.81 335 9.81 362 27 28 28 27 28 28 27 28 27 .8 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 28 27 27 28 27 28 27 27 28 27 27 28 27 27 28 27 27 27 28 27 27 27 28 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 8 8 8 9 8 8 8 9 8 8 8 9 8 8 9 8 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 9 8 8 9 8 9 8 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 60 59 58 57 56 " 28 6 2.8 7 3-3 8 3-7 9 4-2 10 4.7 20 9.3 30 14-0 40 18.7 50 23.3 " 1 6 i 7 2 8 2 9 2 10 3 20 6 30 o 40 12 SOIS 27 20 2.7 2.O 3-2 2.3 3.6 2.7 4-1 3-0 4-5 3-3 9.0 6.7 13.5 10.0 18.0 13.3 22.5 16.7 9 18 .9 1.8 2 2.1 S 2.4 .9 2.7 .2 3.0 .3 6.0 5 9-0 .7 I2.O .8 15.0 B 8 .90.8 .1 0.9 .2 I.I .41.2 5 i-3 .0 2.7 5 4-0 o 5-3 .56.7 5 6 7 S 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 49 48 47 46 15 16 17 18 19 20 21 22 23 24 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 44 43 42 41 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 35 34 33 32 31 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 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 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 // 60 8 i 91 IO I 203 304 40 6 507 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 0.17 401 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 9 8 7 6 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 5 4 3 2 1 60 9.74 756 9.82 899 0.17 101 9.91 857 log cos d. log cot c. d. log tan log sin d. ' Prop. Pts. *I46 236 *326 56 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 34 3 4 1*124 214 *304 1 log sin (1. log tan C. (1. log cot log cos d. Prop. Pts. 1 2 3 4 9.74 756 9.74 775 9.74 794 9.74 812 9.74 831 9 9 8 9 9 9.82 899 9.82 926 9.82 953 9.82 980 9.83 008 27 27 27 28 27 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 8 9 8 9 8 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 8 9 9 8 9 9.83 035 9.83 062 9.83 089 9.83 117 9.83 144 27 27 28 27 27 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 9 8 9 8 9 55 54 53 52 51 " 28 27 1 26 10 11 12 13 14 9.74 943 9.74 961 9.74 980 9.74 999 9.75 017 8 9 9 8 9 9.83 171 9.83 198 9.83 225 9.83 252 9.83 280 27 27 27 28 27 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 9 8 9 8 9 50 49 48 47 46 6 2.8 2.7 2.6 7 3-3 3-2 3.0 8 3-7 3-6 3-5 g 4.2 4.0 3.9 10 4.7 4-5 4-3 15 16 17 18 19 9.75 036 9.75 054 9.75 073 9.75 091 9.75 110 8 9 8 9 8 9.83 307 9.83 334 9.83 361 9.83 388 9.83 415 27 27 27 27 27 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 9 8 9 8 9 45 44 43 42 41 30 14.0 13.5 13.0 40 18.7 18.0 17.3 So 23.3 22.5 21.7 20 21 22 23 24 9.75 128 9.75 147 9.75 165 9.75 184 9.75 202 9 8 9 8 9 9.83 442 9.83 470 9.83 497 9.83 524 9.83 551 28 27 27 . 27 27 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 9 8 9 9 8 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 8 9 8 8 9 9.83 578 9.83 605 9.83 632 9.83 659 9.83 686 27 27 27 27 27 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 9 9 8 9 9 35 34 33 32 31 " 19 18 6 i.g 1.8 7 2.2 2.1 8 2.5 2.4 30 31 32 33 34 9.75 313 9.75 331 9.75 350 9.75 368 9.75 386 8 9 8 8 9 9.83 713 9.83 740 9.83 768 9.83 795 9.83 822 27 28 27 27 27 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 8 9 9 8 9 30 29 28 27 26 9 2.8 2.7 ro 3.2; 3.0 20 6.3 6.0 30 g.sj g.o 40 I2.7;I2.O SO 15.8 IS-0 35 36 37 38 39 9.75 405 9.75 423 9.75 441 9.75 459 9.75 478 8 8 8 9 8 9.83 849 9.83 876 9.83 903 9.83 930 9.83 957 27 27 27 27 27 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 9 9 8 9 9 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 8 9 8 8 8 9.83 984 9.84 Oil 9.84 038 9.84 065 9.84 092 27 27 27 27 27 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 . 8 9 9 9 8 20 19 18 17 16 "98 45 46 47 48 49 9.75 587 9.75 605 9.75 624 9.75 642 9.75 660 8 9 8 8 8 9.84 119 9.84 146 9.84 173 9.84 200 9.84 227 27 27 27 27 27 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 9 9 9 9 8 15 14 13 12 11 6 o.g 0.8 7 i.o o.g 8 1.2 I.I g 1.4 1.2 10 1.5 1.3 20 3.0 2.7 30 4-5 4-0 50 51 52 53 54 9.75 678 9.75 696 9.75 714 9.75 733 9.75 751 8 8 9 8 9.84 254 9.84 280 9.84 307 9.84 334 9.84 361 26 27 27 27 27 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 9 9 9 9 8 10 9 8 7 6 40 6.0 5.3 SO 7-5 6.7 55 56 57 58 59 9.75 769 9.75 787 9.75 805 9.75 823 9.75 841 8 8 8 18 18 9.84 388 9.84 415 9.84 442 9.84 469 9.84 496 27 27 27 27 27 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 9 9 9 9 9 5 4 3 2 1 60 9.75 859 9.84 523 0.15 477 9.91 336 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 45 : ^35 *325 55< 5 63 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 35 *i2 5 ^ 215 * 3 o 5 o i log sin d. log tan c. d. log cot log cos d. Pi-op. Pts. 1 2 3 4 9.75 859 9.75 877 9.75 895 9.75 913 9.75 931 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 17 18 18 18 18 18 i? 18 18 18 17 18 18 18 i? 18 18 i? 18 18 17 18 18 17 18 18 17 18 17 18 17 18 i7 18 17 18 17 18 17 18 17 18 17 17 18 9.84 523 9.84 550 9.84 576 9.84 603 9.84 630 27 26 27 27 27 27 27 27 26 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 27 26 27 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 26 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 8 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 10 9 9 9 9 60 59 58 57 56 " 27 26 6 2.7 2. 7 3-2 3- 8 3-6 3. 9 4-1 3- 10 4.5 4. 20 9.0 8. 30 13.5 13. 40 18.0 17. 5O 22.5 21. " 17 6 1.7 7 2.0 8 2.3 9 2.6 10 2.8 20 5.7 30 8.5 40 11.3 50 14.2 " 9 60.9 7 LI 81.2 9 1-4 10 1.5 20 3.0 304.S 40 6.0 SO 7-5 18 6 1.8 3 2.1 5 2-4 3 2.7 3 3.0 7 6.0 3 9.0 3 12.0 7 iS-o 10 .0 .2 3 .5 7 3-3 5-0 6.7 8-3 8 0.8 3.9 i.i 1.2 1-3 2.7 4.0 5.3 6.7 5 6 7 8 9 10 11 12 13 14 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 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 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *I44 234 * 3 2 4 54 64 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 30 *i26 216 * 3 o6 I log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 5 6 7 8 9 9.76 922 9.76 939 9.76 957 9.76 974 9.76 991 17 18 17 17 18 17 17 18 17 17 17 18 17 17 17 18 17 I? 17 18 17 17 17 17 17 17 17 18 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 I? 17 17 16. 17 17 17 17 17 17 16 17 17 i? 17 16 9.86 126 9.86 153 9.86 179 9.86 206 9.86 232 27 26 27 26 27 26 27 26 27 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 26 27 26 26 27 26 27 26 26 27 26 26 27 26 26 27 26 26 27 .26 26 27 26 26 26 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 9 10 9 9 9 9 10 9 9 9 10 9 9 9 10 9 9 9 10 9 9 10 9 9 9 10 9 9 10 9 9 10 9 10 9 9 10 9 9 10 9 10 9 10 9 9 10 9 10 9 10 9 10 9 10 9 10 9 10 9 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 " 27 2 6 2.7 2 7 3.2 3 8 3.6 3 9 4-0 3 10 4.5 4 20 9.0 8 30 13.5 13 40 18.0 17 SO 22.5121 " 17 6 1.7 7 2.0 8 2.3 9 2.6 10 2.8 20 5-7 30 8.5 40 11.3 50 14.2 " 10 6 i.o 7 1.2 8 1.3 9 i.S 10 1.7 20 3.3 30 5-0 40 6.7 50 8.3 6 IS .6 1.8 .O 2.1 .5 2.4 9 2-7 3 3-0 .7 6.0 .0 9.0 .3 12.0 7 iS.o 16 1.6 1.9 2.1 2-4 2.7 5-3 8.0 10.7 13-3 9 0.9 I.O 1.2 1.4 i-S 3-0 4-5 6.0 7-5 9.77 009 9.77 026 9.77 043 9.77 061 9.77 078 9.86 259 9.86 285 9.86 312 9.S6 338 9.86 365 0.13 741 0.13 715 0.13 688 0.13 662 0.13 635 9.90 750 9.90 741 9.90 731 9.90 722 9.90 713 10 11 12 13 14 IS 16 17 IS 19 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 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 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 29 28 27 26 25 24 23 22 21 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 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 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 9.77 946 9.87 711 0.12 289 9.90 235 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 4 3 233 * 3 2 3 53 65 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 37 i 217 * 3 o 7 ' log sin <1. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.77 946 9.77 963 9.77 980 9.77 997 9.78 013 17 17 17 16 17 9.87 711 9.87 738 9.87 764 9.87 790 9.87 817 27 26 26 27 26 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 10 9 10 9 10 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 17 16' i? 17 16 9.87 843 9.87 869 9.87 895 9.87 922 9.87 948 26 26 27 26 26 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 9 10 9 10 10 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 17 17 16 17 I? 9.87 974 9.88 000 9.88 027 9.88 053 9.88 079 26 27 26 26 26 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 9 10 9 10 10 50 49 48 47 46 15 16 17 IS 19 9.78 197 9.78 213 9.78 230 9.78 246 9.78 263 16 17 16 17 i? 9.88 105 9.88 131 9.88 158 9.88 184 9.88 210 26 27 26 26 26 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 9 10 9 10 10 45 44 43 42 41 " 27 26 17 6 2.7 2.6 1.7 7 3.2 3.0 2.0 8 3-6 3-5 2.3 9 4.0 3-9 2.6 10 4.5 4.3 2.8 20 21 22 23 24 9.78 280 9.78 296 9.78 313 9.78 329 9.78 346 16 17 16 17 16 9.88 236 9.88 262 9.88 289 9.88 315 9.88 341 26 27 26 26 26 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 9 10 IO 9 IO 40 39 38 37 36 20 9.0 8.7 5-7 30 13.5 13.0 8.5 40 18.0 17.3 11.3 So|22.s 21.7 14.2 25 26 27 28 29 9.78 362 9.78 379 9.78 395 9.78 412 9.78 428 17 16 i? 16 i? 9.88 367 9.88 393 9.88 420 9.88 446 9.88 472 26 27 26 26 26 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 IO 9 IO IO 9 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 16 17 16 16 17 9.88 498 9.88 524 9.88 550 9.88 577 9.88 603 26 26 27 26 26 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 IO IO 9 10 10 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 16 i? 16 16 i? 9.88 629 9.88 655 9.88 681 9.88 707 9.88 733 26 26 26 26 26 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 10 9 IO IO IO 25 24 23 22 21 " 16 10 9 40 41 42 43 44 9.78 609 9.78 625 9.78 642 9-78 658 9.78 674 16 i? 16 16 17 9.88 759 9.88 786 9.88 812 9.88 838 9.88 864 27 26 26 26 26 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 9 IO IO IO 9 20 19 18 17 16 7 1.9 .2 i.o 8 2.1 .3 1.2 9 2.4 .5 1.4 10 2.7 .7 1.5 20 5-3 3-3 3-0 30 8.0 5.0 4.5 45 46 47 48 49 9.78 691 9.78 707 9.78 723 9.78 739 9.78 756 16 16 16 17 16 9.88 890 9.88 916 9.88 942 9.88 968 9.88 994 26 26 26 26 26 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 IO IO IO IO 9 15 14 13 12 11 50 13.3 8.317.5 50 51 52 53 54 9.78 772 9.78 788 9.78 805 9.78 821 9.78 837 16 i? 16 16 16 9.89 020 9.89 046 9.89 073 9.89 099 9.89 125 26 27 26 26 26 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 IO IO IO IO IO 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 16 17 16 16 16 9.89 151 9.89 177 9.89 203 9.89 229 9.89 255 26 26 26 26 26 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 9 IO IO IO IO 5 4 3 2 1 60 9.78 934 9.89 281 0.10 719 9.89 653 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *I42 232 *322 52 o 66 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 38 *i2s> 218 * 3 o8 / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.78 934 9.78 950 9.78 967 9.78 983 9.78 999 16 17 16 16 16 16 16 16 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 15 1 6 16 16 16 16 16 16 16 16 15 16 16 16 16 16 15 16 16 16 15 16 16 16 IS 16 16 13 16 16 15 16 16 IS 16 16 15 9. 89 281 9.89 307 9.89 333 9.89 359 9.89 385 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 2S 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 26 25 26 26 26 26 26 26 25 26 26 26 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 o o 9 o o o o o o o o o 9 o o o o o o o o o o I o o o o o o o o o o o I o o o o o I o o o o o I o o o I o 60 59 58 57 56 // 6 8 9 10 20 30 40 5 6 8 9 10 20 30 40 So // 6 7 8 9 10 20 3 40 50 u 6 8 9 10 20 30 40 SO 26 2.6 3-0 3-5 3-9 4-3 8.7 13-0 17-3 21.7 17 1-7 2.O 2-3 2.6 2.8 5-7 8-5 11.3 14.2 15 1.5 1.8 2.O 2.2 2-S S-o 7-5 IO.O 12.5 10 I.O 1.2 1-3 i.S 1.7 3-3 5-o 6.7 8-3 25 2-5 2.9 3-3 3-8 4.2 8.3 12.5 16.7 20.8 16 1.6 i. 9 2.1 2-4 2.7 5-3 8.0 10.7 13-3 11 i.i 1-3 i.S 1.6 1.8 3-7 5-5 7-3 9.2 9 0.9 I.O 1.2 1.4 i-5 3-0 4-5 6.0 7-5 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 ~50 49 48 47 46 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 15 16 17 18 19 9.79 176 9.79 192 9.79 208 9.79 224 9.79 240 9.89 671 9.S9 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 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 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 0.09 447 9.89 203 9.89 193 9.89 183 9.89 173 9.89 162 15 14 13 12 11 50 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 3-14 0.09 318 9.89 152 9.89 142 9.89 132 9.89 122 9.89 112 10 9 8 7 6 :o 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 log cos d. log cot c. d. log tan log sin d. i Prop. Pts. *I 4 I 2 3 I * 3 2I 51 67 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 39 *i29 219 * 3 o 9 ' log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.79 887 9.79 903 9.79 918 9.79 934 9.79 950 16 iS 16 16 IS 16 16 iS 16 IS 16 iS 16 IS 16 IS IS 16 iS 16 16 16 IS 16 iS 16 iS 16 iS IS 16 iS 16 iS IS IS IS IS 16 IS IS IS IS IS IS 15 16 IS IS IS 9.90 837 9.90 863 9.90 889 9.90 914 9.90 940 26 26 25 26 26 26 26 25 26 26 26 26 25 26 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 25 26 26 25 26 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 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 IO IO 10 II IO IO II IO IO IO II IO IO II 10 10 II IO IO II IO IO II IO 10 II 10 II 10 IO II IO II IO II IO IO II IO II IO II IO II IO II 10 II 10 II 10 II II IO II IO II 10 II II 60 59 58 57 56 6 8 9 IO 20 30 40 SO it 6 8 9 IO 20 30 40 50 6 8 9 IO 20 30 40 So 26 2.6 3-0 3-5 3-9 4-3 8-7 13-0 17-3 21.7 16 1.6 1-9 2.1 2.4 2.7 5-3 8.0 10.7 13-3 11 .1 3 5 .6 .8 3-7 5-S 7-3 9.2 25 2.5 2.9 3-3 3-8 4-2 8.3 12.5 16.7 20.8 15 i-S 1.8 2.O 2.2 2.5 5-0 7-5 IO.O 12.5 10 I.O 1.2 1.3 i.S 3-3 S-o 6.7 8-3 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 0.08 853 008 828 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 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 U91 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 9.88 668 988 657 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 55 56 57 58 59 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 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 log cos d. log cot c. d. log tan log sin d. ' Prop. Pts. *I40 230 *320 5O 68 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 4O *I30 220 *3io / log sin (1. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.80 807 9.80 822 9.80 837 9.80 852 9.80 867 15 IS 15 IS IS 15 15 15 IS IS IS IS IS 15 IS IS 14 IS 15 IS 15 15 15 15 14 15 15 15 15 14 IS IS 15 IS 14 IS 13 14 15 is 15 14 15 15 14 15 15 14 IS IS 14 15 14 IS IS 14 15 14 IS 14 9.92 381 9.92 407 9.92 433 9.92 458 9.92 484 26 26 25 26 26 25 26 26 25 ;6 25 26 26 2S 26 26 25 26 25 26 26 25 26 25 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 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 IO II IO II II IO II II IO II II IO II II IO II II IO II II II IO II II II IO II II II IO II II II II IO II II II II II II IO II II II II II II II II IO II II II II II II II II II 60 59 58 57 56 6 8 9 IO 20 30 40 So tt 6 7 8 9 10 20 30 40 SO // 6 8 9 IO 20 30 40 So 26 2.6 3-0 3-S 3-9 4-3 8.7 13-0 17-3 21.7 15 1.5 1.8 2.O 2.3 2.5 5-0 7-5 10.0 12-5 11 .1 3 5 7 .8 3-7 5-5 7-3 9.2 25 2.S 2.9 3.3 3-8 4-2 8-3 12.5 16.7 20.8 14 1-4 1.6 1.9 2.1 2.3 4-7 7-o 9-3 11.7 10 .0 .2 3 5 .7 3-3 S-o 6.7 8-3 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 40 39 38 37 36 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 25 26 27 28 29 30 31 32 33 34 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 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 ~25 24 23 22 21 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 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 0.06 416 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 55 56 57 58 59 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 s 4 3 2 1 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 60 9.81 694 9.93 916 0.06 084 9.87 778 log cos d. log cot c. d. log tan log sin d. ; Prop. Pts. *i 39 229 #319 49 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 41 *I3I 221 *3U / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.81 694 9.81 709 9.81 723 9.81 738 9.81 752 15 x .4 15 14 IS 14 IS 14 IS 14 IS 14 14 IS 14 IS 14 IS 14 14 IS 14 14 IS 14 14 15 J 4 14 14 IS 14 14 IS 14 14 14 14 IS 14 14 14 14 IS J 4 14 14 14 14 14 14 IS 14 14 14 14 14 14 14 14 9.93 916 9.93 942 9.93 967 9.93 993 9.94 018 26 25 26 25 26 25 26 25 26 25 26 25 26 2S 26 25 26 25 26 25 26 25 26 25 26 25 25 26 25 26 25 26 25 26 25 26 25 25 26 25 26 25 26 25 25 26 2S 26 25 26 25 25 26 25 26 25 26 25 25 26 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 ii ii ii ii ii ii ii ii ii ii ii ii ii ii ii 12 II II II II II II II II 12 II II II II II 12 II II II II 12 II II II II 12 II II 12 II II II 12 II II 12 II II 12 II II 12 II II 12 60 59 58 57 56 it 6 8 9 10 20 30 40 50 // 6 8 9 10 20 30 40 SO 6 8 9 10 20 3 40 5 26 2.6 3-0 3-S 3-9 4-3 8.7 13-0 17-3 21.7 15 i-S 1.8 2.O 2-3 2-5 S-o 7-5 IO.O 12.5 12 1.2 1.4 1.6 1.8 2.0 4.0 6.0 8.0 IO.O 25 2.5 2.9 3-3 3-8 4.2 8-3 12.5 16.7 20.8 14 1-4 1.6 1.9 2.1 2-3 4-7 7.0 9-3 11.7 11 .1 3 5 7 .8 3-7 S-S 7-3 9.2 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 9.87 690 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 35.0 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 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 9.87 401 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 0.05 141 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 9.82 424 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 IT 60 9.82 551 9.95 444 0.04 556 9.87 107 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i38 228 *3i8 48 7fl LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 4:2 *I32 222 *3I2 / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9.82 551 9.82 565 9.82 579 9.82 593 9.82 607 14 14 14 14 14 14 14 14 14 14 14 14 14 u 14 14 13 u 14 14 14 14 14 13 14 14 14 14 14 13 14 14 14 13 14 14 14 13 u u 14 13 14 14 13 14 14 13 u 13 14 14 13 u 13 14 14 13 14 J 3 9.95 444 9.95 469 9.95 495 9.95 520 9.95 545 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 26 25 26 25 25 26 25 2S 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 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 i i 2 I 2 I I 2 I 2 I 2 I 2 I 2 I I 2 I 2 2 I 2 I 2 I 2 I 2 I 2 2 I 2 I 2 2 I 2 2 I 2 2 I 2 2 I 2 2 2 I 2 2 I 2 2 2 I 2 60 59 58 57 56 ~5S 54 53 52 51 50 49 48 47 46 6 7 8 9 lo 20 3 40 S // 6 8 9 10 20 30 40 So i> 6 8 9 10 20 3 40 50 26 2.6 3-o 3-5 3-9 4-3 8-7 13-0 17-3 21.7 14 1-4 1.6 1.9 2.1 2-3 4-7 7.0 9-3 11.7 12 1.2 1.4 1.6 1.8 2.O 4.0 6.0 8.0 10.0 26 2.5 2.9 3-3 3-8 4.2 8.3 12-5 16.7 20.8 13 1-3 i-S 1-7 2.O 2.2 4-3 6.5 8-7 10.8 11 i.i 1-3 i.S 1-6 1.8 3-7 5-5 7-3 9.2 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 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 15 16 17 IS 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 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 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 IT 46 47 48 49 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 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 55 56 57 58 59 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 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 0.03 161 003 136 0.03 110 0.03 085 0.03 060 9.86 472 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *i 37 227 * 3 i 7 47 71 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 43 *I33 223 * 3 i 3 / log sin d. log tan c. d. log cot log cos d. Prop. Pts. 1 2 3 4 9. S3 378 9.83 392 9.83 405 9.83 419 9.83 432 14 13 14 13 14 13 14 13 14 13 14 13 14 13 14 13 u 13 13 14 13 13 14 13 14 13 13 14 13 13 14 13 13 13 14 13 13 13 14 13 13 13 u 13 13 13 13 14 13 13 13 13 13 13 14 13 13 13 13 13 9.96 966 9.96 991 9.97 016 9.97 042 9.97 067 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 26 25 25 25 26 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 12 12 12 II 12 12 12 12 12 II 12 12 12 12 12 12 12 II 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 60 59 58 57 56 55 54 53 52 51 // 6 8 9 10 20 30 40 50 6 8 9 10 20 30 40 SO tr 6 8 9 IO 20 30 40 So 26 2.6 3-o 3-5 3-9 4-3 8.7 13-0 17-3 21.7 14 1.4 1.6 1.9 2.1 2-3 4-7 7.0 9-3 11.7 12 1.2 1-4 1.6 1.8 2.O 4.0 6.0 8.0 1 0.0 25 2.5 2-9 3.3 3-8 4.2 8-3 12-5 16.7 20.8 13 1-3 i-5 1-7 2.0 2.2 4-3 6-5 8-7 10.8 11 i.i 1-3 1-5 1-6 1.8 3-7 5-5 7-3 9-2 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 10 11 12 13 14 Is 16 17 18 19 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 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 40 39 38 37 36 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 9.86 164 9.86 152 9.86 140 9.86 128 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 9.86 104 9.86 092 9.86 080 9.86 068 35 34 33 32 31 30 31 32 33 34 9.83 781 9.83 795 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 9.86 056 9.86 044 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 9.97 877 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 12 12 12 12 12 12 13 12 12 12 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 12 12 12 13 12 12 12 12 12 13 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 0.01 617 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 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *I36 226 *3i6 . 46 72 LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 44 D 1 f i34 c 224 *3i4 ? / log sin d. log tan C. (1. log cot log cos d. Prop. Pts. 1 2 3 4 9.84 177 9.84 190 9. 84 203 9.84 216 9.84 229 13 13 13 13 13 9.98 484 9.98 509 9.98 534 9.98 560 9.98 585 25 25 26 25 25 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 2 2 2 2 3 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 13 14 13 13 13 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 25 26 23 25 26 0.01 390 0.01 365 0.01 339 0.01 314 0.01 289 9.85 632 9.85 620 9.85 60S 9.85 596 9.85 583 2 2 3 2 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 13 13 13 13 13 9.98 737 9.98 762 9.98 787 9.98 812 9.98 838 25 25 25 26 25 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 2 2 3 2 2 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 12 13 13 13 9.98 863 9.98 888 9.98 913 9.98 939 9.98 964 25 25 26 25 0.01 137 0.01 112 0.01 087 0.01 061 0.01 036 9.85 510 9.85 497 9.85 485 9.85 473 9.85 460 3 2 2 3 45 44 43 42 41 " 26 25 14 6 2.6 2.5 1.4 7 3-D 2.9' 1.6 8 3-5 3-3 i-9 9 3.9: 3.8 2.1 10 4.3 4.2 2.3 20 21 22 23 24 9.84 437 9.84 450 9.84 463 9.84 476 9.84 489 13 13 13 13 9.98 989 9.99 015 9.99 040 9.99 065 9.99 090 26 25 25 25 26 0.01 Oil 0.00 985 0.00 960 0.00 935 0.00 910 9.85 448 9.85 436 9.85 423 9.85 411 9.85 399 2 3 2 2 3 40 39 38 37 36 20 8.7 8.3; 4.7 30 13.0 12.5 7.0 40 17.3 16.7 9.3 50 21.7120.8,11.7 25 26 27 28 29 9.84 502 9.84 515 9.84 528 9.84 540 9.84 553 13 13 12 13 9.99 116 9.99 141 9.99 166 9.99 191 9.99 217 25 25 25 26 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 2 3 2 2 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 13 13 13 13 9.99 242 9.99 267 9.99 293 9.99 318 9.99 343 25 26 25 25 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 2 3 2 3 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 13 13 13 13 9.99 368 9.99 394 9.99 419 9.99 444 9.99 469 26 25 25 25 26 0.00 632 0.00 606 0.00 581 0.00 556 0.00 531 9.85 262 9.25 250 9.85 237 9.85 225 9.85 212 2 3 2 3 25 24 23 22 21 " 13 12 40 41 42 43 44 9.84 694 9.84 707 9.84 720 9.84 733 9.84 745 13 13 13 12 9.99 495 9.99 520 9.99 545 9.99 570 9.99 596 25 25 25 26 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 3 2 3 2 20 19 18 17 16 7 i-S i-4 8 l.7 : 1.6 9 2.0 1.8 IO 2.2 2.O 20 4.3 4.0 30 6.5 6.0 45 46 47 48 49 9.84 758 9.84 771 9.84 784 9.84 796 9.84 809 13 13 12 13 9.99 621 9.99 646 9.99 672 9.99 697 9.99 722 25 26 25 25 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 2 3 2 3 15 14 13 12 11 50 io.8!io.o 50 51 52 53 54 9.84 822 9.84 835 9.84 847 9.84 860 9.84 873 13 13 12 13 13 9.99 747 9.99 773 9.99 798 9.99 823 9.99 848 26 25 25 25 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 2 3 2 3 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 13 13 12 13 9.99 874 9.99 899 9.99 924 9.99 949 9.99 975 25 25 25 26 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 3 3 2 3 5 4 3 2 1 60 9.84 949 M 0.00 000 25 0.00 000 9.84 949 log cos d. log cot c. d. log tan log sin d. / Prop. Pts. *'35 225 *3i5 45 3 TABLE III NATURAL TRIGONOMETRIC FUNCTIONS FOUR PLACES 75 NATURAL SIXES AXD COSIXES / 1 2 3 4 / sin cos sin cos sin cos sin cos sin cos .0000 1.000 .0175 .9998 .0349 .9994 .0523 .9986 .0698 .9976 60 1 03 .000 77 98 52 94 26 86 0700 75 59 2 06 .000 80 98 55 94 29 86 03 75 58 3 09 .000 83 98 58 94 32 86 06 75 57 4 12 .000 86 98 61 93 35 86 09 75 56 5 .0015 1.000 .0189 .9998 .0364 .9993 .0538 .9986 .0712 .9975 55 6 17 .000 92. 98 66 93 41 85 15 74 54 7 20 .000 95 98 69 93 44 85 18 74 53 8 23 .000 98 98 72 93 47 85 21 74 52 9 26 .000 0201 98 75 93 50 85 24 74 51 10 .0029 1.000 .0204 .9998 .0378 .9993 .0552 .9985 .0727 .9974 50 11 32 .000 07 98 81 93 55 85 29 73 49 12 35 .000 09 98 84 93 58 84 32 73 48 13 38 .000 12 98 87 93 61 84 35 73 47 14 41 .000 15 98 90 92 64 84 38 73 46 15 .0044 1.000 .0218 .9998 .0393 .9992 .0567 .9984 .0741 .9973 45 16 47 .000 21 98 96 92 70 84 44 72 44 17 49 .000 24 97 98 92 73 84 47 72 43 18 52 .000 27 97 0401 92 76 83 50 72 42 19 55 .000 30 97 04 92 79 83 53 72 41 20 .0058 1.000 .0233 .9997 .0407 .9992 .0581 .9983 .0756 .9971 40 21 61 .000 36 97 10 92 84 83 58 71 39 22 64 .000 39 97 13 91 87 83 61 71 38 23 67 .000 41 97 16 91 90 83 64 71 37 24 70 .000 44 97 19 91 93 82 67 71 36 25 .0073 1.000 .0247 .9997 .0422 .9991 .0596 .9982 .0770 .9970 35 26 76 .000 50 97 25 91 99 82 73 70 34 27 79 .000 53 97 27 91 0602 82 76 70 33 28 81 .000 56 97 30 91 05 82 79 70 32 29 84 .000 59 97 33 91 08 82 82 69 31 30 .0087 1.000 .0262 .9997 .0436 .9990 .0610 .9981 .0785 .9969 30 31 90 .000 65 96 39 90 13 81 87 69 29 32 93 .000 68 96 42 90 16 81 90 69 28 33 96 .000 70 % 45 90 19 81 93 68 27 34 99 .000 73 96 48 90 22 81 96 68 26 35 .0102 .9999 .0276 .9996 .0451 .9990 .0625 .9980 .0799 .9968 25 36 05 99 79 96 54 90 28 80 0802 68 24 37 08 99 82 96 57 90 31 80 05 68 23 38 11 99 85 96 59 89 34 80 08 67 22 39 13 99 88 96 62 89 37 80 11 67 21 40 .0116 .9999 .0291 .9996 .0465 .9989 .0640 .9980 .0814 .9967 20 41 19 99 94 % 68 89 42 79 16 67 19 42 22 99 97 96 71 89 45 79 19 66 IS 43 25 99 0300 96 74 89 48 79 22 66 17 44 28 99 02 95 77 89 51 79 25 66 16 45 .0131 .9999 .0305 .9995 .0480 .9988 .0654 .9979 .0828 .9966 15 46 34 99 . 08 95 83 88 57 78 31 65 14 47 37 99 11 95 86 88 60 78 34 65 13 48 40 99 14 95 88 88 63 78 37 65 12 49 43 99 17 95 91 88 66 78 40 65 11 50 .0145 .9999 .0320 .9995 .0494 .9988 .0669 .9978 .0843 .9964 10 51 48 99 23 95 97 88 71 77 45 64 9 52 51 99 26 95 0500 87 74 77 48 64 8 53 54 99 29 95 03 87 77 77 51 64 7 54 57 99 32 95 06 87 80 77 54 63 6 55 .0160 .9999 .0334 .9994 .0509 .9987 .0683 .9977 .0857 .9963 5 56 63 99 37 94 12 87 86 76 60 63 4 57 66 99 40 94 15 87 89 76 63 63 3 58 69 99 43 94 18 87 92 76 66 62 2 59 72 99 46 94 20 86 95 76 69 62 1 60 .0175 .9998 .0349 .9994 .0523 .9986 .0698 .9976 .0872 .9962 cos sin cos sin cos sin cos sin cos sin ; 89 88 87 86 85 / 76 NATURAL TANGENTS AND COTANGENTS 1 1 2 3 4 i tan cot tan cot tan cot tan cot tan cot .0000 Infinite .0175 57.2900 .0349 28.6363 .0524 19.0811 .0699 14.3007 60 1 03 3437.75 77 56.3506 52 3994 27 18.9755 0702 2411 59 2 06 1718.87 80 55.4415 55 1664 30 8711 05 1821 58 3 09 1145.92 83 54.5613 58 27.9372 33 7678 08 1235 57 4 12 859.436 86 53.7086 61 7117 36 6656 11 0655 56 5 .0015 687.549 .0189 52.8821 .0364 27.4899 .0539 18.5645 .0714 14.0079 55 6 17 572.957 92 0807 67 2715 42 4645 17 13.9507 54 7 20 491.106 95 51.3032 70 0566 44 3655 20 8940 53 8 23 429.718 98 50.5485 73 26.8450 47 2677 23 8378 52 9 26 381.971 0201 49.81:7 75 6367 50 1708 26 7821 51 10 .0029 343.774 .0204 49.1039 .0378 26.4316 .0553 18.0750 .0729 13.7267 50 11 32 312.521 07 48.4121 81 2296 56 17.9802 31 6719 49 12 35 286.478 09 47.7395 84 0307 59 8S63 34 6174 48 13 38 264.441 12 0853 87 25.8348 62 7934 37 5634 47 14 41 245.552 15 46.4489 90 6418 65. 7015 40 5098 46 15 .0044 229.182 .0218 45.8294 .0393 25.4517 .0568 17.6106 .0743 13.4566 45 16 47 214.858 21 2261 96 2644 71 5205 46 4039 44 17 49 202.219 24 44.6386 99 0798 74 4314 49 3515 43 IS 52 190.984 27 0661 0402 24.8978 77 3432 52 2996 42 19 55 180.932 30 43.5081 05 7185 80 2558 55 2480 41 20 .0058 171.885 .0233 42.9641 .0407 24.5418 .0582 17.1693 .0758 13.1969 40 21 61 163.700 36 4335 10 3675 85 083 7 61 1461 39 22 64 156.259 39 41.9158 13 1957 88 16.9990 64 0958 38 23 67 149.465 41 4106 16 0263 91 9150 67 0458 37 24 70 143.237 44 40.9174 19 23.8593 94 8319 69 12.9962 36 25 .0073 137.507 .0247 40.4358 .0422 23.6945 .0597 16.74% .0772 12.9469 35 26 76 132.219 50 39.9655 25 5321 0600 6681 75 8981 34 27 79 127.321 53 5059 28 3718 03 5874 78 8496 33 28 81 122.774 56 0568 31 2137 06 5075 81 8014 32 29 84 118.540 59 38.6177 34 0577 09 4283 84 7536 31 30 .0087 114.589 .0262 38.1885 .0437 22.9038 .0612 16.3499 .0787 12.7062 30 31 90 110.892 65 37.7686 40 7519 15 2722 90 6591 29 32 93 107.426 68 3579 42 6020 17 1952 93 6124 28 33 96 104.171 71 36.9560 45 4541 20 1190 96 5660 27 34 99 101.107 74 5627 48 3081 23 0435 99 5199 26 35 .0102 98.2179 .0276 36.1776 .0451 22.1640 .0626 15.9687 .0802 12.4742 25 36 05 95.4895 79 35.8006 54 0217 29 8945 05 4288 24 37 08 92.9085 82 4313 57 21.8813 32 8211 08 3838 23 38 11 90.4633 85 0695 60 7426 35 7483 10 3390 22 39 13 88.1436 88 34.7151 63 6056 38 6762 13 2946 21 40 .0116 85.9398 .0291 34.3678 .0466 21.4704 .'0641 15.6048 .0816 12.2505 20 41 19 83.8435 94 0273 69 3369 44 5340 19 2067 19 42 22 81.8470 97 33.6935 72 2049 47 4638 22 1632 18 43 25 79.9434 0300 3662 75 0747 50 3943 25 1201 17 44 28 78.1263 03 0452 77 20.9460 53 3254 28 0772 16 45 .0131 76.3900 .0306 32.7303 .0480 20.8188 .0655 15.2571 .0831 12.0346 15 46 34 74.7292 08 4213 S3 6932 58 1893 34 11.9923 14 47 37 73.1390 11 1181 86 5691 61 1222 37 9504 13 4S 40 71.6151 14 31.8205 89 4465 64 0557 40 9087 12 49 43 70.1533 17 5284 92 3253 67 14.9898 43 8673 11 50 .0145 68.7501 .0320 31.2416 .0495 20.2056 .0670' 14.9244 .0846 11.8262 10 51 48 67.4019 23 30.9599 98 0872 73 8596 49 7853 9 52 51 66.1055 26 6833 0501 19.9702 76 7954 51 7448 8 53 54 64.8580 29 4116 04 85-16 79 7317 54 7045 7 54 57 63.6567 32 1446 07 7403 82 6685 57 6645 6 55 .0160 62.4992 .0335 29.8823 .0509 19.6273 .0685 14.6059 .0860 11.6248 5 56 63 61.3829 38 6245 12 5156 88 5438 63 5853 4 57 66 60.3058 40 3711 15 4051 90 4823 66 5461 3 58 69 59.2659 43 1220 18 2959 93 4212 69 5072 2 59 72 58.2612 46 28.8771 21 1879 96 3607 72 4685 1 60 .0175 57.2900 .0349 28.6363 .0524 19.0811 .0699 14.3007 .0875 11.4301 cot tan cot tan cot tan cot tan cot tan / 89 88 87 86 85 J i 11 NATURAL SINES AND COSINES 1 5 6 7 QO 9 i sin cos sin cos sin cos sin cos sin cos .0872 .9962 .1045 .9945 .1219 .9925 .1392 .9903 .1564 .9877 60 1 74 62 48 45 22 25 95 02 67 76 59 2 77 61 51 45 24 25 97 02 70 76 58 3 80 61 54 44 27 24 1400 01 73 76 57 4 83 61 57 44 30 24 03 01 76 75 56 5 .0886 .9%! .1060 .9944 .1233 .9924 .1406 .9901 .1579 .9875 55 6 89 60 63 43 36 23 09 00 82 74 54 7 92 60 66 43 39 23 12 00 84 74 53 8 95 60 68 43 42 23 15 9899 87 73 52 9 98 60 71 42 45 22 18 99 90 73 51 10 .0901 .9959 .1074 .9942 .1248 .9922 .1421 .9899 .1593 .9872 50 11 03 59 77 42 50 22 23 98 96 72 49 12 06 59 80 42 53 21 26 98 99 71 48 13 09 59 83 41 56 21 29 97 1602 71 47 14 12 58 86 41 59 20 32 97 05 70 46 15 .0915 .9958 .1089 .9941 .1262 .9920 .1435 .9897 .1607 .9870 45 16 18 58 92 40 65 20 38 96 10 69 44 17 21 58 94 40 68 19 41 96 13 69 43 18 24 57 97 40 71 19 44 95 16 69 42 19 27 57 1100 39 74 19 46 95 19 68 41 20 .0929 .9957 .1103 .9939 .1276 .9918 .1449 .9894 .1622 .9868 40 21 32 56 06 39 79 18 52 94 25 67 39 22 35 56 09 38 82 17 55 94 28 67 38 23 38 56 12 38 85 17 58 93 30 66 37 24 41 56 15 38 88 17 61 93 33 66 36 25 .0944 .9955 .1118 .9937 .1291 .9916 .1464 .9892 .1636 .9865 35 26 47 55 20 37 94 16 67 92 39 65 34 27 50 55 23 37 97 16 69 91 42 64 33 28 53 55 26 36 99 15 72 91 45 64 32 29 56 54 29 36 1302 15 75 91 48 63 31 30 .0958 .9954 .1132 .9936 .1305 .9914 .1478 .9890 .1650 .9863 30 31 61 54 35 35 08 14 81 90 53 62 29 32 64 53 38 35 11 14 84 89 56 62 28 33 67 53 41 35 14 13 87 89 59 61 27 34 70 53 44 34 17 13 90 88 62 61 26 35 .0973 .9953 .1146 .9934 .1320 .9913 .1492 .9888 .1665 .9860 25 36 76 52 49 34 23 12 95 88 68 60 24 37 79 52 52 33 25 12 98 87 71 59 23 38 82 52 55 33 28 11 1501 87 73 59 22 39 85 51 58 33 31 11 04 86 76 59 21 40 .0987 .9951 .1161 .9932 .1334 .9911 .1507 .9886 .1679 .9858 20 41 90 51 64 32 37 10 10 85 82 58 19 42 93 51 67 32 40 10 13 85 85 57 18 43 96 50 70 31 43 09 15 84 88 57 17 44 99 50 72 31 46 09 18 84 91 56 16 45 .1002 .9950 .1175 .9931 .1349 .9909 .1521 .9884 .1693 .9856 15 46 05 49 78 30 51 08 24 83 96 55 14 47 08 49 81 30 54 08 27 83 99 55 13 48 11 49 84 30 57 07 30 82 1702 54 12 49 13 49 87 29 60 07 33 82 05 54 11 50 .1016 .9948 .1190 .9929 .1363 .9907 .1536 .9881 .1708 .9853 10 51 19 48 93 29 66 06 38 81 11 53 9 52 22 48 96 28 69 06 41 80 14 52 8 53 25 47 98 28 72 05 44 80 16 52 7 54 28 47 1201 28 74 05 47 80 19 51 6 55 .1031 .9947 .1204 .9927 .1377 .9905 .1550 .9879 .1722 .9851 5 56 34 46 07 27 80 04 53 79 25 50 4 57 37 46 10 27 83 04 56 78 28 50 3 58 39 46 13 26 86 03 59 78 31 49 2 59 42 46 16 26 89 03 61 77 34 49 1 60 .1045 .9945 .1219 .9925 .1392 .9903 .1564 .9877 .1736 .9848 cos sin cos sin cos sin cos sin cos sin t 84^ 83 82 81 80 / 78 NATURAL TANGENTS AND COTANGENTS / 5 6 7 8 9 / tan cot tan cot tan cot tan cot tan cot .0875 11.4301 .1051 9.5144 .1228 8.1443 .1405 7.1154 .1584 6.3138 60 1 78 3919 54 4878 31 1248 08 1004 87 3019 59 2 81 3540 57 4614 34 1054 11 0855 90 2901 58 3 84 3163 60 4352 37 0860 14 0706 93 2783 57 4 87 2789 63 4090 40 0667 17 0558 96 2666 56 5 .0890 11.2417 .1066 9.3831 .1243 8.0476 .1420 7.0410 .1599 6.2549 55 6 92 2048 69 3572 46 0285 23 0264 1602 2432 54 7 95 1681 72 3315 49 0095 26 0117 05 2316 53 8 98 1316 75 3060 51 7.9906 29 6.9972 08 2200 52 9 0901 0954 78 2806 54 9718 32 9827 11 2085 51 10 .0904 11.0594 .1080 9.2553 .1257 7.9530 .1435 6.9682 .1614 6.1970 50 11 07 0237 83 2302 60 9344 38 9538 17 1856 49 12 10 10.9882 86 2052 63 9158 41 9395 20 1742 48 13 13 9529 89 1803 66 8973 44 9252 23 1628 47 14 16 9178 92 1555 69 8789 47 9110 26 1515 46 15 .0919 10.8829 .1095 9.1309 .1272 7.8606 .1450 6.8969 .1629 6.1402 45 16 22 8483 98 1065 75 8424 53 8828 32 1290 44 17 25 8139 1101 0821 78 8243 56 8687 35 1178 43 18 28 7797 04 0579 81 8062 59 8548 38 1066 42 19 31 7457 07 0338 84 7882 62 8408 41 0955 41 20 .0934 10.7119 .1110 9.0098 .1287 7.7704 .1465 6.8269 .1644 6.0844 40 21 36 6783 13 8.9860 90 7525 68 8131 47 0734 39 22 39 6450 16 9623 93 7348 71 7994 50 0624 38 23 42 6118 19 9387 96 7171 74 7856 53 0514 37 24 45 5789 22 9152 99 6996 77 7720 55 0405 36 25 .0948 10.5462 .1125 8.8919 .1302 7.6821 .1480 6.7584 .1658 6.0296 35 26 51 5136 28 8686 05 6647 83 7448 61 0188 34 27 54 4813 31 8455 08 6473 86 7313 64 0080 33 28 57 4491 33 8225 11 6301 89 7179 67 5.9972 32 29 60 4172 36 7996 14 6129 92 7045 70 9865 31 30 .0963 10.3854 .1139 8.7769 .1317 7.5958 .1495 6.6912 .1673 5.9758 30 31 66 3538 42 7542 19 5787 97 6779 76 9651 29 32 69 3224 45 7317 22 5618 1500 6646 79 9545 28 33 72 2913 48 7093 25 5449 03 6514 82 9439 27 34 75 2602 51 6870 28 5281 06 6383 85 9333 26 35 .0978 10.2294 .1154 8.6648 .1331 7.5113 .1509 6.6252 .1688 5.9228 25 36 81 1988 57 6427 34 4947 12 6122 91 9124 24 37 83 1683 60 6208 37 4781 15 5992 94 9019 23 38 86 1381 63 5989 40 4615 18 5863 97 8915 22 39 89 1080 66 5772 43 4451 21 5734 1700 8811 21 40 .0992 10.0780 .1169 8.5555 .1346 7.4287 .1524 6.5606 .1703 5.8708 20 41 95 0483 72 5340 49 4124 27 5478 06 8605 19 42 98 0187 75 5126 52 3962 30 5350 09 8502 18 43 1001 9.9893 78 4913 55 3800 33 5223 12 8400 17 44 04 9601 81 4701 58 3639 36 5097 15 8298 16 45 .1007 9.9310 .1184 8.4490 .1361 7.3479 .1539 6.4971 .1718 5.8197 15 46 10 9021 87 4280 64 3319 42 4846 21 8095 14 47 13 8734 89 4071 67 3160 45 4721 24 7994 13 48 16 8448 92 3863 70 3002 48 4596 27 7894 12 49 19 8164 95 3656 73 2844 51 4472 30 7794 11 50 .1022 9.7882 .1198 8.3450 .1376 7.2687 .1554 6.4348 .1733 5.7694 10 51 25 7601 1201 3245 79 2531 57 4225 36 7594 9 52 28 7322 04 3041 82 2375 60 4103 39 7495 8 53 30 7044 07 2838 85 2220 63 3980 42 7396 7 54 33 6768 10 2636 88 2066 66 3859 45 7297 6 55 .1036 9.6493 .1213 8.2434 .1391 7.1912 .1569 6.3737 .1748 5.7199 5 56 39 6220 16 2234 94 1759 72 3617 51 7101 4 57 42 5949 19 2035 97 1607 75 3496 54 7004 3 58 45 5679 22 1837 99 1455 78 3376 57 6906 2 59 48 5411 25 1640 1402 1304 81 3257 60 6809 1 60 .1051 9.5144 .1228 8.1443 .1405 7.1154 .1584 6.3138 .1763 5.6713 cot tan cot tan cot tan cot tan cot tan / 84 83 82 81 80 ! 79 NATURAL SINES AND COSINES 1 10 11 12 13V 14 / sin cos sin cos sin cos sin cos sin cos .1736 .9848 .1908 .9816 .2079 .9781 .2250 .9744 .2419 .9703 60 1 39 48 11 16 82 81 52 43 22 02 59 2 12 47 14 15 85 80 55 42 25 02 58 3 45 47 17 15 88 80 58 42 28 01 57 4 48 46 20 14 90 79 61 41 31 00 56 5 .1751 .9846 .1922 .9813 .2093 .9778 .2264 .9740 .2433 .9699 55 6 54 45 25 13 96 78 67 40 36 99 54 7 57 45 28 12 99 77 69 39 39 98 53 8 59 44 31 12 2102 77 72 38 42 97 52 9 62 43 34 11 OS. 76 75 38 45 97 51 10 .1765 .9843 .1937 .9811 .2108 .9775 .2278 .9737 .2447 .9696 50 11 68 42 39 10 10 75 81 36 50 95 49 12 71 42 42 10 13 74 84 36 53 94 48 13 74 41 45 09 16 74 86 35 56 94 47 14 77 41 48 08 19 73 89 34 59 93 46 15 .1779 .9840 .1951 .9808 .2122 .9772 .2292 .9734 .2462 .9692 45 16 82 40 54 07 25 72" 95 33 64 92 44 17 85 39 57 07 27 71 98 32 67 91 43 18 88 39 59 06 30 70 2300 32 70 90 42 19 91 38 62 06 33 70 03 31 73 89 41 20 .1794 .9838 .1965 .9805 .2136 .9769 .2306 .9730 .2476 .9689 40 21 97 37 68 04 39 69 09 30 78 88 39 22 99 37 71 04 42 68 12 29 81 87 38 23 1802 36 74 03 45 67 15 28 84 87 37 24 05 36 77 03 47 67 17 28 87 86 36 25 .1808 .9835 .1979 .9802 .2150 .9766 .2320 .9727- .2490 .9685 35 26 11 35 82 02 53 65 23 26 93 84 34 27 14 34 85 01 56 65 26 26 95 84 33 28 17 34 88 00 59 64 29 25 98 83 32 29 19 33 91 00 62 64 32 24 .2501 82 31 30 .1822 .9833 .1994 .9799 .2164 .9763 .2334 .9724 .2504 .9681 30 31 25 32 97 99 67 62 37 23 07 81 29 32 28 31 99 98 70 62 40 22 09 80 28 33 31 31 2002 98 73 61 43 22 12 79 27 34 34 30 05 97 76 60 46 21 15 79 26 35 .1837 .9830 .2008 .9796 , .2179 .9760 .2349 .9720 .2518 .9678 25 36 40 29 11 96 81 59 51 20 21 77 24 37 42 29 14 95 84 59 54 19 24 76 23 38 45 28 16 95 87 58 57 18 26 76 22 39 48 28 19 94 90 57 60 18 29 75 21 40 .1851 .9827 .2022 .9793 .2193 .9757 .2363 .9717 .2532 .9674 20 41 54 27 25 93 % 56 66 16 35 73 19 42 57 26 28 92 98 55 68 15 38 73 18 43 60 26 31 92 2201 55 71 15 40 72 17 44 62 25 34 91 04 54 74 14 43 71 16 45 .1865 .9825 .2036 .9790 .2207 .9753 .2377 .9713 .2546 .9670 15 46 68 . 24 39 90 10 53 80 13 49 70 14 47 71 23 42 89 13 52 83 12 52 69 13 48 74 23 45 89 15 51 85 11 54 68 12 49 77 22 48 88 18 51 88 11 57 67 11 50 .1880 .9822 .2051 .9787 .2221 .9750 .2391 .9710 .2560 .9667 10 51 82 21 54 87 24 50 94 09 63 66 9 52 85 21 56 86 27 49 97 09 66 65 8 53 88 20 59 86 30 48 99 08 69 65 7 54 91 20 62 85 33 48 .2402 07 71 64 6 55 .1894 .9819 .2065 .9784 .2235 .9747 .2405 .9706 .2574 .9663 5 56 97 18 68 84 38 46 08 06 77 62 4 57 1900 18 71 83 41 46 11 05 80 62 3 58 02 17 73 83 44 45 14 04 83 61 2 59 05 17 76 82 47 44 16 04 85 60 1 60 .1908 .9816 .2079 .9781 .2250 .9744 .2419 .9703 .2588 .9659 cos sin cos sin cos sin cos sin cos sin i 79 78 77 76 75 J / 80 NATURAL TANGENTS AND COTANGENTS / 10 11 12 13 14 / tan cot tan cot tan cot tan cot tan cot .1763 5.6713 .1944 5.1446 .2126 4.7046 .2309 4.3315 .2493 4.0108 60 1 66 6617 47 1366 29 6979 12 3257 96 0058 59 2 69 6521 50 1286 ' 32 6912 15 3200 99 0009 58 3 72 6425 53 1207 35 6845 18 3143 2503 3.9959 57 4 75 6329 56 1128 38 6779 21 3086 06 9910 56 5 .1778 5.6234 .1959 5.1049 .2141 4.6712 .2324 4.3029 .2509 3.9861 55 6 81 6140 62 0970 44 6646 27 2972 12 9S12 54 7 84 6045 65 0892 47 6580 30 2916 15 9763 53 S 87 5951 68 0814 50 6514 33 2859 18 9714 52 9 90 5857 71 0736 53 6448 36 2803 21 9665 51 10 .1793 5.5764 .1974 5.0658 .2156 4.6382 .2339 4.2747 .2524 3.9617 50 11 96 5671 77 0581 59 6317 42 2691 27 9568 49 12 99 5578 80 0504 62 6252 45 2635 30 9520 48 13 1802 5485 83 0427 65 6187 49 2580 33 9471 47 14 05 5393 86 035.0 68 6122 52 2524 37 9423 46 15 .1808 5.5301 .1989 5.0273 .2171 4.6057 .2355 4.2468 .2540 3.9375 45 16 11 5209 92 0197 74 5993 58 2413 43 9327 44 17 14 5118 95 0121 77 5928 61 2358 46 9279 43 IS 17 5026 98 0045 80 5864 64 2303 49 9232 42 19 20 4936 2001 4.9969 83 5800 67 2248 52 9184 41 20 .1823 5.4845 .2004 4.9894 .2186 4.5736 .2370 4.2193 .2555 3.9136 40 21 26 4755 07 9819 89 5673 73 2139 58 9089 39 22 29 4665 10 9744 93 5609 76 2084 61 9042 38 23 32 4575 13 9669 96 5546 79 2030 64 8995 37 24 35 4486 16 9594 99 5483 82 1976 68 8947 36 25 .1838 5.4397 .2019 4.9520 .2202 4.5420 .2385 4.1922 .2571 3.8900 35 26 41 4308 22 9446 05 5357 88 1868 74 8854 34 27 44 4219 25 9372 08 5294 92 1814 77 8807 33 28 47 4131 28 9298 11 5232 95 1760 80 8760 32 29 50 4043 31 9225 14 5169 98 1706 83 8714 31 30 .1853 5.3955 .2035 4.9152 .2217 4.5107 .2401 4.1653 .2586 3.8667 30 31 56 3868 38 9078 20 5045 04 1600 89 8621 29 32 59 3781 41 9006 23 4983 07 1547 92 8575 28 33 62 3694 44 8933 26 4922 10 1493 95 8528 27 34 65 3607 47 8860 29 4860 13 1441 99 8482 26 35 .1868 5.3521 .2050 4. 8788 .2232 4.4799 .2416 4.1388 .2602 3.8436 25 36 71 3435 53 8716 35 4737 19 1335 05 8391 24 37 74 3349 56 8644 38 4676 22 1282 08 8345 23 38 77 3263 59 8573 41 4615 25 1230 11 8299 22 39 80 3178 62 8501 44 4555 28 1178 14 8254 21 40 .1883 5.3093 .2065 4.8430 .2247 4.4494 .2432 4.1126 .2617 3.8208 20 41 87 3008 68 8359 51 4434 35 1074 20 8163 19 42 90 2924 71 8288 54 4373 38 1022 23 8118 18 43 93 2839 74 8218 57 4313 41 0970 27 8073 17 44 96 2755 77 8147 60 4253 44 0918 30 8028 16 45 .1899 5.2672 .2080 4.8077 .2263 4.4194 .2447 4.0867 .2633 3.7983 15 46 1902 2588 83 8007 66 4134 50 0815 36 7938 14 47 05 2505 86 7937 69 4075 53 0764 39 7893 13 48 08 2422 89 7867 72 4015 56 0713 42 7848 12 49 11 2339 92 7798 75 3956 59 0662 45 7804 11 50 .1914 5.2257 .2095 4.7729 .2278 4.3897 .2462 4.0611 .2648 3.7760 10 51 17 2174 98 7659 81 3838 65 0560 51 7715 9 52 20 2092 2101 7591 84 3779 69 0509 55 7671 8 53 23 2011 04 7522 87 3721 72 0459 58 7627 7 54 26 1929 07 7453 90 3662 75 0408 61 7583 6 55 .1929 5.1848 .2110 4.7385 .2293 4.3604 .2478 4.0358 .2664 3.7539 5 56 32 1767 13 7317 96 3546 81 0308 67 7495 4 57 35 1686 16 7249 99 3488 84 0257 70 7451 3 58 38 1606 19 7181 2303 3430 87 0207 73 7408 2 59 41 1526 23 7114 06 3372 90 0158 76 7364 1 60 .1944 5.1446 .2126 4.7046 .2309 4.3315 .2493 4.0108 .2679 3.7321 cot tan cot tan cot tan cot tan cot tan / 79 78 77 76 75 i 81 NATURAL SINES AND COSINES 1 15 16 17 18 19 / sin cos sin cos sin cos sin cos sin cos .2588 .9659 .2756 .9613 .2924 .9563 .3090 .9511 .3256 .9455 60 1 91 59 59 12 26 62 93 10 58 54 59 2 94 58 62 11 29 61 96 09 61 53 58 3 97 57 65 10 32 60 98 08 64 52 57 4 99 56 68 09 3^ 60 3101 07 67 51 56 5 .2602 .9655 .2770 .9609 .2938 .9559 .3104 .9506 .3269 .9450 55 6 05 55 73 08 40 58 07 05 72 49 54 7 08 54 76 07 43 57 10 04 75 49 53 8 11 -53 79 06 46 56 12 03 78 48 52 9 13 52 82 05 49 55 15 02 80 47 51 10 .2616 .9652 .2784 .9605 .2952 .9555 .3118 .9502 .3283 .9446 50 11 19 51 87 04 54 54 21 01 86 45 49 12 22 50 90 03 57 53 23 9500 89 44 48 13 2^ 49 93 02 60 52 26 9499 91 43 47 14 28 49 95 01 63 51 29 98 94 42 46 15 .2630 .9648 .2798 .9600 .2965 .9550 .3132 .9497 .3297 .9441 45 16 33 47 2801 00. 68 49 34 96 3300 40 44 17 36 46 04 9599 71 48 37 95 02 39 43 18 39 46 07 98 74 48 40 94 05 38 42 19 *2 45 09 97 77 '7 43 93 08 37 41 20 .2644 .9644 .2812 .9596 .2979 .9546 .3145 .9492 .3311 .9436 40 21 47 43 15 96 82 45 48 92 13 35 39 22 50 42 18 95 85 44 51 91 16 34 38 23 53 42 21 94 88 43 54 90 19 33 37 24 56 41 23 93 90 42 56 89 22 32 36 25 .2658 .9640 .2826 .9592 .2993 .9542 .3159 .9488 .3324 .9431 35 26 61 39 29 91 96 41 62 87 27 30 34 27 64 39 32 91 99 40 65 86 30 29 33 28 67 38 35 90 3002 39 68 - 85 33 28 32 29 70 37 37 89 04 38 70 84 35 27 31 30 .2672 .9636 .2840 .9588 .3007 .9537 .3173 .9483 .3338 .9426 30 31 75 36 43 87 10 36 76 82 41 25 29 32 78 35 46 87 13 35 79 81 44 24 28 33 81 34 49 86 15 35 81 80 46 23 27 34 84 33 51 85 18 34 84 80 49 23 26 35 .2686 .9632 .2854 .9584 .3021 .9533 .3187 .9479 .3352 .9422 25 36 89 32 57 83 24 32 90 78 55 21 24 37 92 31 60 82 26 31 92 77 57 20 23 38 95 30 62 82 29 30 95 76 60 19 22 39 98 29 65 81 32 29 98 75 63 18 21 40 .2700 .9628 .2868 .9580 .3035 .9528 .3201 .9474 .3365 .9417 20 41 03 28 71 79 38 27 03 73 68 16 19 42 06 27 74 78 40 27 06 72 71 15 18 43 09' 26 76 77 43 26 09 71 74 14 17 44 12 25 79 77 46 25 12 70 76 13 16 45 .2714 .9625 .2882 .9576 .3049 .9524 .3214 .9469 .3379 .9412 15 46 17 24 85 75 51 23 17 68 82 11 14 47 20 23 88 74 54 22 20 67 85 10 13 48 23 22 90 73 57 21 23 66 87 09 12 49 26 21 93 72 60 20 25 66 90 08 11 50 2728 .9621 .2896 .9572 .3062 .9520 .3228 .9465 .3393 .9407 10 51 31 20 99 71 65 19 31 64 96 06 9 52 34 19 2901 70 68 18 34 63 98 05 8 53 37 18 04 69 71 17 36 62 3401 04 7 54 40 17 07 68 74 16 39 61 04 03 6 55 .2742 .9617 .2910 .9567 .3076 .9515 .3242 .9460 .3407 .9402 5 56 45 16 13 66 79 14 45 59 09 01 4 57 48 15 15 66 82 13 47 58 12 00 3 58 51 14 18 65 85 12 50 57 15 9399 2 59 54 13 21 64 87 11 53 56 17 98 1 60 .2756 .9613 .2924 .9563 .3090 .9511 .3256 .9455 .3420 .9397 cos sin cos sin cos sin cos sin cos sin i 74 73 72 71 70 i 82 NATURAL TAX GENTS AND COTANGENTS / 15 16 17 18 19 / tan cot tan cot tan cot tan cot tan cot .2679 3.7321 .2867 3.4874 .3057 3.2709 .3249 3.0777 .3443 2.9042 60 1 83 7277 71 4836 60 2675 52 0746 47 9015 59 2 86 7234 74 4798 64 2641 56 0716 50 8987 58 3 89 7191 77 4760 67 2607 59 0686 53 8960 57 4 92 7148 80 4722 70 2573 62 0655 56 8933 56 5 .2695 3.7105 .2883 3.4684 .3073 3.2539 .3265 3.0625 .3460 2.8905 55 6 98 7062 86 4646 76 2506 69 0595 63 8878 54 7 2701 7019 90 4608 80 2472 72 0565 66 8851 53 8 04 6976 93 4570 83 2438 75 0535 69 8824 52 9 OS 6933 96 4533 86 2405 78 0505 73 8797 51 10 .2711 3.6891 .2899 3.4495 .3089 3.2371 .3281 3.0475 .3476 2.8770 50 11 14 6848 2902 4458 92 2338 85 0445 79 8743 49 12 17 6S06 05 4420 96 2305 88 0415 82 8716 48 13 20 6764 08 4383 99 2272 91 0385 86 8689 47 14 23 6722 12 4346 3102 2238 94 0356 89 8662 46 15 .2726 3.66SO .2915 3.4308 .3105 3.2205 .3298 3.0326 .3492 2.8636 45 16 29 663S 18 4271 08 2172 3301 0296 95 8609 44 17 33 6596 21 4234 11 2139 04 0267 99 8582 43 IS 36 6554 24 4197 15 2106 07 0237 3502 8556 42 19 39 6512 27 4160 18 2073 10 0208 05 8529 41 20 .2742 3.6470 .2931 3.4124 .3121 3.2041 .3314 3.0178 .3508 2.8502 40 21 45 6429 34 4087 24 2008 17 0149 12 8476 39 22 48 6387 37 4050 27 1975 20 0120 15 8449 38 23 51 6346 40 4014 31 1943 23 0090 18 8423 37 24 54 6305 43 3977 34 1910 27 0061 22 8397 36 25 .2758 3.6264 .2946 3.3941 .3137 3.1878 .3330 3.0032 .3525 2.8370 35 26 61 6222 49 3904 40 1845 33 0003 28 8344 34 27 64 6181 53 3868 43 1813 36 2.9974 31 8318 33 28 67 6140 56 3832 47 1780 39 9945 35 8291 32 29 70 6100 59 3796 50 1748 43 9916 38 8265 31 30 .2773 3.6059 .2962 3.3759 .3153 3.1716 .3346 2.9887 .3541 2.8239 30 31 76 6018 65 3723 56 1684 49 9858 44 8213 29 32 80 5978 68 3687 59 1652 52 9829 48 8187 28 33 83 5937 72 3652 63 1620 56 9800 51 8161 27 34 86 5897 75 3616 66 1588 59 9772 54 8135 26 35 .2789 3.5856 .2978 3.3580 .3169 3.1556 .3362 2.9743 .3558 2.8109 25 36 92 5816 81 3544 72 1524 65 9714 61 8083 24 37 95 5776 84 3509 75 1492 69 9686 64 8057 23 38 98 5736 87 3473 79 1460 72 9657 67 8032 22 39 2801 5696 91 3438 82 1429 75 9629 71 8006 21 40 .2805 3.5656 .2994 3.3402 .3185 3.1397 .3378 2.9600 .3574 2.7980 20 41 08 5616 97 3367 88 1366 82 9572 77 7955 19 42 11 5576 3000 3332 91 1334 85 9544 81 7929 18 43 14 5536 03 3297 95 1303 88 9515 84 7903 17 44 17 5497 06 3261 98 1271 91 9487 87 7878 16 45 .2820 3.5457 .3010 3.3226 .3201 3.1240 .3395 2.9459 .3590 2.7852 15 46 23 5418 13 3191 04 1209 98 9431 94 7827 14 47 27 5379 16 3156 07 1178 3401 9403 97 7801 13 48 30 5339 19 3122 11 1146 04 9375 3600 7776 12 49 33 5300 22 3087 14 1115 08 9347 04 7751 11 50 .2836 3.5261 .3026 3.3052 .3217 3.1084 .3411 2.9319 .3607 2.7725 10 51 39 5222 29 3017 20 1053 14 9291 10 7700 9 52 42 5183 32 2983 23 1022 17 9263 13 7675 8 53 45 5144 35 2948 27 0991 21 9235 17 7650 7 54 49 5105 38 2914 30 0961 24 9208 20 7625 6 55 .2852 3.5067 .3041 3.2879 .3233 3.0930 .3427 2.9180 .3623 2.7600 5 56 55 5028 45 2845 36 0899 30 9152 27 7575 4 57 58 4989 48 2811 40 0868 34 9125 30 7550 3 58 61 4951 51 2777 43 0838 37 9097 33 7525 2 59 64 4912 54 2743 46 0807 40 9070 36 7500 1 60 .2867 3.4874 .3057 3.2709 .3249 3.0777 .3443 2.9042 .3640 2.7475 cot tan cot tan cot tan cot tan cot tan i 74 73 72 71 70 / NATURAL SINES AND COSINES / 20 21 22 23 24 i sin cos sin cos sin cos sin cos sin cos .3420 .9397 .3584 .9336 .3746 .9272 .3907 .9205 .4067 .9135 60 1 23 96 86 35 49 71 10 04 70 34 59 2 26 95 89 34 51 70 13 03 73 33 58 3 28 94 92 33 54 69 15 02 75 32 57 4 31 93 95 32 57 67 18 00 78 31 56 5 .3434 .9392 .3597 .9331 .3760 .9266 .3921 .9199 .4081 .9130 55 6 37 91 3600 30 62 65 23 98 83 28 54 7 39 90 03 28 65 64 26 97 86 27 53 8 42 89 05 27 68 63 29 96 89 26 52 9 45 88 08 26 70 62 31 95 91 25 51 10 .3448 .9387 .3611 .9325 .3773 .9261 .3934 .9194 .4094 .9124 50 11 50 86 14 24 76 60 37 92 97 22 49 12 53 85 16 23 78 59 39 91 99 21 48 13 56 84 19 22 81 58 42 90 4102 20 47 14 58 83 22 21 84 57 45 89 05 19 46 15 .3461 .9382 .3624 .9320 .3786 .9255 .3947 .9188 .4107 .9118 45 16 64 81 27 19 89 54 50 87 10 16 44 17 67 80 30 18 92 53 53 86 12 15 43 18 69 79 33 17 95 52 55 84 15 14 42 19 72 78 35 16 97 51 58 83 18 13 41 20 .3475 -9377 .3638 .9315 .3800 .9250 .3961 ,9182 .4120 .9112 40 21 78 76 41 14 03 49 63 81 23 10 39 22 80 75 43 13 05 48 66 80 26 09 38 23 83 74 46 12 08 47 69 79 28 08 37 24 86 73 49 11 11 45 71 78 31 07 36 25 .3488 .9372 .3651 .9309 .3813 .9244 .3974 .9176 .4134 .9106 35 26 91 71 54 08 16 43 77 75 36 04 34 27 94 70 57 07 19 42 79 74 39 03 33 28 97 69 60 06 21 41 82 73 42 02 32 29 99 68 62 05 24 40 85 72 44 01 31 30 .3502 .9367 .3665 .9304 .3827 .9239 .3987 .9171 .4147 .9100 30 31 05 66 68 03 30 38 90 69 50 9098 29 32 08 65 70 02 32 37 93 68 52 97 28 33 10 64 73 01 35 35 95 67 55 96 27 34 13 63 76 00 38 34 98 66 58 95 26 35 .3516 .9362 .3679 .9299 .3840 .9233 .4001 .9165 .4160 .9094 25 36 18 61 81 98 43 32 03 64 63 92 24 37 21 60 84 97 46 31 06 62 65 91 23 38 24 59 87 96 48 30 09 61 68 90 22 39 27 58 89 95 51 29 11 60 71 89 21 40 .3529 .9356 .3692 .9293 .3854 .9228 .4014 .9159 .4173 .9088 20 41 32 55 95 92 56 27 17 58 76 86 19 42 35 54 97 91 59 25 19 57 79 85 18 43 37 53 3700 90 62. 24 22 55 81 84 17 44 40 52 03 89 64 23 25 54 84 83 16 45 .3543 .9351 .3706 .9288 .3867 .9222 .4027 .9153 .4187 .9081 15 46 46 50 08 87 70 21 30 52 89 80 14 47 48 49 11 86 72 20 33 51 92 79 13 48 51 48 14 85 75 19 35 50 95 78 12 49 54 47 16 84 78 18 38 48 97 77 11 50 .3557 .9346 .3719 .9283 .3881 .9216 .4041 .9147 .4200 .9075 10 51 59 45 22 82 83 15 43 46 02 74 9 52 62 44 24 81 86 14 46 45 05 73 8 53 65 43 27 79 89 13 49 44 08 72 7 54 67 42 30 78 91 12 51 43 10 70 6 55 .3570 .9341 .3733 .9277 .3894 .9211 .4054 .9141 .4213 .9069 5 56 73 40 35 76 97 10 57 40 16 68 4 57 76 39 38 75 99 08 59 39 18 67 3 58 78 38 41 74 3902 07 62 38 21 66 2 59 81 37 43 73 05 06 65 37 24 64 1 60 .3584 .9336 .3746 .9272 .3907 .9205 .4067 .9135 .4226 .9063 cos sin cos sin cos sin cos sin cos sin / 69 68 67 66 65 5 / 84 NATURAL TANGENTS AND COTANGENTS / 20 21 22 23 24 / tan cot tan cot tan cot tan cot tan cot .3640 2.7475 .3839 2.6051 .4040 2.4751 .4245 2.3559 .4452 2.2460 60 1 43 7450 42 6028 44 4730 48 3539 56 2443 59 2 46 7425 45 6006 47 4709 52 3520 59 2425 58 3 50 7400 49 5983 50 4689 55 3501 63 2408 57 4 53 7376 52 5961 54 4668 58 3483 66 2390 56 5 .3656 2.7351 .3855 2.5938 .4057 2.4648 .4262 2.3464 .4470 2.2373 55 6 59 7326 59 5916 61 4627 65 3445 73 2355 54 7 63 7302 62 5893 64 4606 69 3426 77 2338 53 8 66 7277 65 5871 67 4586 72 3407 80 2320 52 9 69 7253 69 5848 71 4566 76 3388 84 2303 51 10 .3673 2.7228 .3872 2.5826 .4074 2.4545 .4279 2.3369 .4487 2.2286 50 11 76 7204 75 5804 78 4525 83 3351 91 2268 49 12 79 7179 79 5782 81 4504 86 3332 94 2251 48 13 83 7155 82 5759 84 4484 89 3313 98 2234 47 14 86 7130 85 5737 88 4464 93 3294 4501 2216 46 15 .3689 2.7106 .3889 2.5715 .4091 2.4443 .4296 2.3276 .4505 2.2199 45 16 93 7082 92 5693 95 4423 4300 3257 08 2182 44 17 96 7058 95 5671 98 4403 03 3238 12 2165 43 IS 99 7034 99 5649 4101 4383 07 3220 15 2148 42 19 3702 7009 3902 5627 05 4362 10 3201 19 2130 41 20 .3706 2.6985 .3906 2.5605 .4108 2.4342 .4314 2.3183 .4522 2.2113 40 21 09 6961 09 5583 11 4322 17 3164 26 2096 39 22 12 6937 12 5561 15 4302 20 3146 29 2079 38 23 16 6913 16 5539 18 4282 24 3127 33 2062 37 24 19 6889 19 5517 22 4262 27 3109 36 2045 36 25 .3722 2.6865 .3922 2.5495 .4125 2.4242 .4331 2.3090 .4540 2.2028 35 26 26 6841 26 5473 29 4222 34 3072 43 2011 34 27 29 6818 29 5452 32 4202 38 3053 47 1994 33 28 32 6794 32 5430 35 4182 41 3035 50 1977 32 29 36 6770 36 5408 39 4162 45 3017 54 1960 31 30 .3739 2.6746 .3939 2.5386 .4142 2.4142 .4348 2.2998 .4557 2.1943 30 31 42 6723 42 5365 46 4122 52 2980 61 1926 29 32 45 6699 46 5343 49 4102 55 2962 64 1909 28 33 49 6675 49 5322 52 4083 59 2944 68 1892 27 34 52 6652 53 5300 56 4063 62 2925 71 1876 26 35 .3755 2.6628 .3956 2.5279 .4159 2.4043 .4365 2.2907 .4575 2.1859 25 36 59 6605 59 5257 63 4023 69 2889 78 1842 24 37 62 6581 63 5236 66 4004 72 2871 82 1825 23 38 65 6558 66 5214 69 3984 76 2853 85 1808 22 39 69 6534 69 5193 73 3964 79 2835 89 1792 21 40 .3772 2.6511 .3973 2.5172 .4176 2.3945 .4383 2.2817 .4592 2.1775 20 41 75 6488 76 5150 80 3925 86 2799 96 1758 19 42 79 6464 79 5129 83 3906 90 2781 99 1742 18 43 82 6441 83 5108 87 3886, 93 2763 4603 1725 17 44 85 6418 86 5086 90 3867 97 2745 07 1708 16 45 .3789 2.6395 .3990 2.5065 .4193 2.3847 .4400 2.2727 .4610 2.1692 15 46 92 6371 93 5044 97 3828 04 2709 14 1675 14 47 95 6348 96 5023 4200 3808 07 2691 17 1659 13 48 99 6325 4000 5002 04 3789 11 2673 21 1642 12 49 3802 6302 03 4981 07 3770 14 2655 24 1625 11 50 .3805 2.6279 .4006 2.4960 .4210 2.3750 .4417 2.2637 .4628 2.1609 10 51 09 6256 10 4939 14 3731 21 2620 31 1592 9 52 12 6233 13 4918 17 3712 24 2602 35 1576 8 53 15 6210 17 4897 21 3693 28 2584 38 1560 7 54 19 6187 20 4876 24 3673 31 2566 42 1543 6 55 .3822 2.6165 .4023 2.4855 .4228 2.3654 .4435 2.2549 .4645 2.1527 5 56 25 6142 27 4834 31 3635 38 2531 49 1510 4 57 29 6119 30 4813 34- 3616 42 2513 52 1494 3 58 32 6096 33 4792 38 3597 45 2496 56 1478 2 59 35 6074 37 4772 41 3578 49 2478 60 1461 1 60 .3839 2.6051 .4040 2.4751 .4245 2.3559 .4452 2.2460 .4663 2.1445 cot tan cot tan cot tan cot tan cot tan / 69 68 67 66 66 i 85 NATURAL SINES AND COSINES / 25 26 27 28 29 / sin cos sin cos sin cos sin cos sin cos .4226 .9063 .4384 .8988 .4540 .8910 .4695 .8829 .4848 .8746 60 1 29 62 86 87 42 09 97 28 51 45 59 2 31 61 89 85 45 07 4700 27 53 43 58 3 34 59 92 84 48 06 02 25 56 42 57 4 37 58 94 83 50 05 05 24 58 41 56 5 .4239 .9057 .4397 .8982 .4553 .8903 .4708 .8823 .4861 .8739 55 6 42 56 99 80 55 02 10 21 63 38 54 7 45 54 4402 79 58 01 13 20 66 36 53 8 47 53 05 78 61 8899 15 19 68 35 52 9 50 52 07 76 63 98 18 17 71 33 51 10 .4253 .9051 .4410 .8975 .4566 .8897 .4720 .8816 .4874 .8732 50 11 55 50 12 74 68 95 23 14 76 31 49 12 58 48 15 73 71 94 26 13 79 29 48 13 60 47 18 71 74 93 28 12 81 28 47 14 63 46 20 70 76 92 31 10 84 26 46 15 .4266 .9045 .4423 .8969 .4579 .8890 .4733 .8809 .4886 .8725 45 16 68 43 25 67 81 89 36 08 89 24 44 17 71 42 28 66 84 88 38 06 91 22 43 18 74 41 31 65 86 86 41 05 94 21 42 19 76 40 33 64 89 85 43 03 96 19 41 20 .4279 .9038 .4436 .8962 .4592 .8884 .4746 .8802 .4899 .8718 40 21 81 37 39 61 94 82 49 01 4901 16 39 22 84 36 41 60 97 81 51 8799 04 15 38 23 87 35 44 58 99 79 54 98 07 14 37 24 89 33 46 57 4602 78 56 96 09 12 36 25 .4292 .9032 .4449 .8956 .4605 .8877 .4759 .8795 .4912 .8711 35 26 95 31 52 55 07 75 61 94 14 09 34 27 97 30 54 53 10 74 64 92 17 08 33 28 4300 28 57 52 12 73 66 91 19 06 32 29 02 27 59 51 15 71 69 90 22 05 31 30 .4305 .9026 .4462 .8949 .4617 .8870 .4772 .8788 .4924 .8704 30 31 08 25 65 48 20 69 74 87 27 02 29 32 10 23 67 47 23 67 77 85 29 01 28 33 13 22 70 45 25 66 79 84 32 8699 27 34 16 21 72 44 28 65 82 83 34 98 26 35 .4318 .9020 .4475 .8943 .4630 .8863 .4784 .8781 .4937 .86% 25 36 21 18 78 42 33 62 87 80 39 95 24 37 23 17 80 40 36 61 89 78 42 94 23 38 26 16 83 39 38 59 92 77 44 92 22 39 29 15 85 38 41 58 95 76 47 91 21 40 .4331 .9013 .4488 .8936 .4643 .8857 .4797 .8774 .4950 .8689 20 41 34 12 91 35 46 55 4800 73 52 88 19 42 37 11 93 34 48 54 02 71 55 86 18 43 39 10 96 32 51 53 . 05 70 57 85 17 44 42 08 98 31 54 51 07 69 60 83 16 45 .4344 .9007 .4501 .8930 .4656 .8850 .4810 ,8767 .4962 .8682 15 46 47 06 04 28 59 49 12 66 65 81 14 47 50 04 06 27 61 47 15 64 67 79 13 48 52 03 09 26 64 46 18 63 70 78 12 49 55 02 11 25 66 44 20 62 72 76 11 50 .4358 .9001 .4514 .8923 .4669 .8843 .4823 .8760 .4975 .8675 10 51 60 8999 17 22 72 42 25 59 77 73 9 52 63 98 19 21 74 40 28 57 80 72 S 53 65 97 22 19 77 39 30 56 82 70 7 54 68 96 24 18 79 38 33 55 85 69 6 55 .4371 .8994 .4527 .8917 .4682 .8836 .4835 .8753 .4987 .8668 5 56 73 93 30 15 84 35 38 52 90 66 4 57 76 92 32 14 87 34 40 50 92 65 3 58 78 90 35 13 90 32 43 49 95 63 2 59 81 89 37 11 92 31 46 48 97 62 1 60 .4384 .8988 .4540 .8910 .4695 -8829 .4848 .8746 .5000 .8660 cos sin cos sin cos sin cos sin cos sin i 64 63 62 61 60 / 86 NATURAL TANGENTS AND COTANGENTS / 25 26 27 28 29 / tan cot tan cot tan cot tan cot tan cot .4663 2.1445 .4877 2.0503 .5095 1.9626 .5317 1.8807 .5543 1.8040 60 1 67 1429 81 0488 99 9612 21 8794 47 8028 59 2 70 1413 85 0473 5103 9598 25 8781 51 8016 58 3 74 1396 88 0458 06 9584 28 8768 55 8003 57 4 77 1380 92 0443 10 9570 32 8755 58 7991 56 5 .4681 2.1364 .4895 2.0428 .5114 1.9556 .5336 1.8741 .5562 1.7979 55 6 84 1348 99 0413 17 9542 40 8728 66 7966 54 7 88 1332 4903 0398 21 9528 43 8715 70 7954 53 8 91 1315 06 0383 25 9514 47 8702 74 7942 52 9 95 1299 10 0368 28 9500 51 8689 77 7930 51 10 .4699 2.1283 .4913 2.0353 .5132 1.9486 .5354 1.8676 .5581 1.7917 50 11 4702 1267 17 0338 36 9472 58 8663 85 7905 49 12 06 1251 21 0323 39 9458 62 8650 89 7893 48 13 09 1235 24 0308 43 9444 66 8637 93 7SS1 47 14 13 1219 28 0293 47 9430 69 8624 96 7868 46 15 .4716 2.1203 .4931 2.0278 .5150 1.9416 .5373 1.8611 .5600 1.7856 45 16 20 1187 35 0263 54 9402 77 8598 04 7844 44 17 23 1171 39 0248 58 9388 81 8585 08 7832 43 18 27 1155 42 0233 61 9375 84 8572 12 7820 42 19 31 1139 46 0219 65 9361 88 8559 16 7808 41 20 .4734 2.1123 .4950 2.0204 .5169 1.9347 .5392 1.8546 .5619 1.7796 40 21 38 1107 53 0189 72 9333 96 8533 23 7783 39 22 41 1092 57 0174 76 9319 99 8520 27 7771 38 23 45 1076 60 0160 80 9306 5403 8507 31 7759 37 24 48 1060 64 0145 84 9292 07 8495 35 7747 36 25 .4752 2.1044 .4968 2.0130 .5187 1.9278 .5411 1.8482 .5639 1.7735 35 26 55 1028 71 0115 91 9265 15 8469 42 7723 34 27 59 1013 75 0101 95 9251 18 8456 46 7711 33 28 63 0997 79 0086 98 9237 22 8443 50 7699 32 29 66 0981 82 0072 5202 9223 26 8430 54 7687 31 30 .4770 2.0965 .4986 2.0057 .5206 1.9210 .5430 1.8418 .5658 1.7675 30 31 73 0950 89 0042 09 9196 33 8405 62 7663 29 32 77 0934 93 0028 13 9183 37 8392 65 7651 28 33 80 0918 97 0013 17 9169 41 8379 69 7639 27 34 84 0903 5000 1.9999 20 9155 45 8367 73 7627 26 35 .4788 2.0887 .5004 1.9984 .5224 1.9142 .5448 1.8354 .5677 1.7615 25 36 91 0872 08 9970 28 9128 52 8341 81 7603 24 37 95 0856 11 9955 32 9115 56 8329 85 7591 23 38 98 0840 15 9941 35 9101 60 8316 88 7579 22 39 4802 0825 19 9926 39 9088 64 8303 92 7567 21 40 .4806 2.0809 .5022 1.9912 .5243 1.9074 .5467 1.8291 .5696 1.7556 20 41 09 0794 26 9897 46 9061 71 8278 5700 7544 19 42 13 0778 29 9883 50 9047 75 8265 04 7532 18 43 16 0763 33 9868 54 9034 79 8253 08 7520 17 44 20 0748 37 9854 58 9020 82 8240 12 7508 16 45 .4823 2.0732 .5040 J.9840 .5261 1.9007 .5486 1.8228 .5715 1.7496 15 46 27 0717 44 9825 65 8993 90 8215 19 7485 14 47 31 0701 48 9811 69 8980 94 8202 23 7473 13 48 34 0686 51 9797 72 8967 98 8190 27 7461 12 49 38 0671 55 9782 76 8953 5501 8177 31 7449 11 50 .4841 2.0655 .5059 1.9768 .5280 1.8940 .5505 1.8165 .5735 1.7437 10 51 45 0640 62 9754 84 8927 09 8152 39 7426 9 52 49 0625 66 9740 87 8913 13 8140 43 7414 8 53 52 0609 70 9725 91 8900 17 8127 46 7402 7 54 56 0594 73 9711 95 8887 20 8115 50 7391 6 55 .4859 2.0579 .5077 1.9697 .5298 1.8873 .5524 1.8103 .5754 1.7379 5 56 63 0564 81 9683 5302 8860 28 8090 58 7367 4 57 67 0549 84 9669 06 8847 32 8078 62 7355 3 58 70 0533 88 9654 10 8834 35 8065 66 7344 2 59 74 0518 92 9640 13 8820 39 8053 70 7332 ' 1 60 .4877 2.0503 .5095 1.9626 .5317 1.8807 .5543 1.8040 .5774 1.7321 cot tan cot tan cot tan cot tan cot tan / 64 63 62 61 60 U > 87 NATURAL SIXES AND COSIXES / 30 31 32 33 34 / sin cos sin cos sin cos sin cos sin cos .5000 .8660 .5150 .8572 .5299 .8480 .5446 .8387 .5592 .8290 60 1 03 59 53 70 5302 79 49 85 94 89 59 2 05 57 55 69 04 77 51 84 97 87 58 3 08 56 58 67 07 76 54 82 99 85 57 4 10 54 60 66 09 74 56 80 5602 84 56 5 .5013 .8653 .5163 .8564 .5312 .8473 .5459 .8379 .5604 .8282 55 6 15 52 65 63 14 71 61 77 06 81 54 7 18 50 68 61 16 70 63 76 09 79 53 8 20 49 70 60 19 68 66 74 11 77 52 9 23 47 73 58 21 67 68 72 14 76 51 10 .5025 .8646 .5175 .8557 .5324 .8465 .5471 .8371 .5616 .8274 50 11 28 44 78 55 26 63 73 69 18 72 49 12 30 43 80 54 29 62 76 68 21 71 48 13 33 41 83 52 31 60 78 66 23 69 47 14 35 40 85 51 34 59 80 64 26 68 46 15 .5038 .8638 .5188 .8549 .5336 .8457 .5483 .8363 .5628 .8266 45 16 40 37 90 48 39 56 85 61 30 64 44 17 43 35 93 46 41 54 88 60 33 63 43 18 45 34 95 45 44 53 90 58 35 61 42 19 48 32 98 43 46 51 93 56 38 59 41 20 .5050 .8631 .5200 .8542 .5348 .8450 .5495 .8355 .5640 .8258 40 21 53 30 03 40 51 48 98 53 42 56 39 22 55 28 05 39 53 46 5500 52 45 54 38 23 58 27 08 37 56 45 02 50 47 53 37 24 60 25 10 36 58 43 05 48 50 51 36 25 .5063 .8624 .5213 .8534 .5361 .8442 .5507 .8347 .5652 .8249 35 26 65 22 15 32 63 40 10 45 54 48 34 27 68 21 18 31 66 39 12 44 57 46 33 28 70 19 20 29 68 37 15 42 59 45 32 29 73 18 23 28 71 35 17 40 62 43 31 30 .5075 .8616 .5225 .8526 .5373 .8434 .5519 .8339 .5664 .8241 30 31 78 15 27 25 75 32 22 37 66 40 29 32 80 13 30 23 78 31 24 36 69 38 28 33 83 12 32 22 80 29 27 34 71 36 27 34 85 10 35 20 83 28 29 32 74 35 26 35 .5088 .8609 .5237 .8519 .5385 .8426 .5531 .8331 .5676 .8233 25 36 90 07 40 17 88 25 34 29 78 31 2-1 37 93 06 42 16 90 23 36 28 81 30 23 38 95 04 45 14 93 21 39 26 83 28 22 39 98 03 47 13 95 20 41 24 86 26 21 40 .5100 .8601 .5250 .8511 .5398 .8418 .5544 .8323 .5688 .8225 20 41 03 00 52 10 5400 17 46 21 90 23 19 42 05 8599 55 08 02 15 48 20 93 21 18 43 08 97 57 07 05 14 51 18 95 20 17 44 10 96 60 05 07 12 53 16 98 18 16 45 .5113 .8594 .5262 .8504 .5410 .8410 .5556 .8315 .5700 .8216 15 46 15 93 65 02 12 09 58 13 02 15 14 47 18 91 67 00 15 07 61 11 05 13 13 48 20 90 70 8499 17 06 63 10 07 11 12 49 23 88 72 97 20 04 65 08 10 10 11 50 .5125 .8587 .5275 .8496 .5422 .8403 .5568 .8307 .5712 .8208 10 51 28 85 77 94 24 01 70 05 14 07 9 52 30 84 79 93 27 8399 73 03 17 05 8 53 33 82 82 91 29 98 75 02 19 03 7 54 35 81 84 90 32 96 77 00 21 02 6 55 .5138 .8579 .5287 .8488 .5434 .8395 .5580 .8298 .5724 .8200 5 56 40 78 89 87 37 93 82 97 26 8198 4 57 43 76 92 85 39 91 85 95 29 97 3 58 45 75 94 84 42 90 87 94 31 95 2 59 48 73 97 82 44 88 90 92 33 93 1 60 .5150 .8572 .5299 .8480 .5446 .8387 .5592 .8290 .5736 .8192 cos sin cos sin cos sin cos sin cos sin i 59 58 57 56 55 5 / 88 NATURAL TANGENTS AND COTANGENTS / 30 31 32 33 34 i tan cot tan cot tan cot tan cot tan cot .5774 1.7321 .6009 1.6643 .6249 1.6003 .6494 1.5399 .6745 1.4826 60 1 77 7309 13 6632 53 5993 98 5389 49 4816 59 2 81 7297 17 6621 57 5983 6502 5379 54 4807 58 3 85 7286 20 6610 61 5972 06 5369 58 4798 57 4 89 7274 24 6599 65 5962 11 5359 62 4788 56 5 .5793 1.7262 .6028 1.6588 .6269 1.5952 .6515 1.5350 .6766 1.4779 55 6 97 7251 32 6577 73 5941 19 5340 71 4770 54 7 5801 7239 36 6566 77 5931 23 5330 75 4761 53 8 05 7228 40 6555 81 5921 27 5320 79 4751 52 9 08 7216 44 6545 85 5911 31 5311 83 4742 51 10 .5812 1.7205 .6048 1.6534 .6289 1.5900 .6536 1.5301 .6787 1.4733 50 11 16 7193 52 6523 93 5890 40 5291 92 4724 49 12 20 7182 56 6512 97 5880 44 5282 96 4715 48 13 24 7170 60 6501 6301 5869 48 5272 6800 4705 47 14 28 7159 64 6490 05 5859 52 5262 05 4696 46 15 .5832 1.7147 .6068 1.6479 .6310 1.5849 .6556 1.5253 .6809 1.4687 45 16 36 7136 72 6469 14 5839 60 5243 13 4678 44 17 40 7124 76 6458 18 5829 65 5233 17 4669 43 18 44 7113 80 6447 22 5818 69 5224 22 4659 42 19 47 7102 84 6436 26 5808 73 5214 26 4650 41 20 .5851 1.7090 .6088 1.6426 .6330 1.5798 .6577 1.5204 .6830 1.4641 40 21 55 7079 92 6415 34 5788 81 5195 34 4632 39 22 59 7067 96 6404 38 5778 85 5185 39 4623 38 23 63 7056 6100 6393 42 5768 90 5175 43 4614 37 24 67 7045 04 6383 46 5757 94 5166 47 4605 36 25 .5871 1.7033 .6108 1.6372 .6350 1.5747 .6598 1.5156 .6851 1.4596 35 26 75 7022 12 6361 54 5737 6602 5147 56 4586 34 27 79 7011. 16 6351 58 5727 06 5137 60 4577 33 28 83 6999 20 6340 63 5717 10 5127 64 4568 32 29 87 6988 24 6329 67 5707 15 5118 69 4559 31 30 .5890 1.6977 .6128 1.6319 .6371 1.5697 .6619 1.5108 .6873 1.4550 30 31 94 6965 32 6308 75 5687 23 5099 77 4541 29 32 98 6954 36 6297 79 5677 27 5089 81 4532 28 33 5902 6943 40 6287 83 5667 31 5080 86 4523 27 34 06 6932 44 6276 87 5657 36 5070 90 4514 26 35 .5910 1.6920 .6148 1.6265 .6391 1.5647 .6640 1.5061 .6894 1.4505 25 36 14 6909 52 6255 95 5637 44 5051 99 4496 24 37 18 6898 56 6244 99 5627 48 5042 6903 4487 23 38 22 6887 60 6234 6403 5617 52 5032 07 4478 22 39 26 6875 64 6223 08 5607 57 5023 11 4469 21 40 .5930 1.6864 .6168 1.6212 .6412 1.5597 .6661 1.5013 .6916 1.4460 20 41 34 6853 72 6202 16 5587 65 5004 20 4451 19 42 38 6842 76 6191 20 5577 69 4994 24 4442 18 43 42 6831 80 6181 24 5567 73 4985 29 4433 17 44 45 6820 84 6170 28 5557 78 4975 33 4424 16 45 .5949 1.6808 .6188 1.6160 .6432 1.5547 .6682 1.4966 .6937 1.4415 15 46 53 6797 92 6149 36 5537 86 4957 42 4406 14 47 57 6786 96 6139 40 5527 90 4947 46 4397 13 48 61 6775 6200 6128 45 5517 94 4938 50 4388 12 49 65 6764 04 6118 49 5507 99 4928 54 4379 11 50 .5969 1.6753 .6208 1.6107 .6453 1.5497 .6703 1.4919 .6959 1.4370 10 51 73 6742 12 6097 57 5487 07 4910 63 4361 9 52 77 6731 16 6087 61 5477 11 4900 67 4352 8 53 81 6720 20 6076 65 5468 15 4891 72 4344 7 54 85 6709 24 6066 69 5458 20 4882 76 4335 6 55 .5989 1.6698 .6228 1.6055 .6473 1.5448 .6724 1.4872 .6980 1.4326 5 56 93 6687 33 6045 78 5438 28 4863 85 4317 4 57 97 6676 37 6034 82 5428 32 4854 89 4308 3 58 6001 6665 41 6024 86 5418 37 4844 93 4299 2 59 05 6654 45 6014 90 5408 41 4835 98 4290 1 60 .6009 1.6643 .6249 1.6003 .6494 1.5399 .6745 1.4826 .7002 1.4281 cot tan cot tan cot tan cot tan cot tan i 59 58 57 56 55 / 89 NATURAL SINES AND COSINES / 35 36 37 38 39 / sin cos sin cos sin cos sin cos sin cos .5736 .8192 .5878 .8090 .6018 .7986 .6157 .7880 .6293 .7771 60 1 38 90 80 88 20 85 59 78 95 70 59 2 41 88 83 87 23 83 61 77 98 68 58 3 43 87 85 85 25 81 63 75 6300 66 57 4 45 85 87 83 27 79 66 73 02 64 56 5 .5748 .8183 .5890 .8082 .6030 .7978 .6168 .7871 .6305 .7762 55 6 50 81 92 80 32 76 70 69 07 60 54 7 52 80 94 78 34 74 73 68 09 59 53 8 55 78 97 76 37 72 75 66 11 57 52 9 57 76 99 75 39 71 77 64 14 55 51 10 .5760 .8175 .5901 .8073 .6041 .7969 .6180 .7862 .6316 .7753 50 11 62 73 04 71 44 67 82 60 18 51 49 12 64 71 06 70 46 65 84 59 20 49 48 13 67 70 08 68 48 64 86 57 23 48 47 14 69 68 11 66 51 62 89 55 25 46 46 15 .5771 .8166 .5913 .8064 .6053 .7960 .6191 .7853 .6327 .7744 45 16 74 65 15 63 55 58 93 51 29 42 44 17 76 63 18 61 58 56 96 50 32 40 43 18 79 61 20 59 60 "55 98 48 34 38 42 19 81 60 22 58 62 53 6200 46 36 37 41 20 .5783 .8158 .5925 .8056 .6065 -7951 .6202 .7844 .6338 .7735 40 21 86 56 27 54 67 49 05 42 41 33 39 22 88 55 30 52 69 48 07 41 43 31 38 23 90 53 32 51 71 46 09 39 45 29 37 24 93 51 34 49 74 44 11 37 47 27 36 25 .5795 .8150 .5937 .8047 .6076 .7942 .6214 .7835 .6350 .7725 35 26 98 48 39 45 78 41 16 33 52 24 34 27 5800 46 U 44 81 39 18 32 54 22 33 28 02 45 44 42 83 37 21 30 56 20 32 29 05 43 46 40 85 35 23 28 59 18 31 30 .5807 .8141 .5948 .8039 .6088 .7934 .6225 .7826 .6361 .7716 30 31 09 39 51 37 90 32 27 24 63 14 29 32 12 38 53 35 92 30 30 22 65 13 28 33 14 36 55 33 95 28 32 21 68 11 27 34 16 34 58 32 97 26 34 19 70 09 26 35 .5819 .8133 .5960 .8030 .6099 .7925 .6237 .7817 .6372 .7707 25 36 21 31 62 28 6101 23 39 15 74 05 24 37 24 29 65 26 04 21 41 13 76 03 23 38 26 28 67 25 06 19 43 12 79 01 22 39 28 26 69 23 08 18 46 10 81 00 21 40 .5831 .8124 .5972 .8021 .6111 .7916 .6248 .7808 .6383 .7698 20 41 33 23 74 19 13 14 50 06 85 96 19 42 35 ' 21 76 18 15 12 52 04 88 94 18 43 38 19 79 16 18 10 55 02 90 92 17 44 40 17 81 14 20 09 57 01 92 90 16 45 .5842 .8116 .5983 .8013 .6122 .7907 .6259 .7799 .6394 .7688 15 46 45 14 86 11 24 05 62 97 97 87 14 47 47 12 88 09 27 03 64 95 99 85 13 48 50 11 90 07 29 02 66 93 6401 83 12 49 52 09 93 06 31 00 68 92 03 81 11 50 .5854 .8107 .5995 .8004 .6134 .7898 .6271 .7790 .6406 .7679 10 51 57 06 97 02 36 96 73 88 08 77 9 52 59 04 6000 00 38 94 75 86 10 75 8 53 61 02 02 7999 41 93 77 84 12 74 7 54 64 00 04 97 43 91 80 82 14 72 6 65 .5866 .8099 .6007 .7995 .6145 .7889 .6282 .7781 .6417 .7670 5 56 68 97 09 93 47 87 84 79 19 68 4 57 71 95 11 92 50 85 86 77 21 66 3 58 73 94 14 90 52 84 89 75 23 64 2 59 75 92 16 88 54 82 91 73 26 62 1 60 .5878 .8090 .6018 .7986 .6157 .7880 .6293 .7771 .6428 .7660 cos sin cos sin cos sin cos sin cos sin i 54 53 52 51 50" i 90 NATURAL TANGENTS AND COTANGENTS I 35 36 37 38 39 ; tan cot tan cot tan cot tan cot tan cot .7002 1.4281 .7265 1.3764 .7536 1.3270 .7813 1.2799 .8098 1.2349 60 1 06 4273 70 3755 40 3262 18 2792 8103 2342 59 2 11 4264 74 3747 45 3254 22 2784 07 2334 58 3 15 4255 79 3739 49 3246 27 2776 12 2327 57 4 19 4246 83 3730 54 3238 32 2769 17 2320 56 5 .7024 1.4237 .7288 1.3722 .7558 1.3230 .7836 1.2761 .8122 1.2312 55 6 28 4229 92 3713 63 3222 41 2753 27 2305 54 7 32 4220 97 3705 68 3214 46 2746 32 2298 53 8 37 4211 7301 3697 72 3206 50 2738 36 2290 52 9 41 4202 06 3688 77 3198 55 2731 41 2283 51 10 .7046 1.4193 .7310 1.3680 .7581 1.3190 .7860 1.2723 .8146 1.2276 50 11 50 4185 14 3672 86 3182 65 2715 51 2268 49 12 54 4176 19 3663 90 3175 69 2708 56 2261 48 13 59 4167 23 3655 95 3167 74 2700 61 2254 47 14 63 4158 28 3647 7600 3159 79 2693 65 2247 46 15 .7067 1.4150 .7332 1.3638 .7604 1.3151 .7883 1.2685 .8170 1.2239 45 16 72 4141 37 3630 09 3143 88 2677 75 2232 44 17 76 4132 41 3622 13 3135 93 2670 80 2225 43 18 80 4124 46 3613 18 3127 98 2662 85 2218 42 19 85 4115 50 3605 23 3119 7902 2655 90 2210 41 20 .7089 1.4106 .7355 1.3597 .7627 1.3111 .7907 1.2647 .8195 1.2203 40 21 94 4097 59 3588 32 3103 12 2640 99 2196 39 22 98 4089 64 3580 36 3095 16 2632 8204 2189 38 23 7102 4080 68 3572 41 3087 21 2624 09 2181 37 24 07 4071 73 3564 46 3079 26 2617 14 2174 36 25 .7111 1.4063 .7377 1.3555 .7650 1.3072 .7931 1.2609 .8219 1.2167 35 26 15 4054 82 3547 55 3064 35 2602 24 2160 34 27 20 4045 86 3539 59 3056 40 2594 29 2153 33 28 24 4037 91 3531 64 3048 45 2587 34 2145 32 29 29 4028 95 3522 69 3040 50 2579 38 2138 31 30 .7133 1.4019 .7400 1.3514 .7673 1.3032 .7954 1.2572 .8243 1.2131 30 31 37 4011 04 3506 78 3024 59 2564 48 2124 29 32 42 4002 09 3498 83 3017 64 2557 53 2117 28 33 46 3994 13 3490 87 3009 69 2549 58 2109 27 34 51 3985 18 3481 92 3001 73 2542 63 2102 26 35 .7155 1.3976 .7422 1.3473 .7696 1.2993 .7978 1.2534 .8268 1.2095 25 36 59 3968 27 3465 7701 2985 83 2527 73 2088 24 37 64 3959 31 3457 06 2977 88 2519 78 2081 23 38 68 3951 36 3449 10 2970 92 2512 83 2074 22 39 73 3942 40 3440 15 2962 97 2504 87 2066 21 40 .7177 1.3934 .7445 1.3432 .7720 1.2954 .8002 1.2497 .8292 1.2059 20 41 81 3925 49 3424 24 2946 07 2489 97 2052 19 42 86 3916 54 3416 29 2938 12 2482 8302 2045 18 43 90 3908 58 3408 34 2931 16 2475 07 2038 17 44 95 3899 63 3400 38 2923 21 2467 12 2031 16 45 .7199 1.3891 .7467 1.3392 .7743 1.2915 .8026 1.2460 .8317 1.2024 15 46 7203 3882 72 3384 47 2907 31 2452 22 2017 14 47 08 3874 76 3375 52 2900 35 2445 27 2009 13 48 12 3865 81 3367 57 2892 40 2437 32 2002 12 49 17 3857 85 3359 61 2884 45 2430 37 1995 11 50 .7221 1.3848 .7490 1.3351 .7766 1.2876 .8050 1.2423 .8342 1.1988 10 51 26 3840 95 3343 71 2869 55 2415 46 1981 9 52 30 3831 99 3335 75 2861 59 2408 51 1974 8 53 34 3823 7504 3327 80 2853 64 2401 56 1967 7 54 39 3814 08 3319 85 2846 69 2393 61 1960 6 55 .7243 1.3806 .7513 1.3311 .7789 1.2838 .8074 1.2386 .8366 1.1953 5 56 48 3798 17 3303 94 2830 79 2378 71 1946 4 57 52 3789 22 3295 99 2822 83 2371 76 1939 3 58 57 3781 26 3287 7803 2815 88 2364 81 1932 2 59 61 3772 31 3278 08 2807 93 2356 86 1925 1 60 .7265 1.3764 .7536 1.3270 .7813 1.2799 .8098 1.2349 .8391 1.1918 cot tan cot tan cot tan cot tan cot tan ; 54 53 52 51 50 i 91 NATURAL SINES AND COSINES / 40 41 42 43 44 / sin cos sin cos sin cos sin cos sin cos .6428 .7660 .6561 .7547 .6691 .7431 .6820 .7314 .6947 .7193 60 1 30 59 63 45 93 30 22 12 49 91 59 2 32 57 65 43 96 28 24 10 51 89 58 3 35 55 67 41 98 26 26 08 53 87 57 4 37 53 69 39 6700 24 28 06 55 85 56 6 .6439 .7651 .6572 .7538 .6702 .7422 .6831 .7304 .6957 .7183 55 6 41 49 74 36 04 20 33 02 59 81 54 7 43 47 76 34 06 18 35 00 61 79 53 8 46 45 78 32 09 16 37 7298 63 77 52 9 48 44 80 30 11 14 39 96 65 75 51 10 .6450 .7642 .6583 .7528 .6713 .7412 .6841 .7294 .6967 .7173 50 11 52 40 85 26 15 10 43 92 70 71 49 12 55 38 87 24 17 08 45 90 72 69 48 13 57 36 89 22 19 06 48 88 74 67 47 14 59 34 91 20 22 04 50 86 76 65 46 15 .6461 .7632 .6593 .7518 .6724 .7402 .6852 .7284 .6978 .7163 45 16 63 30 96 16 26 00 54 82 80 61 44 17 66 29 98 15 28 7398 56 80 82 59 43 18 68 27 6600 13 30 % 58 78 84 57 42 19 70 25 02 11 32 94 60 76 86 55 41 20 .6472 .7623 .6604 .7509 .6734 .7392 .6862 .7274 .6988 .7153 40 21 75 21 07 07 37 90 65 72 90 51 39 22 77 19 09 05 39 88 67 70 92 49 38 23 79 17 11 03 41 87 69 68 95 47 37 24 81 15 13 01 43 85 71 66 97 45 36 25 .6483 .7613 .6615 .7499 .6745 .7383 .6873 .7264 .6999 .7143 35 26 86 12 17 97 47 81 75* 62 7001 41 34. 27 88 10 20 95 49 79 77 60 03 39 33 28 90 08 22 93 52 77 79 58 05 37 32 29 92 06 24 91 54 75 81 56 07 35 31 30 .6494 .7604 .6626 .7490 .6756 .7373 .6884 .7254 .7009 .7133 30 31 97 02 28 88 58 71 86 52 11 30 29 32 99 00 31 86 60 69 88 50 13 28 28 33 6501 7598 33 84 62 67 90 48 15 26 27 34 03 96 35 82 64 65 92 46 17 24 26 35 .6506 .7595 .6637 .7480 .6767 .7363 .6894 .7244 .7019 .7122 25 36 08 93 39 78 69 61 % 42 22 20 24 37 10 91 41 76 71 59 98 40 24 18 23 38 12 89 44 74 73 57 6900 38 26 16 22 39 14 87 46 72 75 55 03 36 28 14 21 40 .6517 .7585 .6648 .7470 .6777 .7353 .6905 .7234 .7030 .7112 20 41 19 83 50 68 79 51 07 32 32 10 19 42 21 81 52 66 82 49 09 30 34 08 18 43 23 79 54 64 84 47 11 28 36 06 17 44 25 78 57 63 86 45 13 26 38 04 16 45 .6528 .7576 .6659 .7461 .6788 .7343 .6915 .7224 .7040 .7102 15 46 30 74 61 59 90 41 17 22 42 00 14 47 32 72 63 57 92 39 19 20 44 7098 13 48 34 70 65 55 94 37 21 18 46 96 12 49 36 68 67 53 97 35 24 16 48 94 11 50 .6539 .7566 .6670 .7451 .6799 .7333 .6926 .7214 .7050 .7092 10 51 41 64 72 49 6801 31 28 12 53 90 9 52 43 62 74 47 03 29 30 10 55 88 8 53 45 60 76 45 05 27 32 08 57 85 7 54 47 59 78 43 07 25 34 06 59 83 6 55 .6550 .7557 .6680 .7441 .6809 .7323 .6936 .7203 .7061 .7081 5 56 52 55 83 39 11 21 38 01 63 79 4 57 54 53 85 37 14 19 40 7199 65 77 3 58 56 51 87 35 16 18 42 97 67 75 2 59 58 49 89 33 18 16 44 95 69 73 1 60 .6561 .7547 .6691 .7431 .6820 .7314 .6947 .7193 .7071 .7071 COB sin cos sin cos sin cos sin cos sin i 49 48 47 46 45 i 92 NATURAL TANGENTS AND COTANGENTS 1 40 41 42 43 44 i tan cot tan cot tan cot tan cot tan cot .8391 1.1918 .8693 1.1504 .9004 1.1106 .9325 1.0724 .9657 1.0355 60 1 96 1910 98 1497 09 1100 31 0717 63 0349 59 2 8401 1903 8703 1490 15 1093 36 0711 68 0343 58 3 06 1896 08 1483 20 10S7 41 0705 74 0337 57 4 11 1889 13 1477 25 1080 47 0699 79 0331 56 5 .8416 1.1882 .8718 1.1470 .9030 1.1074 .9352 1.0692 .9685 1.0325 55 6 21 1875 24 1463 36 1067 58 0686 91 0319 54 7 26 1868 29 1456 41 1061 63 0680 96 0313 53 8 31 1861 34 1450 46 1054 69 0674 9702 0307 52 9 36 1854 39 1443 52 1048 74 0668 08 0301 51 10 .8441 1.1847 .8744 1.1436 .9057 1.1041 .9380 1.0661 .9713 1.0295 50 11 46 1840 49 1430 62 1035 85 0655 19 0289 49 12 51 1833 54 1423 67 1028 91 0649 25 0283 48 13 56 1826 59 1416 73 1022 96 0643 30 0277 47 14 61 1819 65 1410 78 1016 9402 0637 36 0271 46 15 .8466 1.1812 .8770 1.1403 .9083 1.1009 .9407 1.0630 .9742 1.0265 45 16 71 1806 75 1396 89 1003 13 0624 47 0259 44 17 76 1799 80 1389 94 0996 18 0618 53 0253 43 18 81 1792 85 1383 99 0990 24 0612 59 0247 42 19 86 1785 90 1376 9105 0983 29 0606 64 0241 41 20 .8491 1.1778 .8796 1.1369 .9110 1.0977 .9435 1.0599 .9770 1.0235 40 21 96 1771 8801 1363 15 0971 40 0593 76 0230 39 22 8501 1764 06 1356 21 0964 46 0587 81 0224 38 23 06 1757 11 1349 26 0958 51 0581 87 0218 37 24 11 1750 16 1343 31 0951 57 0575 93 0212 36 25 .8516 1.1743 .8821 1.1336 .9137 1.0945 .9462 1.0569 .9798 1.0206 35 26 21 1736 27 1329 42 0939 68 0562 9804 0200 34 27 26 1729 32 1323 47 0932 73 0556 10 0194 33 2S 31 1722 37 1316 53 0926 79 0550 16 0188 32 29 36 1715 42 1310 58 0919 84 0544 21 0182 31 30 .8541 1.1708 .8847 1.1303 .9163 1.0913 .9490 1.0538 .9827 1.0176 30 31 46 1702 52 1296 69 0907 95 0532 33 0170 29 32 51 1695 58 1290 74 0900 9501 0526 38 0164 28 33 56 1688 63 1283 79 0894 06 0519 44 0158 27 34 61 1681 68 1276 85 0888 12 0513 50 0152 26 35 .8566 1.1674 .8873 1.1270 .9190 1.0881 .9517 1.0507 .9856 1.0147 25 36 71 1667 78 1263 95 0875 23 0501 61 0141 24 37 76 1660 84 1257 9201 0869 28 0495 67 0135 23 38 81 1653 89 1250 06 0862 34 0489 73 0129 22 39 86 1647 94 1243 12 0856 40 0483 79 0123 21 40 .8591 1.1640 .8899 1.1237 .9217 1.0850 .9545 1.0477 .9884 1.0117 20 41 96 1633 8904 1230 22 0843 51 0470 90 0111 19 42 8601 1626 10 1224 28 0837 56 0464 96 0105 18 43 06 1619 15 1217 33 0831 62 0458 9902 0099 17 44 11 1612 20 1211 39 0824 67 0452 07 0094 16 45 .8617 1.1606 .8925 1.1204 .9244 1.0818 .9573 1.0446 .9913 1.0088 15 46 22 1599 31 1197 49 0812 78 0440 19 0082 14 47 27 1592 36 1191 55 0805 84 0434 25 0076 13 48 32 1585 41 1184 60 0799 90 0428 30 0070 12 49 37 1578 46 1178 66 0793 95 0422 36 0064 11 50 .8642 1.1571 .8952 1.1171 .9271 1.0786 .9601 1.0416 .9942 1.0058 10 51 47 1565 57 1165 76 0780 06 0410 48 0052 9 52 52 1558 62 1158 82 0774 12 0404 54 0047 8 53 57 1551 67 1152 87 0768 18 [0398 59 0041 7 54 62 1544 72 1145 93 0761 23 0392 65 0035 6 55 .8667 1.1538 .8978 1.1139 .9298 1.0755 .9629 1.0385 .9971 1.0029 5 56 72 1531 83 1132 9303 0749 34 0379 77 0023 4 57 78 1524 88 1126 09 0742 40 0373 83 0017 3 58 83 1517 94 1119 14 0736 46 0367 88 0012 2 59 88 1510 99 1113 20 0730 51 0361 94 0006 1 60 .8693 1.1504 .9004 1.1106 .9325 1.0724 .9657 1.0355 1.0000 1.0000 cot tan cot tan cot tan cot tan cot tan / 49 48 47 46 45 / 93 TABLE IV MISCELLANEOUS TABLES 1. RADIAN MEASURE, to 180. 2. NATURAL LOGARITHMS: (4) Numbers from 1 to 200. (B) Numbers from 1 to 9.9. 8. THK HYPERBOLIC FUNCTIONS, SINHOT, COSHX. 95 TABLE IV. 1. RADIAN MEASURE, TO 180, RADIUS = 1 DEGREES ] IINUTES j SECONDS 0.0000000 60 1.04719 76 120 2.09439 51 0' 0.00000 00 0" 0.00000 00 1 2 3 4 5 6 7 8 9 0.01745 33 0.03490 66 0.05235 99 0.06981 32 0.08726 65 0.1047198 0.1221730 0.1396263 0.15707 96 61 62 63 64 65 66 67 68 69 1.06465 08 1.08210 41 1.09955 74 1.11701 07 1.1344640 1.15191 73 1.1693706 1.1868239 1.20427 72 121 122 123 124 125 126 127 128 129 2.11184 84 2.12930 17 2.14675 50 2.1642083 2.18166 16 2.1991149 2.21656 82 2.23402 14 2.25147 47 1 2 3 4 5 6 7 8 9 0.00029 09 0.00058 18 0.00087 27 0.00116 36 0.00145 44 0.00174 53 0.00203 62 0.00232 71 0.00261 80 1 2 3 4 5 6 7 8 9 0.00000 48 0.00000 97 0.00001 45 0.00001 94 0.00002 42 0.00002 91 0.00003 39 0.00003 88 0.00004 36 10 0.17453 29 70 1.22173 05 130 2.26892 80 10 0.00290 89 10 0.00004 85 11 12 13 14 15 16 17 18 19 0.1919862 0.20943 95 0.226S9 28 0.24434 61 0.2617994 0.27925 27 0.29670 60 0.3141593 0.33161 26 71 72 73 74 75 76 77 78 79 1.23918 38 1.25663 71 1.2740904 1.2915436 1.30899 69 1.32645 02 1.34390 35 1.36135 68 1.37881 01 131 132 133 134 135 136 137 138 139 2.28638 13 2.30383 46 2.32128 79 2.33874 12 2.35619 45 2.37364 78 2.39110 11 2.40855 44 2.42600 77 11 12 13 14 15 16 17 18 19 0.00319 98 0.00349 07 0.00378 15 0.00407 24 0.00436 33 0.00465 42 0.00494 51 0.00523 60 0.00552 69 11 12 13 14 15 16 17 18 19 0.00005 33 0.00005 82 0.00006 30 0.00006 79 0.00007 27 0.00007 76 0.00008 24 0.00008 73 0.00009 21 20 0.34906 59 80 1.39626 34 140 2.44346 10 20 0.00581 78 20 0.00009 70 21 22 23 24 25 26 27 28 29 0.36651 91 0.38397 24 0.40142 57 0.41887 90 0.43633 23 0.45378 56 0.47123 89 0.48869 22 0.50614 55 81 82 83 84 85 86 87 88 89 1.41371 67 1.43117 00 1.44862 33 1.46607 66 1.48352 99 1.50098 32 1.51843 64 1.53588 97 1.55334 30 141 142 143 144 145 146 147 148 149 2.46091 42 2.47836 75 2.49582 OS 2.51327 41 2.53072 74 2.5481807 2.5656340 2.58308 73 2.60054 06 21 22 23 24 25 26 27 28 29 0.00610 87 0.00639 95 0.00669 04 0.00698 13 0.00727 22 0.00756 31 0.00785 40 0.00814 49 0.00843 58 21 22 23 24 25 26 27 28 29 0.00010 18 0.00010 67 0.00011 15 0.00011 64 0.00012 12 0.00012 61 0.00013 09 0.00013 57 0.00014 06 30 0.52359 88 90 1.5707963 150 2.61799 39 30 0.00872 66 30 0.00014 54 31 32 33 34 35 36 37 38 39 0.54105 21 0.55850 54 0.57595 87 0.59341 19 0.61086 52 0.62831 85 0.64577 18 0.66322 51 0.68067 84 91 92 93 94 95 96 97 98 99 1.58824 96 1.60570 29 1.62315 62 1.64060 95 1.65806 28 1.67551 61 1.69296 94 1.71042 27 1.7278760 151 152 153 154 155 156 157 158 159 2.63544 72 2.65290 05 2.67035 38 2.68780 70 2.7052603 2.72271 36 2.7401669 2.75762 02 2.77507 35 31 32 33 34 35 36 37 38 39 0.00901 75 0.00930 84 0.00959 93 0.00989 02 0.0101811 0.01047 20 0.01076 29 0.01105 38 0.01134 46 31 32 33 34 35 36 37 38 39 0.00015 03 0.00015 51 0.00016 00 0.00016 48 0.00016 97 0.00017 45 0.00017 94 0.00018 42 0.00018 91 40 0.69813 17 100 1.7453293 160 2.79252 68 40 0.01163 55 40 0.00019 39 41 42 43 44 45 46 47 48 49 0.7155850 0.7330383 0.75049 16 0.7679449 0.78539 82 0.80285 15 0.82030 47 0.83775 80 0.85521 13 101 102 103 104 105 106 107 108 109 1.76278 25 1.78023 58 1.7976891 1.81514 24 1.83259 57 1.85004 90 1.86750 23 1.88495 56 1.9024089 161 162 163 164 165 166 167 168 169 2.80998 01 2.82743 34 2.84488 67 2.86234 00 2.87979 33 2.89724 66 2.91469 99 2.93215 31 2.94960 64 41 42 43 44 45 46 47 48 49 0.01192 64 0.01221 73 0.01250 82 0.01279 91 0.01309 00 0.01338 09 0.01367 17 0.01396 26 0.01425 35 41 42 43 44 45 46 47' 48 49 0.00019 88 0.00020 36 0.00020 85 0.00021 33 0.00021 82 0.00022 30 0.00022 79 0.00023 27 0.00023 76 50 0.87266 46 110 1.91986 22 170 2.96705 97 50 0.01454 44 50 0.00024 24 51 52 53 54 55 56 57 58 59 0.89011 79 0.90757 12 0.92502 45 0.94247 78 0.95993 11 0.97738 44 0.99483 77 1.01229 10 1.02974 43 111 112 113 114 115 116 117 118 119 1.93731 55 1.95476 88 1.97222 21 1.98967 53 2.00712 86 2.02458 19 2.04203 52 2.05948 85 2.07694 18 171 172 173 174 175 176 177 178 179 2.98451 30 3.00196 63 3.01941 96 3.03687 29 3.05432 62 3.07177 95 3.08923 28 3.1066861 3.1241394 51 52 53 54 55 56 57 58 59 0.01483 53 0.01512 62 0.01541 71 0.01570 80 0.01599 89 0.01628 97 0.01658 06 0.01687 15 0.01716 24 51 52 53 54 55 56 57 58 59 0.00024 73 0.00025 21 0.00025 70 0.00026 IS 0.00026 66 0.00027 15 0.00027 63 0.00028 12 0.00028 60 60 1.0471976 120 2.0943951 180 3.1415927 60 0.01745 33 60 0.00029 09 ] DEGREES 1 IINUTES I SECONDS 96 TABLE IV 2. NATURAL LOGARITHMS A. THE NATURAL LOGARITHMS OF INTEGERS FROM 1 TO 200 Base e = 2.7182818284... Conversion Laws : Iog 10 g = 0.4342944819-.. log, 10 = 2. 3025850929-.. log e 77 = 1.1447298858-.. \og e N = log e !0 xlog 10 ^r. lo gio^ r = lo ^io e x lo 8 e N. N loge H loge N loge N loge N loge 1 2 3 oo 40 41 42 43 3.68 888 80 81 82 83 4.38 203 120 121 122 123 4.78 749 160 161 162 163 5.07 517 0.00 000 0.69 315 1.09 861 3.71 357 3.73 767 3.76 120 4.39 445 4.40 672 4.41 884 4 79 579 4.80 402 4.81 218 5.08 140 5.08 760 5.09 375 4 5 6 1.38 629 1.60 944 1.79 176 44 45 46 3.78 419 3.80 666 3.82 864 84 85 86 4.43 082 4.44 265 4.45 435 124 125 126 4.82 028 4.82 831 4.83 628 164 165 166 5.09 987 5.10 595 5.11 199 7 8 9 10 11 12 13 1.94 591 2.07 944 2.19 722 47 48 49 50 51 52 53 3.85 015 3.87 120 3.89 182 87 88 89 90 91 92 93 4.46 591 4.47 734 4.48 864 127 128 129 130 131 132 133 4.84 419 4.85 203 4.85 981 167 168 169 170 171 172 173 5.11 799 5.12 396 5.12 990 2.30 259 3.91 202 4.49 981 4.86 753 5.13 580 2.39 790 2.48 491 2.56 495 3.93 183 3.95 124 3.97 029 4.51 086 4.52 179 4.53 260 4.87 520 4. 88 280 4.89 033 5.14 166 5.14 749 5.15 329 14 15 16 2.63 906 2.70 805 2.77 259 54 55 56 3.98 898 4.00 733 4.02 535 94 95 96 4.54 329 4.55 388 4.56 435 134 135 136 4.89 784 4.90 527 4.91 265 174 175 176 5.15 906 5.16 479 5.17 048 17 18 19 20 21 22 23 2.83 321 2.89 037 2.94 444 57 58 59 60 61 62 63 4.04 305 4.06 044 4.07 754 97 98 99 100 101 102 103 4.57 471 4.58 497 4.59 512 137 138 139 140 141 142 143 4.91 998 4.92 725 4.93 447 177 178 179 180 181 182 183 5.17 615 5.18 178 5.18 739 2.99 573 4.09 434 4.60 517 4.94 164 5.19 296 3.04 452 3.09 104 3.13 549 4.11 087 4.12 713 4.14 313 4.61 512 4.62 497 4.63 473 4.94 876 4.95 583 4.96 284 5.19 850 5.20 401 5.20 949 24 25 26 3.17 805 3.21 888 3.25 810 64 65 66 4.15 888 4.17 439 4.18 965 104 105 106 4.64 439 4.65 396 4.66 344 144 145 146 4.96 981 4.97 673 4.98 361 184 185 186 5.21 494 5.22 036 5.22 575 27 28 29 30 31 32 33 3.29 584 3.33 220 3.36 730 67 68 69 70 71 72 73 4.20 469 4.21 951 4.23 411 107 108 109 110 111 112 113 4.67 283 4.68 213 4.69 135 147 148 149 150 151 152 153 4.99 043 4.99 721 5.00 395 187 188 189 190 191 192 193 5.23 111 5.23 644 5.24 175 3.40 120 4.24 850 4.70 048 5.01 064 5.24 702 3.43 399 3.46 574 3.49 651 4.26 268 4.27 667 4.29 046 4.70 953 4.71 850 4.72 739 5.01 728 5.02 388 5.03 044 5.25 227 5.25 750 5.26 269 34 35 36 3.52 636 3.55 535 3.58 352 74 75 76 4.30 407 4.31 749 4.33 073 114 115 116 4.73 620 4.74 493 4.75 359 154 155 156 5.03 695 5.04 343 5.04 986 194 195 196 5.26 786 5.27 300 5.27 811 37 38 39 40 3.61 092 3.63 759 3.66 356 77 78 79 80 4.34 381 4.35 671 4.36 945 117 118 119 120 4.76 217 4.77 068 4.77 912 157 158 159 160 5.05 625 5.06 260 5.06 890 197 198 199 200 5.28 320 5.28 827 5.29 330 3.68 888 4.38 203 4.78 749 5.07 517 5.29 832 97 TABLE IV. XATTKAL LOIJAKITHMS B. NATURAL LOGARITHMS 1 TO 9 9 The following table ' < shows >- the Natural or Napierian log- arithms, for each tenth, of numbers 1 to 9.9. Interpola- tion may be made for hundredths. ' The logarithms of numbers larger than 9.9 may be found as shown in the following illustration : Let us find the log e 450 : log e 450 = log e (4.5 x 1O*) = log e 4.5 + 2 log e 10 = 1.5041 + 2 (2.3026) = 6.1093. No. lOge No. loge No. lOge No. lOge No. lOge No. loge 1.0 1.1 1.2 1.3 1.4 0.0000 .0953 .1823 .2624 .3365 2.5 2.6 2.7 2.8 2.9 0.9163 .9555 .9933 1.02% .0647 4.0 4.1 4.2 4.3 4.4 1.3863 .4110 .4351 .4586 .4816 5.5 5.6 5.7 5.8 5.9 1.7048 .7228 .7405 .7579 .7750 7.0 7.1 7.2 7.3 7.4 1.9459 .9601 .9741 .9879 2.0015 8.5 8.6 8.7 8.8 8.9 2.1401 .1518 .1633 .1748 .1861 1.5 0.4055 3.0 1.0986 4.5 1.5041 6.0 1.7917 7.5 2.0149 9.0 2.1972 1.6 .4700 3.1 .1314 4.6 .5261 6.1 .8083 7.6 .0282 9.1 .2083 1.7 .5306 3.2 .1632 4.7 .5476 6.2 .8246 7.7 .0412 9.2 .2192 1.8 .5878 3.3 .1939 4.8 .5686 6.3 .8406 7.8 .0541 9.3 .2300 1.9 .6419 3.4 .2238 4.9 .5892 6.4 .8563 7.9 .0669 9.4 .2407 2.0 0.6932 3.5 1.2528 5.0 1.6094 6.5 1.8718 8.0 2.0794 9.5 2.2513 2.1 .7419 3.6 .2809 5.1 .6292 6.6 .8871 8.1 .0919 9.6 .2618 2.2 .7885 3.7 .3083 5.2 .6487 6.7 .9021 8.2 .1041 9.7 .2721 2.3 .8329 3.8 .3350 5.3 .6677 6.8 .9169 8.3 .1163 9.S .2824 2.4 .8755 3.9 .3610 5.4 .6864 6.9 .9315 8.4 .1282 9.9 .2925 98 TABLE IV 3. HYPERBOLIC X sinh j- cosh x x sinhjr coshx O.'.'j 0.0000 1.0000 0.5211 1.1276 .01 .0100 1.0000 .51 .5324 1.1329 .02 .0200 1.0002 .52 .5438 1.1383 .03 .0300 1.0004 .53 .5552 1.1438 .04 .0400 1.0008 .54 .5666 1.1494 .05 .0500 1.0013 .55 .5782 1.1551 .06 .0600 1.0018 .56 .5897 1.1609 .07 .0701 1.0025 .57 .6014 1.1669 .08 .0801 1.0032 .58 .6131 1.1730 .09 .0901 1.0041 .59 .6248 1.1792 .10 .1002 1.0050 .60 .6367 1.1855 .11 .1102 1.0061 .61 .6485 1.1919 .12 .1203 1.0072 .62 .6605 1.1984 .13 .1304 1.0085 .63 .6725 1.2051 .14 .1405 1.0098 .64 .6846 1.2119 .15 .1506 1.0113 .65 .6967 1.2188 .16 .1607 1.0128 .66 .7090 1.2258 .17 .1708 1.0145 .67 .7213 1.2330 .18 .1810 1.0162 .68 .7336 1.2402 .19 .1911 1.0181 .69 .7461 1.2476 .20 .2013 1.0201 .70 .7586 1.2552 .21 .2115 1.0221 .71 .7712 1.2628 .22 .2218 1.0243 .72 .7838 1.2706 .23 .2320 1.0266 .73 .7966 1.2785 .24 .2423 1.0289 .74 .8094 1.2865 .25 .2526 1.0314 .75 . 822 ; 1.2947 .26 .2629 1.0340 .76 .8353 1.3030 .27 .2733 1.0367 .77 .8484 1.3114 .28 .2837 1.0395 .78 .8615 1.3199 .29 .2941 1.0423 .79 .8748 1.3286 .30 .3045 1.0453 .80 .8881 1.3374 .31 .3150 1.0484 .81 .9015 1.3464 .32 .3255 1.0516 .82 .9150 1.3555 .33 .3360 1.0549 .83 .9286 1.3647 .34 .3466 1.0584 .84 .9423 1.3740 .35 .3572 1.0619 .85 .9561 1.3835 .36 .3678 1.0655 .86 .9700 1.3932 .37 .3785 1.0692 .87 .9840 1.4029 .38 .3892 1.0731 .88 .9981 1.4128 .39 .4000 1.0770 .89 1.0122 1.4229 .40 .4108 1.0811 .90 1.0265 1.4331 .41 .4216 1.0S52 .91 1.0409 1.4434 .42 .4325 1.0895 .92 1.0554 1.4539 .43 .4434 1.0939 .93 1.0700 1.4645 .44 .4543 1.0984 .94 1.0847 1.4753 .45 .4653 1.1030 .95 1.0995 1.4862 .46 .4764 1.1077 .96 1.1144 1.4973 .47 .4875 1.1125 .97 1.1294 1.5085 .48 .4986 1.1174 .98 1.1446 1.5199 .49 .5098 1.1225 .99 1.1598 1.5314 0.50 0.5211 1.1276 1.00 1.1752 1.5431 First Course in Differential and Integral Calculus BY WILLIAM F. OSGOOD, PH.D. Professor of Mathematics in Harvard University Revised Edition. Cloth, xv + 462 pages, $2.00 net The treatment of this calculus by Professor Osgood is based on the courses he has given in Harvard College for a number of years. The chief characteristics of the treatment are the close touch between the calculus and those problems of physics, including geometry, to which it owed its origin; and the simplicity and directness with which the principles of the calculus are set forth. It is important that the formal side of the calculus should be thoroughly taught in a first course, and great stress has been laid on this side. But nowhere do the ideas that underlie the calculus come out more clearly than in its applications to curve tracing and the study of curves and surfaces, in definite integrals, with their varied applications to physics and geometry, and in mechanics. For this reason these subjects have been taken up at an early stage and illustrated by many examples not usually found in American text-books. From the beginning the book has been a favorite with the academic classes, and it has now been adopted by some of the best-known and best-thought-of engineering schools in the country. THE MACMILLAN COMPANY 64-66 Fifth Avenue, New York Applied Mechanics for Engineers A Text-book for Engineering Stuaents BY F. L. HANCOCK Professor of Applied Mechanics, Worcester Polytechnic Institute Illustrated, cloth, I2mo, xii + jSS pages, $2.00 net A new edition with typographical corrections In the preparation of this book the author has had in mind the fact that the student finds much difficulty in seeing the applications of theory to practical problems. For this reason each new prin- ciple developed is followed by a number of applications. In many cases these are illustrated, and they all deal with matters that directly concern the engineer. It is believed that the problems in mechanics should be practical engineering work. The author has endeavored to follow out this idea in writing the present volume. Accordingly, the title " Applied Mechanics for Engineers " has been given to the book. The book is intended as a text-book for engineering students of the Junior year. The subject-matter is such as is usually covered by the work of one semester. In some chapters more material is presented than can be used in this time. With this idea in mind the articles in these chapters have been arranged so that those coming last may be omitted without affecting the continuity of the work. The book contains more problems than can usually be given in any one semester. An appendix giving tables for the use of the student is of importance. These tables include the following : Hyperbolic Functions, Trigonometric Functions, Logarithms of Numbers, Squares, Cubes, etc., and Conversion Tables. THE MACMILLAN COMPANY 64-66 Fifth Avenue, New York Co-ordinate Geometry BY HENRY BURCHARD FINE AND HENRY DALLAS THOMPSON Cloth, ismo, x + 316 pages, $1.60 net " It surprised me to see that the authors could put the treat- ment of the plane into less than two hundred pages, and my surprise increased to find that no important topic had been omitted." VIRGIL SNYDER, Ithaca, New York. " After teaching this book one year I want to say that it is the most teachable book of its kind that I have seen in some time. We expect to use it again next year." A. M. HARDING, Fay- etteville, Arkansas. " I am just finishing a half year with a class of twenty-one in Fine and Thompson's Co-ordinate Geometry, having found the book one of the freshest and most teachable I ever put into a student's hands. " Their methods of treatment are a distinct departure from those found in the usual texts, being more direct, more concrete, and more lucid. The examples, solved and unsolved, are beau- tiful and thoroughly representative. In fact, it is every way an uncommonly good beginner's book." IRA M. DELoNG, Boulder, Colorado. THE MACMILLAN COMPANY 64-66 Fifth Avenue, New York MAY MARlb 1948 MAR 14 1949 1950 UNIVERSITY OF CALIFORNIA, LOS ANGELES THE UNIVERSITY LIBRARY DUE on the last date stamped below APR 3 OCT25195I 2 2 1958 ,1* APR ~, AUG13 AUG 1 5 1958 SEP 81 j :363 MAY AUG 5 AUG OCT CT 8 58f A 000 257 690 :'.\ng & latical Sciences Library