£ Irving Stringham © 47 J ■ >LsV\s 30° \M V ■ In n 1 W* i fS \ I Oo ^. 2fs Q y Oo - / bo Oo-/-/ - Oo O*? -'-.Oo OoxOoz O &> - I Oo -o> ^ O t Oo t**X% i b to*** ~ I WENTWORTH'S SERIES OF MATHEMATICS. First Steps in Number. Primary Arithmetic. Grammar School Arithmetic High School Arithmetic. Exercises in Arithmetic. Shorter Course in Algebra. Elements of Algebra. Complete Algebra. ^College Algebra. Exercises in Algebra. Plane Geometry. '"'Plane and Solid Geometry. Exercises in Geometry. VPI. and Sol. Geometry and PI. Trigonometry. Plane Trigonometry and Tables. Plane and Spherical Trigonometry. purveying. Vpi. and Sph. Trigonometry, Surveying, and Tables. Trigonometry, Surveying, and Navigation. Trigonometry Formulas. Logarithmic and Trigonometric Tables (Seven). Log. and Trig. Tables (Complete Edition). Analytic Geometry. Special Terms and Circular on Application. Plane Trigonometry by Qt, A. WENTWOKTH, A.M., PROFESSOR OF MATHEMATICS IN PHILLIPS EXETER A.CADEMY BOSTON, U.S.A.: GINN & COMPANY, PUBLISHEKS. 1891. GK 533 C5*^ Entered, according to Act of Congress, in the year 1883, by G. A. WENT WORTH, in the Office of the Librarian of Congress, at Washington. Typography by J. S. Cushing & Co., Boston, U.'S.A. Prbssworx by Ginn & Co., Boston, U.S.A. PREFACE. IN preparing this work the aim has been to furnish just so much of Trigonometry as is actually taught in our best schools and colleges. Consequently, development of functions in series and all other investigations that are important only for the special student have been omitted. The principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and inter- esting problems have been selected with a view to awaken a real love for the study. Much time and labor have been spent in devis- ing the simplest proofs for the propositions, and in exhibiting the best methods of arranging the logarithmic work. The object of the work on Surveying and Navigation is to pre- sent these subjects in a clear and intelligible way, according to the best methods in actual use ; and also to present them in so small a compass that students in general may find the time to acquire a competent knowledge of these very interesting and important studies. The author is under particular obligation for assistance to G. A. Hill, A.M., of Cambridge, Mass., and to Prof. James L. Patterson, of Law- rencerille, N.J. G. A. WENTWORTH. Phillips Exeter Academy, September, 1882. 800564 CONTENTS. PLANE TRIGONOMETRY. CHAPTER I. Functions of Acute Angles: Definitions, 1 ; representation of functions by lines, 7 ; changes in the functions as the angle changes, 9 ; functions of complementary- angles, 10 ; relations of the functions of an angle, 11 ; formulas for finding all the other functions of an angle when one function of the angle is given, 13; functions of 45°, 30°, 60°, 15. CHAPTER II. The Right Triangle: Solution: Case I., when an acute angle and the hypotenuse are given, 16; Case II., when an acute angle and the opposite leg are given, 17; Case III., when an acute angle and the adjacent leg are given, 17 ; Case IV., when the hypotenuse and a leg are given, 18 ; Case V., when the two legs are given, 18 ; general method of solving the right triangle, 19 ; area of the right triangle, 20 ; the isosceles triangle, 24; the regular polygon, 26. CHAPTER III. Goniometry: Definition of Goniometry, 28; angles of any magnitude, 28; gen- eral definitions of the functions of angles, 29 ; algebraic signs of the functions, 31 ; functions of a variable angle, 32 ; functions of angles larger than 360°, 34 ; formulas for acute angles extended to all angles, 35 ; reduction of the functions of all angles to the functions of angles in the first quadrant, 38 ; functions of angles that differ by 90°, 40 ; functions of a negative angle, 41 ; functions of the sum of two angles, 43 ; functions of the difference of two angles, 45 ; functions of twice an angle, 47 ; functions of half an angle, 47 ; sums and differences of functions, 48. > VI TRIGONOMETRY. CHAPTER IV. The Oblique Triangle: Law of sines, 50 ; law of cosines, 52 ; law of tangents, 52. Solu- tion : Case I., when one side and two angles are given, 54 ; Case II., when two sides and the angle opposite to one of them are given, 56 ; Case III., when two sides and the included angle are given, 60 ; Case IV., when the three sides are given, 64 ; area of a triangle, 68 ; mis- cellaneous problems, 71-88. EXAMINATION PAPERS, 89-102. SPHERICAL TRIGONOMETRY. CHAPTER V. The Right Spherical Triangle: Introduction, 103 ; formulas relating to right spherical triangles, 105 ; Napier's rules, 108. Solution : Case I., when the two legs are given, 110 ; Case II., when the hypotenuse and a leg are given, 110 ; Case III., when a leg and the opposite angle are given, 111 ; Case IV., when a leg and an adjacent angle are given, 111 ; Case V., when the hypotenuse and an oblique angle are given, 111 ; Case VI., when the two oblique angles are given, 111 ; solution of the isosceles spherical triangle, 116 ; solution of a regular spherical polygon, 116. CHAPTER VI. The Oblique Spherical Triangle: Fundamental formulas, 117 ; formulas for half angles and sides, 119 ; Gauss's equations and Napier's analogies, 121. Solution : Case I., when two sides and the included angle are given, 123 ; Case II., when two angles and the included side are given, 125 ; Case III., when two sides and an angle opposite to one of them are given, 127 ; Case IV., when two angles and a side opposite to one of them are given, 129 ; Case V., when the three sides are given, 130; Case VI., when the three angles are given, 131 ; area of a spherical triangle, 133. CHAPTER VII. Applications of Spherical Trigonometry : Problem, to reduce an angle measured in space to the horizon, 136 •, problem, to find the distance between two places on the earth's sur- face when the latitudes of the places and the difference of their longi- tudes are known, 137 ; the celestial sphere, 137 ; spherical co-ordinates, 140 ; the astronomical triangle, 142; astronomical problems, 143-146. PLANE TRIGONOMETRY. CHAPTER I. TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES. § 1. Definitions. The sides and angles of a plane triangle are so related that any three given parts, provided at least one of them is a side, determine the shape and the size of the triangle. Geometry shows how, from three such parts, to construct the triangle and find the values of the unknown parts. Trigonometry shows how to compute the unknown parts of a triangle from the numerical values of the given parts. Geometry shows in a general way that the sides and angles of a triangle are mutually dependent. Trigonometry begins by showing the exact nature of this dependence in the right triangle, and for this purpose employs the ratios of its sides. Let MAN (Fig. 1) be an acute angle. If from any points B, D, F, in one of its sides perpendiculars BC, DF, FG, are let fall to the other side, then the right triangles ABC, ADE, AFG, thus formed have the angle A common, and are there- fore mutually equiangular and similar. Hence, the ratios of their corresponding sides, pair by pair, are equal. That is, AC _AE _AG , AB AD AF' AC^AE^AG BC DF FG ; etc. TRIGONOMETRY. Hence, for every value of an acute angle A there are certain numbers that express the values of the ratios of the sides in all right triangles that have this acute angle A. There are altogether six different ratios : I. The ratio of the opposite leg to the hypotenuse is called the Sine of A, and is written sin A. II. The ratio of the adjacent leg to the hypotenuse is called the Cosine of A, and written cos A III. The ratio of the opposite leg to the adjacent leg is called the Tangent of A, and written tan A IV. The ratio of the adjacent leg to the opposite leg is called the Cotangent of A, and written cot A. V. The ratio of the hypotenuse to the adjacent leg is called the /Secant of A, and written sec A. VI. The ratio of the hypotenuse to the opposite leg is called the Cosecant of A. and written esc A. In the right triangle ABC (Fig. 2) let a, b, c denote the lengths of the sides opposite to the acute angles A, B, and the right angle C, respectively, these lengths being all expressed in terms of a common unit. Then, A a opposite leg sin A — - = ^ — ; -• c hypotenuse . b _ adjacent leg c hypotenuse tan..l- a - ° PP ° siteleg b adjacent leg . . b adjacent leg cot. A = - = — - — ^— -j-^' a opposite leg sec A hypotc b adjacent leg esc A = c hypotenuse a opposite leg These six ratios are called the Trigonometric Functions of the angle A. TRIGONOMETRIC FUNCTIONS. Exercise I. 1. What are the functions of the other acute angle B of the triangle ABC (Fig. 2)? 2. Prove that if two angles, A and B, are complements of each other (i.e., ifA + B = 90°), then, sin A = cos B, tan A = cot B, sec A = esc B ; cos A = sin B, cot A = tan B, esc ^4 = sec B. 3. Find the values of the functions of A, if a, b, c respec- tively have the following values : (i.) 3, 4, 5. (iii.) 8, 15, 17. (v.) 3.9, 8, 8.9. (ii.) 5, 12, 13. (iv.) 9, 40, 41. (vi.) 1.19, 1.20, 1.69. 4. What condition must be fulfilled by the lengths of the three lines a, b, c (Fig. 2) in order to make them the sides of a right triangle ? Is this condition fulfilled in Example 3 ? 5. Find the values of the functions of A, if a, b } c respec- tively have the following values : (i.) 2 raw, m 2 —n 2 , m 2 -\~n 2 . (iii.) pqr, qrs, rsp, (n.) £-, x + y, --OL. (iv.) — , — , — . x — y % — y pq sq ps 6. Prove that the values of a, b, c, in (i.) and (ii.), Example 5, satisfy the condition necessary to make them the sides of a right triangle. 7. What equations of condition must be satisfied by the values of a, b, c, in (iii.) and (iv.), Example 5, in order that the values may represent the sides of a right triangle? Compute the functions of A and B when, 8. a = 24, 5 = 143. 11. a = ^p 2 + q 2 , b = V2pq. 9. a = 0.264, c = 0.265. 12. a = Vp 2 +pq, c = p + q. 10. 5 = 9.5, , = 19.3. 13 b==2 Vfi, c=p + q. 6 TRIGONOMETRY. Compute the functions of A when, 14. a = 2b. 16. a + b = 15. a=t sin 2 A. 20. Given two angles A and B (A + B being less than 90°) ; show that sin (A -f B) < sin A -f sin B. 21. Given sina; in a unit circle; find the length of a line corresponding in position to sina; in a circle -whose radius is r ? 22. In a right triangle, given the hypotenuse c, and also sin A = m, cos A = n ; find the legs. TRIGONOMETRIC FUNCTIONS. § 3. Changes in the Functions as the Angle Changes. If we suppose the Z AOP, or x (Fig. 4) to increase gradtu ally by the revolution of the moving radius OP about 0, the point P will move along the arc AB towards B, T will move along the tangent A T away from A, S will move along the tangent BS towards B, and M will move along the radius OA towards 0. Hence, the lines PM, AT, OT will gradually increase in length, and the lines OM, BS, OS, will gradually decrease. That is, As an acute angle increases, its sine, idngcnt, and secant also in- crease, while its cosine, cotangent, and cosecant decrease. On the other hand, if we suppose x to decrease gradually, the reverse changes in its functions will occur. If we suppose x to decrease to 0°, OP will coincide with OA and be parallel to BS. Therefore, PM and A T will vanish, OM will become equal to OA, while BS said OS will each be infinitely long, and be represented in value by the symbol oo. And if we suppose x to increase to 90°, OP will coincide with OB and be parallel to AT. Therefore, PM and OS will each be equal to OB, OM and BS will vanish, while AT and Twill each be infinite in length. Hence, as the angle x increases from 0° to 90°, sin x increases from to 1, cos x decreases from 1 to 0, tana; increases from to oo, cot x decreases from oo to 0, sec x increases from 1 to co s esc x decreases from oo to 1. 10 TRIGONOMETRY. The values of the functions of 0° and of 90° are the limiting values of the functions of an acute angle. It is evident that (disregarding the limiting values), Sines and cosines are always less than 1 ; Secants and cosecants are always greater than 1 ; Tangents and cotangents have all values between and oo. Remabk. We are now able to understand why the sine, cosine, etc., of an angle are called functions of the angle. By & function of any mag- nitude is meant another magnitude which remains the same so long as the first magnitude remains the same, but changes in value for every change in the value of the first magnitude. This, as we now see, is the relation in which the sine, cosine, etc., of an angle stand to the angle. §4. Functions of Complementary Angles. The general form of two complementary angles is A and 90° - A. In the rt. A ABO (Fig. 5) A + B = 90°, hence B = 90° - A. Therefore (§ 1), sin A = cos B = cos (90° - A), cos A = sin B = sin (90° — A), tan A = cot B = cot (90° - A), cot A = tan B = tan (90° - A), sec A = esc B = esc (90° — A), esc A = sec B = sec (90° - A). Therefore, ■ Each function of an acute angle is equal to the co-named function of the complementary angle. Note. Cosine, cotangent, and cosecant are sometimes called co-functions; the words are simply abbreviated forms of complement's sine, complement s tangent, and complement's secant. Hence, also, Any function of an angle between 45° and 90° may be found by taking the co-named function of the complementary angle between 0° and 45°. TRIGONOMETRIC FUNCTIONS. 11 Exercise III. 1. Express the following functions as functions of the com- plementary angle : sin 30°. tan 89°. esc 18° 10'. cot 82° 19'. cos 45°. cot 15°. cos 37° 24'. esc 54° 46'. 2. Express the following functions as functions of an angle less than 45° : sin60°. tan 57°. csc69°2\ cot89° 59'. cos75°. cot84°. cos85°39'. esc 45° 1'. 3. Given tan 30° = i V3 ; find cot 60°. 4. Given tan A = cot A ; find A:- " 5. Given cos A = sin 2 A ; find A. 6. Given sin A = cos 2 A ; find A. 7. Given cos A = sin (45° — hA); find A. 8. Given cot i A — tan A ; find A. 9. Given tan (45° + A) = cot A ; find A. 10. Find A if sin A = cos 4-4. 11. Find A if cot A = tan 8 A. 12. Find A if cot A = tan nA. § 5. Relations of the Functions of an Angle. Since (Fig. 5) a 2 -\- b 2 = c 2 , therefore, a 2 , 6 2 . fa\* fb^ 2 Therefore (§ 1), (sin ^) 2 + (cos ^L) 2 = 1 ; or, as usually written for convenience, sin 2 A + cos 2 A = l. [1] That is : The sum of the squares of the sine and the cosine of an angle is equal to unity. 12 TRIGONOMETRY. Formula [1] enables us to find the cosine of an angle when the sine is known, and vice versa. The values of sin A and of cos A deduced from [1] are : sin A = Vl — cosM, cos A == Vl — sin 2 A a . abaca Since — j — = _-x 7 = T , c c c b b therefore (§ 1), tan A = —£ [2] That is : The tangent of an angle is equal to the sine divided by the cosine. Formula [2] enables us to find the tangent of an angle when the sine and the cosine are known. a . c b c ~ , a b Since -X- — 1, -X T = 1, and T X- c o b a therefore (§ 1), sin A X esc A = 1 1 cos A x sec A = 1 I • - [3] tan A X cot A = 1 J That is : The sine and the cosecant of an angle, the cosine and secant, and the tangent and cotangent, pair by pair, are reciprocals. The equations in [3] enable us to find an unknown function contained in any pair of these reciprocals when the other func- tion in this pair is known. Exercise IV. 1. Prove Formulas [l]--[3], using for the functions the line Values in unit circle given in § 3. 2. Prove that 1 -f- tan 2 ^. == sec 2 A. Hint. Divide the terms of the equation a 2 + 6* — c 2 by b*. 3. Prove that l + cotM = csc 2 X t . _ . cos ^4 4. Prove that cot A = - — f sin A TRIGONOMETRIC FUNCTIONS. 13 §6. Application of Formulas [l]-[3]. Formulas [1], [2], and [3] enable us, when any one function of an angle is given, to find all the others. A given value of any one function, therefore, determines all the others. Example 1. Given sin A = -§ ; find the other functions. By [1], cos A = Vr=l- = V| = JV5. By [2], ta ^ = !-4v5 = fx-|=A. By |"3l, cot A = ——. secA = , csc.4 = — Example 2. . Given tan A — 3 ; find the other functions. By [2], s i^4 = 3. cos A And by [1], sin 2 A -f- cos 2 ^. = 1. If we solve these equations (regarding sin A and cos A as two unknown quantities), we find that, sin A = 3 V T V cos A = V T V Then by [3], cot A = |, sec ^4 = VIO, esc J. = -J VlO, Example 3. Given sec A = m; find the other functions. 1 By [3], cos A = m By[l], s in^ = Jl-i = J^i=iv: % [2], [3], tan A = Vm 2 - 1 , cot ^ - — ^ esc -4 V- III- 14 TRIGONOMETRY. Exercise V. Find the values of the other functions when, 1. sin A = |f. © tan^4 = |. 9. csc^4=V2 2. sin A = 0.8. 6. cot A = 1. 10. sin.4 = m. 3. cos ^4 = ff 7. cot A = 0.5. 11. sin ^4 = 1 + ra 2 4. cos,4 = 0.28. 8. sec A = 2. 2 mi 12. cos ^4 = — - — • ra a -fn 13. Given tan 45° = 1 ; find the other functions of 45°. 14. Given sin 30° = -J- ; find the other functions of 30°. 15. Given esc 60° = -§ V3 ; find the other functions of 60°. 16. Given tan 15° = 2 - V3 ; find the other functions of 15°. 17. Given cot 22° 30'= V2 + 1; find the other functions of 22° 30'. 18. Given sin 0° = ; find the other functions of 0°. 19. Given sin 90° = 1 ; find the other functions of 90°. 20. Given tan 90° = co ; find the other functions of 90°. 21. Express the values of all the other functions in terms of sin A. 22. Express the values of all the other functions in terms of cos A. 23. Express the values of all the other functions in terms of tan A. 24. Express the values of all the other functions in terms of cot A. 25. Given 2 sin A = cos A ; find sin A and cos A. 26. Given 4 sin A = tan A ; find sin A and tan A. 27. If sin A : cos A — 9 : 40, find sin A and cos A, 28. Transform the quantity tan 2 A -f- cot 2 J. — sin 2 ^4 — cos 2 ^4 into a form containing only cos A. 29. Prove that sin A -f cos ^4 = (1 -f- tan A) cos A. 30. Prove that tan A -j- cot ^4 = sec A X esc A. TRIGONOMETRIC FUNCTIONS. 15 § 7. Functions of 45°. Let ABC (Fig. 6) be an isosceles right triangle, in which the length of the hypotenuse AB is equal to 1 ; then AC is equal to BC, and the angle A is equal to 45°. Since AC + BC = 1, therefore 2AC 2 = 1, and^C = VJ = ^V2. Therefore (§ 1), sin 45° = cos45° = £V2. tan45 = cot45° = l. sec 45° = esc 45° = V2. j \/2 Fig. 6. §8. Functions of 30° and 60°. Let ABC be .an equilateral triangle in which the length of each side is equal to 1 ; and let CD bisect the angle C. Then CD is perpendicular to AB and bisects AB ; hence,. AD = |, and CD = VF^T = V I = tV& In the right triangle ADC, the angle ACD = 30°, and the angle CAD = 60°. Whence (§ 1), sin 30° = cos 60° = i cos 30° = sin 60° tan 30° = cot 60° cot 30° = tan 60° sec 30° = esc 60° iV8. V3. 2 V3 ■ esc 30° = sec 60° = 2. The results for sine and cosine of 30°, 45°, and 60° may be easily remembered by arranging them in the following form : Angle . . . 30° 45° 60° 1 = 0.5 Sine .... } iV2 iVE *V§ = 0.70711 Cosine. . . iV3 lV2 | iV3 = 0.86603 1 CHAPTER II. THE RIGHT TRIANGLE. § 9. The Given Parts. In order to solve a right triangle, two parts besides the right angle must be given, one of them at least being a side. The two given parts may be : I. An acute angle and the hypotenuse. II. An acute angle and the opposite leg. III. An acute angle and the adjacent leg. IV. The hypotenuse and a leg. V. The two legs. § 10. Case I. Given A = 34° 28' and c = 18.75 ; required B, a, b. B 1. J5 = 90°-^4 = 55°32'. 2. 1 =sin^ c '. a = c sin A. b Fig. 8. log a = log c + log sin A logc = 1.27300 log sin A= 9.75276* log a = 1.02576 a = 10.61 cos A ; .*. b = ccosA. log b — log c -f- log cos A logc = 1.27300 log cos A= 9.91617 log& = 1.18917 b = 15.459 * For Logarithms, and directions how to use them, see Wentworth and Hill's Five-place Tables. When —10 belongs to a logarithm or cologarithm, and is not written, it must be remembered that the logarithm or cologarithm is 10 too large. THE RIGHT TRIANGLE. 17 § 11. Case II. Given A = 62* 10', a = 78 ; find B, b, c. 1. .5 = 90°- A = 27° 50'. 2. - = cot^4; .'.b — a cot A 3. - = sin.4. c .'. a =csmA, and a sin ^4 log b = loga + logcot^4 log a = 1.89209 log cot A = 9.72262 loe& = 1.61471 ^g & =41.182 logc log a = log a + colog sin A === 1.89209 colog sin A= 0.05340 log* = 1.94549 c =88.204 § 12. Case III. Given A = 50° 2', b = 88 ; find B, a, c. 1. 5 = 90° - ^4 = 39° 58'. 2. 7 = tan^4; .'. a = Man A b 8.* cos .4. ccos^l, and g< = 1.68358 = 48.259 THE RIGHT TRIANGLE. 19 § 15. General Method of Solving the Right Triangle. From these five cases it appears that the general method of finding an unknown part in a right triangle is as follows : Choose from the equation A-\- B = 90°, and the equations that define the functions of the angles, an equation in which the re- quired part only is unknown; solve this equation, if necessary, to find the value of the unknown part ; then compute the value, using logarithms whenever convenient. Note 1. In Cases IV. and V. the unknown side may also be found by Geometry, from the equation a 2 + b* = c 2 ; whence we obtain (for Case IV.) b = Vc^a* = V(c + a)(c-a) (for Case V.) c = Va 2 + 6 2 . These equations express the values of b and c directly in terms of the two given sides ; and if the values of the sides are simple numbers [e.g. 5, 12, 13), it is often easier to find b or c in this way. But this value of c is not adapted to logarithms, and this value of b is not so readily worked out by logarithms as the value of b given in § 13. Note 2. In Case IV. if the given sides (here a and c) are nearly alike in value, then A is near 90°, and its value cannot be accurately found from the tables, because the sines of large angles differ little in value (as is evident from Fig. 4). In this case it is better to find B first, by means of a formula proved later (see page 47) ; viz., tan^^A/ 1 ^; *c + a and to find b by the method given in Note 1, since the same logarithms are used in both cases. Example. Given a = 49, c = 50 ; find A, B, b. b =H 1 og( c -«)+ 1 °g( c + a )] c — a =1 c + a =99 log (c - a) = 0.00000 log (c + a) = 1.99564 2 )1.99564 log b - 0.99782 b =-■ 9.95 log tan \B = \ [log (c — a) + colog (c + a)] log(c-a) =0.00000 colog (c + a) = 8.00436 2 ) 8.00436 log tan %B =9.00218 %B =5° 44' 21" B =11° 29' A =78^31' 20 TRIGONOMETRY. § 16. Area of the Right Triangle. It is shown in Geometry that the area of a triangle is equal to one-half the product of the base by the altitude. Therefore, if a and b denote the legs of a right triangle, and F the area, F=$ab. By means of this formula the area may always be found when a and b are given or have been computed. For example : Find the area, having given : Case I. (§ 10). .4 = 34° 28', c = 18.75. First find (as in § 10) log a and log&. ]og(F) — \oga-\-logb-{-co\og2 log a = 1.02578 log b = 1.18915 colog2 = 9.69897 \og(F) = 1.91390 F = 82.016 Case IV. (§ 13). a -47.54, c = 58.40. First find (as in § 13) log a and log b. log (F) = log a +log b -f- colog 2 log a =1.67715 log b =1.53025 colog 2 =9.69897 \og(F) = 2.90637 F = 806.06 Exercise VI. 1. In Case II. give another way of finding c, after b has been found. 2. In Case III. give another way of finding c, after a has been found. 3. In Case IV. give another way of finding b, after the angles have been found. 4. In Case V. give another way of finding c, after the angles have been found. 5. Given B and c ; find A, a, b. 6. Given B and b ; find A, a, c. 7. Given B and a; find A, b, c. 8. Given b and c ; find A, B,a. THE RIGHT TRIANGLE. 21 Solve the following triangles : 9 Given : Required : a = 6, c = 12. A = 30°, B = 60°, 6 = 10.392. 10 A - 60°, 6 = 4. B = 30°, c = 8, a = 6.9282. / 11 A - 30°, a = 3. B = 60°, c = 6, 6 = 5.1961. / 12 a = 4, 6 = 4. A = 5 = 45°, c = 5.6568. 13 a = 2, c - 2.82843. ^ = £ = 45°, 6 = 2. 14 c = 627, J. = 23° 30'. 5 = 66° 30', a = 250.02, 6 = 575.0. 15 c - 2280, JL = 28° 5'. B = 61° 55', a = 1073.3, 6 = 2011.6. y 16 c - 72.15, A = 39° 34'. £ = 50° 26', a = 45.958, 6 = 55.620. , 17 c-l f A = 36°. B = 54°, a = 0.58779, 6 = 0.80902. 18 c - 200, £ - 21° 47'. ^ = 68°13', a =185.73, 6 - 74.22. !•- 19 c = 93.4, B =76° 25'. ^1 = 13° 35', a = 21.936, 6 = 90.788. 20 a - 637, ^1= 4° 35'. B = 85° 25', 6 = 7946, c= 7971.5. 21 a = 48.532, 4 = 36° 44'. 5 = 53° 16', 6 = 65.031, c = 81.144. 22 a = 0.0008, A = 86°. B= 4°, 6 = 0.0000559 , c = 0.000802 23 6 = 50.937, £ = 43° 48'. J. = 46° 12', a = 53.116, c - 73.59. 24 6 = 2, B= 3° 38'. A = 86° 22', a = 31.497, c = 31.560. 25 a = 992, £ = 76° 19'. A = 13° 41', 6 = 4074.5. c = 4193.6. 26 a =73, £ = 68° 52'. A = 21° 8, 6 = 188.86, c = 202.47. 27 a = 2.189, B = 45° 25'. ^4 = 44° 35', 6 = 2.2211, c = 3.1185. s 28 6 = 4, A = 37° 56'. 5 = 52° 4', a = 3.1176, c = 5.0714. 29 c = 8590, a = 4476. ^ = 31° 24', £ = 58° 36', 6 = 7332.8. 30 c = 86.53, a = 71.78. 4 = 56° 3', £ = 33° 57'. 6 = 48.324. 31 c = 9.35, a = 8.49. A = 65° 14', B - 24° 46', 6 = 3.917. 32 c = 2194, 6 = 1312.7. A = 53° 15', 5 = 36° 45', a = 1758. 33 c - 30.69, 6 = 18.256. A = 53° 30', B = 36° 30', a = 24.67. 34 a = 38.313, 6 = 19.522. A = 63°, 5 = 27°, c = 43. 35 a = 1.2291, 6 = 14.950. A= 4 C 42', B = 85° 18', c=15. 36 a = 415.38, 6 = 62.080. ^1 = 81° 30', J5= 8° 30', c = 420. 37 a - 13.690, 6 = 16.926. J. = 38° 58', B = 51° 2', c = 21.769. 38 c - 91.92, a = 2.19. J. = 1°21'55", 5 = 88°38'5", 6 - 91.894. Compute the unknown parts and also the area, having given 39. a = 5, 5 = 6. 40. a = 0.615, c = 70 L 41. b = -ty2, c=V3. 42. a = 7, ^ = 18° 14'. 43. 5=12, ^4 = 29° 8'. 44. c = 68, 45. c = 27, 46. a = 47, 47. 5 = 9, ^4 = 69° 54'. ^ = 44° 4'. .5 = 48° 49'. .5 = 34° 44'. 48. c = K462 J9 = 86°4' 22 TRIGONOMETRY. 49. Find the value of Fin terms of c and A. 50. Find the value of .Fin terms of a and A. 51. Find the value of Fin terms of b and A. 52. Find the value of Fin terms of a and c. 53. Given F= 58, a = 10 ; solve the triangle. 54. Given F= 18, b = 5 ; solve the triangle. 55. Given F= 12, ^4 = 29° ; solve the triangle. 56. Given F= 100, c = 22; solve the triangle. 57. Find the angles of a right triangle if the hypotenuse is equal to three times one of the legs. 58. Find the legs of a right triangle if the hypotenuse = 6, and one angle is twice the other. 59. In a right triangle given c, and A — nB ; find a and b. 60. In a right triangle the difference between the hypote- nuse and the greater leg is equal to the difference between the two legs ; find the angles. The angle of elevation of an object (or angle of depression, if the object is below the level of the observer) is the angle which a line from the eye to the object makes with a horizon- tal line in the same vertical plane. 61. At a horizontal distance of 120 feet from the foot of a steeple, the angle of elevation of the top was found to be 60° 30' ; find the height of the steeple. 62. From the top of a rock that rises vertically 326 feet out of the water, the angle of depression of a boat was found to be 24° ; find the distance of the boat from the foot of the rock. 63. How far is a monument, in a level plain, from the eye, if the height of the monument is 200 feet and the angle of ele- vation of the top 3° 30'? 64. In order to find the breadth of a river a distance AB was measured along the bank, the point A being directly op- posite a tree C on the other side. The angle A B C was also measured. If AB = 96 feet, and ABC= 21° W, find the breadth of the river. If ABC= 45°, what would be the breadth of the river? THE BIGHT TRIANGLE. 23 65. Find the angle of elevation of the sun when a tower a feet high casts a horizontal shadow b feet long. Find the angle when a = 120, b = 70. 66. How high is a tree that casts a horizontal shadow b feet in length when the angle of elevation of the sun is A° ? Find the height of the tree when b = 80, A = 50°. 67. What is the angle of elevation of an inclined plane if it rises 1 foot in a horizontal distance of 40 feet ? 68. A ship is sailing due north-east with a velocity of 10 miles an hour. Find the rate at which she is moving due north, and also due east. 69. In front of a window 20 feet high is a flower-bed 6 feet wide. How long must a ladder be to reach from the edge of the bed to the window ? 70. A ladder 40 feet long may be so placed that it will reach a window 33 feet high on one side of the street, and by turn- ing it over without moving its foot it will reach a window 21 feet high on the other side. Find the breadth of the street. 71. From the top of a hill the angles of depression of two successive milestones, on a straight level road leading to the hill, are observed to be 5° and 15°. Find the height of the hill. 72. A fort stands on a horizontal plain. The angle of ele- vation at a certain point on the plain is 30°, and at a point 100 feet nearer the fort it is 45°. How high is the fort? 73. From a certain point on the ground the angles of eleva- tion of the belfry of a church and of the top of the steeple were found to be 40° and 51° respectively. From a point 300 feet farther off, on a horizontal line, the angle of elevation of the top of the steeple is found to be 33° 45'. Find the distance from the belfry to the top of the steeple. 74. The angle of elevation of the top of an inaccessible fort C, observed from a point A, is 12°. At a point B, 219 feet from A and on a line AB perpendicular to AC, the angle ABC is 61* 45'. Find the height of the fort. 24 TRIGONOMETRY. § 17. The Isosceles Triangle. An isosceles triangle is divided by the perpendicular from the vertex to the base into two equal right triangles. Therefore, an isosceles triangle is determined by any two parts that determine one of these right triangles. Let the parts of an isosceles triangle ABC (Fig. 13), among which the altitude CD is to be in- cluded, be denoted as follows : a = one of the equal sides. c = the base. h =3 the altitude. A = one of the equal angles. C= the angle at the vertex. For example : Given a and c quired A, C, h. re- 1 a \c c 1. cos^l = — = - — a 2a 2. (7+ 2.4 = 180°; C= 180° -2 A = 2(90°-^). 3. h may be found directly in terms of a and c from the equation #+!=«*, which gives h = V(a — he) (a -f- £ c). But it is better to find the angles first, and then find h from either one of the two equations, whence or = sin^4. = asin^, or h —^ctdiJiA. The numerical values of A, C, and h may be computed by the aid of logarithms, as in the case of the right triangle. The area F of the triangle may be found, when c and h are given or have been found, by means of the formula F=hch. THE ISOSCELES TRIANGLE. Exercise VII. In an isosceles triangle : 1. Given a and A; find C, c, h. 2. Given a and C; find A, c, h. 3. Given c and A ; find C, a, h. 4. Given c and C; find ^4, a, h. . 5. Given h and J. ; find C, a, c. 6. Given h and C; find J., a, c. 7. Given a and h; find .4, (7, c. 8. Given and in what quadrants do they lie? 6. How many values less than 720° can the angle x have if cos X s= 4" f , an( i m what quadrants do they lie ? 7. If we take into account only angles less than 180°, how many values can x have if sin a; = % ? if cos x = -J- ? if cos a; = -f? if tan£ = f? if cot:r = -7? 8. Within what limits must the angle x lie if cos x = — | ? if cot x = 4 ? if sec x = 80 ? if esc x = — 3 ? (z to be less than 360°.) 9. In what quadrant does an angle lie if sine and cosine are both negative ? if cosine and tangent are both negative ? if the cotangent is positive and the sine negative ? GONIOMETRY. 37 10. Between 0° and 3600° how many angles are there whose sines have the absolute value f ? Of these sines how many are positive and how many negative ? 11. In finding cos# by means of the equation cos# = — Vl — sin 2 o:, when must we choose the positive sign and when the negative sign ? 12. Given cos x = — V^ ; find the other functions when x is an angle in Quadrant II. 13. Given tan#= V3 ; find the other functions when x is an angle in Quadrant III. 14. Given sec x = + 7, and tan x negative ; find the other functions of x. 15. Given cot x = — 3 ; find all the possible values of the other functions. 16. What functions of an angle of a triangle may be nega- tive ? In what case are they negative ? 17. What functions of an angle of a triangle determine the angle, and what functions fail to do so ? 18. Why may cot 360° be considered equal either to -f oo or to — oo ? 19. Obtain by means of Formulas [l]-[3] the other func- tions of the angles given : (i.) tan 90° = oo. (iii.) cot 270° = 0. (ii.) cos 180° = — 1. (iv.) esc 360° =— oo. 20. Find the values of sin 450°, tan 540°, cos G30°, cot 720°, sin 810°, esc 900°. 21. For what an^le in each quadrant are the absolute values of the sine and cosine alike ? tf$ * - / *>$ - $ % $ . -3 / f Compute the values of the following expressions : ^22. a sin 0° + b cos 90° -c tan 180°. 23. acos90 o -Z>tanl80° + ccot90°. O 24. a sin 90° - b cos 360° + (a - b) cos 180°. O 25. (a 2 - & 2 ) cos 360°- 4 a& sin 270°. «' 4. 38 TRIGONOMETRY. < N / M P SV H 8 B' Fig. SI 28. Reduction of Functions to the First Quadrant. In a unit circle (Fig. 25) draw two diameters PR and QS equally inclined to the horizon- tal diameter AA', or so that the angles AOP, A'OQ, A'OR, and -40$ shall be equal. From the points P, Q, P, S let fall per- pendiculars to AA 1 ; the four right triangles thus formed, with a common vertex at 0, are equal ; because they have equal hypote- nuses (radii of the circle) and equal acute angles at 0. There- fore, the perpendiculars PM, QJSf, RN, SM, are equal. Now these four lines are the sines of the angles A OP, AOQ, AOP, and AOS, respectively. Therefore, in absolute value, sin AOP = sin AOQ = sin AOR = sin AOS. And from § 27 it follows that in absolute value the cosines of these angles are also equal ; and likewise the tangents, the cotangents, the secants, and the cosecants.* Hence, for every acute angle (AOP) there is an angle in each of the higher quadrants whose functions, in absolute value, are equal to those of this acute angle. Let A AOP =x, /.POB^y; then #-fy = 90°, and the functions of x are equal to the co-named functions of y (§ 4) ; further, A AOQ (in Quadrant II.) - 180° - x = 90° + y, ZAOR (in Quadrant III.) = 180° + x = 270° - y, Z AOS (in Quadrant IV.) = 360° - x = 270° + y. Hence, if we prefix in each case the proper sign (§ 24), we have the two following series of Formulas : * In future, secants and cosecants will be disregarded. They may be found by [3] if wanted, but are seldom used in computations. GONIOMETRY. 39 Angle in Quadrant II sin (180° — x) = sin x. sin (90° -J- y) = cos y. cos (180° — x) = — cos a?. cos (90° +y) = — sin y. tan (180° - a;) = - tan x. tan (90° + y) = - cot y. cot (180° — x) = — cot a:. cot (90° + y) = — tany. Angle in Quadrant III sin (180° + a?) = — sin x. sin (270° ~y)== — cos y. cos (180° -j- a?) = — cos $. cos (270° - - y) = — sin y. tan (180° + a;)= tanar. tan (270° -- y) = coty. cot (180° + x) = cot x. cot (270° - y) = tan y. Angle in Quadrant IV. sin (360° — x) — — sin x. sin (270° + y) == — cos y. COB (360° — x) = cos 07. cos (270° + y) == sin y. tan (360° - x) = - tan a:. tan (270° + y) = - cot y . cot (360° - a:) = - cot x. cot (270° + y) = - tany. Hence, by selecting the right formulas, The functions of all angles can be reduced to the functions of angles not greater than 45°. Thus, to find the functions of 220° and 230°, we should consider 220° as (180° + 40°), but 230° as (270° - 40°). It is evident from these formulas that, If an acute angle be added to or subtracted from 180° or 360°, the functions of the resulting angle are equal in absolute value to the like-named functions of the acute angle; but if an acute angle be added to or subtracted from 90° or 270°, the functions of the resulting angle are equal in absolute value to the co-named functions of the acute angle. It is evident from the formulas for (180° — x) that, A given value of a sine determines two supplementary angles, one acute, the other obtuse ; a given value of any other function {except the cosecant) determines only one angle : acute if the value is positive, obtuse if the value is negative. 40 TRIGONOMETRY. § 29. Angles whose Difference is 90°. The general form of two such angles is x and 90° + x } and they must lie in adjoining quadrants. The relations between their functions were found in § 28, but only for the case when x is acute. These relations, how- ever, may be shown to hold true for all values of x. In a unit circle (Fig. 26) draw two diameters PB and QS per- pendicular to each other, and let fall to A A' the perpendicu- lars PM, QH, BK, and SJV. The right triangles OMP, O&Q, OKB, and ONS are equal, because they have equal hypote- nuses and equal acute angles POM, OQH, BOK, and OSN. Therefore, OM = QH= OK = iV#, and PM= OH= KB = ON. Hence, taking also into account the algebraic sign, sin AOQ= cos AOP; sin A08 = cos AOB; cos AOQ = — sin AOP; cos AOS = — sin AOB; sin AOB = cos AOQ; sin (360° + A OP) = cos AOS; cos AOB = - sin AOQ ; cos (360° + AOP) = - sin AOS. In all these equations, if x denote the angle on the right-hand side, the angle on the left-hand side will be 90° -\-x. There- fore, if x be an angle in any one of the four quadrants, sin (90° + x) = cos x, cos (90° -\-x) = — sin x. And, by § 27, tan (90° + x) = - cot x, cot (90° + x) = — tan x. In like manner, it can be shown that all the formulas of § 28 hold true, whatever be the value of the angle x. GONIOMETRY. 41 § 30. Functions of a Negative Angle. If the angle A OP (Fig. 25) is denoted by x, the equal angle AOS, generated by a backward rotation of the moving radius from the initial position OA, will be denoted by — x. It is obvious that the position OS of the moving radius for this tngle is identical with its position for the angle 360° — x. Therefore, the functions of the angle — x are the same as those of the angle 360° - x ; or (§ 28), sin (-— x) = — sin x, tan (— x) = — tan x, cos (— x) = cos x, cot (— x) = — cot x. Exercise X. 1. Express sin 250° in* terms of the functions of an acute angle greater than 45°, and also in terms of the functions of an acute angle less than 45°. Ans. 1. sin 250° = sin (180° + 70°) = -sin 70°. 2. sin 250° = sin (270° - 20°) = - cos 20°. Express the following functions in terms of the functions of angles less than 45° : sin 204°. cos 359°. tan 300°. cot 264°. sec 244°. esc 271°. Express all the functions of the following negative angles in terms of those of positive angles less than 45° : 20. -75°. 22. -200°. 24. -52°37 f . 21. -127°. 23. -345°. 25. -196° 54'. 26. Find the functions of 120°. Hint. 120° - 180° - 60°, or, 120° = 90° + 30° ; then apply \ 28. 2. sin 172°. 8. 3. cos 100°. 9. 4. tan 125°. 10. 5. cot 91°. 11. 6. sec 110°. 12. 7. esc 157°. 13. 14. sin 163° 49'. 15. cos 195° 33'. 16. tan269° 15 f . 17. cot 139° 17'. 18. sec 299° 45'. 19. esc 92° 25'. 42 TRIGONOMETRY. Find the functions of the following angles : 27. 135°. 29. 210°. 31. 240°. 33. -30°. 28. 150°. 30. 225°. 32. 300°. 34. -225°. 35. Given sin x = — VJ, and cos# negative ; find the other functions of x, and the value of x. 36. Given cot# = — V3, and x in Quadrant II.; find the other functions of x, and the value of x. 37. Find the functions of 3540°. 38. What angles less than 360° have a sine equal to —J? a tangent equal to — V3? 39. Which of the angles mentioned in Examples 27-34 have a cosine equal to — VT? a cotangent equal to -~-V3? 40. What values of x between 0° and 720° will satisfy the equation sin x = + £ ? 41. In each of the following cases find the other angle be- tween 0° and 360° for which the corresponding function (sign included) has the same value: sin 12°, cos 26°, tan 45°, cot 72°; sin 191°, cos 120°, tan 244°, cot 357°. 42. Given tan 238° = 1.6; find sin 122°. 43. Given cos 333° = 0.89; findtanll7°. Simplify the following expressions : 44. a cos (90° - x) + b cos (90° + ^). 45. m cos (90° -a;) sin (90° — x). 46. (a - b) tan (90° - x) + (a + b) cot (90° + x). 47. a?+b 2 -2abcos(180°-x). 48. sin (90° + x) sin (180° + x) + cos (90° + x) cos (180° - x). 49. cos(180°+a:)cos(270 o -y)~sin(180 o +^) sin (270 o - < y). 50. tan x + tan (- y) — tan (180° — y). 51. For what values of x is the expression sin# + cos a; positive, and for what values negative ? Represent the result by a drawing in which the sectors corresponding to the nega- tive values are shaded. 52. Answer the question of last example for sin x — cos x. 53. Find the functions of (x — 90°) in terms of the functions of x. 54. Find the functions of (x — 180°) in terms of the functions of x. GONIOMETRY. 43 § 31. Functions of the Sum of Two Angles. In a unit circle (Fig. 27) let the angle AOB = x, the angle BOC=y\ then the angle AOC= x + y. In order to express sin(#-f y) and cos (x -f- y) in terms of the sines and cosines of x and y, draw OF A. OA, CB _L OB, BE± OA, BG±CF; then 00 = sin y, OB = cosy, and the angle BCG = the angle GDO = x. Also, sin (x + y ) = CF= DE + CG. BE OB CG CB Therefore, sin (x -f y) = sinx cos y +'cos x sin y Again, cos (a; + y) = OF= OE- BG. OE OB BG i\x ; hence, BE= sin# X OB = sin.x' cosy. = cos x ; hence, CG = cos a; X (7J9 = cos a; siny. m cos:£ CB hence, OE-= cos x X OB = cos a; cosy, hence, BG = sin a; X 6 y Z> = sin a; siny. Therefore , cos(x -j- y) = cos x cos y — sin x sin y. In this proof x and y, and also the sum x-\-y, are assumed to be acute angles. If the sum x-\-y of the acute angles x and y is obtuse, as in Fig. 28, the proof remains, word for word, the same as above, the only dif- ference being that the sign of Oi^will be negative, as BG is now greater than OE. The above formulas, therefore, hold true for all acute angles x and y. 44 TRIGONOMETRY. If these formulas hold true for any two acute angles x and y, they hold true when one of the angles is increased by 90°. Thus, if for x we write x' = 90° + a;, then, by § 29, sin (V -f y) == sin (90° -f- z + y) = cos (a; + y), cos (a;' -j- y) = cos (90° + a; + y) = — sin (a; -f y). Hence, by [5], sin(V+ y) =s cosa; cosy — sina; siny, by [4], cos(a;'+ y) = — sina; cosy — cos a; siny. Now, by § 29, cos a; = sin (90° -f x) = sin x', sin x = — cos (90° -f x) = — cos a;'. Therefore, by putting sina;' for cos a;, and —cos a;' for sina;, in the right-hand members of the above equations, sin (x f + y) = sin a;' cos y -f- cos a;' sin y , cos(a; f + y) = cosa;' cosy — sin x' sin y. Hence, it follows that Formulas [4] and [5] are universally true. For they have been proved true for any two acute angles, and also true when one of these angles is increased by 90° ; hence they are true for each repeated increase of one or the other angle by 90°, and therefore true for the sum of any two angles whatever. By §27, t , N sin (x + y) sin x cos y -f- cos x sin y tan (x + y) = ) — p^< = — y r-^« cos(a? + y) cosa; cosy — sin a; siny If we divide each term of the numerator and denominator of the last fraction by cosa; cosy, and again apply § 27, we obtain , , N tanx + tany rpl tan (x 4- y) = , Z__JL. [6] v rjJ 1 — tanxtany In like manner, by dividing each term of the numerator and denominator of the value of cot (a; -f- y) by sina; siny, we obtain ., , N COtXCOtV — 1 r^T cot(x--Y) = ' • L'J v ' JJ cotx + coty GONIOMETRY. 45 § 32. Functions of the Difference of Two Angles. In a unit circle (Fig. 29) let the angle AOB — x, the angle COB = y ; then the angle AOC= *-y. In order to express sin (x — y) and cos (x — y) in terms of the sines and cosines of x and y, draw CF±OA, CD±OB, DE± OA y DG± FC prolonged ; then CD= siny, OD = cosy, and the angle DCG=the angle EDC=x. And, sin (x - y) == CF= DE- CG. — — • = sin#: hence, DE OD CG — — = cos a; ; hence, CG = cos a* X CD — cos x siny. Therefore, sin (x — y) = sin x cos y — cos x sin y. [8] Again, cos (x - y) = OF== OE+DG. O F --— = coso; ; hence, OE = cos# X OZ> = cos;r cosy. Fig. 29. sin a; X 02) = sin x cosy. CD sin x ; hence, DG = sin a? X OZ) = sin x sin y. Therefore, cos (x — y) = cos x cos y + sin x sin y. [°] In this proof, both x and y are assumed to be acute angles ; but, whatever be the values of x and y, the same method of proof will always lead to Formulas [8] and [9], when due regard is paid to the algebraic signs. The general application of these formulas may be at once shown by deducing them from the general formulas established in § 31, as follows : It is obvious that (x — y)-\~y — x. If we apply Formulas [4] and [5] to (x -- y) 4- y, then 46 TRIGONOMETRY. sin \(x — y) -f y \ or sin a; = sin (x — y) cosy -f- cos {x — y) siny, cos 5 (a; — y)-\-y\ or cos# — cos (a; — y) cosy — sin (x — y) siny. Multiply the first equation by cosy, the second by siny, sin x cosy — sin [x — y) cos 2 y -f cos (x — y) sin y cosy, cos# siny = — sin (x — y) sin 2 y -f cos (x — y) siny cosy ; whence, by subtraction, sin z cosy — cos x sin y = sin (x — y) (sin 2 y -f- cos 2 y). But sin 2 y -f cos 2 y = 1 ; therefore, by transposing, sin (# — y) = sinx cosy - cos x siny. Again, if we multiply the first equation by sin y, the second equation by cosy, and add the results, we obtain, by reducing, cos (x — y) = cos x cos y -f sin x sin y. Therefore, Formulas [8] and [9], like [4] and [5], from which they have been derived, are universally true. From [8] and [9], by proceeding as in § 31, we obtain tan(x j) 1 + tanxtan y [10] .j s cotxcoty + 1 > ni cot (x — y) = — J —- — L 11 v #y coty — cotx Formulas [4] -[11] may be combined as follows: sin (a;±y) = sin x cosy ± cos x siny, cos (x ± y) = cos a; cosy =F sin x siny, ., N tan x -fc tan ?/ tan (a; db y) = ; — — *-i *' ln=tan#tany . / v cot# coty =F 1 cot (a? ± y) = — ^— - — *' coty db cot a; GONIOMETRY. 47 §33. Functions of Twice an Angle. If, in Formulas [4] -[7], y ~ x, they become : sin 2x = 2sinxcosx. [12] cos2x = cos 2 x — sin 2 x. [13] 2tanx --., .„ cot 2 x — 1 ric1 tan2x = - — — -=-■ [141 cot2x = -= — - — . [15] 1 — tan 2 x L J 2cotx By these formulas the functions of twice an angle are found when the functions of the angle are given. § 34. Functions of Half an Angle. Take the formulas cos 2 x -f sin 2 # = 1 [1] cos 2 x — sin 2 x = cos 2x [13] Subtract, 2 sin 2 x = 1 — cos 2 a; Add, 2 cos 2 # = 1 + cos 2 x Whence 11 — cos2# , fl + cos2# sin x = ± <*/ » cos x = d= a) — — These values, if z is put for 2x, and hence \z for re, become *t\ % -±£=*B&. [16] co S i 2 = ±iJ^!i. [17] Hence, by division (§ 27), tan}i = ±Jj^5!i [18] cot^ = ±jLt^^, [19] M + cosz J \1 — cosz By these formulas the functions of half an angle may be computed when the cosine of the entire angle is given. The proper sign to be placed before the root in each case depends on the quadrant in which the angle \z lies. (§ 24.) Let the student show from Formula [18] that tan \ B =J— -• (See page 19, Note 2.' \c-\-a 48 TRIGONOMETRY. § 35. Sums and Differences of Functions. From [4], [5], [8], and [9], by addition and subtraction : sin (x + y) + sin (x — y) = 2 sin x cos y , sin (x + y) — sin (x — y)= 2 cos x sin 3/, cos (x -f- y) + cos (x — y)~ 2 cos 2; cos y, cos (# -f" y) — cos (# — y) = — 2 sin x sin y ; or, by making x-{-y = A, and x — y = B, and therefore, # = J (A + -B), and y = -J (J. — i?), sin A + sin B = 2 sin } (A + B) cos £ (A -B). [20] sin A - sinB = 2cosi(A + B)sin i(A - B). [21] cos A + cosB = 2cos|(A + B)cos £(A - B). [22] cos A - cosB = - 2 sin J(A + B) sin 1 (A - B). [23] From [20] and [21], by division, we obtain sin^ + sini? tan i {A + B)cotuA _ B) . sin A — sin B or, since cot -J (-4 — B) tam$(4— B) sin A -f sin B _ tan^- (A + B) sin A — sin B tan i (A — B)' Exercise XL [24] 1. Find the value of sin (x-\-y) and cos^-f-y), when sinrr = |, cosa; = J, siny = ^, cosy = ff . 2. Find sin (90° -y) and cos (90° -y) by making ar=90° in Formulas [8] and [9]. Find, by Formulas [4] -[11], the first four functions of: 3. 90° + y. 8. 360° -y. 13. -y. 4. 180° -y. 9. 360° + y. 14. 45° -y. 5. 180° + y. 10. 07-90°. 15. 45° + y. 6. 270° -y. 11. a: -180°. - 16. 30° + y. 7. 270° + y. 12. a; -270°. 17. G0°-y. GONIOMETRY. 49 18. Find sin Sx in terms of sin x. 19. Find cos Sx in terms of cos a;. 20. Given tan|-a; = 1 ; find cos x. 21. Given cot \ x = V3 ; find sin a;. 22. Given sins; = 0.2 ; find sin^-a; and cos|-a;. 23. Given cos a; = 0.5 ; find cos 2x and tan 2 a;. 24. Given tan 45° = 1 ; find the functions of 22° 30'. 25. Given sin 30° = 0.5 ; find the functions of 15°. 26. Prove that tan 18° = sin j*j*° + sin )*° . cos 33° + cos 3° Prove the following formulas : 27. sin2a?= , 2tan ^ . 29. tan^ar l + tan 2 a; 1-f-cosa; 28. cos2a;= }~ ta A 30. coUa?: l-f-tan 2 a; 1 — cos as 31. sin-Ja; =t cos-|-a; = Vl ± sina;. QO tan a; ± tan ?/ 6Z. — . _£ = ± tan x tan y. cot x ± cot y ** 33. tan(45°-o;) = 1 ~ tan ^ v , } 1 + tana; If A, B, Care the angles of a triangle, prove that: 34. sin A-\- sin i? + sin C==4cos|-^. cos-^-i? cos-J-C. 35. cos A + cos B + cos (7= 1 + 4 sin -J ^4 sin % B sin \ C. 36. tan A + tani? -f tan C= tan A X tan B X tan C. 37. Qot%A + ooi%B-{-cQt%0=Got%A X cot|i?X cotJC Change to forms more convenient for logarithmic computa- tion : 38. cota; + tana;. 43. 1 + tana; tan y. 39. cot x — tana;. 44. 1— tan x tan y. 40. cot x + tan y. 45. cota;coty+l. 41. cot a;-- tan?/. 46. cota^oty — 1. 49 1 — cos 2 a: 47. tan a; + tan?/ 1 -{-cos 2 a; cot a; + cot y CHAPTER IV. THE OBLIQUE TRIANGLE. § 36. Law of Sines. Let A, B, C denote the angles of a triangle A B (Figs. 30 and 31), and a, b, c, respectively, the lengths of the opposite sides. Draw CD±AB, and meeting AB (Fig. 30) or AB pro- duced (Fig. 31) at D. Let CD = h. B A In both figures, -- = sin^4. In Fig. 30, a = sin B. In Fig. 31, h a = sin (180°- ■B) = sin B. Therefore, we obtain, by whether division, h lief ! within or without the triangle, a b" sin A sinB [25] THE OBLIQUE TRIANGLE. 51 By drawing perpendiculars from the vertices A and B to the opposite sides we may obtain, in the same way, B C sin A sin Hence the Law of Sines, which may be thus stated : The sides of a triangle are proportional to the sines of the opposite angles. If we regard these three equations as proportions, and take them by alternation, it will be evident that they may be writ- ten in the symmetrical form, a sin ^4. sini? Each of these equal ratios has a simple geometrical mean- ing which will appear if the Law of Sines is proved as follows : Circumscribe a circle about the triangle ABO (Fig. 32), and draw the radii OA, OB, 00; these radii divide the triangle into three isosceles triangles. Let R denote the radius. Draw OM Jl BO By Geometry, the angle BOO =2A; hence, the angle BOM=A, then BM=Rsm BOM = RsinA. .'. BO or a = 2Rsm A. In like manner, b = 2 R sin B, and e = 2 R sin O Whence we obtain b 2R = sin A sin B sin O That is : The ratio of any side of a triangle to the sine of the opposite angle is numerically equal to the diameter of the cir- cumscribed circle. 52 TRIGONOMETRY. § 37. Law of Cosines. This law gives the value of one side of a triangle in terms of the other two sides and the angle included between them. In Figs. 30 and 31, a 2 = h 2 + BB 2 . In Fig. 30, BB = c-AB; in Fig. 31, BB =AB-c; in both cases, BBP = ABP -2cxAB+a, then B>A. The formula is still true, but to avoid negative quantities, the formula in this case should be written b - a _ tan j (B - A) b + a ~ tan £ (B + A) Exercise XII. 1. What do the formulas of § 36 become when one of the angles is a right angle ? 2. Prove by means of the Law of Sines that the bisector of an angle of a triangle divides the opposite side into parts pro- portional to the adjacent sides. 3. What does Formula [26] become when A = 90°? when ^1 = 0°? when A = 180° ? What does the triangle become in each of these cases ? Note. The case where A = 90° explains why the theorem of \ 37 is sometimes termed the Generalized Theorem of Pythagoras. 4. Prove (Figs. 30 and 31) that whether the angle B is acute or obtuse c = a cos B -f b cos A. What are the two sym- metrical formulas obtained by changing the letters? What does the formula become when B = 90° ? 5. From the three following equations (found in the last exercise) prove the theorem of § 37 : c = a cos B -}- b cos A, b — a cos C -f- c cos A, a = b cos C + c cos B. Hint. Multiply the first equation by c, the second by b, the third by a ; then from the first subtract the sum of the second and third. 54 TRIGONOMETRY. 6. Iii Formula [27] what is the maximum value of 2 (A—B)? 7. Find the form to which Formula [27] reduces, and describe the nature of the triangle, when (i.) C= 90° ; (ii.) A-B = 90°, and B = C. § 39. The Given Parts of an Oblique Triangle. The formulas established in §§ 36-38, together with the equation A-{-B-\- C= 180°, are sufficient for solving every case of an oblique triangle. The three parts that determine an oblique triangle may be : I. One side and two angles ; II. Two sides and the angle opposite to one of these sides ; III. Two sides and the included angle ; IV. The three sides. In all cases let A, B, O denote the angles, a, b, c the sides opposite these angles respectively. § 40. Case I. Given one side a, and two angles A and B; find the remain- ing parts C, b, and c. 1. C= 180° -(A + JB). o b sin 3. c - = B a sin A c sin_(7 mi \b a&mB sin A a sin O sin A X sin B. X sin C. sin A sin A Example, a = 24.31, A = 45° 18', B = 22° 11'. The work may be arranged as follows : loga = 1.38578 cologsin A = 0.14825 logsin.ff = 9.57700 a= 24.31 A^ 45° 18' B= 22° 11' A + B= 67°29' <7= 112° 31' log& = 1.11103 b = 12.913 1.38578 0.14825 9.96556 log c = 1.49959 c = 31.593 log sin C^- THE OBLIQUE TRIANGLE. 55 Exercise XIII. **1. Given a = 500, .4 = 10° 12', -5 = 46° 36'; find C= 123° 12', b = 2051.48, c = 2362.61. A 2. Given a = 795, J. = 79° 59', j? = 44°41'; find C= 55° 20', b = 567.688, c = 663.986. *3. Given a = 804, A = 99° 55', ^ = 45°r ; find ff= 35° 4', 5 = 577.313, c = 468.933. ^4. Given a = 820, ^4 = 12° 49', J5 = 141° 59'; find C= 25° 12', 5 = 2276.63, c = 1573.89. >-5. Given ^ 8. Given b = 999, ^ = 37° 58', (7= 65° 2' ; find^ = 77°, a = 630.77, c = 929.48. J- 9. In order to determine the distance of a hostile fort A from a place i?, a line BC and the angles ABC and J5C.4 were measured, and found to be 322.55 yards, 60° 34', and 56° 10', respectively. Find the distance AB. 10. In making a survey by triangulation, the angles B and C of a triangle ABC were found to be 50° 30' and 122° 9', respectively, and the length BC is known to be 9 miles. Find AB and AC. 11. Two observers 5 miles apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 55° and 58°, respectively. Find the distance from the balloon to each observer, and also the height of the balloon above the plain. 12. In a parallelogram given a diagonal d and the angles x and y which this diagonal makes with the sides. Find the sides. Compute the results when d= 11.237, #=19° 1', and y = 42°54. 56 TRIGONOMETRY. 13. A lighthouse was observed from a ship to bear N. 34° E. ; after sailing due south 3 miles, it bore N. 23° E. Find the dis- tance from the lighthouse to the ship in both positions. Note. The phrase to bear N. 34° E. means that the line of sight to the lighthouse is in the north-east quarter of the horizon, and makes, with a line due north, an angle of 34°. 14. In a trapezoid given the parallel sides a and b, and the angles x and y at the ends of one of the parallel sides. Find the non-parallel sides. Compute the results when a = 15, &«7, # = 70°, y = 40°. Solve the following examples without using logarithms : 15. Given b = 7.07107, A = 30°, C= 105° ; find a and c. 16. Given sin A a a C= 180° - (A + B). A j c sin C ,i n a sin O And since - = - 7, therefore c = — : — — • a sin A sin A THE OBLIQUE TRIANGLE. 57 When an angle is determined by its sine it admits of two values, which are supplements of each other (§ 28) ; hence, either value of B may be taken unless excluded by the con- ditions of the problem. If a> b, then by Geometry A > B, and B must be acute whatever be the value of A ; for a triangle can have only one obtuse angle. Hence, there is one, and only one, triangle that will satisfy the given conditions. If a = b, then by Geometry A = B ; both A and B must be acute, and the required triangle is isosceles. If a < b, then by Geometry A < B, and A must be acute in order that the triangle may be possible. If A is ^S acute, it is evident from Fig. 33, where Z. BA C=A, /^ AC = b, CB = CB'= a, J^ / that the two triangles A CB ^^ y and ACB' will satisfy the // ■., / given conditions, provided ^ TP^Z P a is greater than the per- ^ pendicular CP; that is, provided a is greater than b sin A (§ 10). The angles ABC and AB'C are supplementary (since Z. ABC— Z. BB'C) ; they are in fact the supplementary angles obtained from the formula • r> b sin A sin Jd = a If, however, a — b sin A = CP (Fig. 33), then sin B = 1 , B = 90°, and the triangle required is a right triangle. If al, and the tri- angle is impossible. These results, for convenience, may be thus stated : Ifa>&, or a = b, or if a = b sin A, One solution. If a < b, but > b sin A, and A < 90°, Two solutions. If«<6 and^L>90°, or if a < b sin A and A < 90°, No solution. 58 TRIGONOMETRY. The number of solutions can often be determined by inspec- tion. If there is any doubt, it may be removed by computing the value of b sin A. Or we may proceed to compute log sin B. If log sin B = 0, the triangle required is a right triangle. If log sin i?>0, the triangle is impossible. If log sin B<0, there is one solution when a>5 ; there are two solutions when a 90° ; therefore the triangle is impossible. 2. Given a = 36, b = 80, A = 30° ; find the remaining parts. Here we have b sin A — 80 X $ — 40; so that a < b sin A, and the triangle is impossible. 3. Given a = 72630, b = 117480, ^4 = 80°0' 50"; find^,C,c. a - 72630 6-117480 A = 80° 0' 50" colog a = 5.13888 log b = 5.06996 log sin A - 9.99337 Here log sin B>0. :. no solution. log sin B = 0.20221 4. Given a = 13.2, 6 = 15.7, ^4 = 57° 13' 15.3"; find B,C,c> a - 13.2 b - 15.7 4 = 57° 13' 15.3' Here log sin B = 0, ■\ a right triangle. colog a - 8.87943 log b = 1.19590 log sin A = 9.92467 log sin B = 0.00000 5=90° .-. C -32° 46' 44.7" c = b cos A log 6 = 1.19590 log cos 4- 9.73352 log c - 0.92942 e-8.5 THE OBLIQUE TRIANGLE. 59 5. Given a a = 767 & = 242 A = 36° 53' 2" 767, 5-242, ^ = 36°53'2 colog a =7.11520 log b = 2.38382 log sin A = 9.77830 log sin B = 9.27732 B =10° 54' 58" .-. (7=132° 12' 0" ; find JB, C, c. log a = 2.88480 log sin O = 9.86970 cologsinJ. = 0.22170 log c = 2.97620 Here a > 5, and log sin B < 0. 5=10° 54' 58 " .°. one solution. 6. Given a = 177.01, 6 = 216.45, ^ = 35°36'20"; find the other parts. a =177.01 b = 216.45 A = 35° 36' 20" Here a 30° 1'23'' 1 log (a -6) = 2.57171 colog(a + b) = 6.94961 log tan \ (A+B) = 0.20766 log tan |(4-^)- 9.72898 I (A-B) = 28° 10' 52" log b = 2.57103 log sin (7=9.95214 colog sin j5 = 0.30073 lose = 2.82690 log .671.27 Note. In the above Example we use the angle B in finding the side c, rather than the angle A, because A is near 90°, and therefore its sine should be avoided. 2. Given a = 4, c = 6, B = 60°; find the third side b. Here Solution II. may be used to advantage. We have b = Va? + d*- 2 ac cos B= Vl6 + 36 - 24 = V28 ; log 28 - 1.44716, log V28 = 0.72358, V28 - 5.2915 ; that is, 6 - 5.2915. I 3- Exercise XV. Given a = 77.99, b = 83.39, C- = 72° 15'; [§38,N. find A = 51° 15', B = 56° 30', c = 95.24. Given b = 872.5, c = 632.7, ^ = 80°; find .£ = 60° 45', C=-- 39° 15', a =984.83. Given a= 17, 6 = 12, (7= 59° 17'; find A = 77° 12' 53", £ = 43° 30' 7", c = 14.987. Given b = V5, c= V3, ^ = 35° 53'; find .£ = 93° 28' 36", C= 50° 38' 24", c* = 1.313. Given a = 0.917, 6 = 0.312, C= 33° 7' 9"; find A = 132° 18' 27", £=14° 34' 24", c = 0.67748. Given a = 13.715, c= 11.214, j5=15°22'36" find^ = 118°55'49", C=45°41'35'\ 6 = 4.1554. Given b = 3000.9, c = 1587.2, ^ = 86° 4' 4"; find .£ = 65° 13' 51", C= 28° 42' 5", a = 3297.2. THE OBLIQUE TRIANGLE. G3 8. Given a = 4527, 6 = 3465, tf=66°6'27"; find A = 68° 29' 15", B = 45° 24 r 18", c = 4449. 9. Given a = 55.14, 6 = 33.09, C= 30° 24'; find A = 117° 24' 33", j? = 32° 11' 27", c = 31.431. 10. Given a = 47.99, 6 = 33.14, C= 175° 19' 10" ; find A = 2° 46' 8", ^ = 1°54'42", c = 81.066. 11. If two sides of a triangle are each equal to 6, and the included angle is 60°, find the third side. 12. If two sides of a triangle are each equal to 6, and the included angle is 120°, find the third side. 13. Apply Solution I. to the case in which a = b or the triangle is isosceles. 14. If two sides of a triangle are 10 and 11, and the in- cluded angle is 50°, find the third side. 15. If two sides of a triangle are 43.301 and 25, and the included angle is 30°, find the third side. 16. In order to find the distance between two objects A and B separated by a swamp, a station C was chosen, and the distances CA = 3825 yards, CB = 3475.6 yards, together with the angle ACB = 62° 31', were measured. Find the distance from A to B. 17. Two inaccessible objects A and B are each viewed from two stations C and D 562 yards apart. The angle ACB is 62° 12', BCD 41° 8', ABB 60° 49', and ABC 34° 51'; required the distance AB. 18. Two trains start at the same time from the same station, and move along straight tracks that form an angle of 30°, one train at the rate of 30 miles an hour, the other at the rate of 40 miles an hour. How far apart are the trains at the end of half an hour? 19. In a parallelogram given the two diagonals 5 and 6, and the angle that they form 49° 18'. Find the sides. 20. In a triangle one angle = 139° 54', and the sides form- ing the angle have the ratio 5:9. Find the other two angles. 64 TRIGONOMETRY. § 43. Case IV. &iven the three sides a, b, c ; find the angles A, £, C. The angles may be found directly from the formulas estab« lished in § 37. Thus, from the formula a 2 = b 2 + c 2 -2bccosA b 2 4- c 2 — a 2 we have cos A = '. ' T 2bc From this equation formulas adapted to logarithmic work are deduced as follows : For the sake of brevity, let a + b -f- c = 2 s ; then b -J- c — a = 2(s — a), a — b -f- c = 2 {s — b), and a + b — c = 2 (s — c). Then the value of 1 — cos A is 1 b 2 + c 2 - a 2 _ 2bc -b 2 - -a' . = (& + g)'-a' 26c 2bc 2bc _ (b + c + a)(b + c - a) ^ 2s(s - a) 2bc be But from Formulas [16] and [17], § 34, it follows that 1 — cos A = 2sm 2 $A, and 1 + cos A =2cos 2 j^4. ; 0^2 1 a 2 (s — b) (s — c) n o 1 /t 2 whence and therefore ■inU^ C- b >fr- °> , [28] V 8 ^ P»] cos 1 A tenU =^ 1 ^r i - rsoi s (s — a) THE OBLIQUE TRIANGLE. G5 By merely changing the letters, • i -n l( S — °) ( S — C ) • i n l( S — a ) ( S ~ ^) (s — a) (5 — /;) (s — b) \ s (s — c) There is then a choice of three different formulas for finding the value of each angle. If half the angle is very near 0°, the formula for the cosine will not give a very accurate result, because the cosines of angles near 0° differ little in value ; and the same holds true of the formula for the sine when half the angle is very near 90°. Hence, in the first case the formula for the sine, in the second that for the cosine, should be used. But, in general, the formulas for the tangent are to be preferred. It is not necessary to compute by the formulas more than two angles ; for the third may then be found from the equation A + B + C=180°. There is this advantage, however, in computing all three angles by the formulas, that we may then use the sum of the angles as a test of the accuracy of the results. In case it is desired to compute all the angles, the formulas for the tangent may be put in a more convenient form. The value of tan 2 A may be written tis-a)(s-b)(s-c) Qr 1 j ( s -g)(s~b)(s -g) Hence, if we put f (s-a)(s-b)(s-c) _ r> [31 -| we have tan 2 A = • [32] s —a GG TRIGONOMETRY. In like manner, tan i B -V tan k C Examples. 1. Given a = 3.41, b = 2.G0, c = 1.58; find the angles. Using Formula [30], and the corresponding formula for tan \ B, wt may arrange the work as follows : a = = 3.41 6 = = 2.60 c = = 1.58 2s- = 7.59 s = = 3 795 s — a = = 0.385 a -b = - 1.195 s — c • = 2.215 colog s = 9.42079 colog(s- a) = 0.41454 log («- b) = 0.07737 log (s-c) = 034537 2 )0.25807 log tan J J. = 0.12903 %A = 53° 23' 20" " A = 106° 46' 40" colog «= 9.42079-10 log(«-a) = 9.58546-10 colog (s- b) = 9.92263-10 log (s - c) = 0.34537 2 )19.27425 - 20 log tan \B = 9.63713-10 \B~ 23° 26' 37" ,5= 46° 53' 14" .:A + B = 153° 39' 54", and (7- 26° 20' 6". 2. Solve Example 1 by finding all three angles by the use of Formulas [31] and [32]. Here the work may be compactly arranged as follows, if we find log tan \ A, etc., by subtracting log(s — a), etc., from logr instead of adding the cologarithm : • a = 3.41 log(s- a) = 9.58546 log tan \A - 10.12903 b - 2.60 log («- b) = 0.07737 log tan \B= 9.63713 c = 1.58 log(«-c) = 0.34537 colog s = 9.42079 log tan \ C = 9.36912 is =7.59 M- 53° 23' 20" s = 3.795 log r 2 = 9.42899 \B= 23° 26' .".7" a = 0385 logr =9.71450 |0- 13° 10' 3" b = 1.195 4 - 106° 46' 40" c = 2.215 J?= 46° 53' 14" s - 7.590 (pre ►of). C = 26° 20' G" Rroof, A + B + C - 180° V 0" Note. Even if no mistakes are made in the work the sum of the three angles found as above may differ very slightly from 180° in conse- quence of the fact that logarithmic computation is at best only a method of close approximation. When a difference of this kind exists it should be divided among the angles according to the probable amount of error for each angle. ^HE OBLIQUE TRIANGLE. Exercise XVI. Solve the following triangles, taking the three sides as the given parts : 1 a 6. c A B 1 Q /- 51 66 20 38° 52' 48" 126° 52' 12" 14° 15' A- 2 78 101 29 32° 10' 54" 136° 23' 50" 11° 25' 16" 3 111 145 40 27° 20' 32" 143° 7' 48" 9° 31 '40" 4 21 26 31 42° 6' 13" 56° 6' 36" 81° 47' 11" 5 19 34 49 16° 25' 36" 30° 24' 133° 10' 24" 6 43 50 57 46° 49' 35" 57° 59' 44" 75° 10' 41" 7 37 58 79 26° 0'29" 43° 25' 20" 110° 34' 11" 8 73 82 91 49° 34' 58" 58° 46' 58" 71° 38' 4" 9 14.493 55.4363 66.9129 8° 20' 33° 40' 138° 10 y/E Vo ^7 51° 53' 12" 59° 31' 48" 68° 35' 11. Given a = 6, 5 = 8, e = 10; find the angles. 12. Given a = 6, 5 = 6, c = 10 ; find the angles. 13. Given a = 6, 5 = 6, is due east from A. In what direction is Cfrom A ? What two answers are admissible? 18. Under what visual angle is an object 7 feet long seen by an observer whose eye is 5 feet from one end of the object and 8 feet from the other end ? 19. When Formula [28] is used for finding the value of an angle, why does the ambiguity that occurs in Case II. not exist ? 20. If the sides of a triangle are 3, 4, and 6, find the sine of the largest angle. 21. Of three towns A, B, and C f A is 200 miles from B and 184 miles from C, B is 150 miles due north from C\ how far is A north of C? g8 trigonometry. § 44. Area of a Triangle. If F denote the area of the triangle ABC (Fig. 30 or 31, page 50), then, by Geometry, F= \c x CD. By § 10, CD — a sin B. Therefore, P = \ ac sinB. [33] And, in like manner, F= $ab sin C and F= Ibcs'mA. That is : TAc area o/* a triangle is equal to half the product of two sides and the sine of the included angle. By Formula [33] the area of a triangle may be found directly when two sides and the included angle are given ; in the other cases the formula may be used when these parts have been computed. When the three sides of a triangle are given, as in Case IV., a formula for its area may be found as follows : By §33, Bin^B=r2siniJ?Xcoe|JB. By substituting for sin § B and cos h B their values in terms of the sides given in § 43, 2 sin B = — Vs (s — a) (s — b) (s — c). By substituting this value of sin B in [33], r-Vs(s-a)(s-b)(s-c). [34] If R denote (as in § 3G) the radius of the circumscribed circle, we have, from § 36, sinJ5 = A. By substituting this value of sin B in [33], ■n abc 4E ^ THE OBLIQUE TRIANGLE. 69 If r denote the radius of the inscribed circle, and we divide the triangle into three triangles by lines from the centre of this circle to the vertices, the altitude of each of the three tri- angles is equal to r. Therefore, P -- j r ( a + b + c) = rs. [3G] By substituting in this formula the value of F given in [34], 4 (s — a)(s — b) (s — c) m whence ? circle. , in [31] § 43, is equal to the radius of the inscribed Exercise XVII. Find the area : 1. Given a = 4474.5, 2. Given b = 21.66, 3. Given a = 510, 4. Given a = 408, 5. Given a = 40, 6. Given a = 624, 7. Given b = 149, 8. Given a = 215.9, 9. Given 5 = 8, 10. Given a =7, 11. Given a =60, B radius of the inscribed circle. 12. Obtain a formula for the area of a parallelogram in terms of two adjacent sides and the included angle. 13. Obtain a formula for the area of an isosceles trapezoid in terms of the two parallel sides and an acute angle. 14. Two sides and included angle of a triangle are 2416, 1712, and 30° ; and two sides and included angle of another triangle are 1948, 2848, and 150° ; find the sum of their areas. 15. The base of an isosceles triangle is 20, and its area is 100 ■+■ V3 ; find its angles. >b = 2164.5 ,P- -116° 30' 20". c = 36.94, A- = 66° 4' 19". c = 173, B -162° 30' 28". b = 41, c- = 401. b = 13, c- = 37. b = 205, c - = 445. A = 70° 42' 30" \ .5 = 39° 18' 28". c = 307.7, A- = 25° 9' 31". c = 5, A-- = 60°. c = 3, A- = 60°. = 40° 35' 12", area = 12 ; find the 70 TRIGONOMETRY. Exercise XVIII. 1. From a ship sailing down the English Channel the Eddy- stone was observed to bear N. 33° 45' "W. ; and after the ship had sailed 18 miles S. 67° 30' W. it bore N. 11° 15' E. Find its distance from each position of the ship. 2. Two objects, A and B, were observed from a ship to be at the same instant in a line bearing N. 15° E. The ship then sailed north-west 5 miles, when it was found that A bore due east and B bore north-east. Find the distance from A to B. 3. A castle and a monument stand on the same horizontal plane. The angles of depression of the top and the bottom of the monument viewed from the top of the castle are 40° and 80° ; the height of the castle is 140 feet. Find the height of the monument. 4. If the sun's altitude is 60°, what angle must a stick make with the horizon in order that its shadow in a horizontal plane may be the longest possible? 5. If the sun's altitude is 30°, find the length of the longest shadow cast on a horizontal plane by a stick 10 feet in length. C. In a circle with the radius 3 find the area of the part comprised between parallel chords whose lengths are 4 and 5 (Two solutions.) 7. A and B, two inaccessible objects in the same horizontal plane, are observed from a balloon at C and from a point B directly under the balloon, and in the same horizontal plane with A and B. If CD = 2000 yards, Z ACD = 10° 15' 10", Z BCD = 6° 7' 20", Z ADB = 49° 34' 50", find AB. 8. A and B are two objects whose distance, on account of intervening obstacles, cannot be directly measured. At the summit C of a hill, whose height above the common horizontal plane of the objects is known to be 517.3 yards, Z ACB id found to be 15° 13' 15". The angles of elevation of C viewed from A and B -are 21° 9' 18" and 23° 15' 34" respectively. Find the distance from A to B. MISCELLANEOUS PROBLEMS. [Selected by permission from " Problems in Plane Trigonometry," prepared by Prof. C. J. White, of Harvard College, and published by Charles W. Sever, Cambridge.] 1. The angular distance of any object from a horizontal plane, as observed at any point of that plane, is the angle which a line drawn from the object to the point of observa- tion makes with the plane. If the object observed be situated above the horizontal plane (that is, if it is farther from the earth's centre than the plane is), its angular distance from the plane is called its angle of elevation. If the object be below the plane, its angular distance from the plane is called its angle of depression. These angles are evidently vertical angles. If two objects are in the same horizontal plane with the point of observation, the angular distance of one object from the other is called its bearing from that object. If two objects are not in the same horizontal plane with cither each other or the point of observation, we may suppose vertical lines to be passed through the two objects, and to meet the horizontal plane of the point of observation in two points. The angular distance of these two points is the bear- ing of either of the objects from the other. It may also be called the horizontal distance of one object from the other. Note. "Problems in Plane Trigonometry" can be obtained in pam- phlet form of Charles W. Sever, Cambridge, Mass. 72 TRIGONOMETRY. Right Triangles. 2. The angle of elevation of a tower is 48° 19' 14", and the distance of its base from the point of observation is 95 ft. Find the height of the tower, and the distance of its top from the point of observation. 3. From a mountain 1000 ft. high, the angle of depression of a ship is 77° 35' 11". Find the distance of the ship from the summit of the mountain. 4. A flag-staff 90 ft. high, on a horizontal plane, casts a shadow of 117 ft. Find the altitude of the sun. 5. When the moon is setting at any place, the angle at the moon subtended by the earth's radius passing through that place is 57' 3". If the earth's radius is 3956.2 miles, what is the moon's distance from the earth's centre? 6. The angle at the earth's centre subtended by the sun's radius is 16' 2", and the sun's distance is 92,400,000 miles. Find the sun's diameter in miles. 7. The latitude of Cambridge, Mass., is 42° 22' 49". What is the length of the radius of that parallel of latitude ? 8. At what latitude is the circumference of the parallel half of that of the equator ? 9. In a circle with a radius of 6.7 is inscribed a regular polygon of thirteen sides. Find the length of one of its sides. 10. A regular heptagon, one side of which is 5.73, is in- scribed in a circle. Find the radius of the circle. 11. A tower 93.97 ft. high is situated on the bank of a river. The angle of depression of an object on the opposite bank is 25° 12' 54". Find the breadth of the river. MISCELLANEOUS PROBLEMS. 73 12. From a tower 58 ft. high the angles of depression of two objects situated in the same horizontal line with the base of the tower, and on the same side, are 30° 13' 18'' and 45° 46' 14". Find the distance between these two objects. 13. Standing directly in front of one corner of a flat-roofed house, which is 150 ft. in length, I observe that the horizontal angle which the length subtends has for its cosine VJ, and that the vertical angle subtended by its height has for its sine 3 — - What is the height of the house? V34 14. A regular pyramid, with a square base, has an edge 150 ft. in length, and the length of a side of its base is 200 ft. Find the inclination of the face of the pyramid to the base. 15. From one edge of a ditch 36 ft. wide, the angle of ele- vation of a wall on the opposite edge is 62° 39' 10". Find the length of a ladder which will reach from the point of observa- tion to the top of the wall. 16. The top of a flag-staff has been broken off, and touches the ground at a distance of 15 ft. from the foot of the staff. The length of the broken part being 39 ft., find the whole length of the staff. 17. From a balloon, which is directly above one town, is observed the angle of depression of another town, 10° 14' 9". The towns being 8 miles apart, find the height of the balloon. 18. From the top of a mountain 3 miles high the angle of depression of the most distant object which is visible on the earth's surface is found to be 2° 13' 50". Find the diameter of the earth. 19. A ladder 40 ft. long reaches a window 33 ft. high, on one side of a street. Being turned over upon its foot, it reaches another window 21 ft. high, on the opposite side of the street. Find the width of the street. 74 TRIGONOMETRY. 20. The height of a house subtends a right angle at a win- dow on the other side of the street ; and the elevation of the top of the house, from the same point, is 60°. The street is 30 ft. wide. How high is the house ? 21. A lighthouse 54 ft. high is situated on a rock. The elevation of the top of the lighthouse, as observed from a ship, is 4° 52', and the elevation of the top of the rock is 4° 2'. Find the height of the rock, and its distance from the ship. 22. A man in a balloon observes the angle of depression of an object on the ground, bearing south, to be 35° 30 f ; the balloon drifts 2| miles east at the same height, when the angle of depression of the same object is 23° 14'. Find the height of the balloon. 23. A man standing south of a tower, on the same horizon- tal plane, observes its elevation to be 54° 16' : he goes east 100 yds., and then finds its elevation is 50° 8'. Find the height of the tower. 24. The elevation of a tower at a place A south of it is 30° ; and at a place B, west of A, and at a distance of a from it, the the elevation is 18°. Show that the height of the tower is a ; the tangent of 18° being ^ ~ l - V(2+2V5) V(io + 2V5) 25. A pole is fixed on the top of a mound, and the angles of elevation of the top and the bottom of the pole are 60° and 30°. Prove that the length of the pole is twice the height of the mound. 26. At a distance (a) from the foot of a tower, the angle of elevation (A) of the top of the tower is the complement of the angle of elevation of a flag-staff on top of it. Show that the length of the staff is 2a cot 2 A. 27. A line of true level is a line every point of which is equally distant from the centre of the earth. A line drawn MISCELLANEOUS PROBLEMS. 75 tangent to a line of true level at any point is a line of appar- ent level. If at any point both these lines be drawn, and extended one mile, find the distance they are then apart. 28. In Problem 2, determine the effect upon the computed height of the tower, of an error in either the angle of eleva- tion or the measured distance. Oblique Triangles. 29. To determine the height of an inaccessible object situ- ated on a horizontal plane, by observing its angles of elevation at two points in the same line with its base, and measuring the distance of these two points. 30. The angle of elevation of an inaccessible tower, situated on a horizontal plane, is 63° 26' ; at a point 500 ft. farther from the base of the tower the elevation of its top is 32° 14'. Find the height of the tower. 31. A tower is situated on the bank of a river. From the opposite bank the angle of elevation of the tower is 60° 13', and from a point 40 ft. more distant the elevation is 50° 19'. Find the breadth of the river. 32. A ship sailing north sees two lighthouses 8 miles apart, in a line due west ; after an hour's sailing, one lighthouse bears S.W., and the other S.S.W. Find the ship's rate. 33. To determine the height of an accessible object situated on an inclined plane. 34. At a distance of 40 ft. from the foot of a tower on an inclined plane, the tower subtends an angle of 41° 19' ; at a point 60 ft. farther away, the angle subtended by the tower is 23° 45'. Find the height of the tower. 35. A tower makes an angle of 113° 12' with the inclined plane on which it stands ; and at a distance of 89 ft. from its base, measured down the plane, the angle subtended by the tower is 23° 27'. Find the height of the tower. 76 TRIGONOMETRY. 36. From the top of a house 42 ft. high, the angle of eleva- tion of the top of a pole is 14° 13' ; at the bottom of the house it is 23° 19'. Find the height of the pole. 37. The sides of a triangle are 17, 21, 28 ; prove that the length of a line bisecting the greatest side and drawn from the opposite angle is 13. 38. A privateer, 10 miles S.W. of a harbor, sees a ship sail from it in a direction S. 80° E., at a rate of 9 miles an hour. In what direction, and at what rate, must the privateer sail in order to come up with the ship in 1-J- hours ? 39. A person goes 70 yds. up a slope of 1 in 3J from the edge of a river, and observes the angle of depression of an ob- ject on the opposite shore to be 2J°. Find the breadth of the river. 40. The length of a lake subtends, at a certain point, an angle of 46° 24', and the distances from this point to the two extremities of the lake are 346 and 290 ft. Find the length of the lake. 41. Two ships are a mile apart. The angular distance of the first ship from a fort on shore, as observed from the second ship, is 35° 14' 10" ; the angular distance of the second ship from the fort, observed from the first ship, is 42° 11' 53". Find the distance in feet from each ship to the fort. 42. Along the bank of a river is drawn a base line of 500 feet. The angular distance of one end of this line from an object on the opposite side of the river, as observed from the other end of the line, is 53° ; that of the second extremity from the same object, observed at the first, is 79° 12'. Find the perpendicular breadth of the river. 43. A vertical tower stands on a declivity inclined 15° to the horizon. A man ascends the declivity 80 ft. from the base of the tower, and finds the angle then subtended by the tower to be 30°. Find the height of the tower. MISCELLANEOUS PROBLEMS. 77 44. The angle subtended by a tower on an inclined plane is, at a certain point, 42° 17'; 325 ft. farther down, it is 21° 47'. The inclination of the plane is 8° 53'. Find the height of the tower. 45. A cape bears north by east, as seen from a ship. The ship sails northwest 30 miles, and then the cape bears east. How far is it from the second point of observation ? 46. Two observers, stationed on opposite sides of a cloud, observe its angles of elevation to be 44° 56' and 36° 4'. Their distance from each other is 700 ft. What is the linear height of the cloud ? 47. From a point B at the foot of a mountain, the eleva- tion of the top A is 60°. After ascending the mountain one mile, at an inclination of 30° to the horizon, and reaching a point C, the angle ACB is found to be 135°. Find the height of the mountain in feet. 48. From a ship two rocks are seen in the same right line with the ship, bearing N. 15° E. After the ship has sailed northwest 5 miles, the first rock bears east, and the second northeast. Find the distance between the rocks. 49. From a window on a level with the bottom of a steeple the elevation of the steeple is 40°, and from a second window 18 ft. higher the elevation is 37° 30'. Find the height of the steeple. 50. To determine the distance between two inaccessible objects by observing angles at the extremities of a line of known length. 51. Wishing to determine the distance between a church A and a tower B, on the opposite side of a river, I measure a line CD along the river (C being nearly opposite A), and ob- serve the angles A CB, 58° 20' ; ACB, 95° 20' ; ABB, ^° SO'; BBC, 98° 45'. CB is 600 ft. What is the distance required ? 78 TRIGONOMETRY. 52. Wishing to find the height of a summit A, I measure a horizontal base line CD, 440 yds. At C, the elevation of A is 37° 18', and the horizontal angle between D and the sum- mit is 76° 18' ; at D, the horizontal angle between C and the summit is 67° 14'. Find the height. 53. A balloon is observed from two stations 3000 ft. apart. At the first station the horizontal angle of the balloon and the other station is 75° 25', and the elevation of the balloon is 18°. The horizontal angle of the first station and the balloon, meas- ured at the second station, is 64° 30'. Find the height of the balloon. 54. Two forces, one of 410 pounds, and the other of 320 pounds, make an angle of 51° 37'. Find the intensity and the direction of their resultant. 55. An unknown force, combined with one of 128 pounds, produces a resultant of 200 pounds, and this resultant makes an angle of 18° 24' with the known force. Find the intensity and direction of the unknown force. 56. At two stations, the height of a kite subtends the same angle (A). The angle which the line joining one station and the kite subtends at the other station is B ; and the distance between the two stations is a. Show that the height of the kite is \a sin A sec B. 57. Two towers on a horizontal plane are 120 ft. apart. A person standing successively at their bases observes that the angular elevation of one is double that of the other ; but, when he is half-way between them, the elevations are complementary. Prove that the heights of the towers are 90 and 40 ft. 58. To find the distance of an inaccessible point C from either of two points A and B, having no instruments to meas- ure angles. Prolong CA to a, and CB to b, and join AB, Ab, and Ba. Measure AB, 500; a A, 100; aB, 560; bB, 100; and Ab, 550. MISCELLANEOUS PROBLEMS. 79 59. Two inaccessible points A and B, are visible from B, but no other point can be found whence both are visible. Take some point C, whence A and D can be seen, and meas- ure CB 200 ft. ; ABC, 89° ; A CD, 50° 30'. Then take some point E, whence D and B are visible, and measure BE, 200 ; BDB, 54° 30' ; BEB, 88° 30'. At D measure ABB, 72° 30'. Compute the distance AB. 60. To compute the horizontal distance between two inac- cessible points A and B, when no point can be found whence both can be seen. Take two points C and D, distant 200 yds. so that A can be seen from C, and B from B. From C meas- ure CF, 200 yds. to B, whence A can be seen ; and from B measure BE, 200 yds. to E, whence B can be seen. Measure AFC, 83°; ACB, 53° 30' ; ACF, 54° 31' ; BBE, 54° 30'; BBC, 156° 25' ; BEB, 88° 30'. 61. A column in the north temperate zone is east-southeast of an observer, and at noon the extremity of its shadow is northeast of him. The shadow is 80 ft. in length, and the elevation of the column, at the observer's station, is 45°. Find the height of the column. 62. From the top of a hill the angles of depression of two objects situated in the horizontal plane of the base of the hill are 45° and 30° ; and the horizontal angle between the two objects is 30°. Show that the height of the hill equals the distance between the objects. 63. Wishing to know the breadth of a river from A to B, I take AC, 100 yds. in the prolongation of BA, and then take CB, 200 yds. at right angles to AC. The angle BBA is 37° 18' 30". Find AB. 64. The sum of the sides of a triangle is 100. The angle at A is double that of B, and the angle at B is double that at C. Determine the sides. 80 TRIGONOMETRY. 65. If sin 7 A + 5 cosM = 3, find A. 66. If sin 2 A = m cos ^4 — n, find cos ^4. 67. Given sin A = m sin B, and tan A — n tan B, find sin A and cos B. 68. If tanM + 4 sinM = 6, find A. 69. If sin A = sin 2^4, find 4. 70. If tan 2 A = 3 tan A, find ^. 71. Prove that tan 50° + cot 50° = 2 sec 10°. 72. Given a regular polygon of n sides, and calling one of them a, find expressions for the radii of the inscribed and the circumscribed circle in terms of n and a. If P, IT, B be the sides of a regular inscribed pentagon, hexagon, and decagon, prove P 2 = H 2 + B 2 . Areas. 73. Obtain the formula for the area of a triangle, given two sides b, c, and the included angle A. 74. Obtain the formula for the area of a triangle, given two angles A and B, and included side c. 75. Obtain the formula for the area of a triangle, given the three sides. 76. If a is the side of an equilateral triangle, its area is 4 77. Two consecutive sides of a rectangle are 52.25 ch. and 38.24 ch. Find its area. 78. Two sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72° 10'. Find the area. 79. Two sides of a parallelogram are 15.36 ch. and 11.46 ch., and the included angle is 47° 30'. Find its area. MISCELLANEOUS PROBLEMS. 81 80. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the inclined angle is 46° 24'. Find the area. 81. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they form a right angle. Find the area. 82. Two angles of a triangle are 76° 54' and 57° 33' 12", and the included side is 9 ch. Find the area. 83. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first bears N. 82° 30' W. ; the second, S. 24° 15' E. Find the area. 84. The three sides of a triangle are 49 ch., 50.25 ch., and 25.69 ch. Find the area. 85. The three sides of a triangle are 10.64 ch., 12.28 ch., and 9 ch. Find the area. 86. The sides of a triangular field, of which the area is 14 acres, are in the ratio of 3, 5, 7. Find the sides. 87. In the quadrilateral ABCD we have AD, 17.22 ch. ; AD, 7.45 ch. ; CD, 14.10 ch. ; DC, 5.25 ch. ; and the diago- nal AC, 15.04 ch. Required the area. 88. The diagonals of a quadrilateral are a and b, and they intersect at an angle D. The area of the quadrilateral is \ ab sin D. 89. The diagonals of a quadrilateral are 34 and 56, inter- secting at an angle of 67°. Find the area. 90. The diagonals of a quadrilateral are 75 and 49, inter- secting at an angle of 42°. Find the area. 91. The area of a regular polygon of n sides, of which one . no* ,180° is a, is — ■ cot 4 n 92. One side of a regular pentagon is 25. Find the area. 93. One side of a regular hexagon is 32. Find the area. 82 TRIGONOMETRY. 94. One side of a regular decagon is 46. Find the area. 95. Find the area of a circle whose circumference is 74 ft. 96. Find the area of a circle whose radius is 125 ft. 97. In a circle with a diameter of 125 ft. find the area of a sector with an arc of 22°. 98. In a circle with a radius of 44 ft. find the area of a sector with an arc of 25°. 99. In a circle with a diameter of 50 ft. find the area of a segment with an arc of 280°. 100. Find the area of a segment (less than a semicircle), of which the chord is 20, and the distance of the chord from the middle point of the smaller arc is 2. 101. If r is the radius of a circle, the area of a regular cir- 180° cumscribed polygon of n sides is nr 2 tan n The area of a regular inscribed polygon is - r 2 sin 2 n 102. If a is a side of a regular polygon of n sides, the area of the inscribed circle is — ■ cot 2 4 n The area of the circumscribed circle is — -esc 2 4 n 103. The area of a regular polygon inscribed in a circle is to that of the circumscribed polygon of the same number of sides as 3 to 4. Find the number of sides. 104. The area of a regular polygon inscribed in a circle is a geometric mean between the areas of an inscribed and a cir- cumscribed regular polygon of half the number of sides. 105. The area of a circumscribed regular polygon is an har- monic mean between the areas of an inscribed regular polygon of the same number of sides, and of a circumscribed regular polygon of half that number. MISCELLANEOUS PROBLEMS. 83 106. The perimeter of a circumscribed regular triangle is double that of the inscribed regular triangle. 107. The square described about a circle is four-thirds the inscribed dodecagon. 108. Two sides of a triangle are 3 and 12, and the included angle is 30°. Find the hypotenuse of an isosceles right tri- angle of equal area. Plane Sailing. 109. Plane Sailing is that branch of Navigation in which the surface of the earth is considered a plane. The problems which arise are therefore solved by the methods of Plane Trigonometry. The following definitions will explain the technical terms which are employed : The difference of latitude of two places is the arc of a merid- ian comprehended between the parallels of latitude passing through those places. The departure between two meridians is the arc of a par- allel of latitude comprehended between those meridians. It evidently diminishes as the distance from the equator at which it is measured increases. When a ship sails in such a manner as to cross successive meridians at the same angle, it is said to sail on a rhumb-line. The constant angle which this line makes with the meridians is called the course, and the distance between two places is measured on a rhumb-line. If we neglect the curvature of the earth, and consider the distance, departure, and difference of latitude of two places to be straight lines, lying in one plane, they will form a right triangle, called the triangle of plane sailing. If ABB be a plane triangle, right-angled at D, and AB represent the dif- ference of latitude of A and B, BAB will be the course from 84 TRIGONOMETRY. A to B, AB the distance, and DB the departure, measured from B, between the meridian of A and that of B. 110. Taking the earth's equatorial diameter to be 7925.6 miles, find the length in feet of the arc of one minute of a great circle.* 111. A ship sails from latitude 43° 45' S., on a course N. by E., 2345 miles. Find the latitude reached, and the departure made. 112. A ship sails from latitude 1° 45' N., on a course S.E. by E., and reaches latitude 2° 31' S. Find the distance, and the departure. 113. A ship sails from latitude 13° 17' EL, on a course N.E. by E. f E., until the departure is 207 miles. Find the dis- tance, and the latitude reached. 114. A ship sails on a course between S. and E., 244 miles, leaving latitude 2° 52' S., and reaching latitude 5° 8' S. Find the course, and the departure. 115. A ship sails from latitude 32° 18' N., on a course be- tween N. and W., making a distance of 344 miles, and a departure of 103 miles. Find the course, and the latitude reached. 116. A ship sails on a course between S. and E., making a difference of latitude 136 miles, and a departure 203 miles. Find the distance, and the course. 117. A ship sails due north 15 statute miles an hour, for one day. What is the distance, in a straight line, from the point left to the point reached ? (Take earth's radius, 3962.8 statute miles.) * The length of the arc of one minute of a great circle of the earth is called a geographical mile, or a knot. In the following problems, this is the distance meant by the term " mile," unless otherwise stated. MISCELLANEOUS PROBLEMS. 85 Parallel and Middle Latitude Sailing. 118. The difference of longitude of two places is the angle at the pole made by the meridians of these two places ; or, it is the arc of the equator comprehended between these two meridians. 119. In Parallel Sailing, a vessel is supposed to sail on a parallel of latitude ; that is, either due east or due west. The distance sailed is, in this case, evidently the departure made ; and the difference of longitude made depends on the solution of the following problem : 120. Given the departure between any two meridians at any latitude, find the angle which those meridians make, or the difference of longitude of any point on one meridian from any point on the other. (The earth is considered to be a perfect sphere, and the solution depends on simple geometric and trigonometric principles. Cf. Problem 7.) The solution gives the following formula : Diff. long. = depart. X sec. lat. 121. A ship in latitude 42° 16' N., longitude 72° 16' W., sails due east a distance of 149 miles. What is the position of the point reached ? 122. A ship in latitude 44° 49' S., longitude 119° 42' E., sails due west until it reaches longitude 117° 16' E. Find the dis- tance made. 123. In Middle Latitude Sailing, the departure between two places, not on the same parallel of latitude, is considered to be, approximately, the departure between the meridians of those places, measured on that parallel of latitude which lies midway between the parallels of the two places. Except in very high latitudes or excessive runs, such an assumption produces no great error. By the formula of Art. 120, then, we shall have — Diff. long. = depart. X sec. mid. lat. 86 TRIGONOMETRY. 124. A ship leaves latitude 31° 14' N., longitude 42° 19 W., and sails E.N.E. 325 miles. Find the position reached. 125. Find the bearing and distance of Cape Cod from Havana. (Cape Cod, 42° 2' N., 70° 3' W. ; Havana, 23° 9' N., 82° 22' W.) 126. Leaving latitude 49° 57' N., longitude 15° 16'.W., a ship sails between S. and W. till the departure is 194 miles, and the latitude is 47° 18' N. Find the course, distance, and longitude reached. 127. Leaving latitude 42° 30' N., longitude 58° 51' W., a ship sails S.E. by S. 300 miles. Find the position reached. 128. Leaving latitude 49° 57' N., longitude 30° W., a ship sails S. 39° W., and reaches latitude 47° 44' N. Find the distance, and longitude reached. 129. Leaving latitude 37° N., longitude 32° 16' W., a ship sails between N. and W. 300 miles, and reaches latitude 41° N. Find the course, and longitude reached. 130. Leaving latitude 50° 10' S., longitude 30° E., a ship sails E.S.E., making 160 miles' departure". Find the distance, and position reached. 131. Leaving latitude 49° 30' N., longitude 25° W., a ship sails between S. and E. 215 miles, making a departure of 167 miles. Find the course, and position reached. 132. Leaving latitude 43° S., longitude 21° W., a ship sails 273 miles, and reaches latitude 40° 17' S. "What are the two courses and longitudes, either one of which will satisfy the data? 133. Leaving latitude 17° N., longitude 119° E., a ship sails 219 miles, making a departure of 162 miles. What four sets of answers do we get ? MISCELLANEOUS PROBLEMS. 87 134. A ship in latitude 30° sails due east 360 statute miles. What is the shortest distance from the point left to the point reached ? Solve the same problem for latitude 45°, 60°, etc. Traverse Sailing. 135. Traverse Sailing is the application of the principles of Plane and Middle Latitude Sailing to cases when the ship sails from one point to another on two or more different courses. Each course is worked up by itself, and these independent results are combined, as may be seen in the solution of the following example : 136. Leaving latitude 37° 16' S., longitude 18° 42' W., a ship sails N.E. 104 miles, then N.N.W. 60 miles, then W. by S. 216 miles. Find the position reached, and its bearing and distance from the point left. We have, for the first course, difference of latitude 73.5 N., departure 73.5 E. We have, for the second course, difference of latitude, 55.4 N., departure 23 W. We have, for the third course, difference of latitude 42.1 S., departure 211.8 W. On the whole, then, the ship has made 128.9 miles of north latitude, and 42.1 miles of south latitude. The place reached is therefore on a parallel of latitude 86.8 miles to the north of the parallel left ; that is, in latitude 35° 49'.2 S. The departure is, in the same way, found to be 161.3 miles W. ; and the middle latitude is 36° 32'.6. With these data, and the formula of Art. 126, we find the difference of longi- tude to be 201 miles, or 3° 21' W. Hence the longitude reached is 22° 3' W. With the difference of latitude 86.8 miles, and the depart- ure 161.3 miles, we find the course to be N. 61° 43' W., and X** 88 TRIGONOMETRY. the distance 183.2 miles. The ship has reached the same point that it would have reached, if it had sailed directly on a course N. 61° 43' W., for a distance of 183.2 miles. 137. A ship leaves Cape Cod (Ex. 125), and sails S.E. by S. 114 miles, N. by E. 94 miles, W.N.W. 42 miles. Solve as in Art. 139. 138. A ship leaves Cape of Good Hope (latitude 34° 22' S., longitude 18° 30' E.), and sails N.W. 126 miles, N. by E. 84 miles, W.S.W. 217 miles. Solve as in Ex. 136. *v fl EXAMINATION PAPERS. * £ V PLANE TRIGONOMETRY. I. (Harvard College, Admission. June, 1881. Time, H hours.) 1. Define a logarithm. What is the logarithm of \ in the system of which 16 is the base? Find the logarithm of 25 in the same system. 2. Compute the value " ^ff^^ J °7 logarithms. 3. Find the functions of 127° 10' from your trigonometric tables. 4. Prove the formula (cos A - cos Bf + (sin A - sin Bf =-- 4 sin 2 ^=^?. 5. Two sides of a triangle are 243 feet and 188 feet, and the angle opposite the second side is 42° 20'. Solve the tri- angle completely. 6. A pine tree growing on the side of a mountain, which is inclined to the horizontal at an angle of 20°, is broken by the wind but not severed at a distance of 40 feet from the ground. The top falls toward the foot of the mountain, and strikes the ground 50 feet from the base of the tree ; find the height of the tree. * Note. In these papers, as in many text-books, the Greek letters a (alpha), ft (bayta), y (gamma), 6 (delta), (thayta),

. Draw a figure for the case where is obtuse, and show that the proof still holds good. Confirm your results by means of the formulas for the sine and cosine of the sum of the two angles. 2. Deduce formulas for the sine, cosine, and tangent of 2 a and \ a, in terms of functions of a. 3. Prove the formula : cos (a -f /?) sin/3 — cos (a + y) siny= sin (a -f- /5) cos/3 — sin (a -f- y) cosy. 4. Prove that in any triangle a 2 = b 2 -f c 2 — 2 be cos A. 5. Deduce the formulas for the tangents of the half angles of a triangle, in terms of the sides. 6. Solve the triangles : C=35°, a = 500, c = 250, B = 22°22', « = 67.0G, 6 = 60.03. 7. The Delta measures 241 yds. on Cambridge St. and 115 yds. on Quincy St., and the angle between these streets is 88° 52'. Find the other angles of the Delta. 8. Find the area of the Delta. 9. A person travelling east in a railroad train observes a tower situated south of a station A, and on the same horizontal plane with the railroad. At the station B he finds that the distance of the tower is 2 miles ; and at C, 3 miles from B, its distance is 4 miles. Find the distance of the tower from A. 92 TRIGONOMETRY. IV. {Harvard College, Freshman Examination. April, 1880. Time, 3 hours.) 1. Deduce the formula sin (a + ft) = , drawing the figure for the case in which a is in the second quadrant and a -f- /3 in the third. 2. Deduce the formulas for cos 2 a, sin * a, and cos-|a, in terms of functions of a. 3. Prove the theorem of the sines. 4. From the fundamental formulas deduce the formula ta,n^(A + B) _ a + b tea±(A — £) a — b 5. Prove that cos (a -j- p -t- y) _ ^ _ tan a tan ^g _ ^ an ^ tan _ tan ^_ an a cos a cos/j cosy 6. In a triangle B = 4° 13.4' and a = 2001, give all the solutions in the following cases : (1) 5 = 150, (2) 5 = 200, (8) 5 = 2001. 7. A, B, and Care the corners of a triangular field. A is 40 ft. W. of B and 400 ft. 8.W. of C. What is the area of the field ? What is the length of the fence which encloses it ? 8. From two corners of the Delta, A and B, lines which make angles of 19° 52' and 57° 32' respectively with the side AB meet directly under Memorial Hall tower. The length of AB is 345.1 ft., and the angle of elevation of the tower at A is 32° 26'. Find the height of the tower, and its angle of elevation at B. EXAMINATION PAPERS. 93 V. {Harvard College, Freshman Examination. April, 1881. Time, 3 hours.) 1. Deduce the formulas which connect the functions of (90° -f 4>) and . 2. Prove the fundamental formula for cos (a + /J). 3. From the formula just found, obtain three values for cos 2 a. 4. Find the values of sin- and cos -, in terms of cos a. 5. Prove the formula (cos A - cos B) 2 + (sin A - sin Bf = 4 sin 2 ^-=^- 6. Solve the following triangles : 6 = 2434, c = 1881, c = 42°22\ a = 0.00543, c = 0.07003, a = 4° 27'. 7. The sides of a triangle are 715, 541, and 368 ; find one angle and the area. 8. The height of Memorial Hall tower is 190 feet. From its top the angles of depression of the corners of the Delta which lie on Cambridge St. are 57° 44' and 16° 59', and the angle subtended by the line joining these corners is 99° 30'. Find the length of the Delta on Cambridge St. VI. (Harvard College, Freshman Examination. April, 1882. Time, 3 hours.) 1. Obtain the formulas which connect the sine, cosine, and tangent of (180° + ) with the functions of <£. 2. Assuming the formulas for the sine and cosine of the sum of two angles, prove that (1) tan (a + /?)=_ (2) sin-^a= V-J(l — cos a). 94 TRIGONOMETRY. 3. Find all the values of x } between 0° and 360°, which will satisfy the equations (1) tan x = 2 sin 2x, (2) (sin x + cos x) 2 = 2 sin 2 x. 4. The length of each side of a regular dodecagon is 24 feet ; find the radius of the inscribed circle and the area of the polygon. 5. In a certain triangle, a = 20, B = 3° 24', C= 85° 31'. Find c by aid of the table containing the values of 8 and T. 6. Prove the Theorem of Sines, and solve the triangles (1) 6 = 468, s A — cos B 1 + tan A tan B (3) sin^ + sin^ = cot ^-^ 102 TRIGONOMETRY. 5. Determine the value of cos 18°, and prove that cos 36° = cos 60° + cos 72°. 6. Show that for certain values of the angles 2cos£v4 = Vl + sin^4. — Vl —sin A. Is this formula true for values of A lying between 200° and 220° ? and if not, how must it be modified ? 7. Prove that in any triangle, with the usual notation, i A ls(s — a) cos £ A = \ l-±-- l i \ be and that the area is equal to Vs (s — a) (s — b)(s — c). Show, also, that ain 2 A = cos 2 B -f cos 2 C-f 2 cos A cos B cos C. 8. When one side of a triangle and the two adjacent angles are given, show how to solve the triangle. Find the greatest side of the triangle, of which one side is 2183 feet, and the adjacent angles are 78° 14' and 71° 24'. APPENDIX. FORMULAS. 1. sin 2 A -f cosM = 1. 2. tan.i=£^. cos A ( sin A X esc ^4 = 1. 3. -j cos A x sec i*L I tan A X cot A = 1. r §5. 4. sin (x -J- 3/) = sin x cosy + cos x sin y. 1 5. cos (# -f- y) — cos x cos y — sin x sin y. 6. tan(* + y)=J«»£+t«2L 1 — tan a: tan y 7. cot(g + y) = cota;cot y- 1 . cot x -f- cot y 8. sin (x —y) = sin a; cos y — cos a; sin y. 9. cos (x —y) = cos a; cosy -j- sin a; sin y. in , f s tana; — tan y 10. tan (x — y) = = £-. 1 -f- tan a: tan y n. / x cot x cot v 4- 1 . cot (x — y) — £— ! — cot y — cot a: ' § 3i. §32. 12. sin 2x = 2 sin a; cos a;. 13. cos 2 a; = cos 2 a; — sin 2 a;. §33. 154 FORMULAS. 14. 15. 16. 17. tan 2 x = cot 2x = 2 tana: 1 — tan' 2 # cot 2 # — 1 2 cot x sin | z = zb-y — cos I z §33. cosz ^ + cosz 2 18. tan J z : \i 1 — cos z + COS 19. 20. 21. 22. 23. 24. 25. 4 26. 27. 28. cot hz -*aJE COS z cosz §34. sin A -f sin B = 2 sin £ (^4 -f i?) cos J (^4 - J5). sin ^ - sin B = 2 cos \ {A + £) sin \(A- B). cos ^4 + cosi? - 2 cos £ (^4 + B) cos £ {A — B). cos J. — cos2? = — 2 sin i (A-\- B) sin £ (-4 — B). sin J. + sin B _ tan £ (^4 -f i?) sin ^4 — sin B tan £ (^4 — B) §35. §36. a sin A b sin J9 a 2 = ^ + c 2 -2focosA §37. a — b = tan £ (^4 — i?) „ 38 a + b toal(A + £) S ' 8m 1^ = >-*)/«"«) . §43. X be FORMULAS. 155 29. 30. Is (s — a) tan ».!=>- *><'-*> \ s(s — a) 31. 32. 33. 34. 35. 36. 37. 38. l (s-a)( S -b)(s-c) _ r tan 2 A = s — a F= \ ac sin B. ^§43. F=\/s(s-a)(s-b)(s- C y tp abc F= h r (a -f b -f- c) = rs. Spherical Trigonometry. cos c = cos a cos b. sin a = sine sin A. sin 6 = sin c sin J9. §44. §46. 40. 41. 42. 43. cos A = tan 6 cot c. cos j5 = tan a cot tan } c. sin £ (J. -f B) 180° §53. 51. tan 2 \ E= tan * (s — a) tan £ (s — 6) tan £ (s — c). §60. FORMULAS. 157 Prof. Blakslee's construction by which the direction ratios for plane right triangles give directly from a figure the analo- gies for a right trihedral or for a right spherical triangle. The construction consists of two parts. (a) Lay off from the vertex V& unit's distance on each edge. (b) Pass through the three extremities of these distances three planes perpendicular to one of the edges, as VA. Now these three parallel planes will cut out three similar right triangles. The first being constructed in either of the two usual ways, the construction of the others is evident. Since the plane angles A if A 2 , A 3 all equal the dihedral A, and the nine right triangles in the three faces give the values in the figure, we have : (1) sin A = sin a : sin h ; similarly, sin B = sin b : sin h. (2) cos A = tan b : tan h ; similarly, cos B = tana : tan h. (3) tan A = tana : sin b ; similarly, tan B = tan b : sin a. (4) cos h = cos a cos b ; (by 3) = cot A cot B. (5) sin A = cos B : cos b ; sin B = cos A : cos a. Note. If a sphere of unit radius be described about Fas a centre, the three faces will cut out a right spherical triangle, having the sides a, b, and h, and angles A, B, and H. The above formulas are thus seen to be the analogies of: 158 FORMULAS. (1) sin A = a : A ; sin i? = Z> : A. (2) cos -4. = 6 : A ; cos 5 = a : A. (3) tan-4 = a ■ 6 ; tan B = b : a. (4) A 2 a 2 + 1 = sin 2 + cos 2 ; 1 = cot A cot B. (5) sin A = cos 5 ; sin B = cos A Napier's rules give only the following, which follow from the analo- gies as numbered : By | sin a = sin A sin A = tan b cot B (1) I sin 6 = sin B sin A = tan a cot ^1 (5){ (3) (2) cos A = sin B cos a — tan b cot A cos 5 = sin A cos & = tan a cot A (4) { cos A = cos a cos b = cot J. cot B } (4) The Gauss Equations. cos %(A + -5) cos | c = cos J-(a + &) sin i O. sin £(.4 -f- B) cos ^ c = cos %{a, — b) cos £ (7. cos \{A — B) sin J c = sin J (a -f- 5) sin £ C. sin J(^L - 5) sin \ c = sin i(a — b) cos } C. /iU±-B) /**= /i(a±i) /la Rule I. sin in (I.) gives — in (3), and conversely, cos in (I.) gives -f in (3), and conversely. Rule II. Functions have same names in (2) and (3). Functions have co-names in (4) and (1). ANSWEKS. PLANE TRIGONOMETRY. Exercise I. 1. sin£=-, cos£ = _, tan£ = _, cotB = ® secB = -, cacB^Z c c a o a b 3. (i.) sin = §, cos = $, (ii.) sin = A, etc. (v.) sin = f f , etc. tan = f , cot = f , (iii.) sin = T 8 T , etc. (vi.) sin — |£$, etc. sec = \ , esc = f. (iv.) sin = ? 9 T , etc. 4. The required condition is that a 2 + b 2 = c 2 . It is. 5. (i.)sin=-^ — -^, etc. (in.) sin = -, etc. v ' m 2 + n 2 v ' s (ii.) sin= . ^, , etc. (iv.) sin = — , etc. 7 x 2 + y 2 qr 7. In (iii.)p 2 ^ 2 + q 2 s 2 =p 2 s 2 \ in (iv.) m 2 n 2 s 2 + m 2 p 2 v 2 = n 2 q 2 r 2 . 8. c = 145 ; whence, sin A = T 2 ¥ % = cos B ; cos J. = ££§ = sin i? ; tan J. = T 2 ¥ 4 j = cot i? ; cot A = -\ 4 ^ «- tan B ; sec J. = Hf = esc B ; eta 9. 5 = 0.023 ; whence, tan J. = cot £ = ^ ; cot A - tan 5 = 3ft*, etc. 10. a = 16.8 ; whence, sin A = |f § = cos B, etc. Vp 2 + g 2 11. c = » + g; whence, sin A =- — ±- = cosj5; etc. 12. 6 = y/q (p + q); whence, tan A = -Vl- = cot -5 ; etc. p — q 13. a = p — q: whence, sin A = — ; — = cosi?; etc. r a p + q 14. sin^4 = fV£ = 0.89443; etc. 15. sin^4 = f ; etc. 16. sin A = $ (5 + Vl) = 0.95572 ; etc. 17. cos A = £ (V3l-1) = 0.57097; sin A -|(VSl + l)- 0.82097; etc. 18. a =12.3. 20. a = 9. 22. c = 40. 19. 6 = 1.54. 21. 6 = 68. 23. c = 229.62. 24. Construct a rt. A with legs equal to 3 and 2 respectively; then construct a similar A with hypotenuse equal to 6. In like manner, 25, 26, 27, may be solved. 28. a - 1.5 miles ; b = 2 miles. 31. 400,000 miles. 30. a - 0.342. 6 = 0.940 ; a = 1.368, 6 = 3.760. 32. 142.926 yards. TRIGONOMETRY. Exercise II. 5. Through A (Fig. 3) draw a tangent, and take AT=>3; the angle A OT is the required angle. 6. From (Fig. 3) as a centre, with a radius = 2, describe an arc cut- ting at S the tangent drawn through B ; the angle SO A is the required angle. 7. In Fig. 3, take OM= %, and erect MP JL OA and intersecting the circumference at P; the angle POM is the required angle. 8. Since sin x = cos a;, OM= PM (Fig. 3), and # = 45° ; hence, construct x = 45°. 9. Construct a rt. A with one leg = twice the other ; the angle opposite the longer leg is the required angle. 10. Divide OA (Fig. 3) into four equal parts ; at the first point of divi- sion from erect a perpendicular to meet the circumference at some point P. Join OP; the angle A OP is the required angle. 21. r sin#. 22. Leg adjacent to A = nc, leg opposite to A = mc. Exercise III. 1. cos 60°. cotl°. sec 71° 50'. tan 7° 41'; sin 45°. . tan 75°. sin 52° 36'. sec 35° 14'. 2. cos 30°. cot 33°. sec 20° 58'. tan0° 1'. sin 15°. tan 6°. sin 4° 21'. sec 44° 59. 3. £V3. 4. tan 4= cot 4 = cot (90° -4); hence, A = 90°- A and 4 = 45°. 5. 30°. 7. 90°. 9. 22° 30'. 11. 10°. 6. 30° 8. 60°. 10. 18°. 12 90° n + 1 Exercise V. 1 . cos A = y\, tan A = ^-, cot A = T \, sec A= ^-, esc A = -j-f. 2. cos A = 0.6, tan -4 = 1.3333, cot 4 = 0.75, sec4 = 1.6667, esc 4 = 1.25. 3. sin A = -H-, tan A - ££, cot 4 = f f , sec A = f £, esc 4 = f f 4. sin A = 0.96, tan J. = 3.42854, cot A = 0.29167, sec A = 3.5714. 5. sin A = 0.8, cos A - 0.6, cot A - 0.75, sec 4 - 1.6667, esc 4 = 1.25. 6. sinA = £V2, cos4=jV2, tanA=l, secA=\/2, csc4 = \/2.~ 7. tan 4 = 2, sin 4 =0.90, cos 4 = 0.45, sec A = 2.22, esc A -1.11. ANSWERS. 8. cos A = £, sin A = $V%, tanA=V3, cot A=^Vs, cscJ. = §V3. 9. sin A = I V% cos A = I V2, tan J. = 1, cot A = 1, sec .A — V2. 10. cos 4= VI -ra 2 , tan4 = — m — Vl-m 2 , cot A = - y/TZZrf 1 — m 2 m ii ^ 1— m 2 ,< 2ra , A 1— m 2 . 1+m 2 11. cosA=- — ^-i tan A= -• cot -4= » sec A = T - l +m 2 1-m 2 2 m 1 • j, m 2 — n 2 >i m 2 — n 2 A m 2 + n 2 12. sin ^4=- 1 tan4 = — • sec A = -— : m? + n 2 2mn zmn 13. cot = l, sin=£V2, cos = £ V2, sec = V2, csc=V2. 14. cos = jV3, tan = ^V3, cot=V3, sec = §V3, esc = 2. 15. sin — £ V3, cos = J, tan = V3, cot = £ V3, sec = 2. 16. sin = ^V2-\/3, cos = ^V2 + V3, cot-2+V£ 17. sin = |V2-V2, cos = £ V2 + V2, tan=V2-l. 18. cos = 1, tan = 0, cot = oo, sec = l, esc = oo. 19. cos = 0, tan = oo, cot = 0, sec = oo, esc = 1. 20. sin = 1, cos =0, cot = 0, sec = oo, csc = l. 21. cosA=Vl-sinM, tan A = — ™±A * C sc4 = -^L-. Vl-smM sin ^ 22. sin^=Vl-cos 2 vl, tan A = Vl ~ C0S ' M , C ot4 = *»■* . , C0 * A Vl-~co~sM sec A = -, esc A cosA Vl-cosM* 23. sin^= tan ^ ^ cos^^ 1 _ , co t^ = -J_, VI + tan 2 A Vl + tan 2 J. tan .4 sec A - Vl+tanM, esc 4 = ^l±5i3. tan ^4 24. tan A = — — , C sc A - Vl + cot 2 4, sin 4 = cot ^4 Vl + cot 2 A VI + cot 2 J. cot ^4 25. s in^ = ^V5, cos^=§V5. 27. sin^ = / T , cosvl = ff 26. sin4 = |Vl5, tanJ.=Vl5. 2g 1 - 3cosM + 3cosM cos 2 A — cos 4 A TRIGONOMETRY. Exercise VI. 1. 6.6 5. - = cos A ; .". c = j. c cos A A = 90°-B, a = ccos£, 6 = c sin B. 39. c- 7.8112, ^1 = 39° 48', B = 50° 12'. 40. 6=69.99, -4 = 30' 12", B = 89° 29' 48". 41. a =1.1886, ^1 = 43° 20', B = 46° 40'. 42. 6 = 21.249, c- 22.372, 5 = 71° 46'. 43. a =6.6882, c= 13.738, £ = 60° 52'. 44. a =63.89, 6 = 23.369, 5 = 20° 6'. 45. a- 19.40, 6 = 18.778, ^4 = 45° 56'. 46. 6 = 53.719, c = 71.377, A = 41° 11'. 47. a =12.981, c = 15.796, ^1 = 55° 16'. 48. a =0.58046, 6 = 8.442, A= 3° 56'. 49. F=$(c 2 sin A cos A). 51. ^=£(6* tan 4). 50. ^=i(a 2 cotA). 52. .P = $(aVc 2 -a 2 ). 53. 6 = 11.6, c - 15.315, A = 40° 45' 48", B = 49° 14' 12". 54. a =7.2, c = 8.766, B =34° 46' 40", A = 55° 13' 20". 55. a = 3.6474, 6 = 6.58, c = 7.5233, 5=61°. 56. a = 10.283, 6 = 19.449. A = 27° 52' B = 62° 8'. 57. 19° 28' and 70° 32'. 65. tan A = £ ^1 = 59° 45'. 6 a = 6 tan ^4, 95.34. 58. 3 and 5.1961. bo. 90° 67. b = c sin n+ 1 90° 68. 7.0712 miles in each direction n + 1 69. 20.88 feet. 60. 36° 52' 12" and 53° 7' 48". 70. 56.65 feet. 61. 212.1 feet. 71. 228.63 yards. 62. 732.22 feet. 72. 136.6 feet. 63. 3270 feet. 73. 140 feet. 64. 37.3 feet, 96 feet. 74. 84.74 feet. Exercise VII. 1. C = 2( 90°-^), c = 2acos^4, A = asinA 2. A = 5(180° -C), c = 2acosA, h-aunA. 3. C=2(90°-A), a = c + 2cosA,?i = asmA. ANSWERS. 4. A = J (180° - C), a = c + 2cosA, h-annA. 5. C = 2 (90° -A), a = Ah- sin A, c-=2acosA. 6. A = £(180°-C), a = h + s'mA, c = 2acosA. 7. sin A = A H- a, C= 2(90° - A), c = 2 a cos A. 8. tanA=AH-ic, C=2(90°~A), a=A + sinA 9. A = 67° 22' 50", C= 45° 14' 20", A = 13.2. 10. e- 0.21943, A = 0.27384, 2?- 0.03004. 11. a = 2.055, A = 1.6852, i^= 1.9819. 12. a = 7.705, c = 3.6676, JP- 13.73, 13. A - 79° 36' 30", 0- 20° 47', c = 2.4206. 14. A = 77° 19' 11", C= 25° 21' 38", a = 20.5. 15. A = 25° 28', C= 129° 4', a = 81.388, A = 35. 16. A = 81° 12' 9", C= 17° 35' 42", a = 17, c = 5.2 17. i^=icV4a 2 -c 2 . 22. 0.76537. 18. .F=a 2 sin£Ccos£C. 23. 94° 20'. 19. F= a? sin A cos A. 24. 2.7261. 20. jF=A 2 tan£C. 25. 38° 56' 33". 21. 28.284 feet, 4525.44 sq. feet. 26. 37.7 Exercise VIII. 1. r - 1.618, A = 1.5388, i^= 7.694. 2. r = 11.269, A = 10.886, i^= 381.04. 3. A = 0.9848, p = 6.2514, F= 3.0782. 4. A = 19.754, c = 6.2537, F= 1236. 5. r = 1.0824, c =0.8284, F= 3.3137. 6. r = 2.592, A = 2.488, c= 1.4615. 7. r- 1.5994, A = 1.441, p = 9.716. 8. 0.6181. 12. 0.2238. 17. 11.636. 9. 0.64984. 13. 0.31. 18. 99.64. 10. 0.51764. 14. 0.82842. 19. 1.0235. c 15. 94.63. 20. 0.635. 11. 6 = 2cos y0_° 16. 415. TRIGONOMETRY. Exercise IX. 5. Two angles : one in Quadrant I., the other in Quadrant II. 6. Four values: two in Quadrant L, two in Quadrant IV. 7. x may have two values in the first case, and one value in each of the other cases. 8. If cos# = — f, x is between 90° and 270°; if cot a = 4, x is between 0° and 90° or 180° and 270° ; if sec x = 80, x is between 0° and 90° or between 270° and 360°; if esc x = -3, x is between 180 c and 360°. 9. In Quadrant III. ; in Quadrant II. ; in Quadrant III. 10. 40 angles ; 20 positive and 20 negative. 11. +, when x is known to be in Quadrant I. or IV. ; — , when x is known to be in Quadrant II. or III. 14. sin x = — f V3, tan x = - 4 V3, cot x =» - ^ 2 V3, esc x = — ^ V3. 15. sino; = ± T VV r To, coso; = =f t 3 q VIO, tan re— - \, sec ar = =f 3 Vlo. esc a; = ± VlO. 10. The cosine, the tangent, the cotangent, and the secant are negative when the angle is obtuse. 17. Sine and cosecant leave it doubtful whether the angle is an acute angle or an obtuse angle ; the other functions, if + determine an acute angle, if — an obtuse angle. 20. sin 450 - sin (360 + 90) = sin 90 = 1 ; tan 540° = tan 180° = ; cos 630° = cos 270° = ; cot 720° = cot 0° = oo ; sin 810° - sin 90° = 1 ; esc 900° = esc 180° - oo. 21. 45°, 135°, 225°, 315°. 22. 0. 23. 0. 24. 0. 25. a 2 -Z> 2 + 4a&. Exercise X. 2. sin 172°- sin 8°. 11. cot 264° = tan G°. 3. cos 100° -- sin 10°. 12. sec 244° = - esc 26°. 4. tan 125° = -cot 35°. 13. esc 271° = - sec 1°. 5. cot 91° = -tan 1°. 14. sin 163° 49'- sin 16° 11'. 6. sec 110° = - esc 20°. 15. cos 195° 33' = - cos 15° 33'. 7. esc 157°= esc 23°. 16. tan 269° 15'= cot 0° 45'. 8. sin 204° -- sin 24°. 17. cot 139° 17' =. -cot 40° 43' . 9. cos 359°- cos 1°. 18. sec 299° 45'= esc 29° 45'. 10. tan 300° --cot 30°. 19. esc 92° 25' - sec 2° 25'. ANSWERS. 20. sin(- 75°)= -sin 75°= -cos 15°, cos(- 75°)= cos 75°= sin 15°, etc. 21. sin(-127°)= -sin 127°= -cos 37°, cos(-127°)= cos 127°= -sin 37°, etc. 22. sin(-200°)= sin 160°= sin20 o ,cos(-200 o )=cos200 o =-cos20°,etc. 23. sin(-345°) = -sin 345° = sin 15°, cos(-345°) = cos 345° = cos 15°. etc. 21. sin(- 52° 37') = -sin 52° 37' = -cos 37° 23', cos (- 52° 37') - cos 52° 37' - sin 37° 23', etc. 25. sin (- 196° 54') = sin 196° 54'= cos 16° 54', cos (- 196° 54') = cos 196° 54' = - cos 16° 54', etc. 26. sin 120° = £ V3, cos 120° = - h, etc. 27. sin 135° = + \ V2, cos 1 35° - - £ V2, etc. 28. sin 150° - + £, cos 150° = - £ V3, etc. 29. sin 210° - - J, cos 210° = - £ V3, etc. B0. sin 225° - - } V2, cos 225° = -\y/% etc. 31. sin 240° = - J\/3, cos 240° -- 1, etc. 32. sin 300° --|V3, cos 300° - + \ % etc. 33. sin (-30°) --|, cos (-30°) = + %V3, etc 34. sin (- 225°) = + h V2, cos (- 225°) - - f V? etc. 35! cos x = - I V2 or - VJ etc., a = 225°. 36. tan x = — \/|~ sin x — J, cos # = — \ y/S, x = 150°. 37. sin 3540° - sin 300° = - sin 60° - -*| V3, cos 3540° - + }, etc. 38. 210° and 330° ; 120° and 300°. 89. 135°, 225°, and -225° ; 150° and -30°. 40. 30°, 150°, 390°, and 510°. 41. sin 168°, cos 334°, tan 225°, cot 252°, sin 349°, cos 240°, tan 64°, cot 177°. 42. 0.848. (Hint: tan 238° = tan 58°, sin 122° - sin 58°). 43. -1.952. 47. a 2 + 6 2 + 2a6cosa;. 44: (a-b)smx. 48. 0. 45. m sin a? cos x. 49. cos x sin y — sin x cos y. 46. (a — b) cot x — (a + b) tan x. 50. tan x. 51. Positive between x = 0° and x = 135°, and between x = 315° and x - 360° ; negative between x = 135° and x = 315°. 52. Positive between x — 45° and x = 225° ; negative between x — 0° and x = 45°, and between x = 225° and x = 360°. 53. sin (x — 90°) = — cos x, cos (a? — 90°) = sin x, etc. 54. sin (x — 180°) = — sin x, cos (x — 180°) = - cos x, etc. Exercises 53 and 54 should be solved by drawing suitable figures, and «mploying a mode of proof sjmilar to that used in \ 28. TRIGONOMETRY. 10. S 11. S 12. s 13. s 14. S 15. s 16. s 17. i Exercise XL n (* + V) = ti cos (x + y) = ||. n(90°+y) = cosy, cos ( 90 + y) = — sin y, etc. n(180— y)= sin y, cos (180 — y) = —cosy, etc. n (180 + y) = — sin y, cos (180 + y) = — cos y, etc. n(270— y) =— cosy, cos (270 -y) = -s:ny, etc. n (270 + y) = - cos y, cos (270° + y)= sin y , etc. n(360°— y)=— siny, cos (360 — y) = cosy, etc. n (360 + y) = sin y, cos (360° + y)= cos y, etc. n (x — 90°) = - cos x, cos (a; — 90°) = sin x, etc. n (x - 180°)= - sin z, cos (a - 180°)= - cos x, etc. n (x - 270°)= cos #, cos (a; — 270°)= -sin #, etc. n ( — 2/) =— siny, cos(— y) = cosy, etc. n(45°-y) = £ V2 (cosy-sin y), cos(45°-y) = J V2(cosy+siny),etc. n(45°+y) = £ V2(cosy+siny), cos (45°+y) = £ V2 (cosy-sin y), etc. n (30°+y) = J (cos y + V% sin y), cos (30° +y) = \ ( V3 cos y-sin y), etc. n (60°— y) = \ ( V3 cos y— sin y), cos (60°— y) = \ (cos y + V3 sin y), etc. 18. 3 sin a; -4 sin 3 a;. 1 9. 4 cos 3 a;- 3 cos a;. 20. 0. 21. £a/3. 22. sin \x ^J 1 ~ 0AV ® = 0.10051 ; cos £ a; --J 1 + °' 4 ^ - 0.99494. 23. cos2a;=--J, tan2a?=»— V3. 24. sin 22£° = £ V2 -V2 = 3827, cos 22£° = ^V2 + V2 = 0.9239. tan 22 £° - V2 -1 = 0.4142, cot 22£° - V2 + 1 = 2.4142. 25. sin 15° = £ V2 - V3 = 0.2588, cos 15° - ^V2+V3 - 0.9659. tan 15° - 2 - V3 - 0.2679, cot 15° = 2 + V3 = 3.7321. 27-33. The truth of these equations is to be established by expressing the given functions in terms of the same function of the same angle. Thus, in Example 27, sin 2x = 2sina;cosa;, and 2tanz = 2 , 1 + tan 2 x = sec 2 x = — =r> COS X COS 1 * X By making these substitutions in the given equation its truth will be evident. 34. sin A + sin B + sin C= sin A + sin B + sin [180 - (A + B)] = sin A + sin B + sin (A + B) = 2sin£M- + ■#) cos${A - B) + 2sinJ(-4 + B) co$$(A + B) « 2sin*U + #) [cos J (4 - B) + cos$(A + B)] = 4 sin \ (A + B) cos £ A cos \B, (see g \ 34 and 35) But cosJC= cos [90°- \{A + B)] = Bm%(A + B). Therefore, sin A + sin B + sin C=~ 4 cos \ A cos | B cos \ C. ANSWERS. 35. Proof similar to that for 34. 00 . a , . t> , l n smAcosBcosA sin B sin 36. tan A + tan B + tan C = — — . + _ + _ cos A cos B cos A cos B cos sin C sin sin C cos C+ cos A cos i? sin C + cos A cos i? cos C Y cos A cos B cos C (cos AcosB + cos C) sin C_ [cos A cos 5 — cos (A + i?)] sin C cos J. cos i? cos cos J. cos B cos C = tan A tan 5 tan C _ sin A sin 5 sin C cos A. cos B cos C 37. Proof similar to that for 36. oo 2 42. tan 2 x. .„ cos (# + y) sin2a: .„ cos(#— y) ' sinrcsiny 39. 2 cot 2 x. cos a; cosy 47. tan x tan y. 40# cos(a:-y) u cosp-f y) sin a; cos y cos # cos y 41. cos ( x + y)_ 45. cos(a;-y) sin a? cosy ' sin x sin y Exercise XII. 1. If, for instance, B= 90°, [25] becomes - - sin A. b 3. a 2 = 6 2 + c 2 , a 2 = 6 2 + c 2 -26c, a 2 = 6 2 + c 2 + 26c. 6. 90° in each case. 7. (i.) ^^ - tan (J. - 45°), and a right triangle. (ii.) a + 6 = (a — 6) (2 + V3), an isosceles triangle with the angles 30 c 30°, 120°. 9. 300. 10. AB = 59.564 miles ; AC = 54.285 miles. 11. 4.6064 miles, 4.4494 m 3.7733 miles. 12. 4.1501 and 8.67. 13. 6.1433 miles and 14. 8 and 5.4723. Exercise XIV. 11. 420. 12. The other diagonal - 124.617 EXERC] [SE XIII. 15. a = 5, c = 9.6592. 16. a = 7, 6=8.573. lies, 17. Sides, 600 feet and 1039.2 feet altitude, 519.6 feet. 18. 855 : 1607. 8 miles. 19. 20. 5.438 and 6.857. 15.588. 10 TRIGONOMETRY. 11. 6. 12. 10.392. 14. 8.9212. Exercise XV. 15. 25. 16. 3800 yards. 17. 729.7 yards. 18. 10.266. 19. a = 5.0032, b = 2.3385. 20. 26°0'10"ar 1 dl4 o 5'50' 11. 4 -36° 5* 12", B 12. ^4 = £ = 33°33'27 13. A = B=C=60°. 14. Impossible. 15. 15°, 45°, 120°. Exercise XVI. = 53° 7' 4S", C = 90°. 16. 45°, 60°, 75°. G'= 112° 53' 0". 17. 4° 23' W. of N., or W. of S. 18. 60°. 20. 0.88873. 21. 54.516 miles. Exercise XVII, 1. 4333600. 2. 365.68. 3. 13260. 4. 8160. 5. 240. 6. 26208. 7. 15540. 8. 29450 or 6983. 9. 10 V3 = 17.3205. 10. 6V3 = 10.3923. 11. 0.19952. 12. ab sin A. 13. £(a 2 -Z> 2 )tanA 14. 2421000. 15. 30°. 30°. 120°. 1. 21.166 miles 2. 6.3399 miles. 3. 119.29 feet. 4. 30°. Exercise XVITT 24.966 miles. 5. 20 feet. 6. 2.6268 or 21.4704. 7. 276.14 yards. 8. 383.35 yards. ANSWERS. 11 Miscellaneous Problems. 2. 107 feet ; 143 feet. 27. 8 inches. 51. 757.5 feet. 3. 1024 feet. 30. 460.45 feet. 52. 520 yards. 4. 37° 44' 5". 31. 88.94 feet. 53. 1366.4 feet. 5. 238,400 miles. 32. 13.657 miles. 54. 658 pounds ; 6 861,800 miles. 34. 56.5 feet. 22° 24' with first 7. 2922.4 miles. 35. 51.6 feet force. 8. 60°. 36. 101.89 feet. 55. 88.33 pounds ; 9. 3.2. 38. N. 76°56'E.; 45° 37' with known 10. 6.6. 13.94 miles an hour. force. 11. 199.56 feet. 39. 422 yards. 58. 500.2; 536.3. 12. 43 feet. 40. 256 feet. 59. 345.47 feet. 13. 45 feet. 41. 3121 feet ; 60. 345.47 yards. 14. 26° 34'. 3633.5 feet. 61. 61.23 feet. 15. 78.37 feet. 42. 529.5 feet. 63. 307.79. 16. 75 feet. 43. 41.41 feet. 64. 19.8; 35.7; 44.5. 17. 1.44 miles. 44. 234.5 feet. 65. 45°. 19. 56.65 feet. 45. 25.43 miles. 68. 60°. 20. 69.28 feet. 46. 294.7 feet. 69. 60°. 21. 260 feet ; 3690 feet. 47. 12,492 feet. 70. 30°. 22. 1.344 miles. 48. 6.34 miles. 73. $ be sin A. 23. 235.8 yards. 49. 210.44 feet. 74. \ c 2 sin A sin B esc (.4 + B). 75. VK< 5 — O) (S — b) (S — c)l 77. 199 a. 3 b. 10 p. 94. 16,281. 114. S. 56° 7' 30" E. ; 78. 210 a. 3 e. 26 p. 95. 435.8 feet. 202.6 miles. 79. 12 a. 3 e. 37 p. 96. 49,089 feet. 115. N. 17°25'W.; 80. 3 a. e. 6 p. 97. 750 feet. 37° 46' N. 81. 12 A. 1 E. 14 P. 98. 422.4 feet. 116. 56° 11' E. ; 244.3 82. 4 a. 2 e. 26 p. 99. 1835 feet. 121. Long. 68° 55' W. 83. 14 a. 2 e. 9 p. 100. 26.88. 122. 103.6 miles. 84. 61 A. 2 E. 103. 6. 124. 33° 18' N. ; 85. 4 A. 2 e. 26 p. 108. 6. 36° 24' W. 86. 13.93, 23.21.32.5ch.no. 6087 feet. 125. N. 28°47'K; 87. 9 a. 111. 5°25'S.; 1293 miles. 89. 876.31. 457.5 miles. 126. S. 50° 40' W. ; 90. 1229.5. 112. 460.8; 383.1 miles. 250.8 ; 20° 9' W. 92. 1075.3. 113. 229 miles ; 127. 38° 21' N. ; 93. 2660.45. lat. 11° 39' S. 55°12'W. 12 TRIGONOMETRY. 128. 171 miles ; 32° 44' W. 129. N. 36° 52' W. ; 36° 8' W. 130. 173 miles ; 51° 16' S. ; 34° 13' E. 131. S. 50° 58' E. ; 47° W N.; 20° 49' W. 132. N. 53° 20' E., 16° 7' W.; or N. 53° 2W W., 25° 53' W 133. N. 47° 42^ E., 19° 27' N., 121° 51' E. ; or N. 47° 42.5' W., 19° 27* N., 116° 9' E. ; or S. 47° 42.5' E., 14° 33' N., 121° 48' E. ; or S 47° 42.5' W., 14° 33' N., 116° 12' E. 137. N. 73' E., 45 miles ; 42° 15* N., 69° & W. 138. N. 72° W., 287 miles : 33° S.. 13° 2' E FIVE-PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES. ARRANGED BY G. A WENTWORTH, A.M., AND G. A HILL, A.M. >'**c BOSTON, U.S.A.: PUBLISHED BY GINN & COMPANY. 1891. Entered according to Act of Congress, in the year 1882, by G. A. WENTWORTH and G. A. HILL, in the office of the Librarian of Congress at Washington. J. S. Cushing & Co., Printers, Boston. INTRODUCTION. >^< 1. If the natural numbers are regarded as powers of ten, the ex- ponents of the powers are the Common or Briggs Logarithms of the numbers. If A and B denote natural numbers, a and b their loga- rithms, then 10 a = A, 10 5 = B ; or, written in logarithmic form, log^l = «, \ogB=b. 2. The logarithm of a product is found by adding the logarithms of its factors. For, AxB = 10« X 10 6 = 10* + \ Therefore, log ( A x B) = a + b = log A + log B. 3. The logarithm of a quotient is found by subtracting the logarithm of the divisor from that of the dividend. For. 4.«g,.M~>. B> 10* A Therefore, log — = a — b = log A — log B. 4. The logarithm of a power of a number is found by multiplying the logarithm of the number b}^ the exponent of the power. For, A n =(10 a ) n =10 an . Therefore, log A n = an — n log A. 5. The logarithm of the root of a number is found by dividing the logarithm of the number b} r the index of the root. For, VJ = \/l0« = 10* logA n Therefore, log \/~A. = - = 6. The logarithms of 1, 10, 100, etc., and of 0.1, 0.01, 0.001, etc, are integral numbers. The logarithms of all other numbers are frac- tions. IV LOGARITHMS. For, 10°= 1, hence log 1 = 0; 10-!= 0.1, hence log 0.1 = —1; 10 1 = 10, hence log 10 = 1 ; 10~ 2 = 0.01, hence log 0.01 = —2 ; 10 2 = 100, hence logl00 = 2; 10" 3 = 0.001, hence log 0.001 = -3; 10 3 = 1000, hence log 1000 = 3 ; and so on. If the number is between 1 and 10, the logarithm is between and 1. If the number is between 10 and 100, the logarithm is between 1 and 2. If the number is between 100 and 1000, the logarithm is between 2 and 3. If the number is between 1 and 0.1, the logarithm is between and —1. If the number is between 0.1 and 0.01, the logarithm is between —1 and —2. If the number is between 0.01 and 0.001, the logarithm is between —2 and —3. And so on. 7. If the number is less than 1, the logarithm is negative (§6), but is written in such a form that the fractional part is always positive. For the number may be regarded as the product of two factors, one of which lies between 1 and 10, and the other is a negative power of 10 ; the logarithm will then take the form of a difference whose minuend is a positive proper fraction, and whose subtrahend is a positive integral number. Thus, 0.48 = 4.8X0.1. Therefore (§2), log 0.48 = log 4.8 + logO.l = 0.68124-1. (Page 1.) Again, 0.0007 = 7X0.0001. Therefore, log 0.0007 = log 7 + log 0.0001 = 0.84510 - 4. 8. Every logarithm, therefore, consists of two parts : a positive or negative integral number, which is called the Characteristic, and a positive proper fraction, which is called the Mantissa. Thus, in the logarithm 3.52179, the integral number 3 is the characteristic, and the fraction .52179 the mantissa. In the logarithm 0.78254 — 2, the inte- gral number — 2 is the characteristic, and the fraction .78254 is the mantissa. 9. If the logarithm is negative, it is customary to change the form of the difference so that the subtrahend shall be 10 or a multiple of 10. This is done by adding to both minuend and subtrahend a number which will increase the subtrahend to 10 or a multiple of 10. Thus, the logarithm 0.78254-2 is changed to 8.78254-10 by adding 8 to both minuend and subtrahend. The logarithm 0.92737 — 13 is changed to 7.92737 — 20 by adding 7 to both minuend and subtrahend. 10. The following rules are derived from § 6 : — If the number is greater than 1, make the characteristic of the logarithm one unit less than the number of figures on the left of the decimal point. If the number is less than 1 , make the characteristic of the loga- rithm negative, and one unit more than the number of zeros between the decimal point and the first significant figure of the given number. INTRODUCTION. If the characteristic of a given logarithm is positive, make the number of figures in the integral part of the corresponding number one more than the number of units in the characteristic. If the characteristic is negative, make the number of zeros between the decimal point and the first significant figure of the corresponding- number one less than the number of units in the characteristic. Thus, the characteristic of log 7849.27 = 3 ; the characteristic of log 0.037 = — 2 = 8.00000 — 10. If the characteristic is 4, the corresponding number has five figures in its integral part. If the characteristic is — 3, that is, 7.00000 — 10, the corre- sponding fraction has two zeros between the decimal point and the first significant figure. 11. The logarithms of numbers that can be derived from one another b} r multiplication or division by an integral power of 10 have the same mantissa. For, multiplying or dividing a number by an integral power of 10 will increase or diminish its logarithm by the exponent of that power of 10 ; and since this exponent is an integer, the mantissa of the logarithm will be unaffected. Thus, log 4.6021 =0.66296. (Page 9.) log 460.21 = log (4.6021 X 10 2 ) = log 4.6021 + log 10 2 = 0.66296 + 2 = 2.66296. log 460210 = log (4.6021 X 10 5 ) = log 4.6021 + log 10 5 = 0.66296 + 5 = 5.66296. log 0.046021 = log (4.6021 -*■ 10 2 ) = log 4.6021 - log 10' = 0.66296 - 2 = 8.66296 - 10. TABLE I. 12. In this table (pp. 1-19) the vertical columns headed N contain the numbers, and the other columns the logarithms. On page 1 both the characteristic and the mantissa are printed. On pages 2-19 the mantissa only is printed. The fractional part of a logarithm can be expressed only approxi- mately, and in a five-place table all figures that follow the fifth are rejected. Whenever the sixth figure is 5, or more, the fifth figure is increased by 1. The figure 5 is written when the value of the figure in the place in which it stands, together with the succeeding figures, is more than 4£, but less than 5. Thus, if the mantissa of a logarithm written to seven places is 5328732, it is written in this table (a five-place table) 53287. If it is 5328751, it is written 53288. If it is 5328461 or 5328499, it is written in this table 53285. Again, if the mantissa is 5324981, it is written 53250; and if it is 4999967, it is written 50000. VI LOGARITHMS. This distinction between 5 and 5, in case it is desired to curtail still further the mantissas of logarithms, removes all doubt whether a 5 in the last given place, or in the last but one followed b}- a zero, should be simply rejected, or whether the rejection should lead us to increase the preceding figure by one unit. Thus, the mantissa 13925, when reduced to four places should be 1392; but 13925 should be 1393. To Find the Logarithm of a Given Number. 13. If the given number consists of one or two significant figures, the logarithm is given on page 1. If zeros follow the significant figures, or if the number is a proper decimal fraction, the character- istic must be determined b} r § 10. 14. If the given number has three significant figures, it will be found in the column headed N (pp. 2-19), and the mantissa of its logarithm in the next column to the right, and on the same line. Thus, Page 2. log Ho = 2.16137, log 14500 = 4.16137. Page 14. log716 = 2.85491, log 0.716 = 9.85491 - 10. 15. If the given number has four significant figures, the first three will be found in the column headed N, and the fourth at the top of the page in the line containing the figures 1, 2, 3, etc. The mantissa will be found in the column headed by the fourth figure, and on the same line with the first three figures. Thus, Page 15. log 7682 =3.88547, log 76.85 =1.88564. Page 18. log 93280 = 4.96979, log 0.9468 = 9.97626 - 10. 16. If the given number has five or more significant figures, a process called interpolation is required. Interpolation is based on the assumption that between two con- secutive mantissas of the table the change in the mantissa is directly proportional to the change in the number. Required the logarithm of 34237. The required mantissa is (§ 11) the same as the mantissa for 3423.7 ; there- fore it will be found by adding to the mantissa of 3423 seven-tenths of the difference between the mantissas for 3423 and 3424. The mantissa for 3423 is 53441. The difference between the mantissas for 3423 and 3424 is 12. Hence, the mantissa for 3423.7 is 53441 + (0.7 X 12) = 53449. Therefore, the required logarithm of 34237 is 4.53449. INTRODUCTION. Vll Required the logarithm of 0.0015764. The required mantissa is the same as the mantissa for 1576.4; therefore it will be found by adding to the mantissa for 1576 four-tenths of the difference between the mantissas for 1576 and 1577. The mantissa for 1576 is 19756. The difference between the mantissas for 1576 and 1577 is 27. Hence, the mantissa for 1576.4 is 19756 + (0.4 x 27) = 19767. Therefore, the required logarithm of 0.0015764 is 7.19767-10. Required the logarithm of 32.6708. The required mantissa is the same as the mantissa for 3267.08 ; therefore it will be found by adding to the mantissa for 3267 eight-huudredths of the difference between the mantissas for 3267 and 3268. The mantissa for 3267 is 51415. The difference between the mantissas for 3267 and 3268 is 13. Hence, the mantissa for 3267.08 is 51415 + (0.08 X 13) = 51416. Therefore, the required logarithm of 32.6708 is 1.51416. 17. When the fraction of a unit in the part to be added to the mantissa for four figures is less than 0.5 it is* to be neglected; when it is 0.5 or more than 0.5 it is to be taken as one unit. Thus, in the first example, the part to be added to the mantissa for 3423 Is 8.4, and the .4 is rejected. In the second example, the part to be added to the mantissa for 1576 is 10.8, and 11 is added. To Find the Number Corresponding to a Given Logarithm. 18. If the given mantissa can be found in the table, the first three figures of the required number will be found in the same line with the mantissa in the column headed N, and the fourth figure at the top of the column containing the mantissa. The position of the decimal point is determined by the character- istic (§10). Find the number corresponding to the logarithm 0.92002. Page 16. The number for the mantissa 92002 is 8318. Therefore, the required number is 8.318. Find the number corresponding to the logarithm 6.09167. Page 2. The number for the mantissa 09167 is 1235. Therefore, the required number is 1235000. Find the number corresponding to the logarithm 7.50325 — 10. Page 6. The number for the mantissa 50325 is 3186. Therefore, the required number is 0.003186. Vlll LOGARITHMS. 19. If the given mantissa cannot be found in the table, find in the table the two adjacent mantissas between which the given mantissa lies, and the four figures corresponding to the smaller of these two mantissas will be the first four significant figures of the required number. If more than four figures are desired, they may be found by interpolation, as in the following examples : Find the number corresponding to the logarithm 1.48762. Here the two adjacent mantissas of the table, between which the given mantissa 48762 lies, are found to be (page 6) 48756 and 48770. The corre- sponding numbers are 3073 and 3074. The smaller of these, 3073, contains the first four significant figures of the required uumber. The difference between the two adjacent mantissas is 14, and the difference between the corresponding numbers is 1. The difference between the smaller of the two adjacent mantissas, 48756, and the given mantissa, 48762, is 6. Therefore, the number to be annexed to 3073 is T 6 T of 1 = 0.428, and the fifth significant figure of the required number is 4. Hence, the required number is 30.734. Find the number corresponding to the logarithm 7.82326 — 10. The two adjacent mantissas between which 82326 lies are (page 13) 82321 and 82328. The number corresponding to the mantissa 82321 is 6656. The difference between the two adjacent mantissas is 7, and the difference between the corresponding numbers is 1. The difference between the smaller mantissa, 82321, and the given mantissa, 82326, is 5. Therefore, the number to be annexed to 6656 is \ of 1 = 0.7, and the fifth significant figure of the required number is 7. Hence, the required number is 0.0066567. In using a five-place table the numbers corresponding to mantissas may be carried to five significant figures, and in the first part of the table to six figures.* 20. The logarithm of the reciprocal of a number is called the Cologarithm of the number. If A denote any number, then cologJ. = log^- = logl — log A (§ 3) = — log^l. A Hence, the cologarithm of a number is equal to the logarithm of the number with the minus sign prefixed, which sign affects the entire logarithm, both characteristic and mantissa. * In most tables of logarithms proportional parts are given as an aid to Interpolation; but, after a little practice, the operation can be performed nearly as rapidly without them. Their omission allows a page with larger- faced type and more open spacing, and consequently less trying to the eyes. INTRODUCTION. IX In order to avoid a negative mantissa in the cologarithm, it is customary to substitute for — log A its equivalent (lO-log^)-lO. Hence, the cologarithm of a number is found by subtracting the logarithm of the number from 10, and then annexing — 10 to the remainder. The best way to perform the subtraction is to begin on the left and subtract each figure of log ^4. from 9 until we reach the last significant figure, which must be subtracted from 10. If log^l is greater in absolute value than 10 and less than 20, then in order to avoid a negative mantissa, it is necessary to write — log A in the form (20-log-4)-20. So that, in this case, colog^. is found by subtracting log^. from 20, and then annexing — 20 to the remainder. Find the cologarithm of 4007. 10 —10 Page 8. log 4007= 3.60282 colog4007 = 6.39718-10 Find the cologarithm of 103992000000. 20 -20 Page 2. log 103992000000 = 11 .01700 colog 103992000000 = 8.98300 - 20 If the characteristic of log A is negative, then the subtrahend, —10 or — 20, will vanish in finding the value of colog-<4. Find the cologarithm of 0.004007. 10 -10 log 0.004007 = 7.60282 - 10 colog 0.004007 = 2.39718 With practice, the cologarithm of a number can be taken from the table as rapidly as the logarithm itself. By using cologarithms the inconvenience of subtracting the loga- rithm of a divisor is avoided. For dividing by a number is equivalent to multiplying by its reciprocal. Hence, instead of subtracting the logarithm of a divisor its cologarithm may be added. X LOGAR1THMS. Computation by Logarithms. 21.(1) Find the value of x, if x = 72214 x 0.08203. Page 14. log 72214 = 4.85862 Page 16. log 0.08203 =8.91397-10 By §2. logx =3.77259 Page 11. x = 5923.63 (2) Find the value of x, if x = 5250 -j- 23487. Page 10. log 5250 =3.72016 Page 4. colog 23487 = 5.62917 - 10 Page 4. log x = 9.34933 — 10 = log 0.22353 .-. x =0.22353 (3) Find the value of x if x - 7 ' 56 X 4667 X 567 Find the value of z, if x- ^ ^ ^^ ^ ^ Page 15. log 7.56 = 0.87852 Page 9. . log 4667 =3.66904 Page 11. log 567 =2.75358 Page 17. ' colog 899. 1 = 7.04619 - 10 Page 6. colog 0.00337 = 2.47237 Page 4. colog 23435 = 5.63013 - 10 Page 5. log x = 2.44983 = log 281.73 .-. x =281.73 (4) Find the cube of 376. Page 7. log 376 =2.57519 Multiply by 3 (§4), 3 Page 10. log 376 3 = 7.72557 = log 53158600 .-. 376 3 =53158600 (5) Find the square of 0.003278. Page 6. log 0.003278 = 7.51561 - 10 2 Page 2. log 0.003278 2 = 15.03122 - 20 = log 0.000010745 .-. 0.003278 2 = 0.000010745 (6) Find the square root of 8322. Page 16. log 8322 = 3.92023 Divide by 2 (§ 5), 2) 3.92023 logV8322 =1.96012 = log 91.226 .-. V8322 =91.226 If the given number is a proper fraction, its logarithm will have as a subtrahend 10 or a multiple of 10. In this case, before dividing the logarithm by the index of the root, both the subtrahend and the num- INTRODUCTION. - XI ber preceding the mantissa should be increased b} T such a number as will make the subtrahend, when divided by the index of the root, 10 or a multiple of 10. (7) Find the square root of 0.000043641. Page 8. log 0.000043641 = 5.63989-10 10 -10 Divide by 2 (§ 5>, 2 )15.63989-^20 Page 13. log v O, 000043641 = 7.81995 - 10 = log 0.0066062 .-. V0.000043641 = 0.0066062 (8) Find the sixth root of 0.076553. Page 15. log 0.076553 = 8.88397-10 50 -50 Divide by 6 (§ 5), 6 )58.88397-60 Page 13 log a/ 0.076553 = 9.81400 - 10 = log 0.65163 .-. ^0.076553 = 0.65163 TABLE II. 22. This table (page 20) contains the value of the number tt, its most useful combinations, and their logarithms. Find the length of an arc of 47° 32' 57" in a unit circle. 47° 32' 57" =171177" log 171177 =5.23344 log-i- =4,68557-10 log arc 47° 32' 57" = 9.91901 - 10 = log 0.82994 . \ length of arc = 0.82994 Find the angle if the length of its arc in a unit circle = 0.54936. log 0.54936 = 9.73986 - 10 colog-L^loga" =5.31443 a" log angle = 5.05429 = log 113316 .-. angle = 113316"= 31° 28' 36" 23. The relations between arcs and angles given in Table II. are readily deduced from the circular measure of an angle. In Circular Measure an angle is defined by the equation angle = ^5_, radius in which the word arc denotes the length of the arc corresponding to the angle, when both arc and radius are expressed in terms of the same linear unit. Xll LOGARITHMS. Since the arc and radius for a given angle in different circles vary in the same ratio, the value of the angle given by this equation is independent of the value of the radius. The angle which is measured by a radius-arc is called a Radian, and is the angular unit iu circular measure. Since C= 2ttR, it follows that- = 2tt, and i^ = tt. Therefore, R R If the arc = circumference, the angle = Zir. If the arc = semicircumference, the angle = w. If the arc = quadrant, the angle = %ir. If the arc = radius, the angle = 1. Therefore, tt = 180°, ^=90°, -^=60°, i* = 45°, 1^ = 30°, \ir = 221°, and so on. Since 180° in common measure equals ir units in circular measure, 1° in common measure = -~- units in circular measure ; , u • • i 180° . 1 unit in circular measure = in common measure. 7T By means of these two equations, the value of an angle expressed in one measure may be changed to its value in the other measure. Thus, the angle whose arc is equal to the radius is an angle of 1 S0° 1 unit in circular measure, and is equal to , or 57° 17' 45", very 7T nearly. TABLE III. 24. This table (pp. 21-49) contains the logarithms of the trigo- nometric functions of angles. In order to avoid negative character- istics, the characteristic of every logarithm is printed 10 too large. Therefore, — 10 is to be annexed to each logarithm. On pages 28-49 the characteristic remains the same throughout each column, and is printed at the top and the bottom of the column. But on pp. 30,49, the characteristic changes one unit in value at the places marked with bars. Above these bars the proper characteristic is printed at the top, and below them at the bottom, of the column. 25. On pages 28-49 the log sin, log tan, log cot, and log cos, of 1° to 89°, are given to every minute. Conversely, this part of the table gives the value of the angle to the nearest minute when log sin, log tan, log cot, or log cos is known, provided log sin or log cos lies between 8.23822 and 9.99992, and log tan or log cot lies between 8.23829 and 11.76171. INTRODUCTION. Xlll If the exact value of the given logarithm of a function is not found in the table, the value nearest to it is to be taken, unless interpolation is employed as explained in § 26. If the angle is less than 45°, the number of degrees is printed at the top of the page, and the number of minutes in the column to the left of the columns containing the logarithm. If the angle is greater than 45°, the number of degrees is printed at the bottom of the page, and the number of minutes in the column to the right of the columns containing the logarithms. If the angle is less than 45°, the names of its functions are printed at the top of the page ; if greater than 45°, at the bottom of the page. Thus, Page 38. log sin 21° 37' = 0.56631-10. Page 45. log cot 36° 53' = 10. 12473 — 10 = 0. 12473. Page 37. log cos 69° 14' = 9.54969-10. Page 49. log tan 45° 59' = 10.01491 - 10 = 0.01491. Page 48. If log cos = 9.87468 — 10, angle = 41° 28'. Page 34. If log cot = 9.39353 — 10, angle = 76° 6'. If log sin = 9.47760 — 10, the nearest log sin in the table is 9.47774 — 10 (page 36), and the angle corresponding to this value is 17° 29'. If log tan = 0.76520= 10.76520 — 10, the nearest log tan in the table is 10.76490 — 10 (page 32), and the angle corresponding to this value is 80° 15'. 26. If it is desired to obtain the logarithms of the functions of angles that contain seconds, or to obtain the value of the angle in degrees, minutes, and seconds, from the logarithms of its functions, interpolation must be employed. Here it must be remembered that, The difference between two consecutive angles in the table is 60". Log sin and log tan increase as the angle increases ; log cos and log cot diminish as the angle increases. Find log tan 70° 46' 8". Page 37. log tan 70° 46' = 0.45731. The difference between the mantissas of log tan 70° 46' and log tan 70° 47' is 41, and^V of 41 = 5. As the function is increasing, the 5 must be added to the figure in the fifth place of the mantissa 45731 ; and Therefore log tan 70° 46' 8" = 0.45736. Find log cos 47° 35' 4". Page 48. log cos 47° 35' = 9.82899 — 10. The difference between this mantissa and the mantissas of the next log cos is 14, and -fa of 14 = 1. As the function is decreasing, the 1 must be subtracted from the figure in the fifth place of the mantissa 82899; and Therefore log cos 47° 35' 4" = 9.82898 - 10. XIV LOGARITHMS. Find the angle for which log sin = 9.45359 — 10. Page 35. The mantissa of the nearest smaller log sin in the table is 45334. The angle corresponding to this value is 1G° 30'. The difference between 45334 and the given mantissa, 45359, is 25. The difference between 45334 and the next following mantissa, 45377, is 43, and ff of 60" = 35". As the function is increasing, the 35" must be added to 1G° 30' ; and the required angle is 16° 30' 35". Find the angle for which log cot = 0.73478. Page 32. The mantissa of the nearest smaller log cot in the table is 73415. The angle corresponding to this value is 10° 27'. The difference between 73415 and the given mantissa is 63. The difference between 73415 and next following mantissa is 71, and ff of 60" = 53". As the function is decreasing, the 53" must be subtracted from 10° 27'; and the required angle is 10° 26' 7". 27. If log sec or log esc of an angle is desired, it may be found from the table by the formulas, sec A = ; hence, log sec A = colog cos A. cos A esc A = ; hence, log esc A = colosr sin A. sin A Page 31. log sec 8° 28' = cologcos 8° 28' =0.00476. Page 42. log esc 59° 36' 44" = colog sin 59° 36' 44" = 0.06418. 28. If a given angle is between 0° and 1°, or between 89° and 90° ; or, conversely, if a given log sin or log cos does not lie between the limits 8.23822 and 9.99992 in the table; or, if a given log tan or log cot does not lie between the limits 8.23829 and 11.76171 in the table ; then pages 21-24 of Table III. must be used. On page 21, log sin of angles between 0° and 0° 3 f , or log cos of the complementary angles between 89° 57' and 90°, are given to every second; for the angles between 0° and 0°3', log tan = log sin, and log cos = 0.00000 ; for the angles between 89° 57' and 90°, log cot = log cos, and log sin = 0.00000. On pages 22-24, log sin, log tan, and log cos of angles between 0° and 1°, or log cos, log cot, and log sin of the complementary angles between 89° and 90°, are given to every 10". Whenever log tan or log cot is not given, they may be found by the formulas, log tan = colog cot. log cot = colog tan. Conversely, if a given log tan or log cot is not contained in the table, then the colog must be found ; this will be the log cot or log tan, as the case may be, and will be contained in the table. INTRODUCTION. XV On pages 25-27 the logarithms of the functions of angles between 1° and 2°, or between 88° and 90°, are given in the manner employed on pages 22-24. These pages should be used if the angle lies between these limits, and if not only degrees and minutes, but degrees, min- utes, and multiples of 10" are given or required. When the angle is between 0° and 2°, or 88° and 90°, and a greater degree of accuracy is desired than that given b} r the table, interpo- lation may be emploj'ed ; but for these angles interpolation does not always give true results, and it is better to use Table IV. Find log tan 0° 2' 47", and log cos 89° 37' 20". Page 21. log tan 0° 2> 47" = log sin 0° 2' 47" = 6.90820-10. Page 23. log cos 89° 37' 20" = 7.81911-10. Find log cot 0° 2' 15". 10 -10 Page 21. log tan 0° 2' 15" = 6.81591 - 10 Therefore, log cot 0° 2' 15" = 3.18409 Find log tan 89° 38' 30". 10 -10 Page 23. log cot 89° 38' 30" = 7.79617 - 10 Therefore, log tan 89° 38' 30" = 2.20383 Find the angle for which log tan = 6.92090 — 10. Page 21. The nearest log tan is 6.92110 — 10. The corresponding angle for which is 0° 2' 52". Find the angle for which log cos = 7.70240 — 1 0. Page 22. The nearest log cos is 7.70261 — 10. The corresponding angle for which is 89° 42' 40". Find the angle for which log cot = 2.37368. This log cot is not contained in the table. The cologcot = 7.62632 — 10 = log tan. The log tan in the table nearest to this is (page 22) 7.62510 — 10, and the angle corresponding to this value of log tan is 0° 14' 30". 29. If an angle x is between 90° and 360°, it follows, from formu- las established in Trigonometiy, that, between 90° and 180°, between 180° and 270 c log sin x = log sin (180° — x) , log sin x = log sin (x — 180°) n , log cos x = log cos ( 1 80° — x) n , log cos x = log cos (x — 1 80°) „, log tan x = log tan ( 1 80° — x) n , log tan x = log tan (x — 1 80°) , log cot x = log cot ( 180° — *) u ; log cot x = log cot (x — 180°) ; XVI LOGARITHMS. between 270° and 360°, log sin x = log sin (360° — x) n , log cos x = log cos (360° — x) , log tan# = log tan (360° — x) n , log cot x = log cot (360° - x) n . The letter n is placed (according to custom) after the logarithms of those functions which are negative in value. . The above formulas show, without further explanation, how to find by means of Table III. the logarithms of the functions of any angle between 90° and 360°. Thus, log sin 137° 45' 22" = log sin 42° 14' 38" = 9.82756 - 10. log cos 137° 45' 22" = log n cos 42° 14' 38" = 9.86940 n - 10. log tan 137° 45' 22" = log„tan 42° 14' 38" = 9.95815 n - 10. log cot 137° 45' 22" = log n cot 42° 14' 38" = 0.04185*. log sin 209° 32' 50" = log n sin 29° 32' 50" = 9.69297„ - 10. log cos 330° 27' 10" = log cos 29° 32' 50" = 9.93949 - 10. Conversely, to a given logarithm of a trigonometric function there correspond between 0° and 360° four angles, one angle in each quad- rant, and so related that if x denote the acute angle, the other three angles are 180°- x, 180° + x, and 360°- x. If besides the given logarithm it is known whether the function is positive or negative, the ambiguity is confined to two quadrants, therefore to two angles. Thus, if the log tan = 9.47451 - 10, the angles are 16° 36' 17" in Quadrant I. and 196° 36' 17" in Quadrant III. ; but if the log tan = 9.47451„ — 10, the angles are 163° 23' 43" in Quadrant II. and 343° 23' 43" in Quadrant IV. To remove all ambiguity, further conditions are required, or a knowledge of the special circumstances connected with the problem in question. TABLE IV. 30. This table (page 50) must be used when great accuracy is desired in working with angles between 0° and 2°, or between 88° and 90°. The values of S and T are such that when the angle a is expressed in seconds, S = log sin a — log a", T = log tan a — log a". Hence follow the formulas given on page 50. The values of S and T are printed with the characteristic 10 too large, and in using them —10 must always be annexed. INTRODUCTION. Xvii Find log sill 0° 58' 17". Find log cos 88° 26' 41. 2". 0° 58' 17" = 3497" 90° -88° 26' 41.2" = 1°33' 18.8" log 3497 =3.54370 = 5598.8" S = 4.68555 -10 log 5598.8 = 3.74809 S = 4.68552 - 10 log sin 0° 58' 17" = 8.22925 - 10 log cos 88° 26' 41.2" = 8.43361 - 10 Find log tan 0° 52' 47.5". Find log tan 89° 54' 37.362". 0° 52' 47.5" = 3167.5" 90° — 89° 54' 37.362" = 0° 5' 22.638' log 3167.5 = 3.50072 = 322.638" T = 4.68561 -10 log 322.638 = 2.50871 T = 4.68558 -10 log tan 0° 52' 47.5" = 8. 18633 - 10 log cot 89° 54' 37.362" = 7. 19429 - 10 log tan 89° 54' 37.362" = 2.80571 Find the angle, if log sin = 6.72306-10. 6.72306-10 S = 4.68557 — 10 Subtract, 2.03749 = log 109.015 109.015" = 0°1' 49.015" Find the angle for which log cot = 1.67604. cologcot = 8.32396 — 10 T = 4.68564 - 10 Subtract, 3.63832 = log 4348.3 4348.3" = 1° 12' 28.3" Find the angle for which log tan = 1.55407. cologtan = 8.44593 — 10 T = 4.68569 - 10 Subtract, 3.76024 = log 5757.6 5757.6" = 1° 35' 57.6", and 90° - 1° 35' 57.6" = 88° 24' 2.4". Therefore, the angle required is 88° 24' 2.4". TABLE V. 31. This table (p. 51), containing the circumferences and areas of circles, does not require explanation. TABLE VI. 32. Table VI. (pp. 52-69) contains the natural sines, cosines, tangents, and cotangents of angles from 0° to 90°, at intervals of l f . If greater accuracy is desired it may be obtained by interpolation. Note. In preparing the preceding explanations, we have made free use of the Logarithmic Tables by F. G. Gauss. For Table VI. we are indebted to D. Carhart. XV111 LOGARITHMS. TABLE VII. 33. This table (pp. 70-75) gives the latitude and departure to three places of decimals for distances from 1 to 10, corresponding to bearings from 0° to 90° at intervals of 15'. If the bearing does not exceed 45° it is found in the left-hand column, and the designations of the columns under "Distance" are taken from the top of the page ; but if the bearing exceeds 45°, it is found in the right-hand column, and the designations of the columns under "Distance" are taken from the bottom of the page. The method of using the table will be made plain by the following examples : — (1) Let it be required to find the latitude and departure of the course N. 35° 15' E. 6 chains. On p. 60, left-hand column, look for 35° 15' ; opposite this bearing, in the vertical column headed "Distance 6," are found 4.900 and 3.463 under the headings "Latitude" and "Departure" respectively. Hence, latitude or northing = 4.900 chains, and departure or easting as 3.463 chains. (2) Let it be required to find the latitude and departure of the course S. 87° W. 2 chains. As the bearing exceeds 45°, we look in the right-hand column of p. 55, and opposite 87° in the column marked " Distance 2 " we find (taking the designa- tions of the columns from the bottom of the page) latitude = .105 chains, and departure = 1.997 chains. Hence, latitude or southing = .105 chains, and departure or westing — 1.997 chains. (3) Let it be required to find the latitude and departure of the course N. 15° 45' W. 27.36 chains. In this case we find the required numbers for each figure of the distance separately, arranging the work as in the following table. In practice, only the last columns under " Latitude " and " Departure " are written. Distance. Latitude. Departure. 20 = 2 X 10 7 0.3 = 3-10 0.06 = 6-100 1.925 X 10 = 19.25 6.737 2.887 + 10 as 0.289 5.775 -100= 0.058 0.543 X 10 = 5.43 1.90 0.814-10 =0.081 1.628 - 100 = 0.016 27.36 26.334 7.427 Hence, latitude = 26.334 chains, and departure = 7.427 chains. TABLE I ■ THE COMMON OR BRIGGS LOGARITHMS , OF THE IsTATTJBAL NUMBEES From 1 to 10000. 1-100 I log V log N log N log *T log 1 0.00 000 21 1. 32 222 41 1.61278 61 1. 78 533 81 1. 90 849 2 0. 30 103 22 1. 34 242 42 1. 62 325 62 1. 79 239 82 1. 91 381 3 0.47 712 23 1.36173 43 1. 63 347 63 1.79 934 83 1. 91 908 4 0. 60 206 24 1.38 021 44 1. 64 345 64 1.80 618 84 1. 92 428 5 0.69 897 25 1. 39 794 45 1.65 321 65 1. 81 291 85 1. 92 942 6 0.77 815 26 1.41497 46 1. 66 276 66 1. 81 954 86 1.93 450 7 0. 84 510 27 1.43136 47 1.67 210 67 1.82 607 87 1.93 952 8 0. 90 309 28 1.44 716 48 1. 68 124 68 1.83 251 88 1.94 448 9 0. 95 424 29 1.46 240 49 1.69 020 69 1.83 885 89 1.94 939 10 1.00000 30 1.47 712 50 1. 69 897 70 1. 84 510 90 1.95 424 11 1.04139 31 1.49136 51 1.70 757 71 1.85 126 91 1.95 904 12 1.07 918 32 1.50 515 52 1. 71 600 72 1. 85 733 92 1.96 379 13 1.11394 33 1.51851 53 1.72 428 73 1.86 332 93 1.96 848 14 1.14 613 34 1. 53 148 54 1. 73 239 74 1.86 923 94 1.97 313 15 1.17609 35 1.54 407 55 1. 74 036 75 1. 87 506 95 1. 97 772 16' 1.20412 36 1.55 630 56 1.74 819 76 1.88 081 •96 1. 98 227 17 1. 23 045 37 1.56 820 57 1. 75 587 77 1.88 649 97 1.98 677 18 1.25 527 38 1.57 978 58 1.76 343 78 1. 89 209 98 1. 99 123 19 1.27 875 39 1. 59 106 59 1.77 085 79 1. 89 763 99 1.99 564 20 I 1.30103 40 1.60 206 60 1.77 815 80 1.90 309 100 2.00 000 log V log N log N log K log 1-100 100-150 N 1 2 3 4 5 6 7 8 9 100 00 000 00 043 00 0S7 00130 00173 00 217 00 260 00 303 00 346 00 389 101 00 432 00 475 00 518 00 561 00 604 00 647 00 6S9 00 732 00 775 00 817 102 00 860 00 903 00 945 00 988 01030 01072 01115 01157 01199 01242 103 01284 01326 01 368 01410 01452 01494 01536 01578 01620 01662 104 01703 01745 01787 01828 01870 01912 01953 01995 02 036 02 078 105 02119 02160 02 202 02 243 02 284 02 325 02 366 02 407 02 449 02 490 106 02 531 02 572 02 612 02 653 02 694 02 735 02 776 02 816 02 857 02 898 107 02 938 02 979 03 019 03 060 03 100 03 141 03 181 03 222 03 262 03 302 108 03 342 03 383 03 423 03 463 03 503 03 543 03 5S3 03 623 03 663 03 703 109 03 743 03 782 03 822 03 862 03 902 03 941 03 981 04 021 04 060 04100 110 04139 04179 04 218 04 258 04 297 04 336 04 376 04 415 04 454 04 493 111 04 532 04 571 04 610 04 650 04 689 04 727 04 766 04S05 04 844 04 883 112 04 922 04 961 04 999 05 038 05 077 05 115 05 154 05 192 05 231 05 269 113 05 308 05 346 05 385 05 423 05 461 05 500 05 538 05 576 05 614 05 652 114 05 690 05 729 05 767 05 805 05 843 05 881 05 918 05 956 05 994 06 032 115 06 070 06108 06145 06183 06 221 06 258 06 296 06 333 06 371 06 408 116 06 446 06 483 06 521 06 558 06 595 06 633 06 670 06 707 06 744 06 7S1 117 06 819 06 856 06 893 06 930 06 967 07 004 07 041 07 07S 07115 07 151 118 07188 07 225 07 262 07 298 07 335 07 372 07 408 07 445 07 482 07 518 119 07 555 07 591 07 628 07 664 07 700 07 737 07 773 07S09 07 846 07 8S2 120 07 918 07 954 07 990 08 027 08 063 08 099 08135 08171 08 207 OS 243 121 08 279 08 314 08 350 08 386 08 422 OS 458 08 493 08 529 08 565 08 600 122 08 636 08 672 OS 707 08 743 08 778 08 814 OS 849 08 884 OS 920 08 955 123 08 991 09 026 09 061 09 096 09 132 09167 09 202 09 237 09 272 09 307 124 09 342 09 377 09 412 09 447 09 482 09 517 09 552 09 587 09 621 09 656 125 09 691 09 726 09 760 09 795 09 830 09 864 09 899 09 934 09 96S 10 003 126 10 037 10 072 10106 10140 10175 10 209 10 243 10 278 10 312 10 346 127 10 380 10 415 10 449 10 483 10 517 10 551 10 5S5 10 619 10 653 10 6S7 128 10 721 10 755 10 789 10 823 10 857 10 890 10 924 10 958 10 992 11025 129- 11059 11093 11126 11160 11193 11227 11261 11294 11327 11361 130 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 131 11727 11760 11793 11826 11860 11893 11926 11959 11992 12 024 132 12 057 12 090 12123 12 156 12 189 12 222 12 254 12 287 12 320 12 352 133 12 385 12 418 12 450 12 483 12 516 12 548 12 581 12 613 12 646 12 67S 134 12 710 12 743 12 775 12 808 12 840 12 872 12 905 12 937 12 969 13 001 135 13 033 13 066 13 098 13 130 13 162 13 194 13 226 13 258 13 290 13 322 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N 1 2 3 4 5 6 7 8 9 400-450 450 - 500 N 1 2 3 4 5 6 7 8 9 450 65 321 65 331 65 341 65 350 65 360 65 369 65 379 65 389 65 398 65 408 451 65 418 65 427 65 437 65 447 65 456 65 466 65 475 65 485 65 495 65 504 452 65 514 65 523 65 533 65 543 65 552 65 562 65 571 65 581 65 591 65 600 453 65 610 65 619 65 629 65 639 65 648 65 658 65 667 65 677 65 686 65 696 454 65 706 65 715 65 725 65 734 65 744 65 753 65 763 65 772 65 782 65 792 455 65 801 65 811 65 820 65 830 65 839 65 849 65 858 65 868 65 877 65 887 456 65 896 65 906 65 916 65 925 65 935 65 944 65 954 65 963 65 973 65 982 457 65 992 66 001 66 011 66 020 66 030 66 039 66 049 66 058 66 068 66 077 458 66087 66 096 66106 66115 66124 66134 66143 66153 66162 66172 459 66181 66191 66 200 66 210 66 219 66 229 66 238 66 247 66 257 66 266 460 66 276 66 285 66 295 66304 66 314 66 323 66 332 66 342 66 351 66 361 461 66 370 66 380 66 389 66 398 66 408 66 417 66427 66 436 66 445 66455 462 66 464 66474 66483 66 492 66 502 66 511 66 521 66 530 66 539 66 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037 69 046 69 055 69 064 69 073 69 082 69 090 69 099 491 69108 69117 69126 69135 69144 69152 69161 69170 69179 69188 492 69197 69 205 69 214 69 223 69 232 69 241 69 249 69 258 69 267 69 276 493 69 285 69 294 69 302 69 311 69 320 69 329 69 338 69 346 69 355 69 364 494 69 373 69 381 69 390 69 399 69 408 69 417 69 425 69 434 69 443 69 452 495 69 461 69 469 69 478 69 487 69 496 69 504 69 513 69 522 69 531 69 539 496 69 548 69 557 69 566 69 574 69 583 69 592 69 601 69 609 69 618 69 627 497 69 636 69 644 69 653 69 662 69 671 69 679 69 688 69 697 69 705 69 714 498 69 723 69 732 69 740 69 749 69 758 69 767 69 775 69 784 69 793 69 801 499 69 810 69 819 69 827 69 836 69 84£ 69 854 69 862 69 871 69 880 69 888 500 69 897 69 906 69914 69 923 69 932 69 940 69 949 69 958 69 966 69 975 N 1 2 3 4 5 6 7 8 9 450 - 500 10 500-550 N 1 2 3 4 5 7 8 9 500 69 897 69 906 69 914 69 923 69 932 69 940 69 949 69 958 69 966 69 975 501 69 984 69 992 70 001 70010 70 018 70 027 70 036 70 044 70 053 70 062 502 70 070 70 079 70 088 70 096 70105 70114 70122 70131 70140 70148 503 70157 70165 70174 70183 70191 70 200 70 209 70 217 70 226 70 234 504 70 243 70 252 70 260 70 269 70 278 70 286 70 295 70 303 70 312 70 321 505 70 329 70 338 70 346 70 355 70 364 70 372 70 381 70 389 70 398 70 406 50$ 70415 70424 70 432 70441 70 449 70 458 70 467 70 475 70 484 70492 507 70 501 70 509 70 518 70 526 70 535 70 544 70 552 70 561 70 569 70 578 508 70 586 70 595 70 603 70 612 70 621 70 629 70 638 70 646 70 65i 70 663 509 70 672 70 680 70 689 70 697 70 706 70 714 70 723 70 731 70 740 70 749 510 70 757 70 766 70 774 70 783 70 791 70 800 70 808 70 817 70 825 70 834 511 70 842 70 851 70 859 70 868 70 876 70 885 70 893 70 902 70 910 70 919 512 70 927 70 935 70 944 70 952 70 961 70 969 70 978 70 986 70995 71003 513 71012 71020 71029 71037 71046 71054 71063 71071 71079 71088 514 71096 71105 71 113 71 122 71130 71139 71147 71155 71164 71172 515 71181 71189 71198 71206 71214 71223 71231 71240 71248 71 257 516 71265 71273 71282 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904 95 617 95 622 95 626 95 631 95 636 95 641 95 646 95 650 95 655 95 660 905 95 665 95 670 95 674 95 679 95 684 95 689 95 694 95 698 95 703 95 708 906 95 713 95 718 95 722 95 727 95 732 95 737 95 742 95 746 95 751 95 756 907 95 761 95 766 95 770 95 775 95 780 95 785 95 789 95 794 95.799 95 804 908 95 809 95 813 95 818 95 823 95 828 95 832 95 837 95 842 95 847 95 852 909 95 856 95 861 95 866 95 871 95 875 95 880 95 885 95 890 95 895 95 899 910 95 904 95 909 95 914 95 918 95 923 95 928 95 933 95 938 95 942 95 947 911 95 952 95 957 95 961 95 966 95 971 95 976 95 980 95 985 95 990 95 995 912 95 999 96 004 96 009 96 014 96 019 96 023 96 028 96 033 96 038 96 042 913 96047 96 052 96 057 96 061 96 066 96 071 96 076 96 080 96 085 96090 914 96 095 96 099 96104 96109 96114 96118 96123 96128 96133 96137 915 96142 96147 96152 96156 96161 96166 96171 96175 96180 96185 916 96190 96194 96199 96 204 96 209 96 213 96 218 96 223 96 227 96 232 917 96 237 96 242 96 246 96 251 96 256 96 261 96 265 96 270 96 275 96 280 918 96 284 96 289 96 294 96 298 96 303 96308 96 313 96 317 96 322 96 327 919 96 332 96 336 96 341 96 346 96 350 96 355 96 360 96 365 96 369 96 374 920 96 379 96 384 96 388 96 393 96 398 96402 96407 96 412 96417 96 421 921 96 426 96 431 96 435 96 440 96 445 96450 96 454 96459 96 464 96 468 922 96 473 96 478 96 483 96 487 96 492 96 497 96 501 96 506 96 511 96 515 923 96 520 96 525 96 530 96 534 96 539 96 544 96 548 96 553 96 558 96 562 924 96 567 96 572 96 577 96 581 96 586 96 591 96 595 96 600 96 605 96 609 925 96 614 96 619 96 624 96 628 96 633 96 638 96 642 96 647 96 652 96 656 926 96 661 96 666 96 670 96 675 96 680 96 685 96 689 96 694 96 699 96 703 927 96 708 96 713 96 717 96 722 96 727 96 731 96 736 96 741 96 745 96 750 928 96 755 96 759 96 764 96 769 96 774 96 778 96 783 96 788 96 792 96 797 929 96 802 96 806 96 811 96 816 96 820 96 825 96 830 96 834 96 839 96 844 930 96 848 96 853 96 858 96 862 96 867 96 872 96 876 96 881 96 886 96 890 931 96 895 96 900 96 904 96 909 96 914 96 918 96 923 96 928 96 932 96 937 932 96 942 96 946 96 951 96 956 96 960 96 965 96 970 96 974 96 979 96 984 933 96 988 96 993 96997 97 002 97 007 97 011 97 016 97 021 97 025 97 030 934 97 035 97 039 97 044 97 049 97 053 97 058 97 063 97 067 97 072 97 077 935 97 081 97 086 97 090 97 095 97100 97104 97109 97114 97118 97123 936 97128 97132 97137 97142 97146 97151 97155 97160 97165 97169 937 97174 97179 97183 97188 97192 97197 97 202 97 206 97 211 97 216 938 97 220 97 225 97 230 97 234 97 239 97 243 97 248 97 253 97 257 97 262 939 97 267 97 271 97 276 97 280 97 285 97 290 97 294 97 299 97 304 97 308 940 97 313 97 317 97 322 97 327 97 331 97 336 97 340 97 345 97 350 97 354 941 97 359 97 364 97 368 97 373 97 377 97 382 97 387 97 391 97 396 97 400 942 97 405 97 410 97414 97 419 97 424 97 428 97 433 97 437 97 442 97 447 943 97 451 97 456 97 460 97 465 97 470 97 474 97 479 97 483 97 488 97 493 944 97497 97 502 97 506 97 511 97 516 97 520 97 525 97 529 97 534 97 539 945 97 543 97 548 97 552 97 557 97 562 97 566 97 571 97 575 97 580 97 585 946 97 589 97 594 97 598 97 603 97 607 97 612 97 617 97 621 97 626 97 630 947 97 635 97 640 97 644 97 649 97 653 97 658 97 663 97 667 97 672 97 676 948 97 681 97 685 97 690 97 695 97 699 97 704 97 708 97 713 97 717 97 722 949 97 727 97 731 97 736 97 740 97 745 97 749 97 754 97 759 97 763 97 768 950 97 772 97 777 97 782 97 786 97 791 97 795 97 800 97 804 97 809 97 813 N 1 2 3 4 5 6 7 8 9 900 - 950 950-1000 19 N 1 2 3 4 5 6 7 8 9 950 97 772 97 777 97 782 97 786 97 791 97 795 97 800 97 804 97 809 97 813 951 97 818 97 823 97 827 97 832 97 836 97 841 97 845 97 850 97 855 97 859 952 97 864 97 868 97 873 97 877 97 882 97 886 97 891 97 896 97 900 97 905 953 97 909 97 914 97 918 97 923 97 928 97 932 97 937 97 941 97 946 97 950 954 97 955 97 959 97 964 97 968 97 973 97 978 97 982 97 987 97 991 97 996 955 98 000 98 005 98 009 98 014 98 019 98 023 98 028 98 032 98 037 98 041 956 98 046 98 050 98 055 98 059 98 064 98 068 98 073 98 078 98 082 98 087 957 98 091 98 096 98100 98105 98109 98114 98118 98123 98127 98132 958 98137 98141 98146 98150 98155 98159 98164 98168 98173 98177 959 98182 98186 98191 98195 98 200 98 204 98 209 98 214 98 218 98 223 960 98 227 98 232 98 236 98 241 98 245 98 250 98 254 98 259 98 263 98 268 961 98 272 98 277 98 281 98 286 98 290 98 295 98 299 98 304 98 308 98 313 962 98 318 98 322 98 327 98 331 98 336 98 340 98 345 98 349 98 354 98 358 963 98 363 98 367 98 372 98 376 98 381 98 385 98 390 98 394 98 399 98 403 964 98 408 98 412 98 417 98421 98 426 98 430 98 435 98 439 98 444 98 448 965 98 453 98 457 98 462 98 466 98 471 98 475 98 480 98 484 98 489 98 493 966 98 498 98 502 98 507 98 511 98 516 98 520 98 525 98 529 98 534 98 538 967 98 543 98 547 98 552 98 556 98 561 98 565 98 570 98 574 98 579 98 583 968 98 588 98 592 98 597 98 601 98 605 98 610 98 614 98 619 98 623 98 628 969 98 632 98 637 98 641 98 646 98 650 98 655 98 659 9S664 98 668 98 673 970 98 677 98 682 98 686 98 691 98 695 98 700 98 704 98 709 98 713 98 717 971 98 722 98 726 98 731 98 735 98 740 98 744 98 749 98 753 98 758 98 762 972 98 767 98 771 98 776 98 780 98 784 98 789 98 793 98 798 98 802 98 807 973 98 811 98S16 98 820 98 825 98 829 98 834 98 838 98 843 98 847 98 851 974 98 856 98S60 98 865 98 869 98 874 98 878 98 883 98 887 98 892 98 896 975 98 900 98 905 98 909 98 914 98 918 98 923 98 927 98 932 98 936 98 941 976 98 945 98 949 98 954 98 958 98 963 98 967 98 972 98 976 98 981 98 985 977 98 989 98 994 98 998 99 003 99 007 99 012 99 016 99 021 99 025 99 029 978 99034 99 038 99 043 99 047 99 052 99 056 99 061 99 065 99 069 99 074 979 99 078 99 083 99 087 99 092 99 096 99100 99105 99109 99114 99118 980 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 981 99167 99171 99176 99180 99185 99189 99193 99198 99 202 99 207 982 99 211 99 216 99 220 99 224 99 229 99 233 99 238 99 242 99 247 99 251 983 99 255 99 260 99 264 99 269 99 273 99 277 99 282 99 286 99 291 99 295 984 99 300 99 304 99 308 99 313 99 317 99 322 99 326 99 330 99 335 99 339 985 99 344 99 348 99 352 99 357 99 361 99 366 99 370 99 374 99 379 99 383 986 99 388 99 392 99 396 99 401 99 405 99 410 99 414 99 419 99 423 99 427 987 99 432 99436 99 441 99 445 99 449 99 454 99 458 99 463 99 467 99 471 988 99 476 99 480 99484 99489 99 493 99 498 99 502 99 506 99 511 99 515 989 99 520 99 524 99 528 99 533 99 537 99 542 99 546 99 550 99 555 99 559 990 99 564 99 568 99 572 99 577 99 581 99 585 99 590 99 594 99 599 99 603 991 99 607 99 612 99 616 99 621 99 625 99 629 99 634 99 638 99 642 99 647 992 99 651 99 656 99 660 99 664 99 669 99 673 99 677 99 682 99 686 99 691 993 99 695 99 699 99 704 99 708 99 712 99 717 99 721 99 726 99 730 99 734 994 99 739 99 743 99 747 99 752 99 756 99 760 99 765 99 769 99 774 99 778 995 99 782 99 787 99 791 99 795 99 800 99 804 99 808 99 813 99 817 99 822 996 99 826 99 830 99 835 99 839 99 843 99 848 99 852 99 856 99 861 99 865 997 99 870 99 874 99 878 99 883 99 887 99 891 99 896 99 900 99 904 99 909 998 99913 99 917 99 922 99 926 99930 99 935 99 939 99 944 99 948 99 952 999 99957 99 961 99 965 99 970 99 974 99 978 99 983 99 987 99 991 99 996 1000 00 000 00 004 00 009 00 013 00 017 00 022 00026 00030 00 035 00039 N 1 2 3 4 5 6 7 8 9 950 - 1000 20 TABLE II, -LOGARITHMS OP CONSTANTS. Circumference of the Circle in degrees = 360 Circumference of the Circle in minutes = 21 600 Circumference of the Circle in seconds . = 1 296 000 If the radius r= 1, half the Circumference of the Circle is 7t = 3. 14 159 265 358 979 323 846 264 338 328 log 2. 55 630 250 4. 33 445 375 6. 11 260 500 0. 49 714 987 Also: 2t = 6.28 318 531 4t = 12. 56 637 061 1.57079 633 2 = 1.04 719 755 — = 4.18 879 020 = 0.78 539 816 = 0.52 359 878 = 0.31830 989 = 0.15 915 494 = 0.95 492 966 = 1.27 323 954 = 0.23 873 241 log 0.79 817 987 1.09 920 986 0.19 611988 0. 02 002 862 0. 62 208 861 9. 89 508 988 - 10 9. 71 899 862 - 10 9. 50 285 013 - 10 9. 20 182 013 - 10 9. 97 997 138 - 10 0.10491012 9.37 791139-10 1 3 At 4 x 3 ±- 8/7T 6 : 9. 86 960 440 0. 10 132 118 1.77 245 385 0.56 418 958 0. 97 720 502 1.12 837 917 1.46 459189 0. 68 278 406 2. 14 502 940 0. 62 035 049 0. 80 599 598 Arc a, whose length is equal to the radius r, is : in degrees «° ...'.. = — = 57.29 577 951° . in minutes a' in seconds a" re 10 800 7T 648 000 = 3 437.74 677' ,= 206 264.806" Arc 2 a, whose length is equal to twice the radius, 2 r, is in degrees 2a° . . . in minutes 2a r in seconds 2a" TV 21 GOO 7T 1 296 000 If the radius r = 1, the length of the arc is : for 1 degree for 1 minute ■ for 1 second • for £ degree....—. for h minute — -, 2a 1 1 2a" TC 180*" 7T 10 800 648 000 TC 360"" 21600 for \ second 114. 59 155 903 6 875.49 354'. 412 529. 612" . = 0.01745 329.. = 0.00029 089.. = 0.00000 485.. -0.00 872 665... .= 0.00014 544.. = 0.00000 242.. 1 296 000 Sin 1" in the unit circle = 0. 00 000 485 log 0. 99 429 975 9.00 570 025-10 0. 24 857 494 9. 75 142 506 - 10 9. 9S 998 569 - 10 0. 05 245 506 0. 16 571 662 9.83 428 338-10 0. 33 143 325 9. 79 263 713 - 10 9. 90 633 287 - 10 log 1. 75 812 263 3. 53 627 388 5.31442 513 2. 05 915 263 3. 83 730 388 5.61545 513 8. 24 187 737 - 10 6.46 372 612-10 4.68 557 487-10 7.94 084 737-10 6. 16 269 612 - 10 4. 38 454 487 - 10 4.68 557 487-10 21 TABLE III, THE LOGAEITHMS TRIGONOMETRIC FUNCTIONS : Prom 0° to 0° 3', or 89° 57' to 90°, for every second ; Prom 0° to 2°, or 88° to 90°, for every ten seconds ; From 1° to 89°, for every minute. Note. To log sin all the logarithms —10 is to be appended. no leg U log tan = log sin cos = 10. 00 000 ft 0' 1' 2' ft tf 0' 1' 2' f r — 6.46 373 6. 76 476 60 30 6. 16 270 6. 63 982 6. 86 167 30 1 4.68557 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. 3S 454 6. 49 S49 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. 5S 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 ft 59' 58' 57' ft ft 59' 58' 57' ft log cot = log COS log sin = 10, 00 000 89 log cos 22 0° t tt log sin log tan log COS tt r t tt log sin log tan log cos ft t _ 10.00000 60 10 7. 46 373 7.46 373 10.00000 50 10 5.68 557 5.68 557 10.00000 50 10 7. 47 090 7. 47 091 10.00000 50 20 5. 98 660 5.98 660 10.00000 40 20 7. 47 797 7. 47 797 10.00000 40 30 6. 16 270 6. 16 270 10.00000 30 30 7. 48 491 7. 48 492 10.00000 30 40 6. 28 763 6. 28 763 10.00000 20 40 7. 49 175 7. 49 176 10.00000 20 50 6. 38 454 6. 38 454 10.00000 10 50 7. 49 849 7. 49 849 10.00000 10 1 6.46 373 6. 46 373 10.00000 59 11 7.50 512 7.50 512 10.00000 49 10 6. 53 067 6. 53 067 10.00000 50 10 7. 51 165 7. 51 165 10.00000 50 20 6. 58 866 6. 58 866 10.00000 40 20 7. 51 808 7. 51 809 10.00000 40 30 6. 63 982 6. 63 982 10.00000 30 30 7. 52 442 7. 52 443 10.00000 30 40 6. 68 557 6. 68 557 10.00000 20 40 7. 53 067 7. 53 067 10.00000 20 50 6. 72 697 6. 72 697 10.00000 10 50 7. 53 683 7. 53 683 10.00000 10 2 6.76476 6. 76 476 10.00000 58 12 7. 54 291 7. 54 291 10.00000 48 10 6. 79 952 6. 79 952 10.00000 50 10 7. 54 890 7. 54 890 10.00000 50 20 6. 83 170 6. 83 170 10.00000 40 20 7. 55 481 7. 55 481 10.00000 40 30 6. 86 167 6. 86 167 10.00000 30 30 7. 56 064 7. 56 064 10.00000 30 40 6. 88 969 6. 88 969 10.00000 20 40 7. 56 639 7. 56 639 10.00000 20 50 6. 91 602 6. 91 602 10.00000 10 50 7. 57 206 7. 57 207 10.00000 10 3 6. 94 085 6. 94 0S5 10.00000 57 13 7. 57 767 7. 57 767 10.00000 47 10 6. 96 433 6. 96 433 10.00000 50 10 7. 58 320 7.58 320 10.00000 50 20 6.98660 6. 98 661 10.00000 40 20 7. 58 866 7. 58 867 10.00000 40 30 7. 00 779 7. 00 779 10.00000 30 30 7.59406 7. 59 406 10.00000 30 40 7. 02 800 7. 02 800 10.00000 20 40 7. 59 939 7.59 939 10.00000 20 50 7. 04 730 7. 04 730 10.00000 10 50 7.60465 7. 60 466 10.00000 10 4 7. 06 579 7. 06 579 10.00000 56 14 7. 60 985 7. 60 986 10.00000 46 10 7.08 351 7.08 352 10.00000 50 10 7. 61 499 7. 61 500 10.00000 50 20 7.10055 7.10 055 10.00000 40 20 7. 62 007 7. 62 008 10.00000 40 30 7.11694 7. 11 694 10.00000 30 30 7. 62 509 7. 62 510 10.00000 30 40 7. 13 273 7. 13 273 10.00000 20 40 7. 63 006 7. 63 006 10.00000 20 50 7. 14 797 7. 14 797 10.00000 10 50 7. 63 496 7. 63 497 10.00000 10 5 7. 16 270 7. 16 270 10.00000 55 15 7. 63 982 7. 63 982 10.00000 45 10 7.17 694 7.17 694 10.00000 50 10 7. 64 461 7.64 462 10.00000 50 20 7.19072 7.19 073 10.00000 40 20 7. 64 936 7. 64 937 10.00000 40 30 7. 20 409 7. 20 409 10.00000 30 30 7. 65 406 7. 65 406 10.00000 30 40 7. 21 705 7. 21 705 10.00000 20 40 7. 65 870 7. 65 871 10.00000 20 50 7. 22 964 7. 22 964 10.00000 10 50 7. 66 330 7. 66 330 10.00000 10 6 7. 24 188 7. 24 188 10.00000 54 16 7. 66 784 7.66 785 10.00000 44 10 7. 25 378 7. 25 378 10.00000 50 10 7. 67 235 7. 67 235 10.00000 50 20 7. 26 536 7. 26 536 10.00000 40 20 7. 67 680 7. 67 680 10.00000 40 30 7. 27 664 7. 27 664 10.00000 30 30 7. 68 121 7. 68 121 10.00000 30 40 7. 28 763 7. 28 764 10.00000 20 40 7. 68 557 7. 68 558 9.99999 20 50 7. 29 836 7. 29 836 10.00000 10 50 7. 68 989 7. 68 990 9.99999 10 7 7. 30 882 7. 30 8S2 10.00000 53 17 7.69 417 7. 69 418 9. 99 999 43 10 7. 31 904 7. 31 904 10.00000 50 10 7. 69 841 7. 69 842 9. 99 999 50 20 7. 32 903 7. 32 903 10.00000 40 20 7. 70 261 7. 70 261 9. 99 999 40 30 7. 33 879 7. 33 879 10.00000 30 30 7. 70 676 7. 70 677 9. 99 999 30 40 7. 34 833 7. 34 833 10.00000 20 40 7. 71 088 7. 71 088 9. 99 999 20 50 7. 35 767 7. 35 767 10.00000 10 50 7. 71 496 7. 71 496 9.99 999 10 8 7. 36 682 7. 36 682 10.00000 52 18 7. 71 900 7.71900 9. 99 999 42 10 7.37 577 7.37 577 10.00000 50 10 7. 72 300 7. 72 301 9. 99 999 50 20 7. 38 454 7.38 455 10.00000 40 20 7. 72 697 7. 72 697 9. 99 999 40 30 7.39 314 7.39 315 10.00000 30 30 7. 73 090 7. 73 090 9.99 999 30 40 7.40158 7. 40 158 10.00000 20 40 7. 73 479 7. 73 480 9. 99 999 20 50 7. 40 985 7. 40 985 10.00000 10 50 7. 73 865 7. 73 S66 9. 99 999 10 9 7.41797 7. 41 797 10.00000 51 19 7. 74 248 7. 74 248 9.99 999 41 10 7. 42 594 7. 42 594 10.00000 50 10 7. 74 627 7. 74 628 9. 99 999 50 20 7.43 376 7. 43 376 10.00000 40 20 7. 75 003 7.75 004 9. 99 999 40 30 7. 44 145 7. 44 145 10.00000 30 30 7. 75 376 7. 75 377 9.99 999 30 40 7. 44 900 7.44 900 10.00000 20 40 7. 76 745 7. 75 746 9.99 999 20 50 7. 45 643 7.45 643. 10.00000 10 50 7.76112 7. 76 113 9.99 999 10 10 7. 46 373 7.46 373 10.00000 50 20 7. 76 475 7.76476 9. 99 999 40 t tt log cos log cot log sin tt t f tt log cos log cot log sin ft f 89 0° 23 t rr log sin log tan log COS tt r t tf log sin log tan log cos ft t 20 7. 76 475 7.76476 9.99 999 40 30 7. 94 084 7.94 086 9. 99 998 30 10 7. 76 836 7.76 837 9. 99 999 50 10 7. 94 325 7.94326 9. 99 998 50 20 7. 77 193 7. 77 194 9. 99 999 40 20 7. 94 564 7.94 566 9. 99 998 40 30 7. 77 548 7. 77 549 9. 99 999 30 30 7. 94 802 7. 94 804 9. 99 998 30 40 7. 77 899 7. 77 900 9.99 999 20 40 7. 95 039 7. 95 040 9. 99 998 20 50 7. 78 248 7. 78 249 9. 99 999 10 50 7. 95 274 7. 95 276 9.99998 10 21 7. 78 594 7. 78 595 9. 99 999 39 31 7. 95 508 7.95 510 9.99998 29 10 7. 78 938 7.78 938 9. 99 999 50 10 7. 95 741 7. 95 743 9. 99 998 50 20 7. 79 278 7. 79 279 9.99 999 40 20 7. 95 973 7. 95 974 9. 99 998 40 30 7.79 616 7.79 617 9. 99 999 30 30 7. 96 203 7. 96 205 9. 99 998 30 40 7. 79 952 7. 79 952 9. 99 999 20 40 7.96432 7. 96 434 9. 99 998 20 50 7. 80 284 7. 80 285 9. 99 999 10 50 7. 96 660 7. 96 662 9.99998 10 22 7. 80 615 7.80 615 9.99 999 38 32 7. 96 887 7. 96 889 9. 99 998 28 10 7. 80 942 7. 80 943 9. 99 999 50 10 7.97113 7.97114 9. 99 998 50 20 7. 81 268 7. 81 269 9. 99 999 40 20 7.97 337 7. 97 339 9. 99 998 40 30 7. 81 591 7. 81 591 9. 99 999 30 30 7. 97 560 7. 97 562 9. 99 998 30 40 7.81911 7.81912 9. 99 999 20 40 7. 97 782 7. 97 784 9. 99 998 20 50 7. 82 229 7. 82 230 9. 99 999 10 50 7. 98 003 7. 98 005 9. 99 998 10 23 7. 82 545 7. 82 546 9. 99 999 37 33 7. 98 223 7. 98 225 9. 99 998 27 10 7. 82 859 7. 82 860 9. 99 999 50 10 7. 98 442 7. 98 444 9. 99 998 50 20 7. 83 170 7. 83 171 9. 99 999 40 20 7. 98 660 7. 98 662 9.99998 40 30 7. 83 479 7. 83 480 9. 99 999 30 30 7. 98 876 7.98 878 9. 99 998 30 40 7. 83 786 7. 83 787 9. 99 999 20 40 7. 99 092 7. 99 094 9. 99 998 20 50 7. 84 091 7. 84 092 9. 99 999 10 50 7. 99 306 7. 99 308 9. 99 998 10 24 7. 84 393 7. 84 394 9. 99 999 36 34 7. 99 520 7. 99 522 9. 99 998 26 10 7. 84 69+ 7. 84 695 9. 99 999 50 10 7. 99 732 7. 99 734 9. 99 998 50 20 7. 84 992 7.84 994 9. 99 999 40 20 7. 99 943 7. 99 946 9. 99 998 40 30 7. 85 289 7. 85 290 9. 99 999 30 30 8. 00 154 8. 00 156 9. 99 998 30 40 7. 85 583 7. 85 584 9. 99 999 20 40 8. 00 363 8. 00 365 9. 99 998 20 50 7. 85 876 7.85 877 9. 99 999 10 50 8. 00 571 8.00 574 9. 99 998 10 25 7. 86 166 7. 86 167 9. 99 999 35 35 8. 00 779 8. 00 781 9. 99 998 25 10 7. 86 455 7. 86 456 9. 99 999 50 10 8. 00 985 8. 00 987 9. 99 998 50 20 7. 86 741 7. 86 743 9. 99 999 40 20 8. 01 190 8. 01 193 9. 99 998 40 30 7. 87 026 7. 87 027 9. 99 999 30 30 8. 01 395 8. 01 397 9.99998 30 40 7. 87 309 7.87 310 9. 99 999 20 40 8. 01 598 8. 01 600 9.99998 20 50 7. 87 590 7. 87 591 9. 99 999 10 50 8. 01 801 8. 01 803 9.99998 10 26 7. 87 870 7. 87 871 9. 99 999 34 36 8. 02 002 8. 02 004 9. 99 998 24 10 7. 88 147 7. 88 148 9. 99 999 50 10 8. 02 203 8. 02 205 9. 99 998 50 20 7. 88 423 7. 88 424 9. 99 999 40 20 8. 02 402 8. 02 405 9. 99 998 40 30 7. 88 697 7. 88 698 9. 99 999 30 30 8. 02 601 8. 02 604 9. 99 998 30 40 7. 88 969 7. 88 970 9. 99 999 20 40 8. 02 799 8. 02 801 9. 99 998 20 50 7. 89 240 7. 89 241 9. 99 999 10 50 8. 02 996 8. 02 998 9. 99 998 10 27 7. 89 509 7. 89 510 9.99 999 33 37 8. 03 192 8. 03 194 9. 99 997 23 • 10 7. 89 776 7. 89 777 9. 99 999 50 10 8. 03 387 8. 03 390 9. 99 997 50 20 7. 90 041 7. 90 043 9. 99 999 40 20 8. 03 581 8. 03 584 9. 99 997 40 30 7. 90 305 7. 90 307 9. 99 999 30 30 8. 03 775 8. 03 777 9. 99 997 30 40 7. 90 568 7. 90 569 9. 99 999 20 40 8. 03 967 8. 03 970 9. 99 997 20 50 7. 90 829 7. 90 830 9. 99 999 10 50 8.04159 8. 04 162 9. 99 997 10 28 7. 91 088 7. 91 0S9 9. 99 999 32 38 8. 04 350 8.04 353 9. 99 997 22 10 7. 91 346 7.91347 9. 99 999 50 10 8. 04 540 8. 04 543 9. 99 997 50 20 7. 91 602 7. 91 603 9. 99 999 40 20 8. 04 729 8. 04 732 9. 99 997 40 30 7. 91 857 7. 91 858 9. 99 999 30 30 8. 04 918 8.04 921 9. 99 997 30 40 7.92 110 7.92 111 9. 99 998 20 40 8. 05 105 8. 05 108 9. 99 997 20 50 7. 92 362 7. 92 363 9. 99 998 10 50 8. 05 292 8. 05 295 9. 99 997 10 29 7. 92 612 7. 92 613 9. 99 998 31 39 8. 05 478 8. 05 481 9. 99 997 21 10 7. 92 861 7. 92 862 9. 99 998 50 10 8. 05 663 8. 05 666 9. 99 997 50 20 7. 93 108 7. 93 110 9. 99 998 40 20 8. 05 848 8. 05 851 9.99 997 40 30 7. 93 354 7. 93 356 9. 99 998 30 30 8. 06 031 8. 06 034 9. 99 997 30 40 7. 93 599 7. 93 601 9. 99 998 20 40 8. 06 214 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210 11 790 90 053 41 20 78 280 88 236 11 764 90 043 40 21 78 296 88 262 11 738 90 034 39 22 78 313 88 289 11 711 90 024 38 23 78 329 88 315 11685 90 014 37 24 78 346 88 341 11659 90 005 36 25 78 362 88 367 11633 89 995 35 26 78 379 88 393 11607 89 985 34 27 78 395 88 420 11580 89 976 33 28 78 412 88 446 11554 89 966 32 29 78 428 88 472 11528 89 956 31 30 78 445 88 498 11502 89 947 30 31 78 461 88 524 11476 89 937 29 32 78 478 88 550 11450 89 927 28 33 78 494 88 577 11423 89 918 27 34 78 510 88 603 11397 89 908 26 35 78 527 88 629 11371 89 898 25 36 78 543 88 655 11345 89 888 24 37 78 560 88 681 11319 89 879 23 38 78 576 88 707 11293 89 869 22 39 78 592 88 733 11267 89 859 21 40 78 609 88 759 11241 89 849 20 41 78 625 88 786 11214 89 840 19 42 78 642 88 812 11 188 89 830 18 43 78 658 88 838 11 162 89 820 17 44 78 674 88 864 11 136 89 810 16 45 78 691 88 890 11 110 89 801 15 46 78 707 88 916 11084 89 791 14 47 78 723 88 942 11058 89 781 13 48 78 739 88 968 11032 89 771 12 49 78 756 88 994 11006 89 761 11 50 78 772 89 020 10 980 89 752 10 51 78 788 89 046 10 954 89 742 9 52 78 805 89 073 10 927 89 732 8 53 78 821 89 099 10 901 89 722 7 54 78 837 89 125 10 875 89 712 6 55 78 853 89 151 10 849 89 702 5 56 78 869 89 177 10 823 89 693 4 57 78 886 89 203 10 797 89 683 3 58 78 902 89 229 10 771 89 673 2 59 78 918 89 255 10 745 89 663 1 60 78 934 89 281 10 719 89 653 n Q —10 — log tan o t log COS log cot y log sin t 38 ( f log sin 9 78 934 log tan 9 89 281 log cot —10— 10 719 log cos g f 89 653 60 1 78 950 89 307 10 693 89 643 59 2 78 967 89 333 10 667 89 633 58 3 78 983 89 359 10 641 89 624 57 4 78 999 89 385 10 615 89 614 56 5 79 015 89 411 10 589 89 604 55 6 79 031 89 437 10 563 89 594 54 7 79 047 89 463 10 537 89 584 53 8 79 063 89 489 10 511 89 574 52 9 79 079 89 515 10 485 89 564 51 10 79 095 89 541 10 459 89 554 50 11 79 111 89 567 10 433 89 544 49 12 79 128 89 593 10 407 89 534 48 13 79 144 89 619 10 381 89 524 47 14 79 160 89 645 10 355 89 514 46 15 79 176 89 671 10 329 89 504 45 16 79 192 89 697 10 303 89 495 44 17 79 208 89 723 10 277 89 485 43 18 79 224 89 749 10 251 89 475 42 19 79 240 89 775 10 225 89 465 41 20 79 256 89 801 10 199 89 455 40 21 79 272 89 827 10 173 89 445 39 22 79 288 89 853 10 147 89 435 38 23 79 304 89 879 10 121 89 425 37 24 79 319 89 905 10 095 89 415 36 25 79 335 89 931 10 069 89 405 35 26 79 351 89 957 10 043 89 395 34 27 79 367 89 983 10 017 89 385 33 28 79 383 90 009 09 991 89 375 32 29 79 399 90 035 09 965 89 364 31 30 79 415 90 061 09 939 89 354 30 31 79 431 90 086 09 914 89 344 29 32 79 447 90 112 09 888 89 334 28 33 79 463 90 138 09 862 89 324 27 34 79 478 90 164 09 836 89 314 26 35 79 494 90 190 09 810 89 304 25 36 79 510 90 216 09 784 89 294 24 37 79 526 90 242 09 758 89 284 23 38 79 542 90 268 09 732 89 274 22 39 79 558 90 294 09 706 89 264 21 40 79 573 90 320 09 680 89 254 20 41 79 589 90 346 09 654 89 244 19 42 79 605 90 371 09 629 89 233 18 43 79 621 90 397 09 603 89 223 17 44 79 636 90 423 09 577 89 213 16 45 79 652 90 449 09 551 89 203 15 46 79 668 90 475 09 525 89 193 14 47 79 684 90 501 09 499 89 183 13 48 79 699 90 527 09 473 89 173 12 49 79 715 90 553 09 447 89 162 11 50 79 731 90 578 09 422 89 152 10 51 79 746 90 604 09 396 89 142 9 52 79 762 90 630 09 370 89 132 8 53 79 778 90 656 09 344 89 122 7 54 79 793 90 682 09 318 89 112 6 55 79 809 90 708 09 292 89 101 5 56 79 825 90 734 09 266 89 091 4 57 79 840 90 759 09 241 89 081 3 58 79 856 90 785 09 215 89 071 2 59 79 872 90 811 09 189 89 060 1 60 79 887 90 837 n 09 163 1 A 89 050 Q f y log cos y log cot — xu — log tan y log sin f 52 51 39 r log sin a log tan Q log cot 10 log cos o f 79 887 90 837 09 163 y 89 050 60 1 79 903 90 863 09 137 89 040 59 2 79 918 90 889 09 111 89 030 58 3 79 934 90 914 09 086 89 020 57 4 79 950 90 940 09 060 89 009 56 5 79 965 90 966 09 034 88 999 55 6 79 981 90 992 09 008 88 989 54 7 79 996 91018 08 982 88 978 53 8 80 012 91043 08 957 88 968 52 9 80 027 91069 08 931 88 958 51 10 80 043 91095 08 905 88 948 50 11 80 058 91 121 08 879 88 937 49 12 80 074 91 147 08 853 88 927 48 13 80 089 91 172 08 828 88 917 47 14 80 105 91 198 08 802 88 906 46 15 80 120 91 224 08 776 88 896 45 16 80 136 91250 08 750 88 886 44 17 80 151 91276 08 724 88 875 43 18 80 166 91301 08 699 88 865 42 19 80 182 91 327 08 673 88 85£ 41 20 80 197 91353 08 647 88 844 40 21 80 213 91379 08 621 88 834 39 22 80 228 91404 08 596 88 824 38 23 80 244 91430 08 570 88 813 37 24 .80 259 91456 08 544 88 803 36 25 80 274 914S2 08 518 88 793 35 26 80 290 91 507 08 493 88 782 34 27 80 305 91533 08 467 88 772 33 28 80 320 91 559 08 441 88 761 32 29 80 336 915S5 08 415 88 751 31 30 80 351 91610 08 390 88 741 30 31 80 366 91636 08 364 88 730 29 32 80 382 91662 08 338 88 720 28 33 80 397 91688 08 312 88 709 27 34 80 412 91713 08 287 88 699 26 35 80 428 91 739 08 261 88 688 25 36 80 443 91 765 08 235 88 678 24 37 80 458 91 791 08 209 88 668 23 38 80 473 91816 08 184 88 657 22 39 80.489 91842 08 158 88 647 21 40 80 504 91868 08 132 88 636 20 41 80 519 91893 08 107 88 626 19 42 80 534 91919 08 081 88 615 18 43 80 550 91945 08 055 88 605 17 44 80 565 91971 08 029 88 594 16 45 80 580 91996 08 004 88 584 15 46 80 595 92 022 07 978 88 573 14 47 80 610 92 048 07 952 88 563 13 48 80 625 92 073 07 927 88 552 12 49 80 641 92 099 07 901 88 542 11 50 80 656 92 125 07 875 88 531 10 51 80 671 92 150 07 850 88 521 9 52 80 686 92 176 07 824 88 510 8 53 80 701 92 202 07 798 88 499 7 54 80 716 92 227 07 773 88 489 6 55 80 731 92 253 07 747 88 478 5 56 80 746 92 279 07 721 88 468 4 57 80 762 92 304 07 696 88 457 3 58 80 777 92 330 07 670 88 447 2 59 80 792 92 356 07 644 88 436 1 60 80 807 92 381 07 619 88 425 n o in o t y log cos y log cot log tan log sin f 40° 47 t log sin 9 log tan 9 92 381 log cot — 10— 07 619 log cos Q t 80 807 88 425 60 1 80 822 92 407 07 593 88 415 59 2 80 837 92 433 07 567 88 404 58 3 80 852 92 458 07 542 88 3^4 57 4 80 867 92 484 07 516 88 383 56 5 80 882 92 510 07 490 88 372 55 6 80 897 92 535 07 465 88 362 54 7 80 912 92 561 07 439 88 351 53 8 80 927 92 587 07 413 88 340 52 9 80 942 92 612 07 388 88 330 51 10 80 957 92 638 07 362 88 319 50 11 80 972 92 663 07 337 88 308 49 12 80 987 92 689 07 311 88 298 48 13 81002 92 715 07 285 88 287 47 14 81017 92 740 07 260 88 276 46 15 81032 92 766 07 234 88 266 45 16 81047 92 792 07 208 88 255 44 17 81061 92 817 07 183 88 244 43 18 81076 92 843 07 157 88 234 42 19 81091 92 868 07 132 88 223 41 20 81 106 92 894 07 106 88 212 40 21 81 121 92 920 07 080 88 201 39 22 81 136 92 945 07 055 88 191 38 23 81 151 92 971 07 029 88 180 37 24 81 166 92 996 07 004 88 169 36 25 81 180 93 022 06 978 88 158 35 26 81 195 93 048 06 952 88 148 34 27 81210 93 073 06 927 88 137 33 28 81225 93 099 06 901 88 126 32 29 81240 93 124 06 876 88 115 31 30 81254 93 150 06 850 88 105 30 31 81269 93 175 06 825 88 094 29 32 81284 93 201 06 799 88 083 28 33 81299 93 227 06 773 88 072 27 34 81314 93 252 06 748 88 061 26 35 81328 93 278 06 722 88 051 25 36 81343 93 303 06 697 88 040 24 37 81358 93 329 06 671 88 029 23 38 81372 93 354 06 646 88 018 22 39 81387 93 380 06 620 88 007 21 40 81402 93 406 06 594 87 996 20 41 81417 93 431 06 569 87 985 19 42 81431 93 457 06 543 87 975 18 43 81446 93 482 06 518 87 964 17 44 81461 93 508 06 492 87 953 16 45 81475 93 533 06 467 87 942 15 46 81490 93 559 06 441 87 931 14 47 81505 93 584 06 416 87 920 13 48 81 519 93 610 06 390 87 909 12 49 81534 93 636 06 364 87 898 11 50 81549 93 661 06 339 87 887 10 51 81563 93 687 06 313 87 877 9 52 81 578 93 712 06 288 87 866 8 53 81592 93 738 06 262 87 855 7 54 81607 93 763 06 237 87 844 6 55 81 622 93 789 06 211 87 833 5 56 81636 93 814 06 186 87 822 4 57 81651 93 840 06 160 87 811 3 58 81665 93 865 06 135 87 800 2 59 81680 93 891 06 109 87 789 1 60 81694 — 9 — log 008 93 916. — 9— log cot 06 084 —10— log tan 87 778 — 9 log sin f ^ 50 49 48 41 42 t log sin 9 81 694 log tan 9 93 916 log cot —10— 06 084 log cos 9 87 778 r 60 t log sin 9 82 551 log tan 9 95 444 log cot log cos Q t 04 556 87 107 60 1 81 709 93 942 06 058 87 767 59 1 82 565 95 469 04 531 87 096 59 2 81 723 93 967 06 033 87 756 58 2 82 579 95 495 04 505 87 085 58 3 81 738 93 993 06 007 87 745 57 3 82 593 95 520 04 480 87 073 57 4 81 752 94 018 05 982 87 734 56 4 82 607 95 545 04 455 87 062 56 5 81767 94 044 05 956 87 723 55 5 82 621 95 571 04 429 87 050 55 6 81 781 94 069 05 931 87 712 54 6 82 635 95 596 04 404 87 039 54 7 SI 796 94 095 05 905 87 701 53 7 82 649 95 622 04 378 87 028 53 8 81810 94 120 05 880 87 690 52 8 82 663 95 647 04 353 87 016 52 9 81 825 94 146 05 854 87 679 51 9 82 677 95 672 04 328 87 005 51 10 81839 94 171 05 829 87 668 50 10 82 691 95 698 04 302 86 993 50 11 81854 94 197 05 803 87 657 49 11 82 705 95 723 04 277 86 982 49 12 81868 94 222 05 778 87 646 48 12 82 719 95 748 04 252 86 970 48 13 81882 94 248 05 752 87 635 47 13 82 733 95 774 04 226 86 959 47 14 81897 94 273 05 727 87 624 46 14 82 747 95 799 04 201 86 947 46 15 81911 94 299 05 701 87 613 45 15 82 761 95 825 04 175 86 936 45 16 81926 94 324 05 676 87 601 44 16 82 775 95 850 04 150 86 924 44 17 81940 94 350 05 650 87 590 43 17 82 788 95 875 04 125 86 913 43 18 81 955 94 375 05 625 87 579 42 18 82 802 95 901 04 099 86 902 42 19 81969 94 401 05 599 87 568 41 19 82 816 95 926 04 074 86 890 41 20 81983 94 426 05 574 87 557 40 20 82 830 95 952 04 048 86 879 40 21 81998 94 452 05 548 87 546 39 21 82 844 95 977 04 023 86 867 39 22 82 012 94 477 05 523 87 535 38 22 82 858 96 002 03 998 86 855 38 23 82 026 94 503 05 497 87 524 37 23 82 872 96 028 03 972 86 844 37 24 82 041 94 528 05 472 87 513 36 24 82 885 96 053 03 947 86 832 36 25 82 055 94 554 05 446 87 501 35 25 82 899 96 078 03 922 86 821 35 26 82 069 94 579 05 421 87 490 34 26 82 913 96 104 03 896 86 809 34 27 82 084 94 604 05 396 87 479 33 27 82 927 96 129 03 871 86 798 33 28 82 098 94 630 05 370 87 468 32 28 82 941 96 155 03 845 86 786 32 29 82 112 94 655 05 345 87 457 31 29 82 955 96 180 03 820 86 775 31 30 82 126 94 681 05 319 87 446 30 30 82 968 96 205 03 795 86 763 30 31 82 141 94 706 05 294 87 434 29 31 82 982 96 231 03 769 86 752 29 32 82 155 94 732 05 268 87 423 28 32 82 996 96 256 03 744 86 740 28 33 82 169 94 757 05 243 87 412 27 33 83 010 96 2S1 03 719 86 728 27 34 82 184 94 783 05 217 87 401 26 34 83 023 96 307 03 693 86 717 26 35 82 198 94 808 05 192 87 390 25 35 83 037 96 332 03 668 86 705 25 36 82 212 94 834 05 166 87 378 24 36 83 051 96 357 03 643 86 694 24 37 82 226 94 859 05 141 87 367 23 37 83 065 96 383 03 617 86 682 23 38 82 240 94 884 05 116 87 356 22 38 83 078 96 408 03 592 86 670 22 39 82 255 94 910 05 090 87 345 21 39 83 092 96 433 03 567 86 659 21 40 82 269 94 935 05 065 87 334 20 40 83 106 96 459 03 541 86 647 20 41 82 283 94 961 05 039 87 322 19 41 83 120 96 484 03 516 86 635 19 42 82 297 94 986 05 014 87 311 18 42 83 133 96 510 03 490 86 624 18 43 82 311 95 012 04 988 87 300 17 43 83 147 96 535 03 465 86 612 17 44 82 326 95 037 04 963 87 288 16 44 83 161 96 560 03 440 86 600 16 45 82 340 95 062 04 938 87 277 15 45 83 174 96 586 03 414 86 589 15 46 82 354 95 088 04 912 87 266 14 46 83 188 96 611 03 389 86 577 14 47 82 368 95 113 04 887 87 255 13 47 83 202 96 636 03 364 86 565 13 48 82 382 95 139 04 861 87 243 12 48 83 215 96 662 03 338 86 554 12 49 82 396 95 164 04 836 87 232 11 49 83 229 96 687 03 313 86 542 11 50 82 410 95 190 04 810 87 221 10 50 83 242 96 712 03 2S8 86 530 10 51 82 424 95 215 04 785 87 209 9 51 83 256 96 738 03 262 86 5 IS 9 52 82 439 95 240 04 760 87 198 8 52 83 270 96 763 03 237 86 507 8 53 82 453 95 266 04 734 87 187 7 53 83 283 96 788 03 212 86 495 7 54 82 467 95 291 04 709 87 175 6 54 83 297 96 814 03 186 86 483 6 55 82 481 95 317 04 683 87 164 5 55 83 310 96 839 03 161 86 472 5 56 82 495 95 342 04 658 87 153 4 56 83 324 96 864 03 136 86 460 4 57 82 509 95 368 04 632 87 141 3 57 83 338 96 890 03 110 86 448 3 58 82 523 95 393 04 607 87 130 2 58 83 351 96 915 03 0S5 86 436 2 59 82 537 95 418 04 5S2 87 119 1 59 83 365 96 940 03 060 S6 425 1 60 82 551 n 95 444 Q 04 556 in 87 107 Q 60 83 378 Q 96 966 g 03 034 in 86 413 f y log cos y — log cot log tan a log sin f f y log cos log cot log tan j log sin r 48 c 47 < 43 r log sin n log tan 9 96 966 log cot 2.0 log cos f 83 378 03 034 86 413 60 1 83 392 96 991 03 009 86 401 59 2 83 405 97 016 02 984 86 389 58 3 83 419 97 042 02 958 86 377 57 4 83 432 97 067 02 933 86 366 56 5 S3 446 97 092 02 908 86 354 55 6 83 459 97 118 02 882 86 342 54 7 83 473 97 143 02 857 86 330 53 8 83 486 97 168 02 832 86 318 52 9 83 500 97 193 02 807 86 306 51 10 83 513 97 219 02 781 86 295 50 11 83 527 97 244 02 756 86 283 49 12 83 540 97 269 02 731 86 271 48 13 83 554 97 295 02 705 86 259 47 14 83 567 97 320 02 680 86 247 46 15 83 581 97 345 02 655 86 235 45 16 83 594 97 371 02 629 86 223 44 17 83 608 97 396 02 604 86 211 43 18 83 621 97 421 02 579 86 200 42 19 83 634 97 447 02 553 86 188 41 20 83 648 97 472 02 528 86 176 40 21 83 661 97 497 02 503 86 164 39 22 83 674 97 523 02 477 86 152 38 23 83 688 97 548 02 452 86 140 37 24 83 701 97 573 02 427 86 128 36 25 83 715 97 598 02 402 86 116 35 26 83 728 97 624 02 376 86 104 34 27 83 741 97 649 02 351 86 092 33 28 83 755 97 674 02 326 86 080 32 29 83 768 97 700 02 300 86 068 31 30 83 781 97 725 02 275 86 056 30 31 83 795 97 750 02 250 86 044 29 32 83 808 97 776 02 224 86 032 28 33 83 821 97 801 02 199 86 020 27 34 83 834 97 826 02 174 86 008 26 35 83 848 97 851 02 149 85 996 25 36 83 861 97 877 02 123 85 984 24 37 83 874 97 902 02 098 85 972 23 38 83 887 97 927 02 073 85 960 22 39 83 901 97 953 02 047 85 948 21 40 83 914 97 978 02 022 85 936 20 41 83 927 98 003 01997 85 924 19 42 83 940 98 029 01971 85 912 18 43 83 954 98 054 01946 85 900 17 44 83 967 98 079 01921 85 888 16 45 83 980 98104 01896 85 876 15 46 83 993 98 130 01870 85 864 14 47 84 006 98 155 01845 85 851 13 48 84 020 98 180 01820 85 839 12 49 84 033 98 206 01 794 85 827 11 50 84 046 98 231 01 769 85 815 10 51 84 059 98 256 01 744 85 803 9 52 84 072 98 281 01719 85 791 8 53 84 085 98 307 01693 85 779 7 54 84 098 98 332 01668 85 766 6 55 84 112 98 357 01643 85 754 5 56 84 125 98 383 01617 85 742 4 57 84 138 98 408 01 592 85 730 3 58 84 151 98 433 01 567 85 718 2 59 84 164 98 458 01 542 85 706 1 60 84 177 y 98 484 Q 01 516 in 85 693 n r log cos y log cot — iu — log tan y log sin ; 44° 49 t log sin g log tan log cot 10 log cos n f 84 177 y 98 484 01 516 85 693 60 1 84 190 98 509 01491 85 681 59 2 84 203 98 534 01466 85 669 58 3 84 216 98 560 01440 85 657 57 4 84 229 98 585 01415 85 645 56 5 84 242 98 610 01 390 85 632 55 6 84 255 98 635 01365 85 620 54 7 84 269 98 661 01339 85 608 53 8 84 282 98 686 01314 85 596 52 9 84 295 98 711 01289 85 583 51 10 84 308 98 737 01263 85 571 50 11 84 321 98 762 01238 85 559 49 12 84 334 98 787 01 213 85 547 48 13 84 347 98 812 01 188 85 534 47 14 84 360 98 838 01 162 85 522 46 15 84 373 98 863 01 137 85 510 45 16 84 385 98 888 01 112 85 497 44 17 84 398 98 913 01087 85 485 43 18 84 411 98 939 01061 85 473 42 19 84 424 98 964 01036 85 460 41 20 84 437 98 989 01011 85 448 40 21 84 450 99 015 00 985 85 436 39 22 84 463 99 040 00 960 85 423 38 23 84 476 99 065 00 935 85 411 37 24 84 489 99 090 00 910 85 399 36 25 84 502 99 116 00 884 85 386 35 26 84 515 99 141 00 859 85 374 34 27 84 528 99 166 00 834 85 361 33 28 84 540 99 191 00 809 85 349 32 29 84 553 99 217 00 783 85 337 31 30 84 566 99 242 00 758 85 324 30 31 84 579 99 267 00 733 85 312 29 32 84 592 99 293 00 707 85 299 28 33 84 605 99 318 00 682 85 287 27 34 84 618 99 343 00 657 85 274 26 35 84 630 99 368 00 632 85 262 25 36 84 643 99 394 00 606 85 250 24 37 84 656 99 419 00 581 85 237 23 38 84 669 99 444 00 556 85 225 22 39 84 682 99 469 00 531 85 212 21 40 84 694 99 495 00 505 85 200 20 41 84 707 99 520 00 480 85 187 19 42 84 720 99 545 00 455 85 175 18 43 84 733 99 570 00 430 85 162 17 44 84 745 99 596 00 404 85 150 16 45 84 758 99 621 00 379 85 137 15 46 84 771 99 646 00 354 85 125 14 47 84 784 99 672 00 328 85 112 13 48 84 796 99 697 00 303 85 100 12 49 84 809 99 722 00 278 85 087 11 50 84 822 99 747 00 253 85 074 10 51 84 835 99 773 00 227 85 062 9 52 84 847 99 798 00 202 85 049 8 53 84 860 99 823 00 177 85 037 7 54 84 873 99 848 00 152 85 024 6 55 84 885 99 874 00 126 85 012 5 56 84 898 99 899 00 101 84 999 4 57 84 911 99 924 00 076 84 986 3 58 84 923 99 949 00 051 84 974 2 59 60 84 936 84 949 99 975 00 025 00 000 84 961 84 949 1 00 000 Q — 10 — log cot —10— log tan o f y log cos log sin f 46 45 50 TABLE IV. For Determining with Greater Accuracy than can be done by means of Table III. : 1. log sin, log tan, and log cot, when the angle is between 0° and 2° ; 2. log cos, log tan, and log cot, when the angle is between 88° and 90° ; 3. The value of the angle when the logarithm of the function does not lie between the limits 8. 54 684 and 11. 45 316. FORMULAS FOR THE USE OF THE NUMBERS S AND T. I. When the angle a is between 0° and 2° : log sin a = log a" + S. log a" = log sin a — &\ log tan a = log a" + T. = log tan a— T, log cot a = colog tan a. = colog cot a — T. II. When the angle a is between 88° and 90° : log COS a = log (90° -a)" + S. log (90°-a)" = log cos a-S, log cot a = log (90°— a)" + T. = log cot a— T, log tan a = colog cot a. = colog tan a—T, and a = 90° -(90° -a). *oX«oo Values of S a^d T, a" S log sin a a" T log tan a a T log tan a ■ 5146 8. 39 713 4. 68 557 4.68 557 4. 68 567 2 409 4.68 556 8.06 740 200 4. 68 558 6. 98 660 5 424 4.68 568 8.41999 3 417 4.68 555 8. 21 920 1726 4. 68 559 7. 92 263 5 689 4. 68 569 8. 44 072 3 823 4. 68 555 8. 26 795 2 432 4. 68 560 8. 07 156 5 941 4. 68 570 8. 45 955 4190 4. 68 554 8. 30 776 2 976 4. 68 561 8. 15 924 6184 4. 68 571 8. 47 697 4 840 4.68 553 8. 37 038 3 434 4. 68 562 8. 22 142 6417 4.68 572 8.49 305 5 414 4. 68 552 8. 41 904 3 838 4. 68 563 8.26 973 6642 4. 68 573 8. 50 802 5 932 4. 68 551 8. 45 872 4 204 4. 68 564 8. 30 930 6 859 4. 68 574 8. 52 200 6 408 4. 68 550 8. 49 223 4 540 4. 68 565 8. 34 270 7070 4. 68 575 8. 53 516 6 633 4. 68 550 8. 50 721 4 699 4. 68 565 8. 35 766 7173 4. 68 575 8. 54 145 6 851 4. 68 549 8. 52 125 4 853 4. 68 566 8. 37 167 7 274 8. 54 753 7 267 8.54 684 5 146 8. 39 713 a" S log sin a a". T log tan a a T log tan a TABLE V.— Circumferences and Areas of Circles, si If N= the radius of the circle, the circumference = 2 irN. If 2f= the radius of the circle, the area = irN2. If JV= the circumference of the circle, the radius = —■#". *7T If X= the circumfereuce of the circle, the area = — iV 2 . 47T N lirN TT.V 2 Lit 2tt 4* I 50 2irtf TriV™ J-2VT 2tt 4tt 0.00 0.0 0.000 0.00 314. 16 7 854 7.96 198. 94 1 6.28 3.1 0.159 0.08 51 320. 44 8171 8.12 206. 98 2 12.57 12.6 0.318 0.32 52 326. 73 8 495 8.28 215. 18 3 18.85 28.3 0.477 0.72 53 333. 01 8 825 8.44 223. 53 4 25.13 50.3 0.637 1.27 54 339. 29 9161 8.59 232. 05 5 31.42 78.5 0.796 1.99 55 345. 58 9 503 8.75 240. 72 6 37.70 113.1 0.955 2.86 56 351. 86 9 852 8.91 249. 55 7 43.98 153.9 1.114 3.90 57 358. 14 10 207 9.07 258. 55 8 50.27 201.1 1.273 5.09 58 364.42 10 568 9.23 267. 70 9 56.55 254.5 1.432 6.45 59 370. 71 10 936 9.39 277. 01 10 62.83 314.2 1. 592 7.96 60 376. 99 11310 9.55 286. 48 11 69.12 380.1 1.751 9.63 61 383. 27 11690 9.71 296. 11 12 75.40 452.4 1.910 11.46 62 389. 56 12 076 9.87 305. 90 13 81.68 530.9 2.069 13.45 63 395. 84 12 469 10.03 315. 84 14 87.96 615.8 2.228 15.60 64 402. 12 12 868 10.19 325.95 15 94.25 706.9 2.387 17.90 65 408. 41 13 273 10.35 336. 21 16 100. 53 804.2 2.546 20.37 66 414. 69 13 685 10.50 346. 64 17 106.81 907.9 2.706 23.00 67 420. 97 14103 10.66 357. 22 18 113. 10 1017.9 2.865 25.78 68 427. 26 14 527 10.82 367. 97 19 119. 38 1 134. 1 3.024 28.73 69 433. 54 14 957 10.98 378. 87 20 125.66 1 256. 6 3.183 31.83 70 439. 82 15 394 11.14 389. 93 21 131.95 1 385. 4 3.342 35.09 71 446. 11 15 837 11.30 401. 15 22 138. 23 1 520. 5 3.501 38.52 72 452. 39 16 286 11.46 412. 53 23 144.51 1661.9 3.661 42.10 73 458. 67 16 742 11.62 424. 07 24 150. 80 1 809. 6 3.820 45.84 74 464.96 17 203 11.78 435. 77 25 157. 08 1 963. 5 3.979 49.74 75 471. 24 17 671 11.94 447. 62 26 163. 36 2 123. 7 4.138 53.79 76 477. 52 18 H6 12.10 459.64 27 169. 65 2 290. 2 4.297 58.01 77 483. 81 18 627 12.25 471. 81 28 175. 93 2 463.0 4.456 62.39 78 490.09 19113 12.41 484. 15 29 182. 21 2 642.1 4.615 66.92 79 496. 37 19 607 12.57 496.64 30 188. 50 2 827. 4 4.775 71.62 80 502. 65 20106 12.73 509. 30 31 194. 78 3 019. 1 4.934 76.47 81 508. 94 20 612 12.89 522. 11 32 201.06 3 217.0 5.093 81.49 82 515.22 21124 13.05 535. 08 33 207. 35 3 421.2 5.252 86.66 83 521. 50 21642 13.21 548. 21 34 213. 63 3 631. 7 5.411 91.99 84 527. 79 22167 13.37 561. 50 35 219. 91 3 848. 5 5.570 97.48 85 534. 07 22 698 13.53 574. 95 36 226. 19 4 071.5 5.730 103. 13 86 540. 35 23 235 13.69 588. 55 37 232. 48 4 300.8 5.889 108. 94 87 546. 64 23 779 13.85 602.32 38 238. 76 4 536. 5 6.048 114.91 88 552. 92 24 328 14.01 616. 25 39 245.04 4 778.4 6.207 121.04 89 559. 20 24 885 14.16 630. 33 40 251.33 5 026. 5 6.366 127.32 90 565.49 25 447 14.32 644.58 41 257. 61 5 281.0 6.525 133. 77 91 571. 77 26 016 14.48 658. 98 42 263. 89 5 541.8 6.685 140. 37 92 578. 05 26 590 14.64 673. 54 43 270. 18 5 808. 8 6.844 147. 14 93 584. 34 27172 14.80 688. 27 44 276. 46 6 082. 1 7.003 154.06 94 590.62 27 759 14.96 703. 15 45 282. 74 6 361.7 7.162 161. 14 95 596. 90 28 353 15.12 718. 19 46 289. 03 6 647.6 7.321 168. 39 96 603. 19 28 953 15.28 733. 39 47 295. 31 6 939. 8 7.480 175. 79 97 609.47 29 559 15.44 748. 74 48 301. 59 7 238. 2 7.639 183. 35 98 615. 75 30 172 15.60 764.26 49 307. 88 7 543. 7.799 191. 07 99 622.04 30 791 15.76 779. 94 50 314. 16 7 854. 7.958 198.94 100 628. 32 31416 15.92 795. 77 N 2 7T.V TT.V 2 2tt 4ir lirN ttN* 2tt 4tt 52 TABLE VI. -NATURAL FUNCTIONS. / o° 1° 2° sin cos 3 ° 4 o r sin cos sin cos sin cos sin COS o 0000 1000 0175 9998 0349 9994 0523 9986 0698 9976 60 1 0003 1000 0177 9998 0352 9994 0526 9986 0700 9975 59 2 0006 1000 0180 9998 0355 9994 0529 9986 0703 9975 58 3 0009 1000 0183 9998 0358 9994 0532 9986 0706 9975 57 4 0012 1000 0186 9998 0361 9993 0535 9986 0709 9975 56 5 0015 1000 0189 9998 0364 9993 0538 9986 0712 9975 55 6 0017 1000 0192 9998 0366 9993 0541 9985 0715 9974 54 7 0020 1000 0195 9998 0369 9993 0544 9985 0718 9974 53 8 0023 1000 0198 9998 0372 9993 0547 9985 0721 9974 52 9 0026 1000 0201 9998 0375 9993 0550 9985 0724 9974 51 io 0029 1000 0204 9998 0378 9993 0552 9985 0727 9974 SO 11 0032 1000 0207 9998 0381 9993 0555 9985 0729 9973 49 12 0035 1000 0209 9998 0384 9993 0558 9984 0732 9973 48 13 0038 1000 0212 9998 0387 9993 0561 9984 0735 9973 47 14 0041 1000 0215 9998 0390 9992 0564 9984 0738 9973 46 15 0044 1000 0218 9998 0393 9992 0567 9984 0741 9973 45 16 0047 1000 0221 9998 0396 9992 0570 9984 0744 9972 44 17 0049 1000 0224 9997 0398 9992 0573 9984 0747 9972 43 18 0052 1000 0227 9997 0401 9992 0576 9983 0750 9972 42 19 0055 1000 0230 9997 0404 9992 0579 9983 0753 9972 41 20 0058 1000 0233 9997 0407 9992 0581 9983 0756 9971 40 21 0061 1000 0236 9997 0410 9992 0584 9983 0758 9971 39 22 0064 1000 0239 9997 0413 9991 0587 9983 0761 9971 38 23 0067 1000 0241 9997 0416 9991 0590 9983 0764 9971 37 24 0070 1000 0244 9997 0419 9991 0593 9982 0767 9971 36 25 0073 1000 0247 9997 0422 9991 0596 9982 0770 9970 35 26 0076 1000 0250 9997 0425 9991 0599 9982 0773 9970 34 27 0079 1000 0253 9997 0427 9991 0602 9982 0776 9970 33 28 0081 1000 0256 9997 0430 9991 0605 9982 0779 9970 32 29 0084 1000 0259 9997 0433 9991 0608 9982 0782 9969 31 30 0087 1000 0262 9997 0436 9990 0610 9981 0785 9969 30 31 0090 1000 0265 9996 0439 9990 0613 9981 0787 9969 29 32 0093 1000 0268 9996 0442 9990 0616 9981 0790 9969 28 33 0096 1000 0270 9996 0445 9990 0619 9981 0793 9968 27 34 0099 1000 0273 9996 0448 9990 0622 9981 0796 9968 26 35 0102 9999 0276 9996 0451 9990 0625 9980 0799 9968 25 36 0105 9999 0279 9996 0454 9990 0628 9980 0802 9968 24 37 0108 9999 0282 9996 0457 9990 0631 9980 0805 9968 23 38 0111 9999 0285 9996 0459 9989 0634 9980 0808 9967 22 39 0113 9999 0288 9996 0462 9989 0637 9980 0811 9967 21 40 0116 9999 0291 9996 0465 9989 0640 9980 0814 9967 20 41 0119 9999 0294 9996 0468 9989 0642 9979 0816 9967 19 42 0122 9999 0297 9996 0471 9989 0645 9979 0819 9966 18 43 0125 9999 0300 9996 0474 9989 0648 9979 0822 9966 17 44 0128 9999 0302 9995 0477 9989 0651 9979 0825 9966 16 45 0131 9999 0305 9995 0480 9988 0654 9979 0828 9966 15 46 0134 9999 0308 9995 0483 9988 0657 9978 0831 9965 14 47 0137 9999 0311 9995 0486 9988 0660 9978 0834 9965 13 48 0140 9999 0314 9995 0488 9988 0663 9978 0837 9965 12 49 0143 9999 0317 9995 0491 9988 0666 9978 0840 9965 11 SO 0145 9999 0320 9995 0494 9988 0669 9978 0843 9964 IO 51 0148 9999 0323 9995 0497 9988 0671 9977 0845 9964 9 52 0151 9999 0326 9995 QSfla 9987 0674 9977 0848 9964 8 53 0154 9999 0329 9995 0503 9987 0677 9977 0851 9964 7 54 0157 9999 0332 9995 0506 9987 0680-9977 0854 9963 6 55 0160 9999 0334 9994 0509 9987 0683 9977 0857 9963 5 56 0163 9999 0337- -9994 0512 9987 0686 9976 0860 9963 4 57 0166 9999 0340 9994 0515 9987 0689 9976 0863 9963 3 58 0169 9999 0343 9994 0518 9987 0692 9976 0866 9962 2 59 0172 9999 0346 9994 0520 9986 0695 9976 0869 9962 1 60 0175 9999 0349 9994 0523 9986 0698 9976 0872 9962 o cos sin 89° cos sin 88° cos sin 87° cos sin 86° cos sin r 85° '1 NATURAL SINES AND COSINES. 53 r 5° 6° 7° 8° 9° r sin cog sin cos sin cos sin cos sin cos O 0872 9962 1045 9945 1219 9925 1392 9903 1564 9877 60 1 0874 9962 1048 9945 1222 9925 1395 9902 1567 9876 59 2 0877 9461 1051 9945 1224 9925 1397 9902 1570 9876 58 3 0880 9961 1054 9944 1227 9924 1400 9901 1573 9876 57 4 0883 9961 1057 9944 1230 9924 1403 9901 1576 9875 56 5 0886 9961 1060 9944 1233 9924 1406 9901 1579 9875 55 6 0889 9960 1063 9943 1236 9923 1409 9900 1582 9874 54 7 0892 9960 1066 9943 1239 9923 1412 9900 1584 9874 53 8 0895 9960 1068 9943 1241 9923 1415 9899 1587 9873 52 9 0898 9960 1071 9942 1245 9922 1418 9899 1590 9873 51 io 0901 9959 1074 9942 1248 9922 1421 9899 1593 9872 50 11 0903 9959 1077 9942 1250 9922 1423 9898 1596 9872 49 12 0906 9959 1080 9942 1253 9921 1426 9898 1599 9871 48 13 0909 9959 1083 9941 1256 9921 1429 9897 1602 9871 47 14 0912 9958 1086 9941 1259 9920 1432 9897 1605 9870 46 15 0915 9958 1089 9941 1262 9920 1435 9897 1607 9870 45 ]6 0918 9958 1092 9940 1265 9920 1438 9896 1610 9869 44 17 0921 9958 1094 9940 1268 9919 1441 9896 1613 9869 43 18 0924 9957 1097 9940 1271 9919 1444 9895 1616 9869 42 19 0927 9957 1100 9939 1274 9919 1446 9895 1619 9868 41 20 0929 9957 1103 9939 1276 9918 1449 9894 1622 9868 40 21 0932 9956 1106 9939 1279 9918 1452 9894 1625 9867 39 22 0935 9956 1109 9938 1282 9917 1455 9894 1628 9867 38 23 0938 9956 1112 9938 1285 9917 1458 9893 1630 9866 37 24 0941 9956 1115 9938 1288 9917 1461 9893 1633 9866 36 25 0944 9955 1118 9937 1291 9916 1464 9892 1636 9865 35 26 0947 9955 1120 9937 1294 9916 1467 9892 1639 9865 34 27 0950 9955 1123 9937 1297 9916 1469 9891 1642 9864 33 28 0953 9955 1126 9936 1299 9915 1472 9891 1645 9864 32 29 0956 9954 1129 9936 1302 9915 1475 9891 1648 9863 31 30 0958 9954 1132 9936 1305 9914 1478 9890 1650 9863 30 31 0961 9954 1135 9935 1308 9914 1481 9890 1653 9862 29 32 9964 9953 1138 9935 1311 9914 1484 9889 1656 9862 28 33 0967 9953 1141 9935 1314 9913 1487 9889 1659 9861 27 34 0970 9953 1144 9934 1317 9913 1490 9888 1662 9861 26 35 0973 9553 1146 9934 1320 9913 1492 9888 1665 9860 25 36 0976 9952 1149 9934 1323 9912 1495 9888 1668 9860 24 37 0979 9952 1152 9933 1325 9912 1498 9887 1671 9859 23 38 0982 9952 1155 9933 1328 9911 1501 9887 1673 9859 22 39 0985 9951 1158 9933 1331 9911 1504 9886 1676 9859 21 40 0987 9951 1161 9932 1334 9911 1507 9886 1679 9858 20 41 0990 9951 1164 9932 1337 9910 1510 9885 1682 9858 19 42 0993 9951 1167 9932 1340 9910 1513 9885 1685 9857 18 43 0996 9950 1170 9931 1343 9909 1515 9884 1688 9857 17 44 0999 9950 1172 9931 1346 9909 1518 9884 1691 9856 16 45 1002 9950 1175 9931 1349 9909 1521 9884 1693 9856 15 46 1005 9949 1178 9930 1351 9908 1524 9883 1696 9855 14 47 1008 9949 1181 9930 1354 9908 1527 9883 1699 9855 13 48 1011 9949 1184 9930 1357 9907 1530 9882 1702 9854 12 49 1013 9949 1187 9929 1360 9907 1533 9882 1705 9854 11 50 1016 9948 1190 9929 1363 9907 1536 9881 1708 9853 IO 51 1019 9948 1193 9929 1366 9906 1538 9881 1711 9853 9 52 1022 9948 1196 9928 1369 9906 1541 9880 1714 9852 8 53 1025 9947 1198 9928 1372 9905 1544 9880 1716 9852 7 54 1028 9947 1201 9928 1374 9905 1547 9880 1719 9851 6 55 1031 9947 1204 9927 1377 9905 1550 9879 1722 9851 5 56 1034 9946 1207 9927 1380 9904 1553 9879 1725 9850 4 57 1037 9946 1210 9927 1383 9904 1556 9878 1728 9850 3 58 1039 9946 1213 9926 1386 9903 1559 9878 1731 9849 2 59 1042 9946 1216 9926 1389 9903 1561 9877 1734 9849 1 60 1045 9945 1219 9925 1392 9903 1564 9877 1736 9848 O cos sin 84° cos sin 83° cor sin 82° cos Bin 81° cos sin 80° f t 54 NATURAL SINES AND COSINES. t io° 11° 12° 13° 14° t sin cos sin cos sin cos sin cos sin cos o 1736 9848 1908 9816 2079 9781 2250 9744 2419 9703 60 1 1739 9848 1911 9816 2082 9781 2252 9743 2422 9702 59 2 1742 9847 1914 9815 2085 9780 2255 9742 2425 9702 58 3 1745 9847 1917 9815 2088 9780 2258 9742 2428 9701 57 4 1748 9846 1920 9814 2090 9779 2261 9741 2431 9700 56 5 1751 9846 1922 9813 2093 9778 2264 9740 2433 9699 55 6 1754 9845 1925 9813 2096 9778 2267 9740 2436 9699 54 7 1757 9845 1928 9812 2099 9777 2269 9739 2439 9698 53 8 1759 9844 1931 9812 2102 9777 2272 9738 2442 9697 52 9 1762 9843 1934 9811 2105 9776 2275 9738 2445 9697 51 10 1765 9843 1937 9811 2108 9775 2278 9737 2447 9696 50 11 1768 9842 1939 9810 2110 9775 2281 9736 2450 9695 49 12 1771 9842 1942 9810 2113 9774 2284 9736 2453 9694 48 13 1774 9841 1945 9809 2116 9774 2286 9735 2456 9694 47 14 1777 9841 1948 9808 2119 9773 2289 9734 2459 9693 46 15 1779 9840 1951 9808 2122 9772 2292 9734 2462 9692 45 16 1782 9840 1954 9807 2125 9772 2295 9733 2464 9692 44 17 1785 9839 1957 9807 2127 9771 2298 9732 2467 9691 43 18 1788 9839 1959 9806 2130 9770 2300 9732 2470 9690 42 19 1791 9838 1962 9806 2133 9770 2303 9731 2473 9689 41 20 1794 9838 1965 9805 2136 9769 2306 9730 2476 9689 40 21 1797 9837 1968 9804 2139 9769 2309 9730 2478 9688 39 22 1799 9837 1971 9804 2142 9768 2312 9729 2481 9687 38 23 1802 9836 1974 9803 2145 9767 2315 9728 2484 9687 37 24 1805 9836 1977 9803 2147 9767 2317 9728 2487 9686 36 25 1808 9835 1979 9802 2150 9766 2320 9727 2490 9685 35 26 1811 9835 1982 9802 2153 9765 2323 9726 2493 9684 34 27 1814 9834 1985 9801 2156 9765 2326 9726 2495 9684 33 28 1817 9834 1988 9800 2159 9764 2329 9725 2498 9683 32 29 1819 9833 1991 9800 2162 9764 2332 9724 2501 9682 31 30 1822 9833 1994 9799 2164 9763 2334 9724 2504 9681 30 31 1825 9832 1997 9799 2167 9762 2337 9723 2507 9681 29 32 1828 9831 1999 9798 2170 9762 2340 9722 2509 9680 28 33 1831 9831 2002 9798 2173 9761 3343 9722 2512 9679 27 34 1834 9830 2005 9797 2176 9760 2346 9721 2515 9679 26 35 1837 9830 2008 9796 2179 9760 2349 9720 2518 9678 25 36 1S40 9829 2011 9796 2181 9759 2351 9720 2521 9677 24 37 1842 9829 2014 9795 2184 9759 2354 9719 2524 9676 23 38 1845 9828 2016 9795 2187 9758 2357 9718 2526 9676 22 39 1848 9828 2019 9794 2190 9757 2360 9718 2529 9675 21 40 1851 9827 2022 9793 2193 9757 2363 9717 2532 9674 20 41 1854 9827 2025 9793 2196 9756 2366 9716 2535 9673 19 42 1857 9826 2028 9792 2198 9755 2368 9715 2538 9673 18 43 1860 9826 2031 9792 2201 9755 2371 9715 2540 9672 17 44 1862 9825 2034 9791 2204 9754 2374 9714 2543 9671 16 45 1865 9825 2036 9790 2207 9753 2377 9713 2546 9670 15 46 1868 9824 2039 9790 2210 9753 2380 9713 2549 9670 14 47 1871 9823 2042 9789 2213 9752 2383 9712 2552 9669 13 48 1474 9823 2045 9789 2215 9751 2385 9711 2554 9668 12 49 1877 9822 2048 9788 2218 9751 2388 9711 2557 9667 11 50 1880 9822 2051 9787 2221 9750 2391 9710 2560 9667 10 51 1882 9821 2054 9787 2224 9750 2394 9709 2563 9666 9 52 1885 9821 2056 9786 2227 9749 2397 9709 2566 9665 8 53 1888 9820 2059 9786 2230 9748 2399 9708 2569 9665 7 54 1891 9820 2062 9785 2233 9748 2402 9707 2571 9664 6 55 1894 9819 2065 9784 2235 9747 2405 9706 2574 9663 , 5 56 1897 9818 2068 9784 2238 9746 240S 9706 2577 9662 4 57 1900 9818 2071 9783 2241 9746 2411 9705 2580 9662 3 58 1902 9817 2073 9783 2244 9745 2414 9704 2583 9661 2 59 1905 9817 2076 9782 2247 9744 2416 9704 2585 9660 1 60 1908 9816 2079 9781 2250 9744 2419 9703 2588 9659 cos sin 79° cos sin 78° cos sin 77° cos sin cos sin 75° ' 76° r NATURAL SINES AND COSINES. 55 f 15° 16° 17° 18° 19° r. sin cos sin cos sin cos sin cos sin cos o 2588 9659 2756 9613 2924 9563 3090 9511 3256 9455 60 1 2591 9659 2759 9612 2926 9562 3093 9510 3258 9454 59 2 2594 9658 2762 9611 2929 9561 3096 9509 3261 9453 58 3 2597 9657 2765 9610 2932 9560 3098 9508 3264 9452 57 4 2599 9656 2768 9609 2935 9560^ 3101 9507 3267 9451 56 5 2602 9655 2770 9609 2938 9559 3104 9506 3269 9450 55 6 2605 9655 2773 9608 2940 9558 3107 9505 3272 9449 54 7 2608 9654 2776 9607 2943 9557 3110 9504 3275 9449 53 8 2611 9653 2779 9606 2946 9556 3112 9503 3278 9448 52 9 2613 9652 2782 9605 2949 9555 3115 9502 3280 9447 51 10 2616 9652 2784 9605 2952 9555 3118 9502 3283 9446 SO 11 2619 9651 2787 9604 2954 9554 3121 9501 3286 9445 49 12 2622 9650 2790 9603 2957 9553 3123 9500 3289 9444 48 13 2625 9649 2793 9602 2960 9552 3126 9499 3291 9443 47 14 2628 9649 2795 9601 2963 9551 3129 9498 3294 9442 46 15 2630 9648 2798 9600 2965 9550 3132 9497 3297 9441 45 16 2633 9647 2801 9600 2968 9549 3134 9496 3300 9440 44 17 2636 9646 2804 9599 2971 9548 3137 9495 3302 9439 43 18 2639 9646 2807 9598 2974 9548 3140 9494 3305 9438 42 19 2642 9645 2809 9597 2977 9547 3143 9493 3308 9437 41 20 2644 9644 2812 9596 2979 9546 3145 9492 3311 9436 40 21 2647 9643 2815 9596 2982 9545 3148 9492 3313 9435 39 22 2650 9642 2818 9595 2985 9544 3151 9491 3316 9434 38 23 2653 9642 2821 9594 2988 9543 3154 9490 3319 9433 37 24 2656 9641 2823 9593 2990 9542 3156 9489 3322 9432 36 25 2658 9640 2826 9592 2993 9542 3159 9488 3324 9431 35 26 2661 9639 2829 9591 2996 9541 3162 9487 3327 9430 34 27 2664 9639 2832 9591 2999 9540 3165 9486 3330 9429 33 28 2667 9638 2835 9590 3002 9539 3168 9485 3333 9428 32 29 2670 9637 2837 9589 3004 9538 3170 9484 3335 9427 31 30 2672 9636 2840 9588 3007 9537 3173 9483 3338 9426 30 31 2675 9636 2843 9587 3010 9536 3176 9482 3341 9425 29 32 2678 9635 2846 9587 3013 9535 3179 9481 3344 9424 28 33 2681 9634 2849 9586 3015 9535 3181 9480 3346 9423 27 34 2684 9633 2851 9585 3018 9534 3184 9480 3349 9423 26 35 26S6 9632 2854 9584 3021 9533 3187 9479 3352 9422 25 36 2689 9632 2857 9583 3024 9532 3190 9478 3355 9421 24 37 2692 9631 2860 9582 3026 9531 3192 9477 3357 9420 23 38 2695 9630 2862 9582 3029 9530 3195 9476 3360 9419 22 39 2698 9629 2865 9581 3032 9529 3198 9475 3363 9418 21 40 2700 9628 2868 9580 3035 9528 3201 9474 3365 9417 20 41 2703 9628 2871 9579 ' 3038 9527 3203 9473 3368 9416 19 42 2706 9627 2874 9578 3040 9527 3206 9472 3371 9415 18 43 2709 9626 2876 9577 3043 9526 3209 9471 3374 9414 17 44 2712 9625 2879 9577 3046 9525 3212 9470 3376 9413 16 45 2714 9625 2882 9576 3049 9524 3214 9469 3379 9412 15 46 2717 9624 2885 9575 3051 9523 3217 9468 3382 9411 14 47 2720 9623 2888 9574 3054 9522 3220 9467 3385 9410 13 48 2723 9622 2890 9573 3057 9521 3223 9466 3387 9409 12 49 2726 9621 2893 9572 3060 9520 3225 9466 3390 9408 11 50 2728 9621 2896 9572 3062 9520 3228 9465 3393 9407 io 51 2731 9620 2899 9571 3065 9519 3231 9464 3396 9406 9 52 2734 9619 2901 9570 3068 9518 3234 9463 3398 9405 8 53 2737 9618 2904 9569 3071 9517 3236 9462 3401 9404 7 54 2740 9617 2907 9568 3074 9516 3239 9461 3404 9403 6 55 2742 9617' 2910 9567 3076 9515 3242 9460 3407 9402 5 56. 2745 9616 2913 9566 3079 9514 3245 9459 3409 9401 4 57 2748 9615 2915 9566 3082 9513 3247 9458 3412 9400 3 58 2751 9614 2918 9565 3085 9512 3250 9457 3415 9399 2 59 2754 9613 2921 9564 3087 9511 3253 9456 3417 9398 1 60 2756 9613 2924 9563 3090 9511 3256 9455 3420 9397 O cog sin 74° cos sin 73° cos sin 72° cos sin 71° cos sin 70° / f 56 NATURAL SINES AND COSINES. t 20° sin cog 21° 22° 23° 24° f sin cos sin cos sin cos sin cos o 3420 9397 3584 9336 3746 9272 3907 9205 4067 9135 60 1 3423 9396 3586 9335 3749 9271 3910 9204 4070 9134 59 2 3426 9395 3589 9334 3751 9270 3913 9203 4073 9133 58 3 3428 9394 3592 9333 3754 9269 3915 9202 4075 9132 57 4 3431 9393 3595 9332 3757 9267 3918 9200 4078 9131 56 5 3434 9392 3597 9331 3760 9266 3921 9199 4081 9130 55 6 3437 9391 3600 9330 3762 9265 3923 9198 4083 9128 54 7 3439 9390 3603 9328 3765 9264 3926 9197 4086 9127 53 8 3442 9389 3605 9327 3768 9263 3929 9196 4089 9126 52 9 3445 9388 3608 9326 3770 9262 3931 9195 4091 9125 51 io 3448 9387 3611 9325 3773 9261 3934 9194 4094 9124 SO 11 3450 9386 3614 9324 3776 9260 3937 9192 4097 9122 49 12 3453 9385 3616 9323 3778 9259 3939 9191 4099 9121 48 13 3456 9384 3619 9322 3781 9258 3942 9190 4102 9120 47 14 3458 9383 3622 9321 3784 9257 3945 9189 4105 9119 46 15 3461 9382 3624 9320 3786 9255 3947 9188 4107 9118 45 16 3464 9381 3627 9319 3789 9254 3950 9187 4110 9116 44 17 3467 9380 3630 9318 3792 9253 3953 9186 4112 9115 43 18 3469 9379 3633 9317 3795 9252 3955 9184 4115 9114 42 19 3472 9378 3635 9316 3797 9251 3958 9183 4118 9113 41 20 3475 9377 3638 9315 3800 9250 3961 9182 4120 9112 40 21 3478 9376 3641 9314 3803 9249 3963 9181 4123 9110 39 22 3480 9375 3643 9313 3805 9248 3966 9180 4126 9109 38 23 3483 9374 3646 9312 3808 9247 3969 9179 4128 9108 37 24 3486 9373 3649 9311 3811 9245 3971 9178 4131 9107 36 25 3488 9372 3651 9309 3813 9244 3974 9176 4134 9106 35 26 3491 9371 3654 9308 3816 9243 3977 9175 4136 9104 34 27 3494 9370 3657 9307 3819 9242 3979 9174 4139 9103 33 28 3497 9369 3660 9306 3821 9241 3982 9173 4142 9102 32 29 3499 9368 3662 9305 3824 9240 3985 9172 4144 9101 31 30 3502 9367 3665 9304 3827 9239 3987 9171 4147 9100 30 31 3505 9366 3668 9303 3830 9238 3990 9169 4150 9098 29 32 3508 9365 3670 9302 3832 9237 3993 9168 4152 9097 28 33 3510 9364 3673 9301 3835 9235 3995 9167 4155 9096 27 34 3513 9363 3676 9300 3838 9234 3998 9166 4158 9095 26 35 3516 9362 3679 9299 3840 9233 4001 9165 4160 9094 25 36 3518 9361 3681 9298 3843 9232 4003 9164 4163 9092 24 37 3521 9360 3684 9297 3846 9231 4006 9162 4165 9091 23 38 3524 9359 3687 9296 3848 9230 4009 9161 4168 9090 22 39 3527 9358 3689 9295 3851 9229 4011 9160 4171 9088 21 40 3529 9356 3692 9293 3854 9228 4014 9159 4173 9088 20 41 3532 9355 3695 9292 3856 9227 4017 9158 4176 9086 19 42 3535 9354 3697 9291 3859 9225 4019 9157 4179 9085 18 43 3537 9353 3700 9290 3862 9224 4022 9155 4181 9084 17 44 3540 9352 3703 9289 3864 9223 4025 9154 4184 9083 16 45 3543 9351 3706 9288 3867 9222 4027 9153 4187 9081 15 46 3546 9350 3708 9287 3870 9221 4030 9152 4189 9080 14 47 3548 9349 3711 9286 3872 9220 4033 9151 4192 9079 13 48 3551 9348 3714 9285 3875 9219 4035 9150 4195 907S 12 49 3554 9347 3716 9284 3878 9218 4038 9148 4197 9077 11 50 3557 9346 3719 9283 3881 9216 4041 9147 4200 9075 10 51 3559 9345 3722 9282 3883 9215 4043 9146 4202 9074 9 52 3562 9344 3724 9281 3886 9214 4046 9145 4205 9073 8 53 3565 9343 3727 9279 3889 9213 4049 9144 4208 9072 7 54 3567 9342 3730 9278 3891 9212 4051 9143 4210 9070 6 55 3570 9341 3733 9277 3894 9211 4054 9141 4213 9069 5 56 3573 9340 3735 9276 3897 9210 4057 9140 4216 906S 4 57 3576 9339 3738 9275 3899 9208 4059 9139 4218 9067 3 58 3578 9338 3741 9274 3902 9207 4062 9138 4221 9066 2 59 3581 9337 3743 9273 3905 9206 4065 9137 4224 9064 1 60 3584 9336 3746 9272 3907 9205 4067 9135 4226 9063 cog sin 69° cos sin 68° cos sin 67° cos sin 66° cos sin 65° f t NATURAL SINES AND COSINES. 57 / 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 4229 9062 4386 8987 4542 8909 4697 8828 4851 8745 59 2 4231 9061 4389 8985 4545 8907 4700 8827 4853 8743 58 3 4234 9059 4392 8984 4548 8906 4702 8825 4856 8742 57 4 4237 9058 4394 8983 4550 8905 4705 8824 4858 8741 56 5 4239 9057 4397 8982 4553 8903 4708 8823 4861 8739 55 6 4242 9056 4399 8980 4555 8902 4710 8821 4863 8738 54 7 4245 9054 4402 8979 4558 8901 4713 8820 4866 8736 53 8 4247 9053 4405 8978 4561 8899 4715 8819 4868 8735 52 9 4250 9052 4407 8976 4563 8898 4718 8817 4871 8733 51 io 4253 9051 4410 8975 4566 8897 4720 8816 4874 8732 50 11 4255 9050 4412 8974 4568 8895 4723 8814 4876 8731 49 12 4258 9048 4415 8973 4571 8894 4726 8813 4879 8729 48 13 4260 9047 4418 8971 4574 8893 4728 8812 4881 8728 47 14 4263 9046 4420 8970 4576 8892 4731 8810 4884 8726 46 15 4266 9045 4423 8969 4579 8890 4733 8809 4886 8725 45 16 4268 9043 4425 8967 4581 8889 4736 8808 4889 8724 44 17 4271 9042 4428 8966 4584 8888 4738 8806 4891 8722 43 18 4274 9041 4431 8965 4586 8886 4741 8805 4894 8721 42 19 4276 9040 4433 8964 4589 8885 4743 8803 4896 8719 41 20 4279 9038 4436 8962 4592 8884 4746 8802 4899 8718 40 21 4281 9037 4439 8961 4594 8882 4749 8801 4901 8716 39 22 4284 9036 4441 8960 4597 8881 4751 8799 4904 8715 38 23 4287 9035 4444 8958 4599 8879 4754 8798 4907 8714 37 24 4289 9033 4446 8957 4602 8878 4756 8796 4909 8712 36 25 4292 9032 4449 8956 4605 8877 4759 8795 4912 8711 35 26 4295 9031 4452 8955 4607 8875 4761 8794 4914 8709 34 27 4297 9030 4454 8953 4610 8874 4764 8792 4917 8708 33 28 4300 9028 4457 8952 4612 8873 4766 8791 4919 8706 32 29 4302 9027 4459 8951 4615 8871 4769 8790 4922 8705 31 30 4305 9026 4462 8949 4617 8870 4772 8788 4924 8704 30 31 4308 9025 4465 8948 4620 8869 4774 8787 4927 8702 29 32 4310 9023 4467 8947 4623 8867 4777 8785 4929 8701 28 33 4313 9022 4470 8945 4625 8866 4779 8784 4932 8699 27 34 4316 9021 4472 8944 4628 8865 4782 8783 4934 8698 26 35 4318 9020 4475 8943 4630 8863 4784 8781 4937 8696 25 36 4321 9018 4478 8942 4633 8862 4787 8780 4939 8695 24 37 4323 9017 4480 8940 4636 8861 4789 8778 4942 8694 23 38 4326 9016 4483 8939 4638 8859 4792 8777 4944 8692 22 39 4329 9015 4485 8938 4641 8858 4795 8776 4947 8691 21 40 4331 9013 4488 8936 4643 8857 4797 8774 4950 8689 20 41 4334 9012 4491 8935 4646 8855 4800 8773 4952 8688 19 42 4337 9011 4493 8934 4648 8854 4802 8771 4955 8686 18 43 4339 9010 4496 8932 4651 8853 4805 8770 4957 8685 17 44 4342 9008 4498 8931 4654 8851 4807 8769 4960 8683 16 45 4344 9007 4501 8930 4656 8850 4810 8767 4962 8682 15 46 4347 9006 4504 8928 4659 8849 4812 8766 4965 8681 14 47 4350 9004 4506 8927 4661 8847 4815 8764 4967 8679 13 48 4352 9003 4509 8926 4664 8846 4818 8763 4970 8678 12 49 4355 9002 4511 8925 4666 8844 4820 8762 4972 8676 11 50 4358 9001 4514 8923 4669 8843 4823 8760 4975 8675 IO 51 4360 8999 4517 8922 4672 8842 4825 8759 4977 8673 9 52 4363 8998 4519 8921 4674 8840 4828 8757 4980 8672 8 53 4365 8997 4522 8919 4677 8839 4830 8756 4982 8670 7 54 4368 8996 4524 8918 4679 8838 4833 8755 4985 8669 6 55 4371 8994 4527 8917 4682 8836 4835 8753 4987 8668 5 56 4373 8993 4530 8915 4684 8835 4838 8752 4990 8666 4 57 4376 8992 4532 8914 4687 8834 4840 8750 4992 8665 3 58 4378 8990 4535 8913 4690 8832 4843 8749 4995 8663 2 59 4381 8989 4537 8911 4692 8831 4846 8748 4997 8662 1 OO 4384 8988 4540 8910 4695 8829 4848 8746 5000 8660 O cos sin 64° cos sin 63° cos sin 62° cos sin 61° cos sin 60° 1' t 58 NATURAL SINES AND COSINES. f 30° 31° 32° 33° 34° f sin cos sin cos sin cos sin cos sin cos O 5000 8660 5150 8572 5299 8480 5446 8387 5592 8290 60 1 5003 8659 5153 8570 5302 8479 5449 8385 5594 8289 59 2 5005 8657 5155 8569 5304 8477 5451 8384 5597 8287 58 3 5008 8656 5158 8567 5307 8476 5454 8382 5599 8285 57 4 5010 8654 5160 8566 5309 8474 5456 8380 5602 8284 56 5 5013 8653 5163 8564 5312 8473 5459 8379 5604 8282 55 6 5015 8652 5165 8563 5314 8471 5461 8377 5606 8281 54 7 5018 8650 5168 8561 5316 8470 5463 8376 5609 8279 53 8 5020 8649 5170 8560 5319 8468 5466 8374 5611 8277 52 9 5023 8647 5173 8558 5321 8467 5468 8372 5614 8276 51 10 5025 8646 5175 8557 5324 8465 5471 8371 5616 8274 SO 11 5028 8644 5178 8555 5326 8463 5473 8369 5618 8272 49 12 5030 8643 5180 8554 5329 8462 5476 8368 5621 8271 48 13 5033 8641 5183 8552 5331 8460 5478 8366 5623 8269 47 14 5035 8640 5185 8551 5334 8459 5480 8364 5626 8268 46 15 5038 8638 5188 8549 5336 8457 5483 8363 5628 8266 45 16 5040 8637 5190 8548 5339 8456 5485 8361 5630 8264 44 17 5043 8635 5193 8546 5341 8454 5488 8360 5633 8263 43 18 5045 8634 5195 8545 5344 8453 5490 8358 5635 8261 42 19 5048 8632 5198 8543 5346 8451 5493 8356 5638 8259 41 20 5050 8631 5200 8542 5348 8450 5495 8355 5640 8258 40 21 5053 8630 5203 8540 5351 8448 5498 8353 5642 8256 39 22 5055 8628 5205 8539 5353 8446 5500 8352 5645 8254 38 23 5058 8627 5208 8537 5356 8445 5502 8350 5647 8253 37 24 5060 8625 5210 8536 5358 8443 5505 8348 5650 8251 36 25 5063 8624 5213 8534 5361 8442 5507 8347 5652 8249 35 26 5065 8622 5215 8532 5363 8440 5510 8345 5654 8248 34 27 5068 8621 5218 8531 5366 8439 5512 8344 5657 8246 33 28 5070 8619 5220 8529 5368 8437 5515 8342 5659 8245 32 29 5073 8618 5223 8528 5371 8435 5517 8340 5662 8243 31 30 5075 8616 5225 8526 5373 8434 5519 8339 5664 8241 30 31 5078 8615 5227 8525 5375 8432 5522 8337 5666 8240 29 32 5080 8613 5230 8523 5378 8431 5524 8336 5669 8238 28 33 5083 8612 5232 8522 5380 8429 5527 8334 5671 8236 27 34 5085 8610 5235 8520 5383 8428 5529 8332 5674 8235 26 35' 5088 8609 5237 8519 5385 8426 5531 8331 5676 8233 25 36 5090 8607 5240 8517 5388 8425 5534 8329 5678 8231 24 37 5093 8606 5242 8516 5390 8423 5536 8328 5681 8230 23 38 5095 8604 5245 8514 5393 8421 5539 8326 5683 8228 22 39 5098 8603 5247 8513 5395 8420 5541 8324 5686 8226 21 40 5100 8601 5250 8511 5398 8418 5544 8323 5688 8225 20 41 5103 8600 5252 8510 5400 8417 5546 8321 5690 8223 19 42 5105 8599 5255 8508 5402 8415 5548 8320 5693 8221 18 43 5108 8597 5257 8507 5405 8414 5551 8318 5695 8220 17 44 5110 8596 5260 8505 5407 8412 5553 8316 5698 8218 16 45 5113 8594 5262 8504 5410 8410 5556 8315 5700 8216 15 46 5115 8593 5265 8502 5412 8409 5558 8313 5702 8215 14 47 5118 8591 5267 8500 5415 8407 5561 8311 5705 8213 13 48 5120 8590 5270 8499 5417 8406 5563 8310 5707 8211 12 49 5123 8588 5272 8497 5420 8404 5565 S30S 5710 8210 11 50 5125 8587 5275 8496 5422 8403 5568 8307 5712 8208 io 51 5128 8585 5277 8494 5424 8401 5570 8305 5714 8207 9 52 5130 8584 5279 8493 5427 8399 5573 8303 5717 8205 8 53 5133 8582 5282 8491 5429 8398 5575 8302 5719 8203 7 54 5135 8581 5284 8490 5432 8396 5577 8300 5721 8202 6 55 5138 8579 5287 8488 5434 8395 5580 8299 5724 8200 5 56 5140 8578 5289 8487 5437 8393 5582 8297 5726 8198 4 57 5143 8576 5292 8485 5439 8391 5585 8295 5729 8197 3 58 5145 8575 5294 8484 5442 8390 5587 8294 5731 8195 2 59 5148 8573 5297 8482 5444 8388 5590 8292 5733 8193 1 60 5150 8572 5299 8480 5446 8387 5592 8290 5736 8192 O cos sin 59° cos sin cos sin 57° cos sin 56° cos sin 55° t 58° / NATURAL SINES AND COSINES. 59 t 35° 30° 37° 38° 39° r sin cos sin cos sin cos sin cos sin cos o 5736 8192 5878 8090 6018 7986 6157 7880 6293 7771 60 1 5738 8190 5880 8088 6020 7985 6159 7878 6295 7770 59 2 5741 8188 5883 8087 6023 7983 6161 7877 629S 7768 58 3 5743 8187 5885 8085 6025 7981 6163 7875 6300 7766 57 4 5745 8185 5887 8083 6027 7979 6166 7873 6302 7764 56 5 5748 8183 5890 8082 6030 7978 6168 7871 6305 7762 55 6 5750 8181 5892 8080 6032 7976 6170 7869 6307 7760 54 7 5752 8180 5894 8078 6034 7974 6173 7868 6309 7759 53 8 5755 8178 5897 8076 6037 7972 6175 7866 6311 7757 52 9 5757 8176 5899 8075 6039 7971 6177 7864 6314 7755 51 io 5760 8175 5901 8073 6041 7969 6180 7862 6316 7753 SO 11 5762 8173 5904 8071 6044 7967 6182 7860 6318 7751 49 12 5764 8171 5906 8070 6046 7965 6184 7859 6320 7749 48 13 5767 8170 5908 8068 6048 7964 6186 7857 6323 7748 47 14 5769 816S 5911 8066 6051 7962 6189 7855 6325 7746 46 15 5771 8166 5913 8064 6053 7960 6191 7853 6327 7744 45 16 5774 8165 5915 8063 6055 7958 6193 7851 6329 7742 44 17 5776 8163 5918 8061 6058 7956 6196 7850 6332 7740 43 18 5779 8161 5920 8059 6060 7955 6198 7848 6334 7738 42 19 5781 8160 5922 8058 6062 7953 6200 7846 6336 7737 41 20 5783 8158 5925 8056 6065 7951 6202 7844 6338 7735 40 21 5786 8156 5927 8054 6067 7950 6205 7842 6341 7733 39 22 5788 8155 5930 8052 6069 7948 6207 7841 6343 7731 38 23 5790 8153 5932 8051 6071 7946 6209 7839 6345 7729 37 24 5793 8151 5934 8049 6074 7944 6211 7837 6347 7727 36 25 5795 8150 5937 8047 6076 7942 6214 7835 6350 7725 35 26 5798 8148 5939 8045 6078 7941 6216 7833 6352 7724 34 27 5S00 8146 5941 8044 6081 7939 6218 7832 6354 7722 33 28 5802 8145 5944 8042 6083 7937 6221 7830 6356 7720 32 29 5805 8143 5946 8040 6085 7935 6223 7828 6359 7718 31 30 5807 8141 5948 8039 6088 7934 6225 7826 6361 7716 30 31 5809 8139 •5951 8037 6090 7932 6227 7824 6363 7714 29 32 5812 8138 5953 8035 6092 7930 6230 7822 6365 7713 28 33 5814 8136 5955 8033 6095 7928 6232 7821 6368 7711 27 34 5S16 8134 5958 8032 6097 7926 6234 7819 6370 7709 26 35 5819 8133 5960 8030 6099 7925 6237 7817 6372 7707 25 36 5821 8131 5962 8028 6101 7923 6239 7815 6374 7705 24 37 5824 8129 5965 8026 6104 7921 6241 7813 6376 7703 23 38 5826 8128 5967 8025 6106 7919 6243 7812 6379 7701 22 39 5828 8126 5969 8023 6108 7918 6246 7810 6381 7700 21 40 5831 8124 5972 8021 6111 7916 6248 7808 6383 7698 20 41 5S33 8123 5974 8020 6113 7914 6250 7806 6385 7696 19 42 5835 8121 5976 8018 6115 7912 6252 7804 6388 7694 18 43 5838 8119 5979 8016 6118 7910 6255 7802 6390 7692 17 44 5840 8117 5981 8014 6120 7909 6257 7801 6392 7690 16 45 5842 8116 5983 8013 6122 7907 6259 7799 6394 7688 15 46 5845 8114 5986 8011 6124 7905 6262 7797 6397 7687 14 47 5847 8112 5988 8009 6127 7903 6264 7795 6399 7685 13 48 5850 8111 5990 8007 6129 7902 6266 7793 6401 7683 12 49 5852 8109 5993 8006 6131 7900 6268 7792 6403 7681 11 SO 5854 8107 5995 8004 6134 7898 6271 7790 6406 7679 IO 51 5857 8106 5997 8002 6136 7896 6273 7788 6408 7677 9 52 5859 8104 6000 8000 6138 7894 6275 7786 6410 7675 8 53 5861 8102 6002 7999 6141 7893 6277 7784 6412 7674 7 54 5864 8100 6004 7997 6143 7891 6280 7782 6414 7672 6 55 5866 8099 6007 7995 6145 7889 6282 7781 6417 7670 5 56 5868 8097 6009 7993 6147 7887 6284 7779 6419 7668 4 57 5871 8095 6011 7992 6150 7885 6286 7777 6421 7666 3 58 5873 8094 6014 7990 6152 7884 6289 7775 6423 7664 2 59 5875 8092 6016 7988 6154 7882 6291 7773 6426 7662 1 60 5878 8090 6018 7986 6157 7880 6293 7771 6428 7660 cos sin 54° cos sin 53° cos sin 52° cos sin 51° cos sin 50° / i 60 NATURAL SINES AND COSINES. / 40° 41° 42° sin cos 43° 44° / sin cos sin cos sin cos sin cos o 6428 7660 6561 7547 6691 7431 6820 7314 6947 7193 60 1 6430 7659 6563 7545 6693 7430 6822 7312 6949 7191 59 2 6432 7657 6565 7543 6696 7428 6824 7310 6951 7189 58 3 6435 7655 6567 7541 6698 7426 6826 7308 6953 7187 57 4 6437 7653 6569 7539 6700 7424 6828 7306 6955 7185 56 5 6439 7651 6572 7538 6702 7422 6831 7304 6957 7183 55 6 6441 7649 6574 7536 6704 7420 6833 7302 6959 7181 54 7 6443 7647 6576 7534 6706 7418 6835 7300 6961 7179 53 8 6446 7645 6578 7532 6709 7416 6837 7298 6963 7177 52 9 6448 7644 6580 7530 6711 7414 6839 7296 6965 7175 51 io 6450 7642 6583 7528 6713 7412 6841 7294 6967 7173 50 11 6452 7640 6585 7526 6715 7410 6843 7292 6970 7171 49 12 6455 7638 6587 7524 6717 7408 6845 7290 6972 7169 48 13 6457 7636 6589 7522 6719 7406 6848 7288 6974 7167 47 14 6459 7634 6591 7520 6722 7404 6850 7286 6976 7165 46 15 6461 7632 6593 7518 6724 7402 6852 7284 6978 7163 45 16 6463 7630 6596 7516 6726 7400 6854 7282 6980 7161 44 17 6466 7629 6598 7515 6728 7398 6856 7280 6982 7159 43 18 6468 7627 6600 7513 6730 7396 6858 7278 6984 7157 42 19 6470 7625 6602 7511 6732 7394 6860 7276 6986 7155 41 20 6472 7623 6604 7509 6734 7392 6862 7274 6988 7153 40 21 6475 7621 6607 7507 6737 7390 6865 7272 6990 7151 39 22 6477 7619 6609 7505 6739 7388 6867 7270 6992 7149 38 23 6479 7617 66J1 7503 6741 7387 6869 7268 6995 7147 37 24 6481 7615 6613 7501 6743 7385 6871 7266 6997 7145 36 25 6483 7613 6615 7499 6745 7383 6873 7264 6999 7143 35 26 6486 7612 6617 7497 6747 7381 6875 7262 7001 7141 34 27 6488 7610 6620 7495 6749 7379 6877 7260 7003 7139 33 28 6490 7608 6622 7493 6752 7377 6879 7258 7005 7137 32 29 6492 7606 6624 7491 6754 7375 6881 7256 7007 7135 31 30 6494 7604 6626 7490 6756 7373 6884 7254 7009 7133 30 31 6497 7602 6628 7488 6758 7371 6886 7252 7011 7130 29 32 6499 7600 6631 7486 6760 7369 6888 7250 7013 7128 28 33 6501 7598 6633 7484 6762 7367 6890 7248 7015 7126 27 34 6503 7596 6635 7482 6764 7365 6892 7246 7017 7124 26 35 6506 7595 6637 7480 6767 7363 6894 7244 7019 7122 25 36 6508 7593 6639 7478 6769 7361 6896 7242 7022 7120 24 37 6510 7591 6641 7476 6771 7359 6898 7240 7024 7118 23 38 6512 7589 6644 7474 6773 7357 6900 7238 7026 7116 22 39 6514 7587 6646 7472 6775 7355 6903 7236 7028 7114 21 40 6517 7585 6648 7470 6777 7353. 6905 7234 7030 7112 20 41 6519 7583 6650 7468 6779 7351 6907 7232 7032 7110 19 42 6521 7581 6652 7466 6782 7349 6909 7230 7034 7108 18 43 6523 7579 6654 7464 6784 7347 6911 7228 7036 7106 17 44 6525 7578 6657 7463 6786 7345 6913 7226 7038 7104 16 45 6528 7576 6659 7461 6788 7343 6915 7224 7040 7102 15 46 6530 7574 6661 7459 6790 7341 6917 7222 7042 7100 14 47 6532 7572 6663 7457 6792 7339 6919 7220 7044 7098 13 48 6534 7570 6665 7455 6794 7337 6921 7218 7046 7096 12 49 6536 7568 6667 7453 6797 7335 6924 7216 7048 7094 11 50 6539 7566 6670 7451 6799 7333 6926 7214 7050 7092 10 51 6541 7564 6672 7449 6801 7331 6928 7212 7053 7090 9 52 6543 7562 6674 7447 6803 7329 6930 7210 7055 7088 8 53 6545 7560 6676 7445 6805 7327 6932 7208 7057 7085 7 54 6547 7559 6678 7443 6807 7325 6934 7206 7059 7083 6 55 6550 7557 6680 7441 6809 7323 6936 7203 7061 7081 5 56 6552 7555 6683 7439 6811 7321 6938 7201 7063 7079 4 57 6554 7553 6685 7437 6814 7319 6940 7199 7065 7077 3 58 6556 7551 6687 7435 6816 7318 6942 7197 7067 7075 2 59 6558 7549 6689 7433 6818 7316 6944 7195 7069 7073 1 60 6561 7547 6691 7431 6820 7314 6947 7193 7071 7071 O cos sin 49° cos sin 48° cos sin 47° cos sin cos sin 45° / 46° / NATURAL TANGENTS AND COTANGENTS. 61 t 0° 1° 2° 3° 4° t tan cot tan cot tan cot tan cot tan cot o 0000 Infinite 0175 57.2900 0349 28.6363 0524 19.0811 0699 14.3007 60 1 0003 3437.75 0177 56.3506 0352 28.3994 0527 18.9755 0702 14.2411 59 2 0006 1718.87 0180 55.4415 0355 28.1664 0530 18.8711 0705 14.1821 58 3 0009 1145.92 0183 54.5613 0358 27.9372 0533 18.7678 0708 14.1235 57 4 0012 859.436 0186 53.7086 0361 27.7117 0536 18.6656 0711 14.0655 56 5 0015 687.549 0189 52.8821 0364 27.4899 0539 18.5645 0714 14.0079 55 6 0017 572.957 0192 52.0807 0367 27.2715 0542 18.4645 0717 13.9507 54 7 0020 491.106 0195 51.3032 0370 27.0566 0544 18.3655 0720 13.8940 53 8 0023 429.718 0198 50.5485 0373 26.8450 0547 18.2677 0723 13.8378 52 9 0026 381.971 0201 49.8157 0375 26.6367 0550 18.1708 0726 13.7821 51 io 0029 343.774 0204 49.1039 0378 26.4316 0553 18.0750 0729 13.7267 50 11 0032 312.521 0207 48.4121 0381 26.2296 0556 17.9802 0731 13.6719 49 12 0035 286.478 0209 47.7395 0384 26.0307 0559 17.8863 0734 13.6174 48 13 0038 264.441 0212 47.0853 0387 25.8348 0562 17.7934 0737 13.5634 47 14 0041 245.552 0215 46.4489 0390 25.6418 0565 17.7015 0740 13.5098 46 15 0044 229.182 0218 45.8294 0393 25.4517 0568 17.6106 0743 13.4566 45 16 0047 214.858 0221 45.2261 0396 25.2644 0571 17.5205 0746 13.4039 44 17 0049 202.219 0224 44.6386 0399 25.0798 0574 17.4314 0749 13.3515 43 18 0052 190.984 0227 44.0661 0402 24.8978 0577 17.3432 0752 13.2996 42 19 0055 180.932 0230 43.5081 0405 24.7185 0580 17.2558 0755 13.24S0 41 20 0058 171.885 0233 42.9641 0407 24.5418 0582 17.1693 0758 13.1969 40 21 0061 163.700 0236 42.4335 0410 24.3675 0585 17.0837 0761 13.1461 39 22 0064 156.259 0239 41.9158 0413 24.1957 0588 16.9990 0764 13.0958 38 23 0067 149.465 0241 41.4106 0416 24.0263 0591 16.9150 0767 13.0458 37 24 0070 143.237 0244 40.9174 0419 23.8593 0594 16.8319 0769 12.9962 36 25 0073 137.507 0247 40.4358 0422 23.6945 0597 16.7496 0772 12.9469 35 26 0076 132.219 0250 39.9655 0425 23.5321 0600 16.6681 0775 12.8981 34 27 0079 127.321 0253 39.5059 0428 23.3718 0603 16.5874 0778 12.8496 33 28 0081 122.774 0256 39.0568 0431 23.2137 0606 16.5075 0781 12.8014 32 29 0084 118.540 0259 38.6177 0434 23.0577 0609 16.4283 0784 12.7536 31 30 0087 114.589 0262 38.1885 0437 22.9038 0612 16.3499 0787 12.7062 30 31 0090 110.892 0265 37.7686 0440 22.7519 0615 16.2722 0790 12.6591 29 32 0093 107.426 0268 37.3579 0442 22.6020 0617 16.1952 0793 12.6124 28 33 0096 104.171 0271 36.9560 0445 22.4541 0620 16.1190 0796 12.5660 27 34 0099 101.107 0274 36.5627 0448 22.3081 0623 16.0435 0799 12.5199 26 35 0102 98.2179 0276 36.1776 0451 22.1640 0626 15.9687 0802 12.4742 25 36 0105 95.4895 0279 35.8006 0454 22.0217 0629 15.8945 0805 12.4288 24 37 0108 92.9085 0282 35.4313 0457 21.8813 0632 15.8211 0808 12.3838 23 38 0111 90.4633 0285 35.0695 0460 21.7426 0635 15.7483 0810 12.3390 22 39 0113 88.1436 0288 34.7151 0463 21.6056 0638 15.6762 0813 12.2946 21 40 0116 85.9398 0291 34.3678 0466 21.4704 0641 15.6048 0816 12.2505 20 41 0119 83.8435 0294 34.0273 0469 21.3369 0644 15.5340 0819 12.2067 19 42 0122 81.8470 0297 33.6935 0472 21.2049 0647 15.4638 0822 12.1632 18 43 0125 79.9434 0300 33.3662 0475 21.0747 0650 15.3943 0825 12.1201 17 44 0128 78.1263 0303 33.0452 0477 20.9460 0653 15.3254 0828 12.0772 16 45 0131 76.3900 0306 32.7303 0480 20.8188 0655 15.2571 0831 12.0346 15 46 0134 74.7292 0308 32.4213 0483 20.6932 0658 15.1893 0834 11.9923 14 47 0137 73.1390 0311 32.1181 0486 20.5691 0661 15.1222 0837 11.9504 13 48 0140 71.6151 0314 31.8205 0489 20.4465 0664 15.0557 0840 11.9087 12 49 0143 70.1533 0317 31.5284 0492 20.3253 0667 14.9898 0843 11.8673 11 50 0146 68.7501 0320 31.2416 0495 20.2056 0670 14.9244 0846 11.8262 10 51 0148 67.4019 0323 30.9599 0498 20.0872 0673 14.8596 0849 11.7853 9 52 0151 66.1055 0326 30.6833 0501 19.9702 0676 14.7954 0851 11.7448 8 53 0154 64.8580 0329 30.4116 0504 19.8546 0679 14.7317 0854 11.7045 7 54 0157 63.6567 0332 30.1446 0507 19.7403 0682 14.6685 0857 11.6645 6 55 0160 62.4992 0335 29.8823 0509 19.6273 0685 14.6059 0860 11.6248 5 56 0163 61.3829 0338 29.6245 0512 19.5156 0688 14.5438 0863 11.5853 4 57 0166 60.3058 0340 29.3711 0515 19.4051 0690 14.4823 0866 11.5461 3 58 0169 59.2659 0343 29.1220 0518 19.2959 0693 14.4212 0869 11.5072 2 59 0172 58.2612 0346 28.8771 0521 19.1879 0696 14.3607 0872 11.4685 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° ' 62 NATURAL TANGENTS AND COTANGENTS / 5° 6° 7° 8° 9° / tan cot tan cot tan cot tan cot tan cot o 0875 11.4301 1051 9.5144 1228 8.1443 1405 7.1154 1584 6.3138 60 1 0878 11.3919 1054 9.4878 1231 8.1248 1408 7.1004 1587 6.3019 59 2 0881 11.3540 1057 9.4614 1234 8.1054 1411 7.0855 1590 6.2901 58 3 0884 11.3163 1060 9.4352 1237 8.0860 1414 7.0706 1593 6.2783 57 4 0887 11.2789 1063 9.4090 1240 8.0667 1417 7.0558 1596 6.2666 56 5 0890 11.2417 1066 9.3831 1243 8.0476 1420 7.0410 1599 6.2549 55 6 0892 11.2048 1069 9.3572 1246 8.0285 1423 7.0264 1602 6.2432 54 7 0895 11.1681 1072 9.3315 1249 8.0095 1426 7.0117 1605 6.2316 53 8 0898 11.1316 1075 9.3060 1251 7.9906 1429 6.9972 1608 6.2200 52 9 0901 11.0954 1078 92806 1254 7.9718 1432 6.9827 1611 6.2085 51 io 0904 11.0594 1080 9.2553 1257 7.9530 1435 6.9682 1614 6.1970 50 11 0907 11.0237 1083 9.2302 1260 7.9344 1438 6.9538 1617 6.1856 49 12 0910 10.9882 1086 9.2052 1263 7.9158 1441 6.9395 1620 6.1742 48 13 0913 10.9529 1089 9.1803 1266 7.8973 1444 6.9252 1623 6.1628 47 14 0916 10.9178 1092 9.1555 1269 7.8789 1447 6.9110 1626 6.1515 46 15 0919 10.8829 1095 9.1309 1272 7.8606 1450 6.8969 1629 6.1402 45 16 0922 ] 0.8483 1098 9.1065 1275 7.8424 1453 6.8828 1632 6.1290 44 17 0925 10.8139 1101 9.0821 1278 7.8243 1456 6.8687 1635 6.1178 43 18 0928 10.7797 1104 9.0579 1281 7.8062 1459 6.8548 1638 6.1066 42 19 0931 10.7457 1107 9.0338 1284 7.7883 1462 6.8408 1641 6.0955 41 20 0934 10.7119 1110 9.0098 1287 7.7704 1465 6.8269 1644 6.0844 40 21 0936 10.6783 1113 8.9860 1290 7.7525 1468 6.8131 1647 6.0734 39 22 0939 10.6450 1116 8.9623 1293 7.7348 1471 6.7994 1650 6.0624 38 23 0942 10.6118 1119 8.9387 1296 7.7171 1474 6.7856 1653 6.0514 37 24 0945 10.5789 1122 8.9152 1299 7.6996 1477 6.7720 1655 6.0405 36 25 0948 10.5462 1125 8.8919 1302 7.6821 1480 6.7584 1658 6.0296 35 26 0951 10.5136 1128 8.8686 1305 7.6647 1483 6.7448 1661 6.0188 34 27 0954 10.4813 1131 8.8455 1308 7.6473 1486 6.7313 1664 6.0080 33 28 0957 10.4491 1134 8.8225 1311 7.6301 1489 6.7179 1667 5.9972 32 29 0960 10.4172 1136 8.7996 1314 7.6129 1492 6.7045 1670 5.9865 31 30 0963 10.3854 1139 8.7769 1317 7.5958 1495 6.6912 1673 5.9758 30 31 0966 10.3538 1142 8.7542 1319 7.5787 1497 6.6779 1676 5.9651 29 32 0969 10.3224 1145 8.7317 1322 7.5618 1500 6.6646 1679 5.9545 28 33 0972 10.2913 1148 8.7093 1325 7.5449 1503 6.6514 1682 5.9439 27 34 0975 10.2602 1151 8.6870 1328 7.5281 1506 6.6383 1685 5.9333 26 35 0978 10.2294 1154 8.6648 1331 7.5113 1509 6.6252 1688 5.9228 25 36 0981 10.1988 1157 8.6427 1334 7.4947 1512 6.6122 1691 5.9124 24 37 0983 10.1683 1160 8.6208 1337 7.4781 1515 6.5992 1694 5.9019 23 38 0986 10.1381 1163 8.5989 1340 7.4615 1518 6.5863 1697 5.8915 22 39 0989 10.1080 1166 8.5772 1343 7.4451 1521 6.5734 1700 5.8811 21 40 0992 10.0780 1169 8.5555 1346 7.4287 1524 6.5606 1703 5.8708 20 41 0995 10.0483 1172 8.5340 1349 7.4124 1527 6.5478 1706 5.8605 19 42 0998 10.0187 1175 8.5126 1352 7.3962 1530 6.5350 1709 5.8502 18 43 1001 9.9893 1178 8.4913 1355 7.3800 1533 6.5223 1712 5.8400 17 44 1004 9.9601 1181 8.4701 1358 7.3639 1536 6.5097 1715 5.8298 16 45 1007 9.9310 1184 8.4490 1361 7.3479 1539 6.4971 1718 5.8197 15 46 1010 9.9021 1187 8.4280 1364 7.3319 1542 6.4846 1721 5.8095 14 47 1013 9.8734 1189 8.4071 1367 7.3160 1545 6.4721 1724 5.7994 13 48 1016 9.8448 1192 8.3863 1370 7.3002 1548 6.4596 1727 5.7894 12 49 1019 9.8164 1195 8.3656 1373 7.2844 1551 6.4472 1730 5.7794 11 50 1022 9.7882 1198 8.3450 1376 7.2687 1554 6.4348 1733 5.7694 IO 51 1025 9.7601 1201 8.3245 1379 7.2531 1557 6.4225 1736 5.7594 9 52 1028 9.7322 1204 8.3041 1382 7.2375 1560 6.4103 1739 5.7495 8 53 1030 9.7044 1207 8.2838 1385 7.2220 1563 6.3980 1742 5.7396 7 54 1033 9.6768 1210 8.2636 1388 7.2066 1566 6.3859 1745 5.7297 6 55 1036 9.6499 1213 8.2434 1391 7.1912 1569 6.3737 1748 5.7199 5 56 1039 9.6220 1216 8.2234 1394 7.1759 1572 6.3617 1751 5.7101 4 57 1042 9.5949 1219 8.2035 1397 7.1607 1575 6.3496 1754 5.7004 3 58 1045 9.5679 1222 8.1837 1399 7.1455 1578 6.3376 1757 5.6906 2 59 1048 9.5411 1225 8.1640 1402 7.1304 1581 6.3257 1760 5.6809 1 60 1051 9.5144 1228 8.1443 1405 7.1154 1584 6.3138 1763 5.6713 O r cot tan cot tan 83° cot tan 82° cot tan 81° cot tan 80° / 84° NATURAL TANGENTS AND COTANGENTS 63 / io° 11° 12° 13° 14° f tan cot tan cot tan cot tan cot tan cot o 1763 5.6713 1944 5.1446 2126 4.7046 2309 4.3315 2493 4.0108 60 1 1766 5.6617 1947 5.1366 2129 4.6979 2312 4.3257 2496 4.0058 59 2 1769 5.6521 1950 5.1286 2132 4.6912 2315 4.3200 2499 4.0009 58 3 1772 5.6425 1953 5.1207 2135 4.6845 2318 4.3143 2503 3.9959 57 4 1775 5.6330 1956 5.1128 2138 4.6779 2321 4.3086 2506 3.9910 56 5 1778 5.6234 1959 5.1049 2141 4.6712 2324 4.3029 2509 3.9861 55 6 1781 5.6140 1962 5.0970 2144 4.6646 2327 4.2972 2512 3.9812 54 7 1784 5.6045 1965 5.0892 2147 4.6580 2330 4.2916 2515 3.9763 53 8 1787 5.5951 1968 5.0814 2150 4.6514 2333 4.2859 2518 3.9714 52 9 1790 5.5857 1971 5.0736 2153 4.6448 2336 4.2803 2521 3.9665 51 10 1793 5.5764 1974 5.0658 2156 4.6382 2339 4.2747 2524 3.9617 50 11 1796 5.5671 1977 5.0581 2159 4.6317 2342 4.2691 2527 3.9568 49 12 1799 5.5578 1980 5.0504 2162 4.6252 2345 4.2635 2530 3.9520 48 13 1802 5.5485 1983 5.0427 2165 4.6187 2349 4.2580 2533 3.9471 47 14 1805 5.5393 1986 5.0350 2168 4.6122 2352 4.2524 2537 3.9423 46 15 1808 5.5301 1989 5.0273 2171 4.6057 2355 4.2468 2540 3.9375 45 16 1811 5.5209 1992 5.0197 2174 4.5993 2358 4.2413 2543 3.9327 44 17 1814 5.5118 1995 5.0121 2177 4.5928 2361 4.2358 2546 3.9279 43 18 1817 5.5026 1998 5.0045 2180 4.5864 2364 4.2303 2549 3.9232 42 19 1820 5.4936 2001 4.9969 2183 4.5800 2367 4.2248 2552 3.9184 41 20 1823 5.4845 2004 4.9894 2186 4.5736 2370 4.2193 2555 3.9136 40 21 1826 5.4755 2007 4.9819 2189 4.5673 2373 4.2139 2558 3.9089 39 22 1829 5.4665 2010 4.9744 2193 4.5609 2376 4.2084 2561 3.9042 38 23 1832 5.4575 2013 4.9669 2196 4.5546 2379 4.2030 2564 3.8995 37 24 1835 5.4486 2016 4.9594 2199 4.5483 2382 4.1976 2568 3.8947 36 25 1838 5.4397 2019 4.9520 2202 4.5420 2385 4.1922 2571 3.8900 35 26 1841 5.4308 2022 4.9446 2205 4.5357 2388 4.1868 2574 3.8854 34 27 1844 5.4219 2025 4.9372 2208 4.5294 2392 4.1814 2577 3.8807 33 28 1847 5.4131 2028 4.9298 2211 4.5232 2395 4.1760 2580 3.8760 32 29 1850 5.4043 2031 4.9225 2214 4.5169 2398 4.1706 2583 3.8714 31 30 1853 5.3955 2035 4.9152 2217 4.5107 2401 4.1653 2586 3.8667 30 31 1856 5.3868 2038 4.9078 2220 4.5045 2404 4.1600 2589 3.8621 29 32 1859 5.3781 2041 4.9006 2223 4.4983 2407 4.1547 2592 3.8575 28 33 1862 5.3694 2044 4.8933 2226 4.4922 2410 4.1493 2595 3.8528 27 34 1865 5.3607 2047 4.8860 2229 4.4860 2413 4.1441 2599 3.8482 26 35 1868 5.3521 2050 4.8788 2232 4.4799 2416 4.1388 2602 3.8436 25 36 1871 5.3435 2053 4.8716 2235 4.4737 2419 4.1335 2605 3.8391 24 37 1874 5.3349 2056 4.8644 2238 4.4676 2422 4.1282 2608 3.8345 23 38 1877 5.3263 2059 4.8573 2241 4.4615 2425 4.1230 2611 3.8299 22 39 1880 5.3178 2062 4.8501 2244 4.4555 2428 4.1178 2614 3.8254 21 40 1883 5.3093 2065 4.8430 2247 4.4494 2432 4.1126 2617 3.8208 20 41 1887 5.3008 2068 4.8359 2251 4.4434 2435 4.1074 2620 3.8163 19 42 1890 5.2924 2071 4;8288 2254 4.4374 2438 4.1022 2623 3.8138 18 43 1893 5.2839 2074 4.8218 2257 4.4313 2441 4.0970 2627 3.8073 17 44 1896 5.2755 2077 4.8147 2260 4.4253 2444 4.0918 2630 3.8028 16 45 1899 5.2672 2080 4.8077 2263 4.4194 2447 4.0867 2633 3.7983 15 46 1902 5.2588 2083 4.8007 2266 4.4134 2450 4.0815 2636 3.7938 14 47 1905 5.2505 2086 4.7937 2269 4.4075 2453 4.0764 2639 3.7893 13 48 1908 5.2422 2089 4.7867 2272 4.4015 2456 4.0713 2642 3.7848 12 49 1911 5.2339 2092 4.7798 2275 4.3956 2459 4.0662 2645 3.7804 11 SO 1914 5.2257 2095 4.7729 2278 4.3897 2462 4.0611 2648 3.7760 10 51 1917 5.2174 2098 4.7659 2281 4.3838 2465 4.0560 2651 3.7715 9 52 1920 5.2092 2101 4.7591 2284 4.3779 2469 4.0509 2655 3.7671 8 53 1923 5.2011 2104 4.7522 2287 4.3721 2472 4.0459 2658 3.7627 7 54 1926 5.1929 2107 4.7453 2290 4.3662 2475 4.0408 2661 3.7583 6 55 1929 5.1848 2110 4.7385 2293 4.3604 2478 4.0358 2664 3.7539 5 56 1932 5.1767 2113 4.7317 2296 4.3546 2481 4.0308 2667 3.7495 4 57 1935 5.1686 2116 4.7249 2299 4.3488 2484 4.0257 2670 3.7451 3 58 1938 5.1606 2119 4.7181 2303 4.3430 2487 4.0207 2673 3.7408 2 59 1941 5.1526 2123 4.7114 2306 4.3372 2490 4.0158 2676 3.7364 1 60 1944 5.1446 2126 4.7046 2309 4.3315 2493 4.0108 2679 3.7321 O t cot tan 79° cot tan 78° cot tan cot tan 76° cot tan 75° f 77° 64 NATURAL TANGENTS AND COTANGENTS. r 15° 16° 17° 18° 19° t tan cot tan cot tan cot tan cot tan cot O 2679 3.7321 2867 3.4874 3057 3.2709 3249 3.0777 3443 2.9042 60 1 2683 3.7277 2871 3.4836 3060 3.2675 3252 3.0746 3447 2.9015 59 2 2686 3.7234 2874 3.4798 3064 3.2641 3256 3.0716 3450 2.8987 58 3 2689 3.7191 2877 3.4760 3067 3.2607 3259 3.0686 3453 2.8960 57 4 2692 3.7148 2880 3.4722 3070 3.2573 3262 3.0655 3456 2.8933 56 5 2695 3.7105 2883 3.4684 3073 3.2539 3265 3.0625 3460 2.8905 55 6 2698 3.7062 2886 3.4646 3076 3.2506 3269 3.0595 3463 2.8878 54 7 2701 3.7019 2890 3.4608 3080 3.2472 3272 3.0565 3466 2.8851 53 8 2704 3.6976 2893 3.4570 3083 3.2438 3275 3.0535 3469 2.8824 52 9 2708 3.6933 2896 3.4533 3086 3.2405 3278 3.0505 3473 2.8797 51 io 2711 3.6891 2899 3.4495 3089 3.2371 3281 3.0475 3476 2.8770 50 11 2714 3.6848 2902 3.4458 3092 3.2338 3285 3.0445 3479 2.8743 49 12 2717 3.6806 2905 3.4420 3096 3.2305 3288 3.0415 3482 2.8716 48 13 2720 3.6764 2908 3.4383 3099 3.2272 3291 3.0385 3486 2.8689 47 14 2723 3.6722 2912 3.4346 3102 3.2238 3294 3.0356 3489 2.8662 46 15 2726 3.6680 2915 3.4308 3105 3.2205 3298 3.0326 3492 2.8636 45 16 2729 3.6638 2918 3.4271 3108 3.2172 3301 3.0296 3495 2.8609 44 17 2733 3.6596 2921 3.4234 3111 3.2139 3304 3.0267 3499 2.8582 43 18 2736 3.6554 2924 3.4197 3115 3.2106 3307 3.0237 3502 2.8556 42 19 2739 3.6512 2927 3.4160 3118 3.2073 3310 3.0208 3505 2.8529 41 20 2742 3.6470 2931 3.4124 3121 3.2041 3314 3.0178 3508 2.8502 40 21 2745 3.6429 2934 3.4087 3124 3.2008 3317 3.0149 3512 2.8476 39 22 2748 3.6387 2937 3.4050 3127 3.1975 3320 3.0120 3515 2.8449 38 23 2751 3.6346 2940 3.4014 3131 3.1943 3323 3.0090 3518 2.8423 37 24 2754 3.6305 2943 3.3977 3134 3.1910 3327 3.0061 3522 2.8397 36 25 2758 3.6264 2946 3.3941 3137 3.1878 3330 3.0032 3525 2.8370 35 26 2761 3.6222 2949 3.3904 3140 3.1845 3333 3.0003 3528 2.8344 34 27 2764 3.6181 2953 3.3868 3143 3.1813 3336 2.9974 3531 2.8318 33 28 2767 3.6140 2956 3.3832 3147 3.1780 3339 2.9945 3535 2.8291 32 29 2770 3.6100 2959 3.3796 3150 3.1748 3343 2.9916 3538 2.8265 31 30 2773 3.6059 2962 3.3759 3153 3.1716 3346 2.9887 3541 2.8239 30 31 2776 3.6018 2965 3.3723 3156 3.1684 3349 2.9858 3544 2.8213 29 32 2780 3.5978 2968 3.3687 3159 3.1652 3352 2.9829 3548 2.8187 28 33 2783 3.5937 2972 3.3652 3163 3.1620 3356 2.9800 3551 2.8161 27 34 2786 3.5897 2975 3.3616 3166 3.1588 3359 2.9772 3554 2.8135 26 35 2789 3.5856 2978 3.3580 3169 3.1556 3362 2.9743 3558 2.8109 25 36 2792 3.5816 2981 3.3544 3172 3.1524 3365 2.9714 3561 2.8083 24 37 2795 3.5776 2984 3.3509 3175 3.1492 3369 2.9686 3564 2.8057 23 38 2798 3.5736 2987 3.3473 3179 3.1460 3372 2.9657 3567 2.8032 22 39 2801 3.5696 2991 3.3438 3182 3.1429 3375 2.9629 3571 2.8006 21 40 2805 3.5656 2994 3.3402 3185 3.1397 3378 2.9600 3574 2.7980 20 41 2808 3.5616 2997 3.3367 3188 3.1366 3382 2.9572 3577 2.7955 19 42 2811 3.5576 3000 3.3332 3191 3.1334 3385 2.9544 3581 2.7929 18 43 2814 3.5536 3003 3.3297 3195 3.1303 3388 2.9515 3584 2.7903 17 44 2817 3.5497 3006 3.3261 3198 3.1271 3391 2.9487 3587 2.7878 16 45 2820 3.5457 3010 3.3226 3201 3.1240 3395 2.9459 3590 2.7852 15 46 2823 3.5418 3013 3.3191 3204 3.1209 3398 2.9431 3594 2.7827 14 47 2827 3.5379 3016 3.3156 3207 3.1178 3401 2.9403 3597 2.7801 13 48 2830 3.5339 3019 3.3122 3211 3.1146 3404 2.9375 3600 2.7776 12 49 2833 3.5300 3022 3.3087 3214 3.1115 3408 2.9347 3604 2.7751 11 50 2836 3.5261 3026 3.3052 3217 3.1084 3411 2.9319 3607 2.7725 IO 51 2839 3.5222 3029 3.3017 3220 3.1053 3414 2.9291 3610 2.7700 9 52 2842 3.5183 3032 3.2983 3223 3.1022 3417 2.9263 3613 2.7675 8 53 2845 3.5144 3035 3.2948 3227 3.0991 3421 2.9235 3617 2.7650 7 54 2849 3.5105 3038 3.2914 3230 3.0961 3424 2.9208 3620 2.7625 6 55 2852 3.5067 3041 3.2880 3233 3.0930 3427 2.9180 3623 2.7500 5 56 2855 3.5028 3045 3.2845 3236 3.0899 3430 2.9152 3627 2.7575 4 57 2858 3.4989 3048 3.2811 3240 3.0868 3434 2.9125 3630 2.7550 3 58 2861 3.4951 3051 3.2777 3243 3.0838 3437 2.9097 3633 2.7525 2 59 2864 3.4912 3054 3.2743 3246 3.0807 3440 2.9070 3636 2.7500 1 60 2867 3.4874 3057 3.2709 3249 3.0777 3443 2.9042 3640 2.7475 O / cot tan 74° cot tan 73° cot tan cot tan 71° cot tan 70° t 72° NATURAL TANGENTS AND COTANGENTS 65 t 20° 21° 22° 23° 24° t tan cot tan cot tan cot tan cot tan cot o 3640 2.7475 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 60 1 3643 2.7450 3842 2.6028 4044 2.4730 4248 2.3539 4456 2.2443 59 2 3646 2.7425 3845 2.6006 4047 2.4709 4252 2.3520 4459 2.2425 58 3 3650 2.7400 3849 2.5983 4050 2.4689 4255 2.3501 4463 2,2408 57 4 3653 2.7376 3852 2.5961 4054 2.4668 4258 2.3483 4466 2.2390 56 5 3656 2.7351 3855 2.5938 4057 2.4648 4262 2.3464 4470 2.2373 55 6 3659 2.7326 3859 2.5916 4061 2.4627 4265 2.3445 4473 2.2355 54 7 3663 2.7302 3862 2.5893 4064 2.4606 4269 2.3426 4477 2.2338 53 8 3666 2.7277 3865 2.5871 4067 2.4586 4272 2.3407 4480 2.2320 52 9 3669 2.7253 3869 2.5848 4071 2.4566 4276 2.3388 4484 2.2303 51 io 3673 2.7228 3872 2.5826 4074 2.4545 4279 2.3369 4487 2.2286 50 11 3676 2.7204 3875 2.5804 4078 2.4525 4283 2.3351 4491 2.2268 49 12 3679 2.7179 3879 2.5782 4081 2.4504 4286 2.3332 4494 2.2251 48 13 3683 2.7155 3882 2.5759 4084 2.4484 4289 2.3313 4498 2.2234 47 14 3686 2.7130 3885 2.5737 4088 2.4464 4293 2.3294 4501 2.2216 46 15 3689 2.7106 3889 2.5715 4091 2.4443 4296 2.3276 4505 2.2199 45 16 3693 2.7082 3892 2.5693 4095 2.4423 4300 2.3257 4508 2.2182 44 17 3696 2.7058 3895 2.5671 4098 2.4403 4303 2.3238 4512 2.2165 43 18 3699 2.7034 3899 2.5649 4101 2.4383 4307 2.3220 4515 2.2148 42 19 3702 2.7009 3902 2.5627 4105 2.4362 4310 2.3201 4519 2.2130 41 20 3706 2.6985 3906 2.5605 4108 2.4342 4314 2.3183 4522 2.2113 40 21 3709 2.6961 3909 2.5583 4111 2.4322 4317 2.3164 4526 2.2096 39 22 3712 2.6937 3912 2.5561 4115 2.4302 4320 2.3146 4529 2.2079 38 23 3716 2.6913 3916 2.5539 4118 2.4282 4324 2.3127 4533 2.2062 37 24 3719 2.6889 3919 2.5517 4122 2.4262 4327 2.3109 4536 2.2045 36 25 3722 2.6865 3922 2.5495 4125 2.4242 4331 2.3090 4540 2.2028 35 26 3726 2.6841 3926 2.5473 4129 2.4222 4334 2.3072 4543 2.2011 34 27 3729 2.6818 3929 2.5452 4132 2.4202 4338 2.3053 4547 2.1994 33 28 3732 2.6794 3932 2.5430 4135 2.4182 4341 2.3035 4550 2.1977 32 29 3736 2.6770 3936 2.5408 4139 2.4162 4345 2.3017 4554 2.1960 31 30 3739 2.6746 3939 2.53S6 4142 2.4142 4348 2.2998 4557 2.1943 30 31 3742 2.6723 3942 2.5365 4146 2.4122 4352 2.2980 4561 2.1926 29 32 3745 2.6699 3946 2.5343 4149 2.4102 4355 2.2962 4564 2.1909 28 33 3749 2.6675 3949 2.5322 4152 2.4083 4359 2.2944 4568 2.1892 27 34 3752 2.6652 3953 2.5300 4156 2.4063 4362 2.2925 4571 2.1876 26 35 3755 2.6628 3956 2.5279 4159 2.4043 4365 2.2907 4575 2.1859 25 36 3759 2.6605 3959 2.5257 4163 2.4023 4369 2.2889 4578 2.1842 24 37 3762 2.6581 3963 2.5236 4166 2.4004 4372 2.2871 4582 2.1825 23 38 3765 2.6558 3966 2.5214 4169 2.3984 4376 2.2853 4585 2.1808 22 39 3769 2.6534 3969 2.5193 4173 2.3964 4379 2.2835 4589 2.1792 21 40 3772 2.6511 3973 2.5172 4176 2.3945 4383 2.2817 4592 2.1775 20 41 3775 2.6488 3976 2.5150 4180 2.3925 4386 2.2799 4596 2.1758 19 42 3779 2.6464 3979 2.5129 4183 2.3906 4390 2.2781 4599 2.1742 18 43 3782 2.6441 3983 2.5108 4187 2.3886 4393 2.2763 4603 2.1725 17 44 3785 2.6418 3986 2.5086 4190 2.3867 4397 2.2745 4607 2.1708 16 45 3789 2.6395 3990 2.5065 4193 2.3847 4400 2.2727 4610 2.1692 15 46 3792 2.6371 3993 2.5044 4197 2.3828 4404 2.2709 4614 2.1675 14 47 3795 2.6348 3996 2.5023 4200 2.3808 4407 2.2691 4617 2.1659 13 48 3799 2.6325 4000 2.5002 4204 2.3789 4411 2.2673 4621 2.1642 12 49 3802 2.6302 4003 2.4981 4207 2.3770 4414 2.2655 4624 2.1625 11 50 3805 2.6279 4006 2.4960 4210 2.3750 4417 2.2637 4628 2.1609 IO 51 3809 2.6256 4010 2.4939 4214 2.3731 4421 2.2620 4631 2.1592 9 52 3812 2.6233 4013 2.4918 4217 2.3712 4424 2.2602 4635 2.1576 8 53 3815 2.6210 4017 2.4897 4221 2.3693 4428 2.2584 4638 2.1560 7 54 3819 2.6187 4020 2.4876 4224 2.3673 4431 2.2566 4642 2.1543 6 55 3822 2.6165 4023 2.4855 4228 2.3654 4435 2.2549 4645 2.1527 5 56 3825 2.6142 4027 2.4834 4231 2.3635 4438 2.2531 4649 2.1510 4 57 3829 2.6119 4030 2.4813 4234 2.3616 4442 2.2513 4652 2.1494 3 58 3832 2.6096 4033 2.4792 4238 2.3597 4445 2.2496 4656 2.1478 2 59 3835 2.6074 4037 2.4772 4241 2.3578 4449 2.2478 4660 2.1461 1 60 3839 2.6051 4040 2.4751 4245 2.3559 4452 2.2460 4663 2.1445 t cot tan 69° cot tan 68° cot tan 67° cot tan 66° cot tan 65° r 66 NATURAL TANGENTS AND COTANGENTS. / 25° 26° 27° 28° 29° t tan cot „ tan cot tan cot tan cot tan cot O 4663 2.1445 4877 2.0503 5095 1.9626 5317 1.8807 5543 1.8040 60 1 4667 2.1429 4881 2.0488 5099 1.9612 5321 1.8794 5547 1.8028 59 2 4670 2.1413 4885 2.0473 5103 1.9598 5325 1.8781 5551 1.8016 58 3 4674 2.1396 4888 2.0458 5106 1.9584 5328 1.8768 5555 1.8003 57 4 4677 2.1380 4892 2.0443 5110 1.9570 5332 1.8755 5558 1.7991 56 5 4681 2.1364 4895 2.0428 5114 1.9556 5336 1.8741 5562 1.7979 55 6 4684 2.1348 4899 2.0413 5117 1.9542 5340 1.8728 5566 1.7966 54 7 4688 2.1332 4903 2.0398 5121 1.9528 5343 1.8715 5570 1.7954 53 8 4691 2.1315 4906 2.0383 5125 1.9514 5347 1.8702 5574 1.7942 52 9 4695 2.1299 4910 2.0368 5128 1.9500 5351 1.8689 5577 1.7930 51 io 4699 2.1283 4913 2.0353 5132 1.9486 5354 1.8676 5581 1.7917 SO 11 4702 2.1267 4917 2.0338 5136 1.9472 5358 1.8663 5585 1.7905 49 12 4706 2.1251 4921 2.0323 5139 1.9458 5362 1.8650 5589 1.7893 48 13 4709 2.1235 4924 2.0308 5143 1.9444 5366 1.8637 5593 1.7881 47 14 4713 2.1219 4928 2.0293 5147 1.9430 5369 1.8624 5596 1.7868 46 15 4716 2.1203 4931 2.0278 5150 1.9416 5373 1.8611 5600 1.7856 45 16 4720 2.1187 4935 2.0263 5154 1.9402 5377 1.8598 5604 1.7844 44 17 4723 2.1171 4939 2.0248 5158 1.9388 5381 1.8585 5608 1.7832 43 18 4727 2.1155 4942 2.0233 5161 1.9375 5384 1.8572 5612 1.7820 42 19 4731 2.1139 4946 2.0219 5165 1.9361 5388 1.8559 5616 1.7808 41 20 4734 2.1123 4950 2.0204 5169 1.9347 5392 1.8546 5619 1.7796 40 21 4738 2.1107 4953 2.0189 5172 1.9333 5396 1.8533 5623 1.7783 39 22 4741 2.1092 4957 2.0174 5176 1.9319 5399 1.8520 5627 1.7771 38 23 4745 2.1076 4960 2.0160 5180 1.9306 5403 1.8507 5631 1.7759 37 24 4748 2.1060 4964 2.0145 5184 1.9292 5407 1.8495 5635 1.7747 36 25 4752 2.1044 4968 2.0130 5187 1.9278 5411 1.8482 5639 1.7735 35 26 4755 2.1028 4971 2.0115 5191 1.9265 5415 1.8469 5642 1.7723 34 27 4759 2.1013 4975 2.0101 5195 1.9251 5418 1.8456 5646 1.7711 33 28 4763 2.0997 4979 2.0086 5198 1.9237 5422 1.8443 5650 1.7699 32 29 4 766 2.0981 4982 2.0072 5202 1.9223 5426 1.8430 5654 1.7687 31 30 4770 2.0965 4986 2.0057 5206 1.9210 5430 1.8418 5658 1.7675 30 31 4773 2.0950 4989 2.0042 5209 1.9196 5433 1.8405 5662 1.7663 29 32 4777 2.0934 4993 2.0028 5213 1.9183 5437 1.8392 5665 1.7651 28 33 4780 2.0918 4997 2.0013 5217 1.9169 5441 1.8379 5669 1.7639 27 34 4784 2.0903 5000 1.9999 5220 1.9155 5445 1.8367 5673 1.7627 26 35 4788 2.0887 5004 1.9984 5224 1.9142 5448 1.8354 5677 1.7615 25 36 4791 2.0872 5008 1.9970 5228 1.9128 5452 1.8341 5681 1.7603 24 37 4795 2.0856 5011 1.9955 5232 1.9115 5456 1.8329 5685 1.7591 23 38 4798 2.0840 5015 1.9941 5235 1.9101 5460 1.8316 5688 1.7579 22 39 4802 2.0825 5019 1.9926 5239 1.9088 5464 1.8303 5692 1.7567 21 40 4806 2.0809 5022 1.9912 5243 1.9074 5467 1.8291 5696 1.7556 20 41 4809 2.0794 5026 1.9897 5246 1.9061 5471 1.8278 5700 1.7544 19 42 4813 2.0778 5029 1.9883 5250 1.9047 5475 1.8265 5704 1.7532 18 43 4816 2.0763 5033 1.9868 5254 1.9034 5479 1.8253 5708 1.7520 17 44 4820 2.0748 5037 1.9854 5258 1.9020 5482 1.8240 5712 1.7508 16 45 4823 2.0732 5040 1.9840 5261 1.9007 5486 1.8228 5715 1.7496 15 46 4827 2.0717 5044 1.9825 5265 1.8993 5490 1.8215 5719 1.7485 14 47 4831 2.0701 5048 1.9811 5269 1.8980 5494 1.8202 5723 1.7473 13 48 4834 2.0686 5051 1.9797 5272 1.8967 5498 1.8190 5727 1.7461 12 49 4838 2.0671 5055 1.9782 5276 1.8953 5501 1.8177 5731 1.7449 11 50 4841 2.0655 5059 1.9768 5280 1.8940 5505 1.8165 5735 1.7437 IO 51 4845 2.0640 5062 1.9754 5284 1.8927 5509 1.8152 5739 1.7426 9 52 4849 2.0625 5066 1.9740 5287 1.8913 5513 1.8140 5743 1.7414 8 53 4852 2.0609 5070 1.9725 5291 1.8900 5517 1.8127 5746 1.7402 7 54 4856 2.0594 5073 1.9711 5295 1.8887 5520 1.8115 5750 1.7391 6 55 4859 2.0579 5077 1.9697 5298 1.8873 5524 1.8103 5754 1.7379 5 56 4863 2.0564 5081 1.9683 5302 1.8860 5528 1.8090 5758 1.7367 4 57 4867 2.0549 5084 1.9669 5306 1.8847 5532 1.8078 5762 1.7355 3 58 4870 2.0533 5088 1.9654 5310 1.8834 5535 1.8065 5766 1.7344 2 59 4874 2.0518 5092 1.9640 5313 1.8820 5539 1.8053 5770 1.7332 1 60 4877 2.0503 5095 1.9626 5317 1.8807 5543 1.8040 5774 1.7321 O cot tan 64° cot tan 63° cot tan 62° cot tan cot tan 60° f f 61° NATURAL TANGENTS AND COTANGENTS. 67 f i 30° 31° 32° 33° 34° t tan cot tan cot tan cot tan cot tan cot o 5774 1.7321 6009 1.6643 6249 1.6003 6494 1.5399 6745 1.4826 60 1 5777 1.7309 6013 1.6632 6253 1.5993 6498 1.5389 6749 1.4816 59 2 5781 1.7297 6017 1.6621 6257 1.5983 6502 1.5379 6754 1.4807 58 3 5785 1.7286 6020 1.6610 6261 1.5972 6506 1.5369 6758 1,4798 57 4 5789 1.7274 6024 1.6599 6265 1.5962 6511 1.5359 6762 1.4788 56 5 5793 1.7262 6028 1.6588 6269 1.5952 6515 1.5350 6766 1.4779 55 6 5797 1.7251 6032 1.6577 6273 1.5941 6519 1.5340 6771 1.4770 54 7 5801 1.7239 6036 1.6566 6277 1.5931 6523 1.5330 6775 1.4761 53 8 5805 1.7228 6040 1.6555 6281 1.5921 6527 1.5320 6779 1.4751 52 9 5808 1.7216 6044 1.6545 6285 1.5911 6531 1.5311 6783 1.4742 51 io 5812 1.7205 6048 1.6534 6289 1.5900 6536 1.5301 6787 1.4733 50 11 5816 1.7193 6052 1.6523 6293 1.5890 6540 1.5291 6792 1.4724 49 12 5820 1.7182 6056 1.6512 6297 1.5880 6544 1.5282 6796 1.4715 48 13 5824 1.7170 6060 1.6501 6301 1.5869 6548 1.5272 6800 1.4705 47 14 5828 1.7159 6064 1.6490 6305 1.5859 6552 1.5262 6805 1.4696 46 15 5832 1.7147 6068 1.6479 6310 1.5849 6556 1.5253 6809 1.4687 45 16 5836 1.7136 6072 1.6469 6314 1.5839 6560 1.5243 6813 1.4678 44 17 5840 1.7124 6076 1.6458 6318 1.5829 6565 1.5233 6817 1.4669 43 18 5844 1.7113 6080 1.6447 6322 1.5818 6569 1.5224 6822 1.4659 42 19 5847 1.7102 6084 1.6436 6326 1.5808 6573 1.5214 6826 1.4650 41 20 5851 1.7090 6088 1.6426 6330 1.5798 6577 1.5204 6830 1.4641 40 21 5855 1.7079 6092 1.6415 6334 1.5788 6581 1.5195 6834 1.4632 39 22 5859 1.7067 6096 1.6404 6338 1.5778 6585 1.5185 6839 1.4623 38 23 5863 1.7056 6100 1.6393 6342 1.5768 6590 1.5175 6843 1.4614 37 24 5867 1.7045 6104 1.6383 6346 1.5757 6594 1.5166 6847 1.4605 36 25 5871 1.7033 6108 1.6372 6350 1.5747 6598 1.5156 6851 1.4596 35 26 5875 1.7022 6112 1.6361 6354 1.5737 6602 1.5147 6856 1.4586 34 27 5879 1.7011 6116 1.6351 6358 1.5727 6606 1.5137 6860 1.4577 33 28 5883 1.6999 6120 1.6340 6363 1.5717 6610 1.5127 6864 1.4568 32 29 5887 1.6988 6124 1.6329 6367 1.5707 6615 1.5118 6869 1.4559 31 30 5890 1.6977 6128 1.6319 6371 1.5697 6619 1.5108 6873 1.4550 30 31 5894 1.6965 6132 1.6308 6375 1.5687 6623 1.5099 6877 1.4541 29 32 5898 1.6954 6136 1.6297 6379 1.5677 6627 1.5089 6881 1.4532 28 33 5902 1.6943 6140 1.6287 6383 1.5667 6631 1.5080 6886 1.4523 Z7 34 5906 1.6932 6144 1.6276 6387 1.5657 6636 1.5070 6890 1.4514 26 35 5910 1.6920 6148 1.6265 6391 1.5647 6640 1.5061 6894 1.4505 25 36 5914 1.6909 6152 1.6255 6395 1.5637 6644 1.5051 6899 1.4496 24 37 5918 1.6898 6156 • 1.6244 6399 1.5627 6648 1.5042 6903 1.4487 23 38 5922 1.6887 6160 1.6234 6403 1.5617 6652 1.5032 6907 1.4478 22 39 5926 1.6875 6164 1.6223 6408 1.5607 6657 1.5023 6911 1.4469 21 40 5930 1.6864 6168 1.6212 6412 1.5597 6661 1.5013 6916 1.4460 20 41 5934 1.6853 6172 1.6202 6416 1.5587 6665 1.5004 6920 1.4451 19 42 5938 1.6842 6176 1.6191 6420 1.5577 6669 1.4994 6924 1.4442 18 43 5942 1.6831 6180 1.6181 6424 1.5567 6673 1.4985 6929 1.4433 17 44 5945 1.6820 6184 1.6170 6428 1.5557 6678 1.4975 o933 1.4424 16 45 5949 1.6808 6188 1.6160 6432 1.5547 6682 1.4966 6937 1.4415 15 46 5953 1.6797 6192 1.6149 6436 1.5537 6686 1.4957 6942 1.4406 14 47 5957 1.6786 6196 1.6139 6440 1.5527 6690 1.4947 6946 1.4397 13 48 5961 1.6775 6200 1.6128 6445 1.5517 6694 1.4938 6950 1.4388 12 49 5965 1.6764 6204 1.6118 6449 1.5507 6699 1.4928 6954 1.4379 11 SO 5969 1.6753 6208 1.6107 6453 1.5497 6703 1.4919 6959 1.4370 IO 51 5973 1.6742 6212 1.6097 6457 1.5487 6707 1.4910 6963 1.4361 9 52 5977 1.6731 6216 1.6087 6461 1.5477 6711 1.4900 6967 1.4352 8 53 5981 1.6720 6220 1.6076 6465 1.5468 6716 1.4891 6972 1.4344 7 54 5985 1.6709 6224 1.6066 6469 1.5458 6720 1.4882 6976 1.4335 6 55 5989 1.6698 6228 1.6055 6473 1.5448 6724 1.4872 6980 1.4326 5 56 5993 1.6687 6233 1.6045 6478 1.5438 6728 1.4863 6985 1.4317 4 57 5997 1.6676 6237 1.6034 6482 1.5428 6732 1.4854 6989 1.4308 3 58 6001 1.6665 6241 1.6024 6486 1.5418 6737 1.4844 6993 1.4299 2 59 6005 1.6654 6245 1.6014 6490 1.5408 6741 1.4835 6998 1.4290 1 60 6009 1.6643 6249 1.6003 6494 1.5399 6745 1.4826 7002 1.4281 O cot tan cot tan cot tan cot tan cot tan f 59° 58° 57° 56° 55° r 68 NATURAL TANGENTS AND COTANGENTS i. t 35° 36° 37° 38° 39° t tan cot tan cot tan cot tan cot tan cot O 7002 1.4281 7265 1.3764 7536 1.3270 7813 1.2799 8098 1.2349 60 l 7006 1.4273 7270 1.3755 7540 1.3262 7818 1.2792 8103 1.2342 59 2 7011 1.4264 7274 1.3747 7545 1.3254 7822 1.2784 8107 1.2334 58 3 7015 1.4255 7279 1.3739 7549 1.3246 7827 1.2776 8112 1.2327 57 4 7019 1.4246 7283 1.3730 7554 1.3238 7832 1.2769 8117 1.2320 56 5 7024 1.4237 7288 1.3722 7558 1.3230 7836 1.2761 8122 1.2312 55 6 7028 1.4229 7292 1.3713 7563 1.3222 7841 1.2753 8127 1.2305 54 7 7032 1.4220 7297 1.3705 7568 1.3214 7846 1.2746 8132 1.2298 53 8 7037 1.4211 7301 1.3697 7572 1.3206 7850 1.2738 8136 1.2290 52 9 7041 1.4202 7306 1.3688 7577 1.3198 7855 1.2731 8141 1.2283 51 10 7046 1.4193 7310 1.3680 7581 1.3190 7860 1.2723 8146 1.2276 50 11 7050 1.4185 7314 1.3672 7586 1.3182 7865 1.2715 8151 1.2268 49 12 7054 1.4176 7319 1.3663 7590 1.3175 7869 1.2708 8156 1.2261 48 13 7059 1.4167 7323 1.3655 7595 1.3167 7874 1.2700 8161 1.2254 47 14 7063 1.4158 7328 1.3647 7600 1.3159 7879 1.2693 8165 1.2247 46 15 7067 1.4150 7332 1.3638 7604 1.3151 7883 1.2685 8170 1.2239 45 16 7072 1.4141 7337 1.3630 7609 1.3143 7888 1.2677 8175 1.2232 44 17 7076 1.4132 7341 1.3622 7613 1.3135 7893 1.2670 8180 1.2225 43 18 7080 1.4124 7346 1.3613 7618 1.3127 7898 1.2662 8185 1.2218 42 19 7085 1.4115 7350 1.3605 7623 1.3119 7902 1.2655 8190 1.2210 41 20 7089 1.4106 7355 1.3597 7627 1.3111 7907 1.2647 8195 1.2203 40 21 7094 1.4097 7359 1.3588 7632 1.3103 7912 1.2640 8199 1.2196 39 22 7098 1.4089 7364 1.3580 7636 1.3095 7916 1.2632 8204 1.2189 38 23 7102 1.4080 7368 1.3572 7641 1.3087 7921 1.2624 8209 1.2181 37 24 7107 1.4071 7373 1.3564 7646 1.3079 7926 1.2617 8214 1.2174 36 25 7111 1.4063 7377 1.3555 7650 1.3072 7931 1.2609 8219 1.2167 35 26 7115 1.4054 7382 1.3547 7655 1.3064 7935 1.2602 8224 1.2160 34 27 7120 1.4045 7386 1.3539 7659 1.3056 7940 1.2594 8229 1.2153 33 28 7124 1.4037 7391 1.3531 7664 1.3048 7945 1.2587 8234 1.2145 31 29 7129 1.4028 7395 1.3522 7669 1.3040 7950 1.2579 8238 1.2138 31 30 7133 1.4019 7400 1.3514 7673 1.3032 7954 1.2572 8243 1.2131 30 31 7137 1.4011 7404 1.3506 7678 1.3024 7959 1.2564 824S 1.2124 29 32 7142 1.4002 7409 1.3498 7683 1.3017 7964 1.2557 8253 1.2117 28 33 7146 1.3994 7413 1.3490 7687 1.3009 7969 1.2549 8258 1.2109 27 34 7151 1.3985 7418 1.3481 7692 1.3001 7973 1.2542 8263 1.2102 26 35 7155 1.3976 7422 1.3473 7696 1.2993 7978 1.2534 8268 1.2095 25 36 7159 1.3968 7427 1.3465 7701 1.2985 7983 1.2527 8273 1.2088 24 37 7164 1.3959 7431 1.3457 7706 1.2977 7988 1.2519 8278 1.2081 23 38 7168 1.3951 7436 1.3449 7710 1.2970 7992 1.2512 8283 1.2074 22 39 7173 1.3942 7440 1.3440 7715 1.2962 7997 1.2504 8287 1.2066 21 40 7177 1.3934 7445 1.3432 7720 1.2954 8002 1.2497 8292 1.2059 20 41 7181 1.3925 7449 1.3424 7724 1.2946 8007 1.2489 8297 1.2052 19 42 7186 1.3916 7454 1.3416 7729 1.2938 8012 1.2482 8302 1.2045 18 43 7190 1.3908 7458 1.3408 7734 1.2931 8016 1.2475 8307 1.2038 17 44 7195 1.3899 7463 1.3400 7738 1.2923 8021 1.2467 8312 1.2031 16 45 7199 1.3891 7467 1.3392 7743 1.2915 8026 1.2460 8317 1.2024 15 46 7203 1.3882 7472 1.3384 7747 1.2907 8031 1.2452 8322 1.2017 14 47 7208 1.3874 7476 1.3375 7752 1.2900 8035 1.2445 8327 1.2009 13 48 7212 1.3865 7481 1.3367 7757 1.2892 8040 1.2437 8332 1.2002 12 49 7217 1.3857 7485 1.3359 7761 1.2884 8045 1.2430 8337 1.1995 11 50 7221 1.3848 7490 1.3351 7766 1.2876 8050 1.2423 8342 1.1988 io 51 7226 1.3840 7495 1.3343 7771 1.2869 8055 1.2415 8346 1.1981 9 52 7230 1.3831 7499 1.3335 7775 1.2861 8059 1.2408 8351 1.1974 8 53 7234 1.3823 7504 1.3327 7780 1.2853 8064 1.2401 8356 1.1967 7 54 7239 1.3814 7508 1.3319 7785 1.2846 8069 1.2393 8361 1.1960 6 55 7243 1.3806 7513 1.3311 7789 1.2838 8074 1.2386 8366 1.1953 5 56 7248 1.3798 7517 1.3303 7794 1.2830 8079 1.2378 8371 1.1946 4 57 7252 1.3789 7522 1.3295 7799 1.2822 8083 1.2371 8376 1.1939 3 58 7257 1.3781 7526 1.3287 7803 1.2815 8088 1.2364 8381 1.1932 2 59 7261 1.3772 7531 1.3278 7808 1.2807 8093 1.2356 8386 1.1925 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 / 64° 53° 52° 51° 50° t NATURAL TANGENTS AND COTANGENTS. 69 ; 40° 41° 42° 43° 44° f tan cot tan cot tan cot tan cot tan cot O 8391 1.1918 8693 1.1504 9004 1.1106 9325 1.0724 9657 1.0355 60 1 8396 1.1910 8698 1.1497 9009 1.1100 9331 1.0717 9663 1.0349 59 2 8401 1.1903 8703 1.1490 9015 1.1093 9336 1.0711 9668 1.0343 58 3 8406 1.1896 8708 1.1483 9020 1.1087 9341 1.0705 9674 1.0337 57 4 8411 1.1889 8713 1.1477 9025 1.1080 9347 1.0699 9679 1.0331 56 5 8416 1.1882 8718 1.1470 9030 1.1074 9352 1.0692 9685 1.0325 55 6 8421 1.1875 8724 1.1463 9036 1.1067 9358 1.0686 9691 1.0319 54 7 8426 1.1868 8729 1.1456 9041 1.1061 9363 1.0680 9696 1.0313 53 8 8431 1.1861 8734 1.1450 9046 1.1054 9369 1.0674 9702 1.0307 52 9 8436 1.1854 8739 1.1443 9052 1.1048 9374 1.0668 9708 1.0301 51 10 8441 1.1847 8744 1.1436 9057 1.1041 9380 1.0661 9713 1.0295 50 11 8446 1.1840 8749 1.1430 9062 1.1035 9385 1.0655 9719 1.0289 49 12 8451 1.1833 8754 1.1423 9067 1.1028 9391 1.0649 9725 1.0283 48 13 8456 1.1826 8759 1.1416 9073 1.1022 9396 1.0643 9730 1.0277 47 14 8461 1.1819 8765 1.1410 9078 1.1016 9402 1 .0637 9736 1.0271 46 15 8466 1.1812 8770 1.1403 9083 1.1009 9407 1.0630 9742 1.0265 45 16 8471 1.1806 8775 1.1396 9089 1.1003 9413 1.0624 9747 1.0259 44 17 8476 1.1799 8780 1.1389 9094 1.0996 9418 1.0618 9753 1.0253 43 18 8481 1.1792 8785 1.1383 9099 1.0990 9424 1.0612 9759 1.0247 42 19 8486 1.1785 8790 1.1376 9105 1.0983 9429 1.0606 9764 1.0241 41 20 8491 1.1778 8796 1.1369 9110 1.0977 9435 1.0599 9770 1.0235 40 21 8496 1.1771 8801 1.1363 9115 1.0971 9440 1.0593 9776 1.0230 39 22 8501 1.1764 8806 1.1356 9121 1.0964 9446 1.0587 9781 1.0224 38 23 8506 1.1757 8811 1.1349 9126 1.0958 9451 1.0581 9787 1.0218 37 24 8511 1.1750 8816 1.1343 9131 1.0951 9457 1.0575 9793 1.0212 36 ^5 8516 1.1743 8821 1.1336 9137 1.0945 9462 1.0569 9798 1.0206 35 26 8521 1.1736 8827 1.1329 9142 1.0939 9468 1.0562 9804 1.0200 34 27 8526 1.1729 8832 1.1323 9147 1.0932 9473 1.0556 9810 1.0194 33 28 8531 1.1722 8837 1.1316 9153 1.0926 9479 1.0550 9816 1.0188 32 29 8536 1.1715 8842 1.1310 9158 1.0919 9484 1.0544 9821 1.0182 31 30 8541 1.1708 8847 1.1303 9163 1.0913 9490 1.0538 9827 1.0176 30 31 8546 1.1702 8852 1.1296 9169 1.0907 9495 1.0532 9833 1.0170 29 32 8551 1.1695 8858 1.1290 9174 1.0900 9501 1.0526 9838 1.0164 28 33 8556 1.1688 8863 1.1283 9179 1.0894 9506 1.0519 9844 1.0158 27 34 8561 1.1681 8868 1.1276 9185 1.0888 9512 1.0513 9850 1.0152 26 35 8566 1.1674 8873 1.1270 9190 1.0881 9517 1.0507 9856 1.0147 25 36 8571 1.1667 8878 1.1263 9195 1.0875 9523 1.0501 9861 1.0141 24 37 8576 1.1660 8884 1.1257 9201 1.0869 9528 1.0495 9867 1.0135 23 38 8581 1.1653 8889 1.1250 9206 1.0862 9534 1.0489 9873 1.0129 22 39 8586 1.1647 8894 1.1243 9212 1.0856 9540 1.0483 9879 1.0123 21 40 8591 1.1640 8899 1.1237 9217 1.0850 9545 1.0477 9884 1.0117 20 41 8596 1.1633 8904 1.1230 9222 1.0843 9551 1.0470 9890 1.0111 19 42 8601 1.1626 8910 1.1224 9228 1.0837 9556 1.0464 9896 1.0105 18 43 8606 1.1619 8915 1.1217 9233 1.0831 9562 1.0458 9902 1.0099 17 44 8611 1.1612 8920 1.1211 9239 1.0824 9567 1.0452 9907 1.0094 16 45 8617 1.1606 8925 1.1204 9244 1.0818 9573 1.0446 9913 1.0088 15 46 8622 1.1599 8931 1.1197 9249 1.0812 9578 1.0440 9919 1.0082 14 47 8627 1.1592 8936 1.1191 9255 1.0805 9584 1.0434 9925 1.0076 13 48 8632 1.1585 8941 1.1184 9260 1.0799 9590 1.0428 9930 1.0070 12 49 8637 1.1578 8946 1.1178 9266 1.0793 9595 1.0422 9936 1.0064 11 50 8642 1.1571 8952 1.1171 9271 1.0786 9601 1.0416 9942 1.0058 10 51 8647 1.1565 8957 1.1165 9276 1.0780 9606 1.0410 9948 1.0052 9 52 8652 1.1558 8962 1.1158 9282 1.0774 9612 1.0404 9954 1.0047 8 53 8657 1.1551 8967 1.1152 9287 1.0768 9618 1.0398 9959 1.0041 7 54 8662 1.1544 8972 1.1145 9293 1.0761 9623 1.0392 9965 1.0035 6 55 8667 1.1538 8978 1.1139 9298 1.0755 9629 1.0385 9971 1.0029 5 56 8672 1.1531 8983 1.1132 9303 1.0749 9634 1.0379 9977 1.0023 4 57 8678 1.1524 8988 1.1126 9309 1.0742 9640 1.0373 9983 1.0017 3 58 8683 1.1517 8994 1.1119 9314 1.0736 9646 1.0367 9988 1.0012 2 59 8688 1.1510 8999 1.1113 9320 1.0730 9651 1.0361 9994 1.0006 1 60 8693 1.1504 9004 1.1106 9325 1.0724 9657 1.0355 1000 1.0000 O / cot tan 49° cot tan 48° cot tan 47° cot tan cot tan 45° t 46° A TABLE OF THE ANGLES Which every Point and Quarter Point of the Compass makes with the Meridian. North. Points. o- 3 I O / (/ Points. South. N. by E. N. by W. 2 48 45 5 37 30 8 26 15 11 15 o-V 4 o-v 2 0-34 S. by E. S. by W. N.N.E. N.N.W. 14 3 45 16 52 30 19 41 15 22 30 S.S.E. S.S.W. N.E. by N. N.W. by N. 3 3 25 18 45 28 7 30 30 56 15 33 45 S.E. by S. S.W. by S. N.E. N.W. 3-V4 3-% 3-% 4 36 33 45 39 22 30 42 11 15 45 3-V* 3-% 3-% 4 S.E. S.W. N.E. by E. N.W. by W. Sit 5 47 48 46 50 37 30 53 26 15 56 15 4 -V4 4-? 5 S.E. by E. S.W. by W. E.N.E. W.N.W. 6 59 3 45 61 62 30 64 41 15 07 30 5-% 5-% 5-sJ 6 E.S.E. W.S.W. E. by N. W. by N. j:f 70 18 45 73 7 30 75 56 15 78 45 6-% 6-V0 6-% 7 E. by S. W. by S. East. West. P 81 33 46 84 22 30 87 11 15 90 East. West. o V w <* BOOK UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. DEC 24 1^ 7 R EC'D l.q 'MAY 4 12 190 MAY 2 2 1953 LU