/ i^\t'0 IN MEMORIAM FLORIAN CAJORl Vla^i Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/fiveplacelogaritOOwentrich MATHEMATICAL TEXT-BOOKS By G. A. WENTWORTH, A.M. Mental Arithmetic. Elementary Arithmetic. Practical Arithmetic. Primary Arithmetic. Grammar School Arithmetic. High School Arithmetic. High School Arithmetic (Abridged). First Steps in Algebra. School Algebra. College Algebra. Elements of Algebra. Complete Algebra. Shorter Course in Algebra. Higher Algebra. New Plane Geometry. New Plane and Solid Geometry. Syllabus of Geometry. Geometrical Exercises. Plane and Solid Geometry and Plane Trigonometry. New Plane Trigonometry. New Plane Trigonometry, with Tables. New Plane and Spherical Trigonometry. New Plane and Spherical Trig., with Tables. New Plane and Spherical Trig., Surv., and Nav, New Plane Trig, and Surv., with Tables. New Plane and Spherical Trig., Surv., with Tables. Analytic Geometry. PLANE AND SPHERICAL TEIGONOMETOY AND TABLES BY G. A. WENTWOETH, A.M. AUTHOR OF A SERIES OF TEXT -BOOKS IN MATHEMATICS REVISED EDITION Boston, U.S.A., and London GINN & COMPANY, PUBLISHEES 1897 Entered, according to Act of Congress, in tM year 1882, by G. A. WENTWORTH in the Office of the Librarian of Congress, at Washington. Copyright, 1895, by G. A. Wentwobth. (QJ\^5 ( PEEFACE. "TN 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, all investigations that are important only for the special student have been omitted, except the development of functions in series. 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 devising the simplest proofs for the propositions, and in exhibiting the best methods of arranging the logarithmic work. The author is under particular obligation for assistance to G. A. Hill, A.M., of Cambridge, Mass., to Prof. James L. Patterson, of Schenectady, N.Y., to Dr. F. N". Cole, of Ann Arbor, Mich., and to Prof. S. F. Norris, of Baltimore, Md. G. A. WENTWORTH. Exeter, N.H., July, 1895. iv]S05100 COJ^TEISTTS. PLANE TRIGONOMETRY. CHAPTER I. Functions op Acute Angles : Angular measure, page 1 ; trigonometric functions, 3 ; representation of functions by lines, 7 ; changes in the functions as the angle changes, 10 ; functions of complementary angles, 11 ; relations of the functions of an angle, 12 ; formulas for finding all the other functions of an angle, when one function of the angle is given, 15 ; functions of 45°, 30°, 60°, 17. CHAPTER 11. The Right Triangle: Given parts of a triangle, 19. Solutions without logarithms, 19 Case I., when an acute angle and the hypotenuse are given, 19 Case II., when an acute angle and the opposite leg are given, 20 Case III., when an acute angle and an adjacent leg are given, 20 Case IV., when the hypotenuse and a leg are given, 21 Case v., when the two legs are given, 21. General method of solving a right triangle, 22 ; solutions by logarithms , 24 ; area of the right triangle, 26 ; the isosceles triangle, 31 ; the regular polygon, 33. CHAPTER III. Goniometry: Definition of goniometry, 36 ; angles of any magnitude, 36 ; general definitions of the functions of angles, 37 ; algebraic signs of the func- tions, 39 ; functions of a variable angle, 40 ; functions of angles greater than 360°, 42 ; formulas for acute angles extended to all angles, 43 ; reduction of the functions of all angles to the functions of angles in the first quadrant, 46 ; functions of angles that differ by 90°, 48 ; functions of a negative angle, 49 ; functions of the sum of two angles, 51 ; func- tions of the difference of two angles, 53 ; functions of twice an angle, 55; functions of half an angle, 55 ; sums and differences of functions, 56. CHAPTER IV. The Oblique Triangle : Law of sines, 60 ; law of cosines, 62 ; law of tangents, 62. Solu- tions: Case I., when one side and two angles are given, 64 ; Case II., VI TRIGONOMETRY. when two sides and the angle opposite to one of them are given, 66 ; Case III., when two sides and the included angle are given, 71 ; Case IV., when the three sides are given, 74 ; area of a triangle, 78. CHAPTER V. Miscellaneous Examples : Plane Trigonometry, 82 ; Goniometry, 99. Examination Papers, 106. CHAPTER VI. Construction of Tables : Logarithms, 117; exponential and logarithmic series, 120; trigo- nometric functions of small angles, 125 ; Simpson's method of con- structing a trigonometric table, 127 ; De Moivi-e's theorem, 128 ; expansion of sinx, cos(c, and tanx, in infinite series, 132. SPHERICAL TRIGONOMETRY. CHAPTER VII. The Right Spherical Triangle : Introduction, 135 ; formulas relating to right spherical triangles, 137 ; Napier's rules, 141. Solutions: Case I., when the two legs are given, 142 ; Case II., when the hypotenuse and a leg are given, 142 ; Case III., when a leg and the opposite angle are given, 143 ; Case IV., when a leg and an adjacent angle are given, 143 ; Case V., when the hypotenuse and an oblique angle are given, 144; Case VI., when the two oblique angles are given, 144. The isosceles spherical triangle, 149. CHAPTER VIII. The Oblique Spherical Triangle : Fundamental formulas, 150 ; formulas for half angles and sides, 152 ; Gauss's equations and Napier's analogies, 154. Solutions : Case I., when two sides and the included angle are given, 156; Case II., when two angles and the included side are given, 158 ; Case III., when two sides and an angle opposite to one of them are given, 160 ; Case IV. , when two angles and a side opposite to one of them are given, 162; Case V., when the three sides are given, 163; Case VI., when the three angles are given, 164. Area of a spherical triangle, 166. CHAPTER IX. Applications of Spherical Trigonometry: To reduce an angle measured in space to the horizon, 170; to find the distance between two places on the earth's surface, when the latitudes of the places and the difference in their longitudes are known, 171 ; the celestial sphere, 171 ; spherical co-ordinates, 174 ; the astro- nomical triangle, 176 ; astronomical problems, 177. PLAICE TEIGONOMETEY. CHAPTER I. TRIGONOMETRIC FUNCTIONS OP ACUTE ANGLES. § 1. Angular Measure. As lengths are measured in terms of various conventional units, as the foot, meter, etc., so different units for measuring angles are employed, or have been proposed. In the common or sexagesimal system the circumference of a circle is divided into 360 equal parts. The angle at the centre subtended by each of these parts is taken as the unit angle and is called a degree. The degree is subdivided into 60 minutes, and the minute into 60 seconds. A right angle is equal to 90 degrees. Note. The sexagesimal system was invented by the early Babylonian astronomers in conformity with their year of 360 days. In the circular system an arc of a circle is laid off equal in length to the radius. The angle at the centre subtended by this arc is taken as the unit angle and is called a radian. The number of radians in 360° is equal to the number of times the length of the radius is contained in the cir- cumference. It is proved in Geometry that this number is 2 7r(7r = 3.1416) for all circles; therefore the radian is the same angle in all circles. 2 TRIGONOMETRY. Since the circumference of a circle is 2 tt times the radius, 27r radians = 360°, and tt radians = 180° ; 180° therefore, 1 radian = = 57° 17' 45" TT IT and 1 degree = -r-^ radian = 0.017453 radian. By the last two equations the measure of an angle can be changed from radians to degrees or from degrees to radians. 180° Thus, 2 radians = 2 X -^— = 2 X (57° 17' 45") = 114° 35' 30". ' TT Note. The circular system came into use early in tlie last century. It is found more convenient in the higher mathematics, where the radians are expressed simply as numbers. Thus the angle tt means tt radians, and the angle 3 means 3 radians. On the introduction of the metric system of weights and measures at the close of the last century, it was proposed to divide the right angle into 100 equal parts called grades^ which were to be taken as units. The grade was subdivided into 100 minutes and the minute into 100 seconds. This French or centesimal system, however, never came into actual use. Exercise I. [Assume ;r = 3.1416.] 1. Keduce the following angles to circular measure, express- ing the results as fractions of tt.. 60°, 45°, 150°, 195°, 11° 15', 123° 45', 37° 30'. 2 3 2. How many degrees are there in - ir radians ? t tt radians ? - TT radians ? 77; tt radians ? 3-r tt radians ? o lb lo 3. What decimal part of a radian is 1° ? 1'? 4. How many seconds in a radian ? ^"" TRIGONOMETRIC FUNCTIONS. 3 5. Express in radians one of the interior angles of a regular octagon ; dodecagon. 6. On a circle of 50 ft. radius an arc of 10 ft. is laid off ; how many degrees does the arc subtend at the centre ? 7. The earth^s equatorial radius is approximately 3963 miles. If two points on the equator are 1000 miles apart, what is their difference in longitude ? 8. If the difference in longitude of two points on the equator is 1°, what is the distance between them in miles ? 9. What is the radius of a circle, if an arc of 1 foot sub- tends an angle of 1° at the centre ? 10. In how many hours is a point on the equator carried by the earth's rotation on its axis through a distance equal to the earth's radius ? 11. The minute hand of a clock is 3|- ft. long ; how far does its extremity move in 25 minutes? [Take 7r = ^.] 12. A wheel makes 15 revolutions a second ; how long does it take to turn through 4 radians ? [Take ir = ^-.'] § 2. The Trigonometric Functions. 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 riffht triangle, and for this purpose employs the ratios of its sides. TRIGONOMETRY. Let MAN (Fig. 1) be an acute angle. If from any points B, D, F, in one of its sides perpendiculars BC, DE, 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 C Fig. 1 pair, are equal. That is, AC^AE^AG AC _AE _AG AB~ AD~ AF' BC~ DE~ FG' These ratios, therefore, remain unchanged so long as the angle A remains unchanged. 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 all together 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. These six ratios are called the Trigonometric Functions of the angle A. / TRIGONOMETRIC FUNCTIONS. O To these six ratios are often added the two following func- tions, which also depend only on the angle A : VII. The versed sine of ^ is 1 — cos A and is written vers A. VIII. The cover sed sine oi A is 1 — sin A and is written covers A. In the right triangle ABC (Fig. 2) let a, b, c denote the lengths of the sides opposite to the acute an- gles A, B, and the right angle C, respectively, these lengths being all expressed in terms of a common unit. Then, a opposite leg sm J = -=r 7 • c hypotenuse tan J = a opposite leg b adjacent leg' cos^ = cot^ = b adjacent leg c hypotenuse ' b adjacent leg a opposite leg' c hypotenuse b adjacent leg' c hypotenuse esc -4 vT 1 ■ a opposite leg h c-b vers A=^L = . G c a c — covers ^=1 = c c Exercise IT. 1. What are the functions of the other acute angle B of the triangle ABC (Fig. 2) ? 2. If ^ + ^ = 90°, prove sin A = cos Bf cos A = sin B, tan A = cot B, cot A =ta,nB, sec A = CSC B, esc A = sec B, vers A = covers B, covers A = vers B. 6 TRIGONOMETRY. 3. Find the values of the functions of A, if a, b, c respec- tively have the following values : (i.) 3, 4, 5. (iii.) S, 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.) 2mn, m^ — n^, m^-\-n^, (iii.) pqr, qrs, rsp. ... ^ 2x7/ , x^-\-ip' .. . mn mv nr (ii.) — ^,x-\-y, — ^^- (iv.) — , — , — ^ ^ X — y X — 7/ ^ ^ 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, & = 143. 11. a=4^^, b=-J 2pq. 9. a = 0.264, c = 0.26^. 12. a=V^+^, c=p^q. 10. & = 9.5, c = 19.3. 13. b=2^^, c=p-\-q. Compute the functions of A when, 14. a = 2b, 16. a-\-b = ^c. 15. a = fc. 17. a — b = ^. 18. Find a if sin ^ = f and c = 20.5. 19. Find b if cos A = 0.44 and c = 3.5. 20. Find a if tan ^ = -y- and b = 2j\. TRIGONOMETRIC FUNCTIONS. 1 21. Find 5 if cot ^ = 4 and a = 17. 22. Find c if sec ^ = 2 and b = 20. 23. Find c if esc A = 6.45 and a = 35.6. Construct a right triangle : given, 24. c = 6, tan^ = |. '26. b = 2, sin^ = 0.6. 25. a = 3.5, cos^=:f -27. b = ^, csc^ = 4. 28. In a right triangle, c = 2.5 miles, sin ^ = 0.6, cos^ = 0.8 ; compute the legs. - 29. Construct (with a protractor) the A 20°, 40°, and 70°; determine their functions by measuring the necessary lines, and compare the values obtained in this way with the more nearly correct values given in the following table : 20° 40° 70° sin cos tan cot sec CSC 0.342 0.643 0.940 0.940 0.766 0.342 0.364 0.839 2.747 2.747 1.192 0.364 1.064 1.305 2.924 2.924 1.556 1.064 30. Find, by means of the above table, the legs of a right triangle if ^ == 20°, c = l; also if ^ == 20°, c = 4. 31. In a right triangle, given a = S and c = 5; find the hypotenuse of a similar triangle in which a = 240,000 miles. 32. By dividing the length of a vertical rod by the length of its horizontal shadow, the tangent of the angle of elevation of the sun at the time of observation was found to be 0.82. How high is a tower, if the length of its horizontal shadow at the same time is 174.3 yards ? § 3. Representation of the Functions by Lines. The functions of an angle, being ratios, are numbers ; but we may represent them by lines if we first choose a unit of length, and then construct right triangles, such that the 8 TRIGONOMETRY. Fig. 3. denominators of the ratios shall be equal to this unit. The most convenient way to do this is as follows : About a point (Fig. 3) as a centre, with a radius equal to one unit of length, describe a circle and draw two diameters AA^ and BB^ perpendicular to each other. The circle with radius equal to 1 is called a unit circle, AA^ the horizontal^ and BB^ the vertical diameter. Let AOP be an acute angle, and let its value (in degrees, etc.) be denoted by x. We may regard the Z a; as generated by a radius OP that revolves about from the position OA to the position shown in the figure ; viewed in this way, OP is called the moving radius. Draw PM J. to OA, PN ± to OB. In the rt. A 0PM the hypotenuse 0P = 1; therefore, sinx =^ PM-j cos x^=OM. Since P3I is equal to ON, and ON is the projection of OP on BB', and since OM is the projection of OP on AA', there- fore, in a unit circle, sinic = projection of moving radius on vertical diameter; cos 07 = projection of moving radius on horizontal diameter. Through A and B draw tangents to the circle meeting OP, produced in T and S, respectively; then, in the rt. A OAT, the leg 0A = 1, and in the rt. A OBS, the leg 0B = 1', while the Z OSB = Zx. Therefore, tainx = AT] seca: = OT: cot x = BS', CSC x = OS; veTsx = AM'j covers x = BN. These eight line values (as they may be termed) of the functions are all expressed in terms of the radius of the circle as a unit ; and it is clear that as the angle varies in value the TRIGONOMETRIC FUNCTIONS. 9 line values of the functions will always remain equal numer- ically to the ratio values. Hence, in studying the changes in the functions as the angle is supposed to vary, we may employ the simpler line values instead of the ratio values. Exercise III. U^ 1. Eepresent by lines the functions of a larger angle than that shown in Fig. 3. If X is an acute angle, show that 2. since is less than tancc. 3. seca; is greater than tana;. 4. csccc is greater than cot jr. Construct the angle x if 5. tanir = 3. 7. cosa? = ^. 9. sin a; = 2 cos ic. 6. cscx = 2. 8. sin ic^ cos a;. 10. 4 sin a; = tan a;. 11. Show that the sine of an angle is equal to one-half the chord of twice the angle. 12. Find X if sin x is equal to one-half the side of a regular inscribed decagon. 13. Given x and y, x-\-y being less than 90°; construct the value of sin (x-\-y) — sin x. 14. Given x and y, x-\-y being less than 90°; construct the value of tan (a? + y) — sin (x-\-y)-\- tana? — sina;. Given an angle x ; construct an angle y such that 15. sin?/ = 2 sin a?. 17. tan?/ = 3 tan a?. ! 16. cos ?/ =: J cos ar. 18. sec?/ = csca;. 19. Show by construction that 2 sin ^ > sin 2 A. 20. Given two angles A and^, A-\-B being less than 90°; show that sin (A-\- B)<, sin A -\- sin B . 21. Given sin a; in a unit circle; find the length of a line corresponding in position to sin a; 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. 10 TRIGONOMETRY. § 4. Changes in the Functions as the Angle Changes. If we suppose the /_ AOP, or x (Fig. 4) to increase gradu- ally by the revolution of the moving radius OP about 0, the point F will move along the arc AB towards B, T will move along the tangent AT 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 FM, 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, tangent, and secant also increase, while its cosine, cotangent, and cose- cant 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°, OF will coincide with OA and be parallel to BS. Therefore, FM and AT will vanish, OM will become equal to OA, while BS and OS will each be infinitely long, and be represented in value by the symbol oo. And if we suppose x to increase to 90°, OF will coincide with OB and be parallel to AT. Therefore, PJf and OS will each be equal to OB, OM and BS will vanish, while AT and OT will each be infinite in length. ^ Hence, as the angle x increases from 0° to 90°, sinx increases from to 1, cos x decreases from 1 to 0, tan X increases from to oo, cotcc decreases from go to 0, sec a; increases from 1 to go, CSC a; decreases from oo to 1. TRIGONOMETRIC FUNCTIONS. 11 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. Remark. We are now able to understand why the sine, cosine, etc., of an angle are called functions of the angle. By a 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. /- § 5. Functions of Complementary Angles. The general form of two complementary angles is A and 90° -A In the rt. A ABC (Fig. 5), ^ -f ^ = 90°; hence ^ = 90° — A. Therefore (§ 2), sin A = cos B = cos (90° — A), cos ^ = sin ^ == sin (90° — A), tan ^ = cot i? =: cot (90° — A), cot ^ = tan ^ = tan (90° — A), sec ^ = CSC ^ = esc (90° — A), CSC A = sec B = sec (90° — A). Therefore, ^ach 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 complemenVs sine, complemenVs tangent, and complemenVs secant. Hence, also, Ang function of an angle between 45° and 90° may he found hy taking the co-named function of the complementary angle between 0° and 45°. 12 TRIGONOMETRY. Exercise IV. 1. Express the following functions as functions of the complementary 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°: sin 60°. tan 57°. esc 69° 2'. cot 89° 59'. cos 75°. cot 84°. cos 85° 39'. esc 45° 1'. 3. Given tan 30° = -J V3 ; find cot 60°. 4. Given tan ^ = cot ^ ; find A. 5. Given cos ^ = sin2^; find A ^:, 6. Given sin ^ = cos 2 ^ ; find A. :■ 7. Given cos A = sin (45° — ^A); find A. 8. Given cot ^ ^ = tan A ; find A.^ 9. Given tan (45° -\-A)=GotA', find A. 10. Find^if sin^ = cos4A ^ 11. Find A if cot ^ = tan 8 ^. : 12. Find A if cot A = tan nA. § 6. Eelations of the Functions of an Angle. Formula [1]. Since (Fig. 5) a^ + b^=c^, therefore, 7^+? = ^ ^^ (!)■-(')■ Therefore (§ 2), (sin Af + (cos Ay = l'j or, as usually written for convenience, sin2A + cos2A = l. [1] That is : The sum of the squares of the sine and the cosine of an angle is equal to unity. TRIGONOMETRIC FUNCTIONS. 13 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 : sm ^ = V 1 — cos^^, cos ^ = V 1 — sin^A Formula [2], Since a h a c a c ' c c b b therefore (§ 2), 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. Formula [3]. Since -X- = l, -X- = l, and -X- = l, c a c b a a therefore (§ 2), sin A X esc A = 1 ] cosAXsecA=l> C^] tan AX cotA=l J That is : The sine and the cosecant of an angle, the cosine and the secant, and the tangent and the cotangent, pair by pair, are reciprocals. The equations in [3] enable us to find an unknown func- tion contained in any pair of these reciprocals when the other function in this pair is known. zy\ 14 TRIGONOMETRY. Exercise V. 1. Prove Formulas [1] - [3], using for the functions the line values in the unit circle given in § 3. Prove that 2. l + tan2A = sec2A. 3. l + cot2A = csc2A. Note. Equations 2 and 3 should be remembered. 4. cot A = — -' sm^ 5. sin ^ sec ^ = tan A. 6. sin A cot A = cos A. 7. cos ^ esc ^ = cot A. 8. tan ^ cos ^ = sin A. 9. sin A sec A cot ^ = 1. 10. cos A CSC A tan J = 1. 11. (l-sin2^)tan2J = sin2^. 12. Vl — cos^^ cot A = cos A. 13. (l + tan2^)sin2^ = tan2A 14. csc2^(l — sin2^) = cot2J. 15. tan2^cos2^+cos2^ = l. 16. (sin^^ — cos^Af = 1 — 4 sin^^ cos»^ ■ 17. (1 — tanM)2 = sec*^--4tan2A ^o sin^ , cos^ lo- - — 7 + - — 7 = sec A CSC J. cos^ sm^ 19. sin^^ — cos^^ = sin^^ — cos^^. 20. sec^ — cos^=sin^ tanA 21. esc A — sin A = cos A cot A, cos A 1 + sin^ ^/ ^/22. 1 — sin^ cos^ / \'' TRIGONOMETRIC FUNCTIONS. 15 § 7. Application of Formulas [1] - [3]. Formulas [1], [2], and [3] enable us, when any one func- tion of an angle is given, to find all the others. A given value of any one function, therefore, determines all the others. Example L G-iven sin ^ = | ; find the other functions. By [IJ, cos ^ = Vl - 1 - ^i=h-^^' By [2], tan^ = |^|V5^?X^^^ = |V5. By [3], cot ^ = -y-, sec^= - V5, csc^ = -- Example 2. Given tan A = 3 -, find the other functions. By [2], ^ = 3. •^ •- -" COS ^ . And by [1], sinM + cos^^ = 1. If we solve these equations (regarding sin A and cos A as two unknown quantities), we find that, sin^ = 3V^, cos^ = V^- Then by [3], cot A = ^, seGA = VlO, csc yl = ^ VlO. Example 3. Given sec A = m; find the other functions. By [3], COS A = sin^ = tan^ = CSC A = m _1 m 1 =v>- 1 cot 4 '■ — -1_ By [1], Vm2- -1. By [2], [3], m -1, Vm2- ^' Vm^-l 16 TRIGONOMETRY. Exercise VI. Eind the values of the other functions, when 1. sin A = If. 5. tan A = ^. 9. esc A = V2. 2. sin ^ = 0.8. 6. cot^=::l. 10. sin ^ = w. 2 m 3. cos A = If. ^ 7. cot A = 0.5. 11. sin A = —j- — - • 4. cos ^ = 0.28. 8. sec ^==2. 12. cos ^ = -IT^* nr -f- n^ Given tan 45° = 1 ; find the other functions of 45°. Given sin 30° = i; find the other functions of 30°. Given esc 60° = § V3. find the other functions of 60°. 'Vie. Given tan 15°= 2— V3; find the other functions of 15°. 17. Given cot 22° 30'= V24-I; find the other functions of 22° 30'. 18. Given sin 0° =0; find the other functions of 0°. 19. Given sin 90° = 1; find the other functions of 90°. T 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. 4. 26. Given 4 sin A = tan A ; find sin A and tan A. -V 27. If sin A : cos A = 9 : 40, find sin A and cos A. ■ 28. Transform the quantity tanM + cot^^ — sin^^l — cos^^ into a form containing only cos A. \29. Prove that sin A -{-cos ^ = (1 + tan A) cos A. 30. Prove that tan ^ + cot ^ = sec ^ X esc A. TRIGONOMETRIC FUNCTIONS. 17 § 8. Functions of 45°. Let ABC (Fig. 6) be an isosceles right triangle, in whicli 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 2 AC^ = 1, and ^C = V| = i V2. Therefore (§ 2), sin45°=:cos45° = iV2. tan45° = cot45°=:l._ sec 45° = esc 45° = V2. § 9. Functions of 30° and 60' Let ABC (Fig. 7) 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 = i, and Ci)= Vl -i== Vf = i V3. In the right triangle ADC, the angle ACD = SO°, and the angle CAD = 60°. Whence (§ 2), sin 30° = cos 60° = -J. cos 30° = sin 60° = J V3. tan 30° = cot 60° = -^^ = i^/S. V3 cot 30° = tan 60° =V3. sec 30° = esc 60° = 4= = f V3. V3 CSC 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 : 18 TRIGONOMETRY. Angle .... 30° 45° 60° ^ Vi = 0.5 Sine iVi iV2 ^V3 iV2 = 0.70711 Cosine . . . ^V3 ^V2 Wi ^V3 = 0.86603 Exercise VII. Solve the following equations : 1. 2 cos X = sec X. 2. 4 sin X = CSC x. 3. tan x =2 sin x. 4. sec X = V2 tan x. 5. sin^ ic = 3 cos^ x. 6. 2 sin^a; + cos^a; = |. 7. 3 tan^a: — sec2a; = l. 8. tan x + cot £c = 2. 9. sin^ a:; — cos x = ^. 10. tan^ ic — sec x = l. 11. sin a? + V3 cos ic = 2. 12. tan^ X -{- csc^ x = 3. ^\\ 13. 2 cos X -\- sec cc = 3. 14. cos^ X — sin^ x = sin x. A 15. 2 sin cc -j- cot £c = 1 + 2 cos x. 16. sin^ X -\- tan^ a: = 3 cos^ x. ^ 17. tan X -\-2 cot a; = | esc a;. Note. Went worth & Hill's Five-place trigonometric and logarithmic tables have full explanations, and directions for using them. Before pro- ceeding to Chapter II. the student should learn how to use these tables. Table VI. is to be used in solutions without logarithms. This four- place table contains the natural functions of angles at intervals of Y. The decimal point must be inserted before each value given, except where it appears in the values of the table. CHAPTER II. THE RIGHT TRIANGLE. § 10. 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 : 1. 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. y. The two legs. § 11. Solution without Logarithms. The following examples illustrate the process of solution when logarithms are not employed. Case I. Given ^ = 43° 17', c = 26 ; find B, a, h, \. ^ = 90° -J = 46° 43'. 2. - = sin^: .•.« = ' 21. A barn is 40 X 80 feet, the pitch of the roof is 45°; find the length of the rafters and the area of both sides of the roof. 22. In a unit circle what is the length of the chord corre- sponding to the angle 45° at the centre ? o 23. If the radius of a circle is 30, and the length of a chord is 44, find the angle at the centre. ;. 24. Find the radius of a circle if a chord whose length is 5 subtends at the centre an angle of 133°. THE REGULAR POLYGON. 33 25. What is the angle at the centre of a circle if the corre- sponding chord is equal to f of the radius ? V 26. Find the area of a circular sector if the radius of the circle = 12, and the angle of the sector = 30°. i § 16. The Regular Polygon. Lines drawn from the centre of a regular polygon (Fig. 19) to the vertices are radii of the circumscribed circle; and lines drawn from the centre to the middle points of the sides are radii of the inscribed circle. These lines divide the polygon into equal right triangles. Therefore, a regular polygon is determined by a right triangle whose sides are the radius of the circumscribed circle, the radius of the inscribed circle, and half of one side of the polygon. If the polygon has n sides, the angle of this right triangle at the centre is equal to 1 /360°\ 180° 2\-^) ^^ ^- If, also, a side of the polygon, or one of the above-men- tioned radii, is given, this triangle may be solved, and the solution gives the unknown parts of the polygon. Let, 71 = number of sides, c = length of one side, r = radius of circumscribed circle, h = radius of inscribed circle, p = the perimeter, i^=the area. Then, by Geometry, F = ^hp, Fig. 19. 34 TRIGONOMETRY, Exercise XI. ~- 1. Given n = 10, c = l; find r, h, F. 2. Given 71 = 12, ^ = 70; find r, h, F. ^ 3. Given w = 18, r = 1 ; find h, p, F. 4. Given w = 20, r = 20; find h, c, F. ~^ 5. Given n = S, h = l; find r, c, F. 6. Given 71 = 11, 7^=20; find r, h, c. ~^7. Given w = 7, F=7; find r, h, p. 8. Find the side of a regular decagon inscribed in a unit circle. I^ W .^ \(^ ~~ 9. Find the side of a regular decagon circumscribed about a unit circle. 10. If the side of an inscribed regular hexagon is equal to 1, find the side of an inscribed regular dodecagon. 11. Given n and c, and let b denote the side of the inscribed regular polygon having 27i sides; find b in terms of Tiand c. 12. Compute the difference between the areas of a regular octagon and a regular nonagon if the perimeter of each is 16. 13. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12. N-il 14. From a square whose side is equal to 1 the corners are cut away so that a regular octagon is left. Find the area of this octagon. 15. Find the area of a regular pentagon if its diagonals are each equal to 12. "t m < (^ "^ 16. The area of an inscribed regular pentagon is 331.8; find the area of a regular polygon of 11 sides inscribed in the same circle. THE REGULAR POLYGON. 35 17. The perimeter of an equilateral triangle is 20 ; find the area of the inscribed circle. H . i 1 h 18. The area of a regular polygon of 16 sides, inscribed in a circle, is 100; find the area of a regular polygon of 15 sides, inscribed in the same circle. 19. A regular dodecagon is circumscribed about a circle, the circumference of which is equal to 1 ; find the perimeter of the dodecagon. /• ^ "^ "3 J^ 20. The area of a regular polygon of 25 sides is equal to 40; find the area of the ring comprised between the circum- ferences of the inscribed and the circumscribed circles. CHAPTER III. GONIOMETRY. § 17. Definition of Goniometry. In order to prepare the way for the solution of an oblique triangle, we now proceed to extend the definitions of the trigonometric functions to angles of all magnitudes, and to deduce certain useful relations of the functions of diiferent angles. That branch of Trigonometry which treats of trigono- metric functions in general, and of their relations, is called Goniometry. § 18. Angles of any Magnitude. Let the radius OP of a circle (Fig. 20) generate an angle by turning about the centre 0. This angle will be measured by the arc described by the point P; and it may have any magnitude, because the arc described by P may have any magnitude. Let the horizontal line OA be the initial position of OP, and let OP revolve in the direction shown by the arrow, or opposite to the way 'clock hands revolve. Let, also, the four quadrants into which the circle is divided by the horizontal and vertical diameters AA\ BB'j be numbered I., II., III., IV., in the direction of the motion. GONIOMETRT. * 37 During one revolution OF will form with OA all angles from 0° to 360°. Any particular angle is said to be an angle of the quadrant in which OP lies; so that, Angles between 0° and 90° are angles of Quadrant I. Angles between 90° and 180° are angles of Quadrant II. Angles between 180° and 270° are angles of Quadrant III. Angles between 270° and 360° are angles of Quadrant IV. If OP make another revolution, it will describe all angles from 360° to 720°, and so on. If OP, instead of making another revolution in the direc- tion of the arrow, be supposed to revolve backwards about 0, this backward motion tends to undo, or cancel, the original forward motion. Hence, the angle thus generated must be regarded as a negative angle; and this negative angle may, obviously, have any magnitude. Thus we arrive at the con- ception of an angle of any magnitude, positive or negative. § 19. General Definitions of the Functions. The definitions of the trigonometric functions may be extended to all angles, by making the functions of any angle equal to the line values in a unit circle drawn for the angle in question, as explained in § 4. But the lines that represent the sine, cosine, tangent, and cotangent must he regarded as negative, if they are opposite in direction to the lines that repre- sent the corresponding functions of an angle in the first quad- rant; and the lines that represent the secant and cosecant must be regarded as negative, if they are opposite in direction to the moving radius. Figs. 21-24 show the functions drawn for an angle ^OP in each quadrant, taken in order. In constructing them, it must be remembered that the tangents to the circle are always drawn through A and B, never through A' or B'. Let the angle AOP be denoted by x ; then, in each figure, 38 TRIGONOMETRY. the absolute values of the functions (that is, their values without regard to the signs + and — ) are as follows : Bm.x=MP, t2inx = AT, secx= OT, cos X = OMj cot 03 = BSf CSC x = OS. B c ^ B Fig. 23. Fig. 24. Keeping in mind the position of the points A and B, we may define in words the first four functions of the angle x thus : sin a? = the vertical projection of the moving radius; cosic= the horizontal projection of the moving radius; the distance measured along a tangent to the circle tana; = ^ from the beginning of the first quadrant to the moving radius produced; '-{ Kj\ 60NI0METRY. 39 r the distance measured along a tangent to the circle cotx ^< from the end of the first quadrant to the moving L radius produced. Secx and esc a: are the distances from the centre of the circle measured along the moving radius produced to the tangent and cotangent, respectively. § 20. Algebraic Signs of the Functions. The lengths of the lines, defined above as the functions of any angle, are expressed numerically in terms of the radius of the circle as the unit. But, before these lengths can be treated as algebraic quantities, they must have the sign + or — prefixed, according to the condition stated in § 19. The reason for this condition lies in that fundamental relation between algebraic and geometric magnitudes, in virtue of which contrary signs in Algebra correspond to opposite directions in Geometry. The sine MP and the tangent AT always extend from the horizontal diameter, but sometimes upwards ai^i sometimes downwards ; the cosine OM and the cotangent BS always extend from the vertical diameter, but sometimes towards the right and sometimes towards the left. The functions of an angle in the first quadrant are assumed to be positive. There- fore, 1. Sines and tangents extending from the horizontal diameter upwards, are positive ; downwards, negative. 2. Cosines and cotangents extending from the vertical diameter towards the right, are positive ; towards the left^ are negative. The signs of the secant and cosecant are always made to agree with those of the cosine and sine, respectively. This agreement is secured if secants and cosecants extending from the centre, in the direction of the moving radius, are considered positive ; in the opposite direction, negative. 40 TRIGONOMETRY. Hence, the signs of the functions for each quadrant are : In Quadrant I. all the functions are positive. In Quadrant II. the sine and cosecant only are positive. In Quadrant III. the tangent and cotangent onlysiTe positive. In Quadrant IV. the cosine and secant only are positive. § 21. Functions of a Variable Angle. Let the angle x increase continuously from 0° to 360°; what changes will the values of its functions undergo? It is easy, by reference to Fig. 25, to trace these changes throughout all the quadrants. Fig. 25. 1. The Sine. In the first quadrant, the sine MP increases from to 1; in the second it remains positive, and decreases from 1 to 0; in the third it is negative, and increases in absolute value from to 1 ; in the fourth it is negative, and decreases in absolute value from 1 to 0. GONIOMETRY. 41 2. The Cosine. In the first quadrant, the cosine OM de- creases from 1 to ; in the second it becomes negative, and increases in absolute value from to 1; in the third it is negative, and decreases in absolute value from 1 to ; in the fourth it is positive, and increases from to 1. 3. The Tangent. In the first quadrant, the tangent AT increases from to oc ; in the second quadrant, as soon as the angle exceeds 90° by the smallest conceivable amount, the moving radius OF', prolonged in the direction opposite to that of OF', will cut AT Sit Q, point T' situated very far below A; hence, the tangents of angles near 90° in the second quadrant have very large negative values. As the angle increases, the tangent AT' continues negative, but diminishes in absolute value. When x = 180°, then T' coincides with A, and tan 180° = 0. In the third quadrant, the tangent is positive, and increases from to oo ; in the fourth it is negative, and decreases in absolute value from O) to 0. 4. The Cotangent In the first quadrant, the cotangent ^^S^ decreases from oo to 0; in the second quadrant it is negative, and increases in absolute value from to oo ; in the third and fourth quadrants it has the same sign, and undergoes the same changes as in the first and second quadrants, respectively. 5. The Secant. In the first quadrant, the secant OT in- creases from 1 to GO ; in the second it is negative (being measured in the direction opposite to that of OF'), and decreases in absolute value from oo to 1; in the third it is negative, and increases in absolute value from 1 to oo ; in the fourth it is positive, and decreases from go to 1^ 6. The Cosecant. In the first quadrant, the cosecant OS decreases from c» to 1 ; in the second it is positive, and increases from 1 to go ; in the third it is negative, and decreases in absolute value from oo to 1 ; in the fourth it is negative, and increases in absolute value from 1 to go. 42 TRIGONOMETRY. The limiting values of the functions are as follows : Sine 0° 90' 180- 270° 360- ±0 1 ±0 -1 ±0 Cosine 1 ±0 -1 ±0 1 Tangent ±0 ±cc ±0 ± 00 ±0 Cotangent .... ±00 ±0 d= CC -±o d=oo Secant 1 ±0) -1 rtoo 1 Cosecant dboo 1 =b 00 - 1 iO) Sines and cosines extend from -|-1 to —1; tangents and cotangents from -f- ^ to — oo ; secants and cosecants from -f- 00 to -\-l, and from — 1 to — oo. In the table given above the double sign ± is placed before and oo. From the preceding investigation it appears that the functions always change sign in passing through and oo ; and the sign + or — prefixed to or 00 simply shows the direction from which the value is reached. Take, for example, tan 90° : The nearer an acute angle is to 90°, the greater the positive value of its tangent ; and the nearer an obtuse angle is to 90°, the greater the negative value of its tangent. When the angle is 90°, OP (Fig. 25) is parallel to A T, and cannot meet it. But tan 90° may be regarded as extending either in the positive or in the negative direction; and according to the view taken, it will be + oo or — oo. § 22. Functions of Angles Larger than 360°. The functions of 360° + x are the same in sign and in absolute value as those of x ; for the moving radius has the same position in both cases. If w is a positive integer, The functions of (n X 360° + x) are the same as those of x. For example: The functions of 2200° (6 X 360° -f 40°) are equal to the functions of 40°. goniometrt. 43 § 23. Extension of Formulas [1]-[3] to all Angles. The Formulas established for acute angles in § 6 hold true for all angles. Thus, Formula [1], sin^x + cos^x = 1, is universally true ; for, whether MP and OM (Fig. 25) are positive or negative, MP^ and OM^ are always positive, and in each quadrant JZp'+ 057'= (jF = l. Also, r21 tan x = j •■ -■ cos a: {sin X X CSC x^l, cos a- X sec a: = 1, tan X X cot x = l, are universally true ; for they are in harmony with the alge- braic signs of the functions, given at the end of § 20 ; and we have in each quadrant from the similar triangles OMP, OAT, OBS, (Fig. 25) the proportions AT : OA = MP: OM, MP: OP=OB : OSy OM: OP=OA : OT, AT : OA=OB : BS, which, by substituting 1 for the radius, and the right names for the other lines, are easily reduced to the above formulas. Formulas [l]-[3] enable us, from a given value of one function, to find the absolute values of the other five func- tions, and also the sign of the reciprocal function. But in order to determine the proper signs to be placed before the other four functions, we must know the quadrant to which the angle in question belongs ; or the sign of any one of these four functions ; for, by § 20, it will be seen that the signs of any two functions that are not reciprocals determine the quadrant to which the angle belongs. 44 TRIGONOMETRY. Example. Given sin cc = -f |, and tan x negative ; find the values of the other functions. Since sin x is positive, x must be an angle in Quadrant I. or II. ; but, since tan x is negative. Quadrant I. is inadmissible. By [1], cosa^ = ±Vl-i| = ±t- Since the angle is in Quadrant II. the minus sign must be taken, and we have cos ic = — f . By [2] and [3], tanx^ — I, cotx = — f, seca: = — |, cscic:=|. Exercise XII. 1. Construct the functions of an angle in Quadrant II. What are their signs ? 2. Construct the functions of an angle in Quadrant III. What are their signs ? 3. Construct the functions of an angle in Quadrant IV. What are their signs ? 4. What are the signs of the functions of the following angles : 340°, 239°, 145°, 400°, 700°, 1200°, 3800° ? 5. How many angles less than 360° have the value of the sine equal to+ f, and in what quadrants do they lie ? 6. How many values less than 720° can the angle x have if cos a;:= + f, and in 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; = f ? if cos x=^\ ? if cos x = — I? iftana; = |? if cota! = — 7? 8. Within wliat limits must the angle x lie if cos cc = — | ? ifcot(r = 4? ifsec2c = 80? ifcscx = — 3? (if a^ < 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 ? GONIOMETRT. 45 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 cc by means of the equation cosaj = ± V 1 — sui^x, when must we choose the positive sign and when the negative sign ? 12. Given cos a: = — V ^ ; find the other functions when X is an angle in Quadrant II. 13. Given tan ic = V 3 ; 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 functians. 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 + oo or to — 00 ? 19. Obtain by means of Formulas [l]-[3] the other func- tions of the angles given : (i.) tan 90° = 00. (iii.) cot 270° = 0. (ii.) cos 180° = — 1. (iv.) esc 360° = — oo. 20. Find the values of sin 450°, tan 540°, cos 630°, cot 720°, sin 810°, CSC 900°. 21. For what angle in each quadrant are the absolute values of the sine and cosine equal ? Compute the values of the following expressions : 22. a sin 0°-\-b cos 90° — c tan 180°. 23. a cos 90° — b tan 180° + c cot 90°. 24. a sin 90° — b cos 360° -\-(a — b) cos 180°. 25. (a^ — b^) cos 360° — Aab sin 270°. 46 TRIGONOMETRY. § 24. Reduction of Functions to the First Quadrant. In a unit circle (Fig. 26) draw two diameters PR and QS equally inclined to the horizon- tal diameter AA', or so that the angles AOP, A'OQ, A'OH, and AGS shall be equal. From the points F, Q, B, S let fall perpen- diculars to AA^ ; 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, QN, EN, SM, are equal. Now these four lines are the sines of the angles AOP, AOQ, AOR, and AOS, respectively. Therefore, in absolute value, BuiAOP = ^mAOQ = &inAOR = ^mAOS. And from § 23 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 ZAOP = x,/_ POP = y ; then x-\-y = 90°, and the functions of x are equal to the co-named functions of y (§ 5) ; and Z ^0^ (in Quadrant 11.) =180° — :r= 90° + ?/, Z.AOR (in Quadrant III.) == 180° -^x = 270° — y, Z.AOS (in Quadrant IV.) = 360° — x = 270° + y. Hence, prefixing the proper sign (§ 20), we have : * In future, secants, cosecants, versed sines, and coversed sines will be disregarded. Secants and cosecants may be found by [3], versed sines and coversed sines by VII. and VIII., page 5, if wanted, but they are seldom used in computations. GONIOMETRT. 47 Angle in Quadrant II. sin (180° — ic) = sin x. sin (90° + ?/) = cos y. cos (180° — if) = — cos X. cos (90° + y) = — sin y. tan (180° — x) = — tan x. tan (90° + y) = — cot y. cot (180° — x) = — cot X. cot (90° + 2/) = — tan y. Angle in Quadrant HI. sin (180° -\~x) = — sin ic. sin (270° — ?/)=-- cos y. cos (180° -^x) = — cos cc. cos (270° — y) = — sin y. tan (180° + x) = tan x. tan (270° — v/) = cot y. cot (180° + x) = cot a;. cot (270° — y)= tan y. Angle in Quadrant IV. sin (360° — x) = — sin x. sin (270° -\-y)= — cos y. cos (360° — x)= cos a-. cos (270° + ij) = sin 2/. tan (360° — x)= — tan a;. tan (270° -f- ?/) = — cot 2/. cot (360° — x)= — cot a:. cot (270° + ?/)= — tan 3/. Remark. The tangents and cotangents may be found directly from the figure, or by formula [2]. It is evident from these formulas, 1. The functions of all angles can be reduced to the functions of angles not greater than 45°. 2. If an acute angle he 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 func- tions of the resulting angle are equal in absolute value to the co-named functions of the acute angle. 3. A given value of a sine or cosecant determines two supple- mentary angles, one acute, the other obtuse ; a given value of any other function determines only one angle: acute if the value is positive, obtuse if the value is negative. [See functions of (180° -x).] ' f 48 TRIGONOMETRY. § 25. Angles whose Difference is 90°. The general form of two such angles is x and 90° + ^j and they must lie in adjoining quadrants. The relations between their functions were found in § 24, but only for the case when x is acute. These relations, however, may be shown to hold true for all values of x. In a unit circle (Fig. 27) draw two diameters FB and QS per- pendicular to each other, and let fall to AA' the perpendiculars FM, QH, RK, and SN. The right triangles OMF, OHQ, OKR, and ONS are equal, because they have equal hypotenuses and equal acute angles FOM, OQH, ROK, and OSN. Therefore, OM=QH=OK=NS, and FM=OH = KR=ON. Hence, taking into account the algebraic sign, sin^O^= cos^OP; ^inAOS = cos AOR cos AOQ = — sin AOF; cos AOS = — sinAOR: sin AOR = cos AOQ; sin (360° + ^ OP) = cos^O^; cos^OP = — sin^O^; cos (360° + ^ OP) = — sin ^ 0^. 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. Therefore, if x be an angle in any one of the four quadrants, sin (90° -\-x)= cos x, tan (90° -f cc) = — cot x, cos (90° -\-x) = — sin x, cot (90° -]-x) = — tan x. In like manner, it can be shown that all the formulas of § 24 hold true, whatever be the values of the angles x and y. Hence, in every case the algebraic sign of the function of the resulting angle will he the same as when x and y are both acute. (fl/- goniometrt. 49 § 26. Functions of a Negative Angle. If the angle AOF (Fig. 26) is denoted by x, the equal angles 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 angle 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 (§ 24), sin ( — x) = — sin cc, tan ( — x) = — tan x, cos (— x) = cos X, cot (— x) = — cot X. Exercise XIII. 1. Express sin 250° in terms of the functions of an acute angle less than 45°. Ans. sin 250° = sin (270° — 20°) = - cos 20°. Express the following functions in terms of the functions of angles less than 45° : 2. sin 172°. 8. sin 204°. 14. sin 163° 49'. 3. cos 100°. 9. cos 359°. 15. cos 195° 33'. 4. tan 125°. 10. tan 300°. 16. tan 269° 15'. 5. cot 91°. 11. cot 264°. 17. cot 139° 17'. 6. sec 110°. 12. sec 244°. 18. sec 299° 45'. 7. CSC 157°. 13. CSC 271°. 19. esc 92° 25'. 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'. 21. —127°. 23. —345°. 25. —196° 54'. 26. Find the functions of 120°. Hint. 120° - 180°- 60°, or, 120° = 90° + 30° ; then apply § 24. 60 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 = — V-J-, and cos x negative ; find the other functions of x, and the value of x. 36. Given cota:; = — V3, and x in Quadrant II.; ^nd 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 — ^ ? a tangent equal to — V3 ? 39. Which of the angles mentioned in Examples 27-34 have a cosine equal to — V-J ? a cotangent equal to — V3 ? 40. What values of x between 0° and 720° will satisfy the equation sin x = + ^ ? 41. Find the other angle between 0° and 360° for which the corresponding function (sign included) has the same value as 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 ; find tan 117°. Simplify the following expressions : 44. acos(90° — ^) + ^>cos(90° + a^). 45. m cos (90° — x) sin (90°- x). 46. {a — b) tan (90° -x)-\- (a + b) cot (90° + x). 47. a^-\-P — 2abcos(lS0° — x). 48. sin (90° + x) sin (180° + ^) + cos (90° + x) cos (180° — x). 49. cos(180°+cc)cos(270°— ?/) — sin(180°+cc)sin(270°-?/). 50. tanicH-tan(— ?/) — tan(180° — y). 51. For what values of x is the expression sincc + cosa? positive, and for what values negative ? Eepresent the result by shading the sectors corresponding to the negative values. 52. Answer the question of last example for sin x — cos x. 53. Find the functions of (x — 90°) in functions of x. 54. Find the functions of (x — 180°) in functions of x. GONIOMETRY. 51 § 27. Functions of the Sum of Two Angles. In a unit circle (Fig. 28) let the angle AOB^=x, the angle BOC=y', then the angle AOC = x-\-y. In order to express sin {x -\- y) and cos (x -\- y) in terms of the sines and cosines of x and y, draw CFA_OA, CD1_0B, DE^OA, DG± CF; then CD = smy, OB = cos?/, and the angle BCG = the an^lc 6^Z>0 = ic. Also, sin (x + yy= CF= I)F-\- CG. BE —— = sm X ; hence, BF = sin x X OB = sin x cos y. CG CB = cos X ; hence, CG = cos x X CB = cos x sin y. m Therefore, sin (x + y) = sin x cos y + cos x sin y . Again, cos (x + y) = OF= OF —BG. OF -— = cos X ; hence, OF = cos x X OB = cos x cos y. DC -—- = sin X ; hence, BG = since X CB = sin x sin y. uB Therefore, cos (x + y) = cos x cos y — sin x sin y . [5] In this proof x and y, and also the sum x-\-y, are assumed to be acute angles. If the sum x-{-y ot the acute angles x and y is obtuse, as in Fig. 29, the proof remains, word for word, the same as above, the only dif- ference being that the sign of OF will be negative, as BG is now greater than OF. The above formulas^ therefore, hold true for all acute angles x and y. 52 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 cc' = 90° -|- x, then, by § 25, sin (x' -\-y) = sin (90° -\-x-{-y)= cos (x -\- ?/), cos (x' -J- ?/) = cos (90° + ic + 2/) = — sin (x -{- y). Hence, by [5], sin (x' -[- 2/) = cos x cos y — sin x sin ?/, by [4], cos (x' + y) = — sin x cos y — cos x sin y, Now, by § 25, cos x = sin (90° -\-x)= sin x\ sin cc = — cos (90° -\-x) = — cos x\ Substitute these values of cos x and sin x, then sin (x^ -\-y^=z sin £c' cos y -\- cos x^ sin y, cos (cc' + y) = cos £c' cos y — sin ic'"sin y. It follows that Formulas [4] and [5] hold true if either angle is repeatedly increased by 90° ; therefore they apply to all angles whatever. By § 23, , , , . sin (x -f y) sin x cos y + cos x sin ?/ tan (x-^y) = ) — p^ = '^—^ — : ^' cos {X -\- y) cos X cos ?/ — sm x sm y If we divide each term of the numerator and denominator of the last fraction by cos x cos y, and again apj)ly § 23, we obtain , / , . tanx+tany ^^^ *^°^^+y)= l-tanxtany ' ^ In like manner, by dividing each term of the numerator and denominator of the value of cot (x -\- t/) by sin x sin y, we obtain .... cotxcoty— 1 cot(x + y) = — - — r-^4 — r71 ^ ^^^ coty + cotx "-'J GONIOMETRY. 53 § 28. Functions of the Difference of Two Angles. Iji a unit circle (Fig. 30) let tlie angle AOB = x, COB = y, then the angle AOC = x — ?/. In order to express sin (x — y) and cos (x — ?/) in terms of the sines and cosines of x and y, draw CF _L OA, CD ± OB, DE J_ OA, DG _L FC prolonged; then CD= sin y, OD = cos y, and the angle I)CG = the angle FDC=x. And, sin (x — y)= CF=DE — CG. ^ '^^ Fig. 30. DE -r-r = sin X ; hence, DE = sin x X OD = sin x cos y. CC — — = cos X ; hence, CG = cos x X CD = cos x sin y. Therefore, sin(x — y) = sin x cos y — cos x sin y. [8] Again, cos(x — y)= OF=OE -\-DG. OE OD DG = cos x ; hence, OE = cos x X OD = cos x cos y. hence, DG = sin x X -CD = sin x sin y. Therefore, cos (x — y) = cos x cos y + sin x sin y . [9] ^ = sm^;_ 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 estab- lished in § 27, as follows : It is obvious that (x — y)-\-y = X' If we apply Formulas [4] and [5] to (x — y)-\-y, then 64 TRIGONOMETRY. sin J (a? — y)-\-yl or sin x = sin (x — ?/) cos y + cos (x — ?/) sin y, cos \(x — y)-{-yi or cos x = cos (x — y) cos y — sin (x — y) sin y. Multiply the first equation by cos y, the second by sin j/, sin X cos y= * sin (x — ?/) cos^y + cos (x — y) sin y cos 2^, cos a; sin y = — sin (x — y) sin'^y -\- cos (x — y) sin y cos y ; whence, by subtraction, sin X cos ?/ — cos x sin ?/ = sin (x — y) (sin^y + cos^y). But sin^y + cos^^ = 1 5 therefore, by transposing, sin (x — y) = sin x cos y — cos x sin y. Again, if we multiply the first equation by sin y, the second equation by cos y, and add the results, we obtain, by reducing, cos (x — y) = cos x cos y -}- 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 § 27, we obtain . . - tanx — tany ^_^ *'^°("-y)= l + tanxtany - t^^^ , . . cotxcoty+1* ^^^^ Formulas [4]-[ll] may be combined as follows: sin (xdzy) = sin x cos ?/ d= cos x sin ?/, cos (x ± ?/) = cos cc cos y :+: siu a^ sin y, tan a; ± tan y tan (a; ± ?/) tan x tan ?/ ^ , , cot x cot ?/ =F 1 cot (a! ± ?/) = --^—^ — cot y ± cot x goniometry. 55 § 29. Functions of Twice an Angle. If y = x, Formulas [4]-[7], become sin 2 X = 2 sin X cos X. [12] cos 2 x = cos^x — sin^x. [13] . _ 2tanx ,. ,_, ._ cot^x— 1 _^_ By these formulas the functions of twice an angle are found when the functions of the angle are given. § 30. Functions of Half an Angle. Take the formulas cos^a; + sin^j; = 1 [1 ] cos^aj — sin^x = cos 2x ~ [1 3] Subtract, 2 sin^j? = 1 — cos 2 x Add, 2 cos^^ = 1 + cos 2 x Whence /l — cos2a^ /l + . = ±\ , cosa; = ±^— ^ cos 2 X sm X " . - . . 2 ~ >/ 2 These values, if z is put for 2x, and hence ^;^ for x, become • 1 , ^/l — cosz ^,„^ , /l + cosz ^^„^ siniz = it:^ [16] cosiz=:±^ — — [17] Hence, by division (§ 23), ta„iz = ±Vf^^' [18] cotiz=±Vi±^. [19] ^l + cosz •- -" ^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 -J z lies. (§ 21.) Let the student show from Formula [18] that tan I- ^ = \ — — • (See page 22, Note.) 56 tkigonometrt. § 31. Sums and Differences of Eunctions. From [4], [5], [8], and [9], by addition and subtraction : sin (ic + ?/) + sin (x— y)=^ 2 sin x cos i/, sin (x-{- y) — sin (x — ?/) = 2 cos x sin y, cos (x-\-y)-{- cos (x — y)=^ 2 cos x cos y, cos (x-{-y) — cos (x — y) = — 2 sin x sin ?/ ; or, by making £c + ?/ = ^, and x — y^=B, and therefore, x^=^{A-\- B), and ?/ = |- (y1 — B), sinA+sinB=: 2siiii(A + B)cosi-(Ar-B). [20] sin A — sin B= 2cosi(A + B)sini(A— B). [21] cos A + cos B = 2 cos ^ (A + B) cos i (A — B). [22] cosA — cosB=-2sinJ(A + B)sini(A — B). [23] From [20] and [21], by division, we obtain sin^ + sin^ i / ^ i t>n 4.1/1 t>\ — — ^ — - = tan \(A-\-B) cot 1 (A — B) : sm^ — sm^ 2v I y 2v y> or, since cot -^(-4 — B)^= tan ^(yl—i?) sin A -[- sin B _ tan j^ (A + B) sin A — sin B tan ^ (A — B) Exercise XIV. [24] 1. Find the value of sin (x -\- y) and cos (x -j- y), when sin x = f, cosa; = f, smy = j% cos?/ = f|. 2. Find sin (90° — 3/) and cos(90° — ?/) by making 0^ = 90° in Formulas [8] and [9]. Find, by Formulas [4:]-[ll], the first four functions of: 3. 90° + 2/. 8. 360° -y. 13. -y. 4. 180°-?/. 9. 360° + ?/. 14. 45° — ?/. 5. 180° + ?/. 10. ic — 90°. 15. 45°+?/. 6. 270°-?/. 11. 0^-180°. 16. 30° + ?/. 7. 270° + ?/. 12. X — 270°. 17. 60° — ?/. GONIOMETRT. 67 18. Find sinSic in terms of since. 19. Find cos 3 a? in terms of cos x. 20. Given tan ^cc = 1 ; find cos x. 21. Given cot-J-ic=: V3; find sin a?. 22. Given since = 0.2 ; find sin^cc and cos^-cc. 23. Given cos x = 0.5 ; find cos 2x and tan 2ic. 24. Given tan 45° = 1 ; find the functions of 22° 30'. 25. Given sin 30° = 0.5 ; find the functions of 15°. sin 33° + sin 3° ., 26. Prove that tan 18° = cos 33° + cos 3*^ Prove the following formulas : 1 ^ nrr n 2tanx _^ ^ . sin a; ^-^ 27. sin2a; = ^_^^_,^ ' 29. tan Jo; = l + tan^oj ' i_|_cosi» oo o 1 ~ tan^a; _ . . ^ sin x 28. cos2a; = — -rr — 5— 30. coti-a; = :; 1 + tannic ^ 1 — cos x 31. sin ^x =b cos -J-a: = Vl ± sin x. __. tana^rttany 32. — 7 —^ = =t tan x tan y. cot X ± cot y 33. tan(45°-a^) = iq^^^- ^ ^ 1 + tan X If Aj Bj C are the angles of a triangle, prove that : 34. sin^ + sin^ + sin (7==4cos-j-^ cos^i?cos-|-C. 35. cos ^ + cos J5 + cos C = l+4sin^^ sin ^^ sin -^ C. 36. tan^ + tan B + tan C = t^nA X tan^ X tan C. 37. cot |^ + cot i^ + cot ^C = Got^AX cot ^B X cot 1 C. Change to forms more convenient for logarithmic computa- tion : 38. cot a; + tan aj. 43. 1 + tan a; tan y. ^ 39. cot X — tan x. 44. 1 — tan x tan y. _ 40. cot a? + tan 2/. 45. cota; cot^z + l. 41. cot a? — tany. 46. cot a? cot y — 1. ^ 1 — cos 2a? tan a: + tan 2/ 14- cos 2a; ' cot a; + cot 1/ 58 trigonometry. § 32. Anti-Trigonometric Functions. If y is any trigonometric function of an angle x, then x is said to be the corresponding anti-trigonometric function of y. Thus, if 2/ = sin x, x is the anti-sine of y, or inverse sine of y. The anti-trigonometric functions of y are written sin-^y, tan-^?/, sec~^2/j vers"^?/, cos~^2/, cot~^?/, csc~^?/, covers"^ y. These are read, the angle whose sine is y, etc. For example, sin 30° = ^; hence 30° = sin"' ^. Similarly 90° = cos-^ =: sin-i ;i^ . ^nd 45° = tan-^ 1 = sin-^ —-= ; etc. V2 The symbol -^ must not be confused with the exponent — 1. Thus sin-ix is a very different expression from - — > which would be written (sinx)— 1. On the Continent of Europe mathematical writers employ the notation arc sin, arc cos, etc. , for sin— i, cos-i, etc. But the latter symbols are most common in England and America. There is an important difference between the trigonometric and the anti-trigonometric functions. When an angle is given, its functions are all completely determined; but when one of the functions is given the angle may have any one of an indefinite number of values. Thus, if sin ?/= |, y may be 30°, or 150°, or either of these increased or diminished by any integral multiple of 360° or 2'jr, but cannot take any other values. Accordingly sin~^ ^ = 30° ± 2 mr, or 150° =b 2 titt, where n is any positive integer. Similarly, tan~^l = 45°d=27i7r or 225° + 2 WTT ; i.e., tan-^ 1 = 45° ± utt. Since one of the angles whose sine is x and one of the angles whose cosine is x together make 90°, and since similar rela- tions hold for the tangent and cotangent, for the secant and cosecant, and for the versed sine and coversed sine, we have sin~^ x -\- cos~^ ^ = o ' sec~^ ic + csc~^ ^ = o ' GONIOMETRY. 69 tan~^ X -\- cot~^ ^ = 9 ' vers~^ x -\- covers "^ cc = - , where it must be understood that each equation is true only for a particular choice of the various possible values of the functions. For example, if x is positive, and if the angles are always taken in the first quadrant, the equations are correct. Exercise XV. 1. Find all the values of the following functions : sin-iiV3, tan-i^Va, vers-^^, cos-i(— iV2), csc-i(V2), tan-i Qc^ ggg-i 2^ cos-^ (— \ V3) . 2. Prove that sin~^(—£c)=—sin-^ a;; cos~^(— ic)=7r— cos~^ic. ^ 3. If sin~^x -|- sin"^?/ = tt, prove that x^y. ^ 4. If 2/ = sin~^-J, find tan 3/. 5. Prove that cos (sin-^ic) = Vl — x^. 6. Prove that cos (2 sin~^ cc) = 1 — 2 icl X ~T~ II 7. Prove that tan (tan~^ x + tan-^ ?/) = -- ^ ^ 1 — xy 8. If ic= \l^, find all the values of sin~^ic + cos"~^a;. 9. Prove that tan~^ I , \ = sin~^a;. 10. Find the value of sin (tan~\\). 11. Find the value of cot (2 sin-if ). 12. Find the value of sin (tan-^^ + tan"^^). 13. If sin~^ic = 2 cos~^x, find x. 2x t 14. Prove that tan (2 tan"^ x) = _ g - _L X 2x 15. Prove that sin (2 tan"^ x) = ■^ - CHAPTER IV. THE OBLIQUE TRIANGLE. § 33. Law of Sines. Let a, B, C denote the angles of a triangle ABC (Figs. 31 and 32), and a, b, c, respectively, the lengths of the opposite sides. Draw CD _L AB, and meeting AB (Fig. 31) or AB pro- duced (Fig. 32) at D. Let CD = h. C A c Fig. D 31. B A~ c B Fig. 32. In both figures, ^ ' A In Fig. 31, h - = sm B. a In Fig, 32, ^ = sin (180°- -B) = smB. Therefoi-e, whether h lies within or without the triangle, we obtain, by division, a _ sin A b sinB [25] THE OBLIQUE TRIANGLE. 61 By drawing perpendiculars from the vertices A and B to the opposite sides we may obtain, in the same way, b sinj5 a sin^ c sin C sm C 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 written in the symmetrical form, a h c sin A sin ^ sin G Each of these equal ratios has a simple geometrical meaning which will appear if the Law of Sines is proved as follows : Circumscribe a circle about the triangle ABC (Fig. 33), and draw the radii OA, OB, OC] these radii divide the triangle into three isosceles triangles. Let B denote the radius. Draw 031 J_BC. By Geometry, the angle BOC = 2 A; hence, the angle BOM=A, then BM=EsmBOM = BsinA. .-.BCoT a = 2BsmA. In like manner, h = 2 B sin B, and G = 2 B sin C. Whence we obtain a h Fig. 33. 2B sinvl sin^ sin (7 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. 62 trigonometry. § 34. 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 rigs. 31 and 32, w" = h' + RD\ In Fig. 31, BB =c — AD', in Fig. 32, BD =AD~c ; in both cases/ BD" = AB^ — 2cXAB-\-c\ Therefore, in all cases, a^ = h^ + Ajf -\-c^ — 2cX AB. Now, h''-}-AF = b% and AlB ^hcosA. Therefore, B,^ = h^+c^ — 2\)CG0sA. 1262 In like manner, it may be proved that i2^^2_j_^2 — 2ac cos B, c^ = a^-\-P — 2abG0sC. The three formulas have precisely the same form, and the law may be stated as follows : The square of any side of a triangle is equal to the sum of the squares of the other two sides, dinmiished by twice their product into the cosine of the included angle. § 35. Law op Tangents. By § 33, a :b = sin A : sin ^ ; whence, by the Theory of Proportion, a — b sin ^ — sin B a-\-b sin ^4" sin ^ But by [24], page 56, sin A — sin B tan ^(A — B) sin A + sin B tan ^(A-\-B) Therefore, a-b_ tani(A-B) a + b tan^(A4-B) L^^-J THE OBLIQUE TRIANGLE. 63 By merely changing the letters, a — c tan \{A — C) h — c tan \{B — (7) a4-c"~tani(^+C)' H^'~tan^(^+C) Hence the Law of Tangents : The difference of two sides of a triangle is to their sum as the tangent of half the difference of the opjiosite angles is to the tangent of half their sum. Note. If in [27] 6> a, then B'^A. The formula is still true, but to avoid negative numbers, the formula in this case should be written h — a _ tani(E — J.) Exercise XVI. 1. What do the formulas of § 33 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 proportional to the adjacent sides. 3. What does Formula [26] become when A = 90° ? when j; = 0° ? when J. = 180° ? What does the triangle become in each of these cases ? Note. The case when A = 90° explains why the theorem of § 34 is sometimes termed the Generalized Theorem of Pythagoras. 4. Prove (Figs. 31 and 32) that whether the angle B is acute or obtuse, c = a cos B-{-h 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 example) prove the theorem of § 34 ; c:=a cos B -{-h cos A, h=^a cos C -\- c cos A, a=^h cos C -\rG cos B. Hint. Multiply the first equation by c, the second by 6, the third by a ; then from the first subtract the sum of the second and third. 64 TRIGONOMETRY. 6. In Eormula [27] what is the maximum value of -J (A — B) ? 7. Find the form to which Formula [27] reduces, and describe the nature of the triangle, when (i.) (7 = 90° ; (ii.) A — B = 90°, and B=a § 36. The Solution of an Oblique Triangle. The formulas established in §§ 33-35, 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. Let A, B, C denote the angles, a, h, c the sides respectively. § 37. Case I. Given one side a, and two angles A and B; find the remain- ing parts C, b, and c. 1. C=180°-(^ + J5). „ b smB ^ asinB a . „ 2. - = - — -; .■.b = — — - = - — rXsm^. a sm^ sm^ sm^ ^ c sin (7 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 07ie, and only one^ triangle that will satisfy the given conditions. li a = h, then by Geometry A = B', both A and B must be acute, and the required triangle is isosceles. If a<,h, then by Geometry A 1, and the tri- angle is impossible. 68 TRIGONOMETRY. These results, for convenience, may be thus stated : Two solutions J if A is acute and the value of a lies between b and h sin A. No solution ; if ^ is acute and sin A ; or if A is obtuse and a < Z>. One solution ; in all other cases. The number of solutions can often be determined by inspec- tion. In case of doubt, find the value of b sin A. Or we may proceed to compute log sin B. If log sin ^ = 0, the triangle required is a right triangle. If log sin^>0, the triangle is impossible. If log sin i> < 0, there is one solution when a'>b; there are two solutions when a<^b. When there are two solutions, let B\ C\ c', denote the unknown parts of the second triangle j then, ^' = 180°-^, C' = lSO° — (A-\-B')=B — A, a sin C c' sin^ Examples. 1. Given a = 16, b = 20, ^==106°; find the remaining parts. In this case a<6, and ^ >90° ; therefore the triangle is impossible. 2. Given a = 36, ^^ = 80, ^ = 30°; find the remaining parts. Here we have &sin^ = 80 X ^ = 40 ; so that a<6sin^, and the triangle is impossible. 3. Givena = 72630, ^» = 117480, J = 80°0'50"; find^, C,c. a = 72630 b= 117480 ^ = 80°0'50' cologa= 5.13888 log&= 5.06996 log sin A = 9.99337 log sin 1?= 0.20221 Here logsinB>0. .-. no solution. THE OBLIQUE TRIANGLE. 69 4. Given a = 13.2, b = 15.7, A = 57° 13' 15" ; find B, C, c. a= 13.2 6=15.7 ^ = 57° 13' 15" Here log sin B = 0, .-. a right triangle. cologa = 8.87943 log 6= 1.19590 logsin^ = 9.92467 log sin 5 =0.00000 5=90° .-. C = 32°46'45' c = 6 cos J. log 6 =1.19590 logcos^ = 9.73352 logc = 0.92942 c=8.5 5. Given a = 767, h = 2A2, ^ = 36° 53' 2"; find B, C, c. a = 767 6=242 A = 36° 53' 2' Here a > 6, and log sin 5 <; 0. .-. one solution. 6. Given a other parts. a =177.01 6 = 216.45 A = 35° 36' 20" cologa= 7.11520 log6= 2.38382 logsinJ. = 9.77830 log sin J5= 9.27732 B = 10° 54' 58' .-. C = 132° 12' 0' loga = 2.88480 logsinC = 9.86970 colog sin A = 0.22170 logc = 2.97620 c = 946.675 = 177.01, Z* = 216.45, ^ = 35° 36' 20"; find the Here a < 6, and log sin B ^ = 0.72358, V28 = 5.2915; thatis, & = 5,2915. THE OBLIQl-^ TRIANGLE. 73 Exercise XIX. ^ 1. Given a = 77.99, i» = 83.39, C= 72*15'; fiiid.4=5ri5', ^=56*30', «?=95.24. 2. Given ft = 872.5, r= 632.7, ^=80**; find ^ = 60** 45', C= 39° 15', a = 984.83. _ 3. Given (7 = 17, ft = 12, C=59°17'; find .4 =77° 12' 53", ^ = 43" 30' 7", r= 14.987. 4. Given ft = VB, «?= V3, ^ =35° 53'; find ^ = 93° 28' 36", C= 50° 38' 24", a = 1.313. 5. Given a =0.917, ft = 0.312, C=33°7'9"; find .4 =132° 18' 27", ^=14° 34' 24", c=0.6775. 6. Given CI = 13.715, and 2gos^^A = — H ^> bG ^ be whence smiA = yj ^^~^^^^^~''\ [28] C08iA = V^-^^' [29] and by [2] tan i A = V^'t^Z^' C^^] By merely changing the letters,- sin 1 -r. (s — ^) (^ — ^) • 1 ^ l(s — <^) (s — b) ^ ^^ 5(S— ^) ^ ^^ 5(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. 76 TRIGONOMETRY. It is not necessary to compute by the formulas more than two angles ; for the third may then be found from the equation 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^^ may be written l (s — a)(s — b ^ s (s-ay Hence, if we put !')(^-o) or 1 (s-a)(s-l,)(s — a^ s 1 V (s— a) (s — b) (s — c) r, we have Likewise, tan^A: tan ^ B ■■ r [31] [32] tan^ C = Examples. Given c^ = 3.41, h = 2m, c o\ 1.58 : find the ansrles. as' ^ Using Formula [30], and the corresponding formula for tan ^JB, we 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 s-b= 1.195 s — c = 2.215 cologs = 9.42079 colog {s— a) = 0.41454 log (s - 6) = 0.07737 log (s - c) = 0.34537 2 )0.25807 log taniJ. = 0.12903 iA= 53° 23' 20" A = 106° 46' 40'' . J. 4- 5 =153° 39' 64", and colog s= 9.42079—10 log(s-a)= 9.58546-10 colog(s-6)= 9.92263-10 log (s - c) = 0.34537 2 )19.27425-20 log tan i I? = 9.63713-10 B = C = 26° 20' 6". 23= 46° 26' 37' 53' 14' THE OBLIQUE TRIANGLE. 77 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, log tan ^^, etc., by subtracting log(s — a), etc., from log adding the cologarithm : a = 3.41 b = 2.60 c= 1.58 2s =7.59 s = 3.795 s-a = 0.385 5-6=1.195 5— c = 2.215 log (s- a) = 9.58546 log (s- 6) = 0.07737 log {s- c) = 0.34537 colog s = 9.42079 logr2 = 9.42899 logr =9.71450 2s =7.590 (proof). log tan iA = log tan i 5 = log tan i C = iA = iB = iC = A = B = C = Proof, A + B+ C = , if we find r instead of 10.12903 9.63713 9.30912 53° 23' 20'' 23° 26' 37" 13° 10' 3" 106° 46' 40" 46° 53' 14" 26° 20' 6" 180° 0' 0" Note. Even if no mistakes are n^ade 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. Exercise XX. Solve the following triangles, taking the three sides as the given parts : 1 a 6 c yi B C 51 65 20 38° 52' 48" 126° 52' 12" 14° 15' 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 34 50 31 49 57 42° 6' 13" 16° 25' 36" 46° 49' 35" 56° 6' 36" 30° 24' 57° 59' 44" 81° 47' 11" 133° 10' 24" 75° 10' 41" 5 6 19 43 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 I 8° 20' 33° 40' 138° 10 V5 V6 V7 ! 51° 53' 12" 59° 31' 48" 68° 35' 78 TRIGONOMETRY. 11. Given a = 6, b = S, c = 10; find the angles. 12. Given a = 6, b = 6, c = 10 ; find the angles. 13. Given a = 6, b = 6, c = 6; find the angles. 14. Given a = 6, b = 5, c = 12 ; find the angles. 15. Given a = 2, b= VC, c = VS — 1 ; find the angles. 16. Given a = 2, b= V6, c = V3 + 1 ; find the angles. 17. The distances between three cities A, B, and C are as follows : ^^ = 165 miles, ^C= 72 miles, and BC^= 185 miles. B is due east from A. In what direction is C from 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, 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? § 41. Area of a Triangle. Case I. When two sides and the i7icluded angle are given: In the triangle ABC (Fig. 31 or 32), the area F=icXCB. By §11, CD = a sin B. Therefore, F = i ac sin B. [33] Also, F= ^ab sin C and F=^ bo sin A. Case II. When a side and the two adjacent angles are given: By § 33, sinA:s\nC::a\c. a sin C Therefore, c = sin^ THE OBLIQUE TRIANGLE. 79 Putting this value of c in Formula [33], we have a^sinBsinC 2sin(B + C)' ^^ Case III. When the three sides of a triangle are given: By § 29, sini? = 2sini^Xcosi-^. By substituting for sin ^ B and cos |- B their values in terms of the sides given in § 40^ ,*: 2 /- sin i> = — V^ (s — a)(s — b) {s — c). By putting this value of sin B in [33], we have F=Vs(s-a)(s — b)(s — c). [35] Case IY. When the three sides and the radius of the circum- scribed circle, or the radius of the inscribed circle, are given : If R denotes the radius of the circumscribed circle, we have, from § 33, By putting this value of sin B in [33], we have F = ^- [36] If r denotes the radius of the inscribed circle, divide the triangle into three triangles by lines from the centre of this circle to the vertices; then the altitude of each of the three triangles is equal to r. Therefore, F = ir(a + b + c)=js. [37] By putting in this formula the value of F given in [35], =4 (s — a) (s — b) (s — c) whence r, in [31] § 40, is equal to the radius of the inscribed circle. 80 TRIGONOMETRY. Exercise XXI. Find the area : ^\. Given a = 4474.5, ^> = 2164.5, (7=116° 30' 20". 2. Given Z» = 21.66, c = 36.94, ^ = 66° 4' 19". 3. Given a = 510, c = 173, ^ = 162° 30' 28". 4. Given a = 408, ^^ = 41, c = 401. 5. Given a = 40, ^^ = 13, c = 37. _^^6. Given a = 624, ^^ = 205, c = 445. ^^1. Given & = 149, ^ = 70° 42' 30", ^ = 39° 18' 28". 8. Given 61 = 215.9, c = 307.7, ^1 = 25° 9' 31". 9. Given ^^ = 8, c = 5, ^ = 60°. 10. Given a = 7, c = 3, ^ = 60°. ^ 11. Given a = 60, i? = 40° 35' 12", area = 12 ; find the 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. 16. Show that the area of a quadrilateral is equal to one half the product of its diagonals into the sine of their included angle. Exercise XXII. 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 K 11° 15' E. Find its distance from each position of the ship. THE OBLIQUE TRIANGLE. 81 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. 6. 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 (7, and from a point D directly under the balloon and in the same horizontal plane with A and B. If CI) = 2000 yards, Z^ CD = 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 is 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, CHAPTER V. MISCELLANEOUS EXAMPLES. Problems in Plane Trigonometry. 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 is 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 ayigle of elevation. If the object is 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 hearing from that object. If two objects are not in the same horizontal plane with either 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 bearing 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 " are selected from those published by Mr. Charles W. Sever, Cambridge, Mass. The full set can be obtained from him in pamphlet form. MISCELLANEOUS EXAMPLES. 83 Eight 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 IG' 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 of latitude 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 inscribed 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. 84 TRIGONOMETRY. 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 V^, 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 a lateral 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 elevation of a wall on the opposite edge is 62° 39' 10". Find the length of a ladder which will reach from the point of observation 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 lehgth 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. MISCELLANEOUS EXAMPLES. 85 20. The height of a house subtends a right angle at a window 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'; the balloon drifts 2J 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 elevation is 18°. Show that the height of the tower is the tangent of 18° being V(2 + 2V5) V(10+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° respectively. Prove that the length of the pole is twice the height of the mound. 2Q. 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 2 a 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 86 TRIGONOMETRY. tangent to a line of true level at any point is a line of apparent level. If at any point both these lines are 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 elevation or the measured distance. Oblique Triangles. 29. To determine the height of an inaccessible object situated on a horizontal plane, by observing its angles of elevation at two points in the same line with its base, and measuring the distance between these two points. X 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. MISCELLANEOUS EXAMPLES. 87 36. From the top of a house 42 ft. high, the angle of elevation 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^ 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 object on the opposite bank to be 2^°. 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. 88 TRIGONOMETRY. 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° ^^^ 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 elevation of the top A is 60°. After ascending the mountain one mile, at an inclination of 30° to the horizon, and reaching a point (7, 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 IST. 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 ((7 being nearly opposite A), and observe the angles ACB, 58° 20'; ACD, 95° 20'; ADB, 53° 30'; BDC, 98° 45'. CD is 600 ft. What is the distance required ? MISCELI.ANEOUS EXAMPLES. 89 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 summit 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, measured 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. 5S. To find the distance of an inaccessible point C from either of two points A and B, having no instruments to measure angles. Prolong CA to a, and CB to b, and join AB, Ah, and Ba. Measure AB, 500 j aA, 100; aB, 560; hB, 100 ; and Ab, 550. 90 TRIGONOMETRY. 59. Two inaccessible points A and B, are visible from D, 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 CD, 200 ft. ; ADC, 89° ; ACD, 50° 30'. Then take some point JS, whence D and B are visible, and measure DE, 200 ; BDE, 54° 30' ; BBD, 88° 30'. At D measure ADD, 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 D. From C meas- ure CF, 200 yds. to F, whence A can be seen ; and from D measure DF, 200 yds. to F, whence B can be seen. Measure AFC, 83°; ACD, 53° 30'; ACF, 54° 31'; BDF, 54° 30'; BDC, 156° 25'; DFB, 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 is equal to 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 CD, 200 yds. at right angles to AC. The angle BDA 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. MISCELLANEOUS EXAMPLES. 91 65. If sin^A -f 5 cos^^ = 3, find A. 66. If sinM = m cos A — n, find cos A. 67. Given sin ^ = m sin B, and tan A = n tan ^, find sin A and cos ^. 68. If tanM + 4 sin^^ = 6, find ^. 69. If sin ^ = sin 2 A, find A. 70. If tan 2 ^ = 3 tan ^, find A. 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 circles in terms of n and a. If P, H, D are the sides of a regular inscribed pentagon, hexagon, and decagon, prove P^^H^-\-D\ 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, show that 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. 92 TRIGONOMETRY. 80. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the included 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. S2° 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 AB, 17.22 ch. ; AD, 7.45 ch. ; CD, 14.10 ch. ; BC, 5.25 ch. ; and the diagonal AC, 15.04 ch. Find the area. 88. The diagonals of a quadrilateral are a and h, and they intersect at an angle D. Show that the area of the quadri- lateral is ^ah 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. Show that the area of a regular polygon of n sides, of ^. ^ . . nw^ ,180° wnicn one is a, is ~r- 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. MISCELLANEOUS EXAMPLES. 93 94. One side of a regular decagon is 46. Eind 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 circumscribed polygon of n sides is nr^ tan The area of a regular inscribed polygon is - r^ sin ^1 ft 102. If a is a side of a regular polygon of 7t. sides, the area of the inscribed circle is — — cot^ 4 n TTd The area of the circumscribed circle is —r- csc^ 180^ 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 circumscribed regular polygon of half the number of sides. 105. The area of a circumscribed regular polygon is an harmonic mean between the areas of an inscribed regular 94 TRIGONOMETRY. polygon of the same number of sides, and of a circumscribed regular polygon of half that number. 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 meridian comprehended between the parallels of latitude passing through those places. The departure between two meridians is the arc of a parallel 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 MISCELLANEOUS EXAMPLES. 95 be straight lines, lying in one plane, they will form a right triangle, called the triangle of plane sailing. If ABD be a plane triangle, right-angled at Z>, and AD represent the dif- ference of latitude of A and B, DAB will be the course from 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' S., on a course N.E. by E. f E., until the departure is 207 miles. Find the distance, 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 between 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 * 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. 96 TRIGONOMETRY. point left to the point reached ? (Take earth's radins, 3962.8 statute miles.) 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 distance 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 MISCELLANEOUS EXAMPLES. 97 very high latitudes or excessive runs, such an assumption produces no great error. By the formula of Example 120, then, we shall have Diff. long. = depart. X sec. mid. lat. 124. A ship leaves latitude 31° 14' JST., 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° K, 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' K, 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? 98 TRIGONOMETRY. 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 ? 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 by itself, and these inde- pendent 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 KN.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 K, 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. MISCELLANEOUS EXAMPLES. 99 and the formula of Example 123, we find the difference of longitude to be 201', or 3° 21' W. Hence the longitude reached is 22° 3' W. With the difference of latitude 86.8 miles, and the departure 161.3 miles, we find the course to be N. 61° 43' W., and 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.KW. 42 miles. Solve as in Ex. 136. 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. Problems in Goniometry. Prove that 1. sin X -|- cos £c = V2 cos (x — Jtt). 2. sinic — cos £c = —V2 cos (cc + iTr). 3. sina;+ V3cosic = 2sin(a? + -J-7r). 4. sin (x + -J-tt) + sin (cc — •J7r) = sinx. 5. cos (^ + ^ tt) + cos (x — ^ 7r) = V3 cos X, 6. tancc + seca: = tan(icc4-:|^7r). 1 7. tan X -\- sec x 8. sec X — tan x 1 — tan X cot x—1 1 + tan x cot x-\-l ■ sin X , 1 + cos X 9. zr-x -• = 2csca;. 1 + cos X sm X 10. tan£c + cotcc = 2csc2ic. 12. 1 +tan£ctan2a; = sec2x. sec cc 11. cotic — tana3 = 2cot2ic. 13. sec2ic = ^r 5— 2 — sec^ic 100 TRIGONOMETRY. Prove that 14. 2sec2a^ = sec(cc + 45°)sec(a; — 45°). cos X -\- sin X 15. tan 2 a; + sec 2 a:; = cos X — sm X • ^ 2 tan a? ^rr o • . • o 2sin^a7 16. sm 2a;— ■ , . — n— 17. 2sina;4-sin2a; 18. sin 3 a; = 1 + tan^a; 1 — cos a; sin^ 2 a; — sin^a; sma; 3 tan a; — tan^x tan 2a; + tan a; sin 3a; 1 — Stanza; * tan 2a; — tana; sin a; 21. sin (x-\-y)-\- cos (x — y) = '^ sin (a; + :J-7r) sin {y -\- Jtt). 22. sin (x-{-y) — cos (a; — ?/) = — 2 sin (x — ^ir) sin {y — ^ tf). 23. tanx + tany^ ^'"(^+y> - COS a; cos 3/ ^. ^ / . N sin 2a; + sin 2y 24. tan (a; + 2/)= ^^ — ^• ^ "^^ cos 2 a; 4- cos 2?/ 2^ sin a; + cosy _ tan j j-(a;+?/)4-45°^ sin a; — cos?/ tanJ-J(a; — y) — 45° J 26. sin 2a; + sin 4a; = 2 sin 3a; cos X. 27. sin 4 a; = 4 sin a; cos a; — 8 sin^a; cos x = 8 cos^a; sin x — 4 cos x sin x. 28. cos 4 a; = 1 — 8 cos^a; + 8 cos^a; = 1 — 8 sin^ic + 8 sin*a;. 29. cos 2 a; + cos 4 a; = 2 cos 3 a; cos a;. 30. sin 3a; — sin a; = 2 cos 2a; sin a;. 31. sin^a; sin 3 a; + cos^a; cos 3 a; = cos' 2 x. 32. cos^a; — sin*a; = cos 2 x. 33. cos^a; -f- sin*a; = 1 — ^ sin^ 2 x. 34. cos^a; — sin^a; = cos 2 a; (1 — sin^a; cos^a;). 35. cos^a; -j- sin^a; = 1 — 3 sin^a; cos^x. „„ sin 3 a; 4" sin 5 a; 06. — - = cot x. cos3x — cos 5a; MISCELLANEOUS EXAMPLES. 101 Prove that ^„ Sin3a7 + sm5x 37. —. r— 7— 7-- = 2cos2ic. smx + sm3a; 38. csccc — 2 cot 2 cc cos ic^ 2 since. 39. (sin 2 cc — sin 2 ?/) tan (cc -j- ?/) = 2 (sin^aj — sin^i/). sec 07 csccc csc^a? sec^ic sin2 3£c 40. (1 + cot £c 4" tan x) (sin x — cos x) 41. sina; + sin3cc + sin5ic = smic ,^ 3cosic + cos3ic ,, 42. 77-^ ■ — r— :— = cot^aj. 3sinic — smSic 43. sin 3 ic = 4 sin x sin (60° + x) sin (60° — x). 44. sin4a; = 2siniccos3ic + sin2ic. 45. sinic + sin(£c — f 7r)4-sin(^7r — a7) = 0. 46. cos X sin (?/ — ^) + cos y sin (z — x)-\- cos z sin (x — y)=^ 0. 47. cos (x + 2/) sin y — cos (x + z) sin « = sin (x -\- y) cos ?/ — sin (x + ^) cos z. 48. cos (cc + ?/ + ^) -f- cos (cc + 2/ — «) + cos (x — y-}-z) + cos (2/ + « — cc) ^ 4 cos X cos y cos «. 49. sin (x + ?/) cos (ic — ?/) + sin (y -\- z) cos {y — z) + sin {z + a?) cos (^ — a;) = sin 2 x -|- sin 2y-\- sin 2^. 60. si°7y + siniy ^^^ sin75° — sinl5° 51. cos 20° + cos 100° + cos 140° = 0. 52. cos 36° + sin 36° =V2 cos 9°. 53. tan 11° 15' + 2 tan 22° 30' + 4 tan 45° = cot 11° 15'. If A, B, C are the angles of a plane triangle, prove that 54. sin 2^ + sin 2^ + sin 2 C = 4 sin^ sin^ sin C. 55. cos 2 ^ + cos 2 .B + cos 2 C = — 1 — 4 cos A cos B cos C. 102 TRIGONOMETRY. If A, B, C are the angles of a plane triangle, prove that 56. sin 3^ + sin 3^+ sin 3 C = — 4cos -;^cos -^ cos -^' Z Z 2i 57. Gos^A + cos^-B + cos^ C = 1 — 2 cos A cos B cos C. If A^B+C = 90°, prove that 58. tan^ tan J5 + tan^ tan C + tan C tsmA = 1. 59. sin^^ + sin2^ + sin^ (7 = 1 — 2 sin^ sin jB sin C. 60. sin 2^ + sin 2^ + sin 2 C = 4 cos^ cos J5 cos a Prove that 61. sin (sin-^ x + sin~^ y) z=x \ll — if -\- ysl 1 — x\ 62. tan (tan-^ x + tan-^^/) = ^^^ - 63. 2tan-ix = tan-i:r^^- 1—x^ 64. 2 sin-^a; = sin-i (2a; Vl — x^, 65. 2cos-^x = cos-i(2ic2 — 1). 3£c — a;^ QQ. 3 tan~^ x = tan~^ 1-3x2 67. sin-iV- = tan-i\/ 68. sin-i\/^^^ = tan-i-\/^::^. y — x l-2x + 4x2 ' — l + 2aj+4x2-^'^ii 2a;'' 1 69. tan-i -— — — — -^ + tan-i :7-n-7^ — r-r-5= tan 70. sin-^a; = sec~^ Vl — a;2 71. 2sec-^a; = tan-^ ^y^'~^ . 2— or 72. tan-ii + tan-ii = 45°. MISCELLANEOUS EXAMPLES. 103 73. tan-i ^ _|_ ^^^j^-i ^ _ ^^j^-i ^^ 74. sin-^ I H- sin-^ || = sin-^ f |. 75. sin-i-^+siii-i-4^=45°. V82 V41 76. sec-i I + sec-i || ^ 75° 45'. 77. tan-i (2 + V3) — tan-^ (2 — V3) = sec-^ 2. 78. tan-i ^ _^ ^^j^-i ^ _^ ^an-i i + tan-^ ^ = 45°. 79. Given cos ic = f , find sin ^cc and cos ^x, 80. Given tan x = ^, find tan ^x. 81. Given sin x -\- cos a: = V^, find cos 2x. 82. Given tan 2x = \^- , find sin x. 83. Given cos 3 x = |f , find tan 2 a;. 84. Given 2 cscx — cotcc= V3, find sin -J a;. 85. Find sin 18°, cos 36°. Solve the following equations : 86. sin£c = 2sin (-J-TT + a;). 90. sinic + cos2ic = 4sin2a;. 87. sin 2ic = 2cosa7. 91. 4 cos 2a; + 3 cos a? = 1. 88. cos 2 £c = 2 sin a;. 92. sin a; + sin 2 a; = sin 3 a;. 89. sin a; + cos a; = 1. 93. sin 2a;:=3sin^a; — cos'^a;. 94. tan a; + tan 2 a; = tan 3 a;. 95. cot a; — tanx=:sina; + cosa;. 96. tan^a; = sin2a;. 99. sina;+sin2a;=l — cos2a:. 97. tana; + cota;:=tan2a;. 100. sec2a^ + l = 2cosa;. 98. -^~l^'^^ = cos2x. 101. tan 2a; + tan 3a; = 0. 1 + tan X 102. tan(i7r + a;)+tan(i7r — a;) = 4. 103. Vl + sina; — Vl — sina; = 2cosx. 104 TRIGONOMETRT. Solve the following equations : 104. tanxtan3ic = — f. 105. sin(45°+ic) + cos(45° — ic) = l. 106. tSinx-{-seGx = a. 107. cos2a; = a(l — coscc). 108. cos2£c (1 — tancc) = a(l-[-tanic). 109. sin^ X + cos^x = j\ siu^ 2 x. 110. cosScc + Scos^x^O. 111. sec (x + 120°) + sec (x — 120°) = 2 cos x. 112. cscic = cota:4- V3. 114. coscc — cos 2a; = l. 113. 4cos2ic + 6sinic = 5. 115. sin 4 a? — sin 2 a: = sin ic. 116. 2sin2x + sin2 2x = 2. 117. cos5x + cos3a3 + cosa: = 0. 118. secic — cot x = CSC a: — tana;. 119. t&-n.^x-i-cot^x = i^^-. 120. sin4a; — cos3a; = sin2a;. 121. sina; + cosa; = secx. 122. 2cosa;cos3a^ + l = 0. 123. cos3a; — 2cos2a; + cosx = 0. 124. tan 2 a; tan a: :=1. 125. sin (x + 12°) + sin (x — 8°) = sin 20°. 126. tan(60° + a;)tan(60° — a;) = — 2. 127. sin (a; + 120°) + sin (a; + 60°) = 1. 128. sin(a; + 30°)sin(a; — 30°)==i. 129. sin*a; + cos*a; = |. 131. tan (a; + 30°) = 2 cos cc. 130. sin'^a; — cos^x = ^5^. 132. sec a; = 2 tana; + J. 133. sin (x — y)= cos x, cos (x-{-y) = sin x. 134. tan x + tan y = a, cot x + cot y = b. 135. sin (a; + 12°) cos (a; — 12°)= cos 33° sin 57°. 136. sin-^ X + sin-i ix = 120°. 137. tan-ia; + tan-i2a; = tan-i3V3. 138. sin-ix + 2cos-ia; = |7r. MISCELLANEOUS EXAMPLES. 105 Solve the following equations : 139. sin-ia; + 3cos-i£c = 210°. 140. tan-ia; + 2cot-iaj = 135°. 141. tan-i (^ _j_ 1) _|. tan-i (^^_i^^ tan-^ 2 x. 142. tan-i^ + tan-i^^^fTT. x-\-l X — 1 143. tan-i3^^„-::60°. 1 — x^ Find the value of : 144. asecic + ^cscic, when tancc^AZ- 145. sin 3 x, when sin 2 cc = Vl — ml pso iT* ^"^ sec cc / — 146. — 5 — i r- , when tan x = \l\. 147. sin X, when tan^cc + 3 cot^cc = 4. 148. cos X, when 5 tan x + sec a? = 5. a 149. sec 07, when tan x = V2a+1 Simplify the following expressions : (cos X + cos ijY + (sin x + sin y)' cos'^ i (»^ — 2/) sin (a; H- 2y) — 2 sin (x-^y)-\- sin .t cos (.T 4- 2?/)— 2 cos (£c + 2/) + cosa; sin (g; — ^) + 2 sina; 4- sin (x + g) * sin(?/ — ;^) + 2sin2/+sin(2/+2;) cosGa? — cos 4a; 153. -: — -^ — j — : — -. — sm 6a; + sin4a7 154. tan-i (2a; + 1) + tan-i ^2x- 1). 1 1^^- 1 4_ sin2 ^ + 1 -I- cos^a; "^ 1 + sec^x "^ 1 + csc^x 156. 2sec2a; — sec*a3 — 2csc2a:; + csc'*a;. EE"TEA]^OE EXAMII^ATIOK" PAPEES.* PLANE TRIGONOMETRY AND LOGARITHMS. {Cornell, June, 1889.) (One question may be omitted.) 1. Prove that cos co-^ = sin ^ ; sec (^V + ^) = — CSC ^ ; tan (—0)= — tan 6 j CSC (7r — 0)= CSC $. 2. Draw the curve of tangents, and show the changes in the value of this function as the arc increases from 0° to 360°. 3. In terms of functions of positive angles less than 45°, express the values of sin — 250°, esc j|7r, tan — J / tt. Also find all the values of 6 in terms of a when cos ^ = Vsin^a. 4. (a) Given cos x = 0.5, find cos 2x and tan 2ic. (b) Prove that vers (180° — A)-\- vers (360° — A) = 2. 5. Prove the check formulae : a-]rb : c = cos^ (A — B) : sin^ C; a — b:c = sini(A — B):cosiC. * Note. In these papers, as in many text-books, the Greek letters a (alpha), ^ (bayta), y (gamma), 5 (delta), 6 (thayta), and (phee), are occasionally used to denote angles. ENTRANCE EXAMINATION PAPERS. 107 6. In a right triangle, r (the hypotenuse) is given, and one acute angle is n times the other ; find the sides about the right angle in terms of r and n. 7. The tower of McGraw Hall is 125 ft. high, and from its summit the angles of depression of the bases of two trees on the campus, which stand on the same level as the Hall, are respectively 57° 44' and 16° 59', and the angle subtended by the line joining the trees is 99° 30'. Find the distance between the trees. 11. {Cornell^ June, 1890.) (Omit one question.) 1. Prove that cot ( — 0) = — cot 6 ; esc tt — = esc 6 ; sin(7r4-^)= — sin^; sec co-^ = esc ^ ; cos (^7r-\-6)=^ — sm6. 2. Show that in any plane triangle sin iA = ^ ^ ^• 3. Find the value of sin (6 ± 0') in terms of sin 6, cos 6, sin O'j and cos 0'. 4. Given tan45° = l ; find all the functions of 22° 30'. 5. Determine the number of solutions of each of the tri- angles: a = 13.4, ^» = 11.46, A = 77°20'', c = 58, a=75, 0=60°^ b = 109, a = 9A, A = 92°10'; c = S09, b = 360, (7=21° 14'25". 6. In a parallelogram, given side a, diagonal d, and the angle A formed by the diagonals ; find the other diagonal and the other side. 7. A and B are two objects whose distance, on account of intervening obstacles, cannot be directly measured. At the summit of a hill, whose height above the common horizontal *-^-.^ 108 TKIGONOMETRY. plane of the objects is known to be 517.3 yds., angle ACB is 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. Pind the distance from A to B. III. {Cornell, September, 1891.) 1. Trace the value of tan and that of esc 6, as increases from 0° to 360°. 2. (a) Find the remaining functions of ^ when cos 6 = — |- V3. (b) Determine all the values of $ that will satisfy the relation cot ^ — 2 cos 0. 3. Prove the identity tan ^ — cot ^ = — : — y~ = — 2 cot 2 A. sm ^ cos ^ 4. Derive an expression for the sine of half an angle in a triangle in terms of the sides of the triangle. 5. Construct a figure and explain fully (giving formulae) how you would find the height above its base, and the distance from the observer, of an inaccessible vertical object that is visible from two points whose distance apart is known, and which can be seen from one another. 6. Given two sides of a plane triangle equal respectively to 121.34 and 216.7, and the included angle 47° 21' 11", to find the remaining parts of the triangle. 7. In a right triangle, if the difference of the base and the perpendicular is 12 yds., and the angle at the base is 38° 1' 8", what is the length of the hypotenuse ? ENTRANCE EXAMINATION PAPERS. 109 IV. {Cornell, June, 1892.) 1. By means of an equilateral triangle, one of whose angles is bisected, find the numerical values of the functions of 30° and 60°. 2. If 6 be any angle, prove that sin 6 = tan ^ : Vl + tan^ 0, cos 6 = Vcsc^ 9 — 1: esc 0. 3. Prove that ^^^^+^^^^ = - cot ^ (6-0'), where 6 and 0' cos ^ — cos ^' ^ ^ ^' are any angles. 4. Find sin 2 ^, cos 2 0, and tan 2 ^, in terms of functions of 0. 5. Assuming the law of sines for a plane triangle, prove that (a-\-b) : c = cosi(A — B) : sin ^ C, (a — b) : c = sin ^(A — B) : cos -J- C. 6. At 120 feet distance, and on a level with the foot of a steeple, the angle of elevation of the top is 62° 27'; find the height. 7. Solve the plane triangle given the three sides, a = 48.76, ^'^ 62.92, c = 80.24. V. {Harvard, June, 1889.) 1. In how many years will a sum of money double itself at 4 per cent., interest being compounded semi-annually ? 2. Given sin^a? = ? find sin 2x and tan 2 x. z 3. Find all values of x, under 360°, which satisfy the equation V8 cos 2 ic = 1 — 2 sin ic. 110 TRIGONOMETRY. 4. What is always the value of 2 sin^ic sin^y + 2 cos^a; cos^y — cos 2x cos 2y? 5. Find the area of a parallelogram, if its diagonals are 2 and 3, and intersect each other at an angle of 35°. 6o Find the bearing and distance from Cape Horn (55° 55' S., 67° 40' W.) to Falkland Island (51° 40' S., 59° W.). VI. {Earmrd^ June, 1890.) 1. In a certain system of logarithms 1.25 is the logarithm of -J. What is the base ? Be careful to remember what 1.25 means. 2. Find the tangent of 3 a; in terms of the tangent of x. 3. One angle of a triangle is 35°, and one of the sides including this angle is 24. What are the smallest values the other sides can have ? 4. Find all values of ic, under 360°, which satisfy the equation tan 2x (tSLTi^x — 1) = 2 sec^x — 6. 5. Two ships leave Cape Cod (42° N., 70° W.), one sailing E., the other sailing N.E. How many miles must each sail to reach longitude 65° W. ? 6. UA-{-B+C = 180°, find the value of « tan A -\- tan B + tan C — tan A tan B tan C. ENTRANCE EXAMINATION PAPERS. Ill VII. {Harvard, September, 1891.) 1. What is the base, when log 0.008 = — 1.5? 2. If cos (a—b) = 3 cos (a-\-b), find the value of I • ^ ^ ^ ^ sec a sec b 3. The area of an oblique-angled triangle is 50. One angle is 30°, and a side adjacent to that angle is 12. Solve the triangle. 4. Find all values of x, less than 360°, which satisfy the equation sin 2x — cos x = cos^x. 5. Find, by Middle Latitude Sailing, the course and the distance from Cape Cod (Lat. 42° 2' N., Long. 70° 4' W.) to Fayal (Lat. 38° 32' N., Long. 28° 39' W.). 6. In any triangle ABC, prove tan ^ A tan ^B -\- tan i A tan ^ C + tan -J B tan -J- C = 1. VIIL {Harvard, September, 1892.) (Take the questions in any order. One of the starred questions may he omitted.) 1. What is the base of a system of logarithms in which logGi^) = 2.33i? *2. Given the area of a right triangle, and the smallest angle, find the legs of the triangle in terms of the data. ^,^ -r^. -, -, ^ . sina /- , tana ,- =^3. Find a and B, given -7—7: = V2, and ^ = V3. ^' * sm/3 tan/? 9 i 112 TRIGONOMETRY. 4. One angle of an oblique-angled triangle is 45°, and an adjacent side is V2. What is the smallest value which the opposite side can have ? Solve the triangle when the opposite side is |. 5. A ship leaves Cape Cod (42° 2' N., 70° 4' W.) and sails 200 knots on a course S. 40° E. Find the latitude and longi- tude reached. *6. If 2 tan 2a = tan 25 sin 2 J, find the relation between the tangents of a and k IX. {Harvard, June, 1893.) (Take the problems in any order. One of the starred problems may be omitted.) 1. What is the base of the system of logarithms, when log 3 = 0.3976? *2. Solve the right-angled triangle in which one angle is 30°, and the difference of the legs is 4. =^3. Find x, given sec a; = 2 tan x-\-2. *4. One angle of a triangle is double another angle. The side opposite the first angle is three-halves of the side opposite the second angle. Find the angles. tV"'^ 6. Find, by Middle Latitude sailing, the course and distance from Funchal (32° 38' N., 16° 54' W.) to Gibraltar (36° 7' K, 5° 21' W.). *6. Eeduce to its simplest form cos 2 x tan (45° + ic) — sin 2 x. ENTRANCE EXAMINATION PAPERS. 113 {Harvard, September, 1893.) * (One of the starred problems may be omitted.) 1. If the base of our system of logarithms were 20 instead of 10, what would be the logarithm of one tenth? *2. The area of a right triangle is 6, and the sum of the three sides is 12. Solve the triangle, *3. Eeduce to its simplest form cos2J5 + sin2^cos2^ — sin2^cos2^. *4. Two angles of a triangle are 40° 14' and 60° 37'. The sum of the two opposite sides is 10. Find these sides. 5. A ship leaves Cape of Good Hope (34° 22' S., 18° 30' E.), and sails K 40° W. to Latitude 30° S. Find, by Middle Lati- tude Sailing, the Longitude reached and the distance sailed. *6. The base angles of a triangle are 22° 30' and 112° 30'. Find the ratio between the base and the height of the triangle. {Harvard, June, 1894.) (Arrange your work neatly.) 1. What is meant by the logarithm of a number n in the system whose base is 8 ? What will be the logarithm of 4 in this system? 2. Establish the formula : ,^ , ^ N /l — cos a; sin|x = ±(l+2cosa;)-Y Which sign should be used when x lies in the first quadrant ? When X lies in the second quadrant? 114 TRIGONOMETRY. 3. In a triangle two angles are equal to 32° 47' and 49° 28' respectively and the length of the included side is 0.072. Solve the tftangle. 4. A circular tent 30 feet in diameter subtends at a certain point an angle of 15°. Find the distance of this point from the centre of the tent. 5. A ship leaves Latitude 42° 2' N., Longitude 70° 3' W., and sails K 40° E. a distance of 420 miles. Find by Middle Latitude Sailing the position reached. XII. {Sheffield Scientific School, September, 1892.) 1. Express an angle of 60° in radians. 2. Kepresent geometrically the different trigonometric functions of an angle. State the signs of each function for each quadrant. 3. Express tan <^ and sec <^ in terms of sin <^. 4. Derive the formula sin a + sin /3 = 2 sin ^ (a-f- ^) cos i (a — (3). 5. Show that, if a, h and c are the sides of a triangle and A is the angle opposite the side a, then a^=h'^-[- 0^—21)6 cos A. 6. Given cos 2 a? = 2 sin x, to find the value of sin x. 7. Given two sides of a triangle a = 450.2, ^ = 425.4, and the included angle C = 62° 8'j find the remaining parts. xin. (Sheffield Scientific School, June, 1893.) 1. Express an angle of 15° in radians. 2. Write the simplest equivalents for sin (7r+<^), tan (27r — cf>). C0S(|7r— «^), Sec(7r+<^). ENTRANCE EXAMINATION PAPERS. 115 3. Express tan (f> in terms of sin (f>, cos <^ and cot , respect- ively; and cos in terms of tan <^, sec and cosec , respectively. 4. Show (a) that sin (a + y8) + sin (a — y8) = 2 sin a cos y8 ; (b) that cos (a + /?) + cos (a — )8) = 2 cos a cos 13. 5. Assume the formula cosa= tt-, and show that 2 00 sin^-J-a= ^^ y^^ -, when s^i(a-\-b-\-c). oc 6. Obtain a formula for tan -J a in terms of cos a. 7. The base of a triangle c = 556.7, and the two adjacent angles a = 65° 20.2', /8 = 70° 00.5'; calculate the area of the triangle. 8. Given < a < 90°, and log cos a = 1.85254, to determine a. XIV. (Sheffield Scientific School, September, 1893.) 1. Eeduce an angle of 3.5 radians to degrees. 2. Define the different trigonometrical functions of an angle and give their algebraic signs for an angle in each quadrant. 3. Write simple equivalents for the following functions : sin (—a); cos (—a); tan(|7r + a); sec(|7r — a). 4. Express cosec a in terms, respectively, of sin a, cos a, tan a, cot a, sec a. 5. Reduce (cos a cos /? — sin a sin ^8)^+ (sin a cos /? + cos a sin py to its simplest equivalent. » r^-, .^ . . ( "^ \ 1— tan a 6. Show that tan y — a = . , ^ V4 y 1 + tan a 116 TRIGONOMETRY. 7. The sum of two sides, a and h, of a triangle is 546.7 ft., tlie sum of the opposite angles, a and p, is 124°, and sin a: sin j8 = 1.003 ; find the angles and sides of the triangle. 8. Given < a< 90°, and log cot a = 0.03293, to determine a. XV. {Sheffield Scientific School, June, 1894.) 1. Express (a) an angle of 2 radians in degrees ; (h) an angle of 30° in radians. 2. Give simple equivalents for the following functions : tan (— x), cosec (— x), sin {x-\-\ tt), sin (« — ^ tt), tan (| tt — ic), sin(27r — x). 3. Given tan a; = -? to express sin x, cos x, cot ic, sec x, and cosec X in terms of a and b, . ct, ,, , , , , sin (a dtzb) 4. (Show that tan a ± tan b = ^ 7* cos a cos 6. Derive the formulae , /l + cosa . , / cosia = ±yl , sin|-a = ±-y 1 •— cos a 2 ' ^ - - \ 2 6. Given 180° < < 270°, and log cot <^ = 0.03232, find 4,. 7. The sides of a triangle are a = 32.5 ft., J = 33.1 ft., c = 32.4 ft. : calculate the area of the triangle and the angle C opposite the side c, using the following formulae : S='^p(p — a)(p — b) (p — c) = iab sin C, in which S denotes the area of the triangle, smd p=i(a-\-b-\-c). CHAPTER VI. CONSTRUCTION OF TABLES. § 42. Logarithms. Properties of Logarithms. Any positive Dumber being selected as a base, the logarithm of any other positive number is the exponent of the power to which the base must be raised to produce the given number. Thus, if a" = JV, then n = logaJV. This is read, n is equal to log N to the base a. Let a be the base, M and N any positive numbers, m and n their logarithms to the base a ; so that a'^ = M, a'' = N, Then, in any system of logarithms : 1. The logarithm of 1 is 0. For, a« = l. .•.0 = log„L 2. The logarithm of the base itself is 1. For, a^ = a. .•.l=log„a. 3. The logarithm of the reciprocal of a positive number is the negative of the logarithm of the number. For, if a" = iV, then —= — = «-». ' ' N a"" (^)— logahv? =-^ = -10g„iV. H^ TRIGONOMETRY. 4. The logarithm of the product of two or more positive numbers is found by adding together the logarithms of the several factors. For, jf xiV^==a"*Xa« = a'" + ^ ' Similarly for the product of three or more factors. 5. The logarithm of the quotient of two positive numbers is found by subtracting the logarithm of the divisor from the logarithm of the dividend. J?Or, zz= — fjm — n jsr a- "" ' 6. The logarithm of a power of a positive number is found by multiplying the logarithm of the number by the exponent of the power. For, ]srP — (^a''y = a''P. 7. The logarithm of the real positive value of a root of a positive number is found by dividing the logarithm of the number by the index of the root. For, 7]i^=7^^^7 r r Change of System. Logarithms to any base a may be converted into logarithms to any other base b as follows : Let iVbe any number, and let n = log„ iV^ and m = logj, iV. Then, ]sr= a" and ]V= b"". .-. a" = b"^. CONSTRUCTION OF TABLES. 119 Taking logarithms to any base whatever, n log a=^m log b, or, log a X log„ N^ log b X log^ N, from which log^, N may be found when log a, log b, and log^ N are given ; and conversely, log„ N may be found when log a, log &, and logb N are given. Two Important Systems. Although the number of different systems of logarithms is unlimited, there are but two systems which are in common use. These are : 1. The common system, also called the Briggs, denary, or decimal system, of which the base is 10. 2. The natural system of which the base is the fixed value which the sum of the series 1 1 1 1 1 I 1 I 1 . ■ "^ 1 "^ 1.2 "^ 1.2.3 "^ 1.2.3.4 "^ approaches as the number of terms is indefinitely increased. This fixed value, correct to seven places of decimals, is 2.7182818, and is denoted by the letter e. The common system is used in actual calculation ; the natural system is used in the higher mathematics. Exercise XXIII. 1. Given Iogio2=0.30103,logio3=0.47712, logio 7 =0.84510 find logio6, logiol4, logio21, logio4, logiol2, logio5, logioi, logioi, logiol, logiofj. 2. With the data of example 1, find logalO, logaS, loggS, log^i, logs^l^. 120 TRIGONOMETRY. 3. Given logio e = 0.43429 find log,2, log,3, log,5, logj, logeS, l0g,9, log.l, l0g4, logelf, loge/^. 4. Find x from the equations 5^ = 12, 16^=: 10, 27^ = 4. § 43. Exponential and Logarithmic Series. Exponential Series. By the binomial theorem (-=)■ . , 1 , nx (nx — 1) ^ ^ 1 n 1-2 n^ nx (nx — 1) (nx — 2) 1 ^^ 1-2-3 ' / 1\ / IV 2\ x[ X X\ X )[ X = 1 + ^' + [2 + g + • (1) This equation is true for all real values of x, since the binomial theorem may readily be extended to the case of incommensurable exponents (College Algebra, § 264); it is, however, true only for values of ?i numerically greater than 1, since - must be numerically less than 1 (College Algebra, § 375). As (1) is true for all values of x, it is true when x = 1. n \ ^/ \ ^/ 1 + 1 + -^+^ f - + • (2) ■■(-=)' - [(-r.)TK-9' CONSTRUCTION OF TABLES. Hence, from (1) and (2), 1 121 1 + 14 \2 ' \3 x[ X X\ X X l + o: 12 ^ This last equation is true for all values of n numerically greater than 1. Taking the limits of the two members as n increases without limit we obtain ( 1+1+1+1+ )x ^2 ^3 =l+x+|+|+ (3) and this is true for all values of x. It is easily seen that both series are convergent for all values of x. The sum of the infinite series in parenthesis is the natural base e. Hence by (3), ^2 _3 (4) To calculate the value 1 + ^ f e we 2 3 4 5 6 7 8 9 x^ x^ 0^^ ^|2^|3^ proceed as follows 1.000000 1.000000 0.500000 0.166667 0.041667 0.008333 0.001388 0.000198 0.000025 Adding, To ten places, e = e = 0.000003 = 2.71828. :: 2.7182818284, 122 TRIGONOMETRY. Limit of f 1 + - J . By the binomial theorem, I n(n-l)(n-2) ^^x' 1-2-3 This equation is true for all values of n greater than x (College Algebra, § 375). Take the limit as n increases with- out limit, X remaining finite ; then limit /i I A"_i , 1^'.^', nmMii^y-^n) ~ "^ "" "^ [2 "*" [3 "^ limit f 1 _i_ 1 1 ^ ~ n infinite \ n) ' ^^^ Logarithmic Series. then i+^ = ^.^limit A , lA" n mfinite V nj If n is merely a large number, but not infinite, ( l + f)=l+cc + c, where e is a variable number which approaches the limit 0, when n increases without limit. Hence y — y = n\ll-{-x-\-e — n. CONSTRUCTION OF TABLES. 123 If now n becomes oo, and consequently c becomes 0, we have limit ^ n infinite n\ll -\-x — n \ Assuming that x is less than 1, we may expand the right- hand member of this equation by the binomial theorem. The result is =:Ti.ehG-O^G-OG-^)l+ ] 2^ [3 li /v*^ /y»" /y»4 .-. log,(l + a?) = a3 — - + - — j + This series is known as the logarithmic series. It is con- vergent only if X lies between — 1 and +1, or is equal to +1. Even within these limits it converges rather slowly, and for these reasons it is not well adapted to the computation of logarithms. A more convenient series is obtained as follows. Calculation of Logarithms. The equation log.(H-2/) = 2/-f + f-f + (1) holds true for all values of y numerically less than 1 ; there- fore, if it holds true for any particular value of y less than 1, it will hold true when we put — ?/ for ?/ ; this gives iog.(i-y)=-2/-f-f-J- (2) 124 TRIGONOMETRY. Subtracting (2) from (1), since log. (1 + y)- log. (1 - 2/) = log. ([±|) , wefind log.(i±f) = 2(.+ f + f + )• Put ,= ^; then^ = -^. and log, f ^-^ J = loge (^ + 1) - logc« ^o? 1 I 1 I 1 Y V2^ + 1^3(2^ + l)3^5(2^ + l)«^ y This series is convergent for all positive values of z. Logarithms to any base a can be calculated by the series : loga(^ + l) — log„« '^log,aV2;^4-l"*"3(2^ + l)3"'"5(2^ + iy"^ / ^ ^^' Calculate log, 2 to five places of decimals. Let z=\; then z+ 1 = 2, 2z+l = 3, A 1 o 2 , 2 , 2 , 2 . and iog,2 = 5 + r— — + — — + 3 3 X 33 ■ 5 X 35 7 X 37 The work may be arranged as follows : 2.000000 0.666667 -^ 1=0.666667 0.074074 -f- 3 = 0.024691 0.008230 -f 5 = 0.001646 0.000914 -^ 7 = 0.000131 0.000102 -f 9 = 0.000011 0.000011^11 = 0.000001 loge2 = 0.693147 Note, In calculating logarithms the accuracy of the work may be tested every time we come to a composite number by adding together the logarithms of the several factors. In fact, the logarithms of composite numbers are best found in this way, and only the logarithms of prime numbers need be computed by the series. CONSTRUCTION OF TABLES. 125 Exercise XXIY. 1. Calculate to five places of decimals logg3, logg5, log^T. 2. Calculate to ten places of decimals log^lO. 3. Calculate to five places of decimals logio2, logioe, logioll. § 44. Trigonometric Functions of Small Angles. Let A OF be any angle less than 90° and x its circular measure. Describe a circle of unit radius about as a centre and take Z.AOP' = — AA OP. Draw the tangents to the circle at P and P\ meeting OA in T. Then from Geometry chord PP'< arc PP' smic since X < sec X, > cos X. (1) Then since lies between cos x and 1. If now the angle x is constantly diminished, cos x approaches the value 1. sma? Accordingly, the limit of , as x approaches 0, is 1 ; since or, in other words, if cc is a very small angle differs from X 1 by a small value c, which approaches as cc approaches 0. 126 TKIGONOMETRY. To find the sine and cosine of V. If X is the circular measure of Y, * = ad^O = 411^ = 0-00029088+, the next figure in x being either 7 or 8. Now sina:> but <« ; hence sin V lies between and 0.000290889. Again cos 1' = Vl — sin2 V >Vl- (0.0003)2 > 0.9999999. Hence cos V - 0. 9999999+. But, from (1), sin ic > ic cos x .-. sin l'> 0.000290887 X 0.9999999 > 0.000290887 (1 — 0.0000001) > 0.000290887 — 0.0000000000290887 > 0.000290886. Hence sinl' lies between 0.000290886 and 0.000290889; that is, to eight places of decimals sin 1' = 0.00029088 + , the next figure being 6, 7, or 8. Exercise XXV. Given 7r = 3.141592653589, 1. Compute sin 1', cos V, and tan 1' to eleven places of decimals. 2. Compute sin 2' by the same method, and also by the formula sin 2 a; = 2 sin x cos x. Carry the operations to nine places of decimals. Do the two results agree ? 3. Compute sin 1° to four places of decimals. 4. From the formula cos a; = 1 — 2 sin^ -, show that cosa;> 1 — — • Li CONSTRUCTION OF TABLES. 127 5. Show by aid of a table of natural sines that sin x and x agree to four places of decimals for all angles less than 4° 40'. 6. If the values of log x and log sin x agree to five decimal places, find from a table the greatest value x can have. § 45. Simpson's Method of Constructing a Trigono- metric Table. By § 31 (Plane Trigonometry) we have* sin (^-f^) + sin (^ — ^) =2 sin A cos B. If we put A = x-\-'^ij, B = ^j, this becomes sin {x-\-^y)-\r sin (cc + ?/) = 2 sin (x-\-2y) cos y, or sin (x + 3?/) = 2 sin (ic + 2?/) cos y — sin (x + y). Similarly cos (ic + 3 y) = 2 cos (x-\r2y') cos y — cos (x-\-y). (1) li y = V, the last two equations become sin (x + 3') = 2 sin (x + 2') cos 1' — sin {x + 1% cos (x + 3') = 2 cos (x + 2') cos 1' — cos \x-\-V). Hence, taking x successively equal to — V, 0', 1', 2\ we obtain sin 2' = 2 sin 1' cos 1', sin3' = 2sin2'cosl' — sinl', sin 4' = 2 sin 3' cos 1' — sin 2', cos2' = 2cosn' — 1, cos 3' = 2 cos 2' cos 1' — cos 1', cos 4' = 2 cos 3' cos 1' — cos 2', Since the sinl' and cosl' are known, these equations enable us to compute step by step the sine and cosine of any angle. The tangent may then be found in each case as the quotient of the sine divided by the cosine. 128 TRIGONOMETRY. This process need be carried only as far as 30°. For sin (30° + cc) + sin (30° — x)=2 sin 30° cos x = cos x, cos (30° -i-x) — cos (30° — 4 = — 2 sin 30° sin ic = — sin x, . • . sin (30° + ic) = cos X — sin (30° ~ x), cos (30° -\-x)== — sin x + cos (30° — x). Moreover the sines and cosines need be calculated only to 45°, since sin (45° + x) = cos (45° — x), cos (45° -{-x) = sin (45° - x). In using this method the multiplication by cos 1', which occurs at each step, can be simplified by noting that cos V = 0.9999999 = 1 - 0.0000001. Simpson's method is superseded in actual practice by much more rapid and convenient processes in which we employ the expansions of the trigonometric functions in infinite series. Exercise XXVI. 1. Compute the sine and cosine of 6' to seven decimal places. 2. In the formula (1) let 2/=:r. Assuming sin 1°=0.017454 + , cos 1°= 0.999848 + , compute the sines and cosines from degree to degree as far as 4°. § 46. De Moivre's Theorem. Expressions of the form cos x-^i sin x, when i = V — 1, play an important part in modern analysis. Given two such expressions cos x-\-i sin x, cos y-\-i sin y, their product is (cos x-{-i sin x) (cos y-\-i sin y) = cos X COS y — sin x sin 7/-{-i (cos x sin y -}- sin x cos y) = cos (x-\-y')-\- i sin (x-\-y). CONSTRUCTION OF TABLES. 129 Hence, the product of two expressions of the form cos x -\- i sin X, cos y-\-i sin y is an expression of the same form in which a; or ?/ is replaced by x-{-y. In other words, the angle which enters into such a product is the sum of the angles of the factors. If X and y are equal, we have at once from the preceding (cos x-]ri sin xy = cos 2x-\-ism2x] and again (cos x-\-i sin x)^ = (cos x-\-i sin xy (cos x-^i sin x) = (cos 2x-{-ism2x) (cos x-\-i sin x) = cos 3 a? 4" ^ sin 3 cc. Similarly (cos x-{-i sin xy = cos 4:X-{-i sin 4a;, and in general if ?^ is a positive integer (cos x-\- i sin a;)" = cos nx -\- i sin tix. (1) Hence To raise the expression cos x -\- i sin x to the nth power when n is a positive integer, we have only to multiply the angle x by n. Again, if ^ is a positive integer as before, / X . . . x\ ... I cos- + iSin- ) = cos a; + * sm a; \ n nj .*. (cos a; + i sin aj> = cos - + ^ sin — n n Since, however, x may be increased by any integral multiple of 2 TT without changing cos x-\-i sin x, it follows that all the n expressions x , . . x a + 27r , . . x-\-2Tr cos- + isin-> cos f-^sln j n n n n x-\-4:'ir . . . x-\-4:'jr cos h t sm } ) cos x-\-(n — l)27r , . . x-\-(n — l)2'jr — ^-^ J- ^ sm ^ 130 TRIGONOMETRY. are nth. roots of cos x-j-i sin x. There are no other roots, since x-\-n27r , . . x-\-n2'7r cos \- 1 sm X isin-7 n = cos ( - + 27r ) +isin ( - + 27r ) — cos- + \n J y^ J ^ and cos ^ h * sm ^ ^ — = cos ( — ' [-27r 1 4-^Sln ( — ' K^tt 1 = cos h t sm > n n and so on. Hence, if w is a positive integer, 1 (cos x-\-i sin ic)« = cos — ' f-*sm— ! (k — 0,1,2, n — l). (2) From (1) and (2) it follows at once that if m and n are positive integers (cos x-\-{^mxY =^\ (cos £c + / sin a?)« !- =cos-(a;+2A:7r)+*'sin — (£c-l-27c7r)(A;=0,l,2, w— 1). (3) Finally, if is a negative fraction, 71) TO 1 (cos £c + ^ sin a?) « But (cosa;-}-^' since)" 1 cos X — { sin X cos x-\-i sin x (cos x-\-i sin £c) (cos a; — i sin a;) cosic — -isinx cos^ a? + i sin^ a; = cos X — i sin x , = cos ( — a?) + ^ sin (— x). CONSTRUCTION OF TABLES. 131 Hence (cos x-\-i sin x) « ^ k cos ( — x) + ^ sin (— x) p 771 TTl = cos — (— cc + 2A;7r) + *'sin — (— a: + 2A;7r), (k = 0, 1, 2, n — 1) '■ (x-{- 2kir) ^ + *' sin -^ {x-\-2 Jctt) j> , (k = 0,l,2, 71-1). (4) Consequently if ti is a positive or negative integer or fraction (cos x-jri sin af)" = cos [n(x-\^2 kir)^ -{- i sin [w (cc + 2 A^tt)], (7c = 0, 1, 2, n-1). (5) Example : Find the three cube roots of — 1. We have — 1 = cos 180° + i sin 180° , ,,, 180° + 2A:7r , . . 180° + 2A;7r,, ^ , „, . •. (— 1)^ = cos + I sm {k = 0, 1, 2). For the three cube roots of — 1 we find therefore cos 60° + i sin 60°, cos 180° + i sin 180°, cos 300° + i sin 300°, l + iVs , l-iVs -^— ' -1' -^- By aid of De Moivre's Theorem we may express sin 716 and cos nO, when n is an integer, in terms of sin and cos 6. Thus cos n6 + i sin tiO = (cos 0-\-i sin Oy 71 {71 — 1 I = COS" 0-{-i7i cos"-^ ^ sin ^ + P -^ — ^ cos"-^ sin^ _^.3 ^(n-l)(.-2) ^^^^_3^^.^3^^ Or, sincei^= — 1, i^= — i, {*=-}-l, cos w ^ 4" *' sin w ^ = cos" 6-{-i7i cos"~^ ^ sin ^ 2 |o 132 TRIGONOMETRY. Equating now the real parts and the imaginary parts separately, we obtain 2 Gosne = cos" — '^^^ — ^ cos"-2 sin^ , n(n-l)(n-2)(n-S) ., . ,. + — ^^- — ^-^^ — ^ — ^cos**-^^sin*^— sin n$ = n cos"~^ ^ sin ^ ^^^ — r^ cos""^ 6 sin^ 6 H ^^ ^^ r~ — ^ cos**-^ ^ sm^ ^ — Exercise XXVII. 1. Find the six 6th roots of — 1 ; of + 1. 2. Find the three cube roots of i. 3. Find the four 4th roots of — i. 4. Express sin 4 and cos 4 ^ in terms of sin and cos 0. § 47. Expansion of Sin x, Cos x, and Tan x in Infinite Series. Let one radian be denoted simply by 1, and let cos 1 + i sin l = k. Then cos x-\-i sin x = (cos 1 + i sin 1)^ = k^, and putting — x for x cos ( — x)-\-i sin ( — x) = cos x — i sin x = k~^. That is cosa;4-isina;= A:^ and cos x — i sin x = k~^ By taking the sum and difference of these two equations, and dividing the sum by 2 and the difference by 2 i, we have cos a? = ^ (7c^ + k-^), sin x = ~ (k"^ — k-^). But k^ = (e '^^ ^') * = e^ ^^ *, k'"" = e"^ ^^ *, CONSTRUCTION OF TABLES. 133 and 2i \o 1 /7r . 7 rX -. . ^^^ (log A;)^ , iC* (log A^V , .-. cosx = i(A:^ + A:-^) = l + — ^^-^ + — ^T^^ + sinx = -^ a: log a; -|- ,0 r .^ It only remains to find the value of ^, and this can be obtained by dividing the last equation through by x and letting X approach indefinitely, when we have limit X limit /sin icX 1 . . • . log li = i, k^ e\ Therefore we have X X X COS a; = i (e-+ e— ■) :=!_- + _ --4- From the last two series we obtain by division since , x^ , 2x^ , 11 x' , cosa: 15 olo By the aid of these series the trigonometric functions of any angle are readily calculated. In the computation it must be remembered that x is the circular measure of the given angle. 134 TRIGONOMETRY. Exercise XXVIII. Verify by the series just obtained that 1. sin^ic + cos^x = 1. 2. sin(— cc) = — sinic, cos (— x) = cos a;. 3. siii2£c = 2siiia;cosic. 4. cos 2 ic = 1 — 2 sin^ic. 5. Find the series for sec x as far as the term containing the 6th power of x. X 6. Find the series for x cot x, noting that a; cot ic = — — cos x. 7. Calculate sin 10° and cos 10° to 6 places of decimals. 8. Calculate tan 15° to 5 places of decimals. From the exponential values of sin x and cos x show that 9. cos3cc = 4cos^a; — 3coscc. 10. sin 3 X = 3 sin ic — 4 sin^x. SPHEEICAL TEIGONOMETEY. CHAPTER VII. THE RIGHT SPHERICAL TRIANGLE. § 48. Introduction. The object of Spherical Trigonometry is to show how spherical triangles are solved. To solve a spherical triangle is to compute any three of its parts when the other three parts are given. The sides of a spherical triangle are arcs of great circles. They are measured in degrees, minutes, and seconds, and therefore by the plane angles formed by radii of the sphere drawn to the vertices of the triangle. Hence, their measures are independent of the length of the radius, which may be assumed to have any convenient numerical value; as, for example, unity. The angles of the triangle are measured by the angles made by the planes of the sides. Each angle is also measured by the number of degrees in the arc of a great circle, described from the vertex of the angle as a pole, and included between its sides. The sides may have any values from 0° to 360°; but in this work only sides that are less than 180° will be considered. The angles may have any values from 0° to 180°. If any two parts of a spherical triangle are either both less than 90° or both greater than 90°, they are said to be alike in kind; but if one part is less than 90°, and the other part greater than 90°, they are said to be unlike in kind. 136 SPHERICAL TRIGONOMETRY. Spherical triangles are said to be isosceles, equilateral, equiangular, right, and oblique, under the same conditions as plane triangles. A right spherical triangle, however, may have one, two, or three right angles. When a spherical triangle has one or more of its sides equal to a quadrant, it is called a quadrantal triangle. It is shown in Solid Geometry, that in every spherical triangle I. If two sides of a spherical triangle are unequal, the angles opposite them are unequal, and the greater angle is opposite the greater side; and conversely. II. The sum of the sides is less than 360°. III. The sum of the angles is greater than 180° and less than 540°. IV. If, from the vertices as poles, arcs of great circles ai'e described, another spherical triangle is formed so related to the first triangle that the sides of each triangle are supplements of the angles opposite them in the other triangle. Two such triangles are called ^oZar triangles, or supplemental triangles. Let A, B, C (Fig. 37) denote the angles of one triangle ; a, h, c the sides opposite these angles respectively ; and let A\ B', C and a', b', c' denote the corresponding angles and sides of the polar triangle. Then the above theorem gives the six following equations : ^ + ^' = 180°, ^+^^' = 180°, C+^'==180°, ^' + ^=180°, B'-^b = 180°, C'+c= 180°. THE RIGHT SPHERICAL TRIANGLE. 137 Exercise XXIX. 1. The angles of a triangle are 70°, 80°, and 100°; find the sides of the polar triangle. 2. The sides of a triangle are 40°, 90°, and 125°; find the angles of the polar triangle. 3. Prove that the polar of a quadrantal triangle is a right triangle. 4. Prove that, if a triangle has three right angles, the sides of the triangle are quadrants. 5. Prove that, if a triangle has two right angles, the sides opposite these angles are quadrants, and the third angle is measured by the number of degrees in the opposite side. 6. How can the sides of a spherical triangle, given in degrees, be found in units of length, when the length of the radius of the sphere is known? 7. Find the lengths of the sides of the triangle in Example 2, if the radius of the sphere is 4 feet. § 49. Formulas Kelating to Right Spherical Triangles. As is evident from § 48, Examples 4 and 5, the only kind of right spherical triangle requiring further investigation is that which contains 07ily one right angle. Let ABC (Fig. 38) be a right spherical triangle having only one right angle; and let A, B, C denote the angles of the triangle; a, b, c, respectively, the opposite sides. Let CJoeJih^jdght angle ; and for the present suppose that each of the other parts is less than 90°, and that the radius of the sphere is 1. Let planes be passed through the sides, intersecting in the radii OA, OB, and OC, 138 SPHERICAL TRIGONOMETRY. Also, let a plane perpendicular to OA be passed through B, cutting OA at ^ and OC at D. Draw BE, BD, and BE. BE and BE are each J_ to OA (Geom. § 462) ; therefore Z BEB = A. The plane BBE is _L to the plane AOC (Geom. § 518); hence BB, which is the intersection of the planes BBE and BOC, is ± to the plane AOC (Geom. § 520), therefore _L to OC and BE. Now cos c = 0E= OB X cos b, and OB = cos a. cos c = COS a cos b. sin a = BB — BE X sin A, and BE = sin c. sin a = sin c sin A sin b = sin c sin B BE OE tsiiib Fig. 39. Therefore, Therefore, changing letters. } [38] [39] Hence, changing letters. """"^^-BE-OEt'^nG cos A = tan b cot c cos B == tan a cot c . } [40] Again, cos^ BE OB sin b By substituting for changing letters. Also, Hence, changing letters, BE sinb sine = cos a sin b sm c its value from [39], we obtain } [41] sm c COS A = COS a sin B cos B = COS b sin A sin b = -r— = BE BB cot A OB OB sin b = tan a cot A sin a = tan b cot B = tan a cot A. } [42] If in [38] we substitute for cos a and cos b their values from [41], we obtain cos c = cot A cot B. [43] Note. In order to deduce the second formulas in [39]-[42] geometri- cally, the auxiliary plane must be passed through -4 _L to OB. THE RIGHT SPHERICAL TRIANGLE. 139 These ten formulas are sufficient for the solution of any right spherical triangle. In deducing these formulas, it has been assumed that all the parts of the triangle, except the right angle, are less than 90°. But the formulas also hold true when this hypoth- esis is not fulfilled. Let one of the legs a be greater than 90°, and construct a figure for this case (Fig. 40) in the same manner as Fig. 38. Fig. 40. , The auxiliary plane BDE will now cut both CO and AO pro- duced beyond the centre ; and we have gosc = —OjE=—OD cos DOE = (— cos a) (— cos b) = cos a cos b. Likewise, the other formulas, [39]-[43], hold true in this case. Again, suppose that both the legs a and b are greater than 90°. In this case the plane BDE (Fig. 41) will cut CO pro- duced beyond 0, and A between A and ; and we have cos c=OE = OD cos DOE = (— cos a) (— cos b) = cos a cos b, a result agreeing with [38]. And the remaining formulas may be easily shown to hold true. Like results follow in all cases ; in other words, Formulas [38]-[43] are universally true. 140 SPHERICAL TRIGONOMETRY. Exercise XXX. 1. Prove, by aid of Formula [38], that the hypotenuse of a right spherical triangle is less than or greater than 90°, according as the two legs are alike or unlike in kind. 2. Prove, by aid of Formula [41], that in a right spherical triangle each leg and the opposite angle are always alike in kind. 3. What inferences may be drawn from Formulas [38]-[43] respecting the values of the other parts: (i.) if c = 90°; (ii.) iia = 90° ; (iii.) if c = 90° and a = 90° ; (iv.) \i a = 90° and ft = 90°? Deduce from [38] -[43] and [18] -[23] the following formulas : 4. tan2^& = tan^(c — a) tan|-(c4-^). Hint. Use Formula [18] and substitute in it the value of cos 6 in [38]. 5. tan^ (45° — I ^) = tan i (c — a) cot ^ (c + tt) . 6. tan^ iB = sin (c — a) esc (c -f- cos A = cos a sin B. 144 SPHERICAL TRIGONOMETRY. Case V. Given the hypotenuse c and the angle A. From Formulas [39], [40], and [43] it follows that sin a = sin c sin Aj tan b = tan c cos A, cot B = cos c tan A. Here a is determined by sin a, since a and A must be alike in kind (see Exercise XXX., Example 2). Case VI. Given the two angles A and B. By means of Formulas [43] and [41] we obtain cos c = cot ^ cot B, cos a = cos ^ CSC B, cos b = COS B CSC A. Note 1. In Case I. (a and 6 given) the formula for computing c fails to give accurate results when c is very near 0° or 180°; in this case it may- be found with greater accuracy by first computing B, and then computing c, as in Case IV. Note 2. In Case II. (c and a given), if 6 is very near 0° or 180°, it may be computed more accurately by means of the derived formula tan2 i 6 = tan i (c + a) tan \ (c — a). (Ex. 4, § 49. ) And if A is so near 90° that it cannot be found accurately in the Tables, it may be computed from the derived formula tan2 (45° - i^) = tan i (c - a) coti (c + a). (Ex. 5, § 49.) In like manner, when B cannot be accurately found from its cosine we may make use of the formula tan2 ^ ^ =r sin (c — a) esc (c + a). (Ex. 6, § 49. ) Note 3. In Case III. (a and A given), when the formulas for the required parts do not give accurate results, we may employ the derived formulas tan2 (45° - i c) = tan i {A-a) cot i (^ + a), (Ex. 9, § 49.) tan2 (45° - i 6) = sin {A — a) esc {A + a), (Ex. 10, § 49.) tan2 (45° - i E) = tan i (^ - a) tan i (^ + a). (Ex. 11, § 49.) THE RIGHT SPHERICAL TRIANGLE. 145 Note 4. In Case IV. (a and B given), if A is near 0° or 180°, it may be more accurately found by first computing b and tlien finding A. Note 5. In Case V. (c and A given), if a is near 90°, it may be found by first computing 6, and then computing a by means of Formula [42]. Note 6. In Case VI. {A and B given), for unfavorable values of the sides greater accuracy may be obtained by means of the derived formulas tan2 ic = -cos{A + B) sec (A — B), (Ex. 7, § 49.) tan2 i a = tan [i {A + B)- 45°] tan [45° + ^{A-B)'\, (Ex. 8, § 49.) tan2 ^ 6 = tan [J {A + B)- 45°] tan [45° - i(J. - B)'\. Note 7. In Cases I., IV., and V., the solution is always possible. In the other Cases, in order that the solution may be possible, it is necessary and sufficient that in Case II. sin a sin a ; in Case VI. , that A+B + C be > 180°, and the difference of A and i? be < 90°. Note 8. It is easy to trace analogies between the formulas for solving right spherical triangles and those for solving right plane triangles. The former, in fact, become identical with the latter if we suppose the radius of the sphere to be infinite in length ; in which case the cosines of the sides become each equal to 1, and the ratios of the sines of the sides and of the tangents of the sides must be taken as equal to the ratios of the sides themselves. Note 9. In solving spherical triangles, the algebraic sign of the functions must receive careful attention. If the sign of each function is written just above it, the sign of the function in the first member will be + or — according to the rule that like signs give + and unlike signs give — . If the function is a cos, tan, or cot, the + sign shows that the angle is less than 90°; the — sign shows that the angle is greater than 90°, and the supplement of the angle obtained from the table must be taken. If the function is a sine, since the sine of an angle and its supplement are the same, the acute angle obtained from the table and its supplement must be considered as solutions, unless there are other conditions that remove the ambiguity. For the conditions that remove the ambiguity, in case of right spherical triangles see examples 1 and 2 in Exercise XXX., and in case of oblique spherical triangles see I. of § 48. 146 SPHERICAL TRIGONOMETRY. Note 10. The solutions of a spherical triangle may conveniently be tested by substituting them in the formula containing the three required parts. If the formula required for any case is not remembered, it is always easy to find it by means of Napier's Rules. In applying these Eules we must choose for the middle part that one of the three parts considered — the two given and the one required — which will make the other two either adjacent parts or opposite parts. For example : given a and B ; solve the triangle. First, represent the parts as in Fig. 42, and to prevent mistakes mark each of the given parts with a cross. To find h, take a as the middle part ; then b and co. B are adjacent parts ; and by Eule I., sin a = tan b cot B ; whence, tan b = sin a tan B. To find c, take co. B as middle part ; then a and CO. c are adjacent parts ; and by Eule I., cos ^= tan a cot e\ whence, tan c = tan a sec B. To find A, take co. A as middle part ; and CO. B are the opposite parts ; and by Eule II., cos A == cos a sin B. then a In like manner, every case of a right spherical triangle may be solved. Exercise XXXII. Solve the following right triangles, taking for the given parts in each case those printed in columns I. and II. : THE RIGHT SPHERICAL TRIANGLE. 147 Note. The values in the last three columns of example 9 cannot be combined promiscuously with those given in columns I. and II. Since a<90°, with the value of 6>90° must be taken angle 5>90° and c > 90° ; while with the value of 6 < 90° must be taken, for the same reason, angle 5<90° and c<90°. Exercise XXX., 1 and 2. 148 SPHERICAL TRIGONOMETRY. 23. Define a quadrantal triangle, and show how its solution may be reduced to that of the right triangle. 24. Solve the quadrantal triangle whose sides are : «=: 174° 12' 49.1", ^» = 94°8'20", c = 90°. 25. Solve the quadrantal triangle in which G = 90% ^ = 110° 47' 50", ^ = 135° 35' 34.5". 26. Given in a spherical triangle A, C, and c each equal to 90°; solve the triangle. 27. Given A = 60°, C = 90°, and c = 90° ; solve the triangle. 28. Given in a right spherical triangle, A = 42° 24' 9", ^ = 9° 4' 11"; solve the triangle. 29. In a right spherical triangle, given a = 119° 11', B = 126° 54'; solve the triangle. 30. In a right spherical triangle, given c=50°, ^'==44°18'39"; solve the triangle. 31. In a right spherical triangle, given A = 156° 20' 30", a = 65° 15' 45"; solve the triangle. 32. If the legs a and 6 of a ri ght s pherical triangle are equal, prove that cos a = cot A = Vcos c. 33. In a right spherical triangle prove that cosM X sin^c = sin (c — a) sin (c + a). 34. In a right spherical triangle prove that tan a cos c = sin h cot B. 35. In a right spherical triangle prove that sin^ A = cos^ B -\- sin^ a sin^ B. 36. In a right spherical triangle prove that sin (b-{- c) :=2 cos^ -J- A cos b sin c. 37. In a right spherical triangle prove that sin (c — b) = 2 sin^ -J A cos b sin c. 38. If, in a right spherical triangle, p denotes the arc of the great circle passing through the vertex of the right angle and perpendicular to the hypotenuse, m and n the segments of the hypotenuse made by this arc adjacent to the legs a and b, prove that (i.) tan^ a = tan c tan m, (ii.) sin^^ = tan m tan n. the eight spherical triangle. 149 § 52. Solution of the Isosceles Spherical Triangle. If an arc of a great circle is passed through the vertex of an isosceles spherical triangle and the middle point of its base, the triangie will be divided into two symmetrical right spherical triangles. In this way the solution of an isosceles spherical triangle may be reduced to that of a right spherical triangle. In a similar manner the solution of a regular spherical polygon* may be reduced to that of a right spherical triangle. Arcs of great circles, passed through the centre of the polygon and" its vertices, divide it into a series of equal isosceles tri- angles ; and each one of these -may be divided into two equiv- alent right triangles. Exercise XXXIII. 1. In an isosceles spherical triangle, given the base h and the side a ; find A the angle at the base, B the angle at the vertex, and h the altitude. 2. In an equilateral spherical triangle, given the side a ; find the angle A. 3. Given the side a of a regular spherical polygon of n sides ; find the angle A of the polygon, the distance R from the centre of the polygon to one of its vertices, and the dis- tance r from the centre to the middle point of one of its sides. 4. Compute the dihedral angles made by the faces of the five regular polyhedrons. 5. A spherical square is a regular spherical quadrilateral. Find the angle A of the square, having given the side a. * A regular spherical polygon is the polygon formed by the intersec- tions of the spherical surface by the faces of a regular pyramid whose vertex is at the centre of the sphere. CHAPTER VIII. THE OBLIQUE SPHERICAL TRIANGLE. § 53. Fundamental Formulas. Let ABC (Fig. 45) be an oblique spherical triangle, a, b, c its three sides, A, B, C the angles opposite to them, respectively. Through C draw an arc CD of a great circle, perpendicular to the side AB, meeting AB at D. For brevity let CD = p, AD = m, BD = 7i, ZACD = x, ZBCD = 7/. 1. By § 49 [39], in the right triangles BBC and ABC, sinj9 = sin asin B, and sin p = sin b sin A. sin a sin B = sin b sin A ^ sin a sin C = sin c sin A I [44] sin b sin C = sine sin B J Therefore, similarly, and These equations may also be written in the form of proportions sin a : sin b : sin c = sin ^ : sin J? : sin C. That is, the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. In Fig. 45 the arc of the great circle CD cuts the side AB within the triangle. In case it cuts AB produced without the triangle, sin (180°-^), sin (180°— ^), or sin (180°- (7), would THE OBLIQUE SPHERICAL TRIANGLE. 151 be employed in the above proof instead of sin^, sin^, or sin C. These sines, however, are equal to sin A, sin B, and sin C, respectively, so that the Formulas [44] hold true in all cases. 2. In the right triangle IWC, by § 49 [38], cos a = cos j9 cos n = cos^ cos (c — m), or (§ 28) cos a = cosp cos c cos m + cos^ sin c sin m. Now, [38] cos^ cos m = cosbj whence cos^ ^ cos & seem, and cos^sinm = cosft tanm [40] = cos b tan b cos A = sin b cos A. Substituting these values of cos^cosm and cos^sinm in the value of cos a, we obtain cos a = COS b cos c + sin b sin c cos A"j and similarly, cos b = cos a cos c + sin a sin c cos B > [45] cos c = cos a cos b + sin a sin b cos C J 3. In the right triangle ADC, by [41], gos A = cosp sin x = cosp sin (C — y), or (§ 28) cos A = cosp sin (7 cos y — cosp cos Csin y. Now, [41] cosp sin y = cos B ; whence, cosp ^ cos^cscy, and cos p cos y = cos B cot y [43] = cos B tan B cos a = sin B cos a. Substituting these values of cosp sin y and cosp cos y in the value of cos A, we obtain cos A = — cos B cos C + sin B sin C cos a~^ and similarly, cos B — — cos A cos C + sin A sin C cos b V [46] cosC = — cos A cos B + sin A sin B cose J 152 SPHERICAL TRIGONOMETRY. Formulas [45] and [46] are also universally true ; for the same equations are obtained when the arc CD cuts the side AB without the triangle. Exercise XXXIV. 1. What do Formulas [44] become if ^ = 90° ? if ^ = 90° ? if C=90°? if a = 90°? if ^ = ^ = 90°? if a ==^ = 90° ? 2. What does the first of [45] become if ^===0°? if ^=90°? if Jt = 180°? 3. From Formulas [45] deduce Formulas [46], by means of the relations between polar triangles (§ 48). § 54. Formulas for the Half Angles and Sides. From the first equation of [45], cos^ I? cos a — cos h cos c whence, 1 — cos ^ = sin b sin c ' sin h sin c + cos h cos c — cos a sin b sin c cos (b — c) — cos a ^ sin b sin c ' . , . sin b sin c — cos b cos c 4- cos a 1 + cos -4 — : — --; ' sm b sm c cos a -^ cos (b + c) sin b sin c Hence, by § 30 [16] and [17], and § 31 [23], sin^l" A == sin i(a-\-b — c) sin ^(a — b-\-c) esc b esc c, cos^^ A = sin ^(a-\-b-\-c) sin ^(b-\-c—a) esc b esc c. Now let ^(a-{-b-^c) = s; whence, i(b-\-c — a) = s — a, i(a—b-\-c) = s — b, ^(a-^b — c) = s — c. THE OBLIQUE SPHERICAL TRIANGLE. 153 Then, by substitution and extraction of the square root, sin ^ A = Vsin (s — b) sin (s — c) esc b esc c cos ^ A = Vsin s sin (s — a) esc b esc c tan| A = Vese s esc (s — a) sin (s — b) sin (s — e) In like manner, it may be shown that sin 1^ B = Vsin (s — a) sin (s — c) esc a esc e cos ^ B = Vsin s sin (s — b) esc a esc e tan -j- B = Vese s esc (s — b) sin (s — a) sin (s — e) sin 1^ C = Vsin (s — a) sin (s — b) esc a esc b cos J C = Vsin s sin (s — e) esc a esc b tan i C = Vese s esc (s — c) sin (s — a) sin (s — b) Again, from the first equation of [46], whence, cos a C0S(X = 1 -}- cos 1 / 1 1.x COS 4- (A— B) ^ tani(a + b)^ ^^^|;^^^j tani a 1^ tan "I- (a — b) sin^(A— B) sini(A+B) tan^^c f\|lM' [50] There will be other forms in each case, according as other elements of the triangle are used. These equations are called Napier's Analogies. In the first equation the f actors, cos J (ct — b) and cot|-C are always positive : therefore, tan ^(A -\- B) and cos i(a-\-b) must always have like signs. Hence, if (X -}- J < 180°, and therefore cos ^(a -\-b)>0, then, also, tan ^(A-\- B)> 0, and therefore A + B < 180°. Similarly, it follows that if a + b> 180°, then, also, A'{-B> 180°. If a-}-b = 180°, and therefore cos ^ (a-\-b)=^0, then tan ^ (A -{- B) =^ cg -^ whence i(A-{-B) = 90°, and ^ + ^ == 180°. Conversely, it may be shown from the third equation, that a-\-b is less than, greater than, or equal to 180°, according as ^ + ^ is less than, greater than, or equal to 180°. 156 SPHERICAL TRIGONOMETRY. § 56. Case I. Given two sides, a and b, and the included angle C. The angles A and B may be found by the first two of Napier's Analogies ; viz. : cos ^(a — b) tan J- (A + B) tSini(A — B) cos ^(a-\- b) sin ^(a — b) cot I- a cot^a sin ^ (a -\- b) After A and B have been found, the side c may be found by [44] or by [50] ; but it is better to use for this purpose Gauss's Equations, because they involve functions of the same angles that occur in working Napier's Analogies. Any one of the equations may be used; for example, from the first we have Q,o^\(a-\-b) . , ^ GOS^{A-{-B) ^ Example. a =73° 58' 64' 6= 38° 45' 0' C=46°33'4r log cos i (a — 6) =9. 97914 log sec i (a + 6) = 0. 25658 log cot jC = 0.36626 log tan i(^ + B)= 0.60198 log sec ^{A + B) = 0.61515 logcosi(a + 6) =9.74342 log sin ^C = 9.59686 log cos ^ c = 9.95543 = 25° 31' therefore, ^ (a - 6) = 17° 36' 57" i (a +6) = 56° 21' 57" iC =23° 16' 50. 5' log sin |(a — 6) =9. 48092 log CSC 1 (a +6) =0.07956 coti C = 0.36626 log tan |(^ — B) = 9.92674 ^{A + B)= Ib^bTAO.r \{A-B)= 40° 11' 25.6' A = 116° 9' 6.3' B= 35° 46' 15.1' c= 51° 2' If the side c only is desired, it may be found from [45], without previously computing A and B. But the Formulas [45] are not adapted to logarithmic work. Instead of changing them to forms suitable for logarithms, we may use the following method, which leads to the same results, and has the advantage that, in applying it, nothing has to be remembered except Napier's Eules : THE OBLIQUE SPHERICAL TRIANGLE. 157 Make the triangle (Fig. 46), as in § 53, equal to the sum (or the difference) of two right triangles. For this purpose, through B (or A, but not C) draw an arc of a great, circle perpendicular to AC, cutting AC at D. Let BD^p, CD=m, AD = n; and mark with crosses the given parts. By Rule I., cos C = tan m cot a, whence tan m = tan a cos C. By Rule II., cos a = cos m cos^^, whence cosp=^ cos a sec m. cos c = cos 71 cos^, whence cos^ = cos c sec n. Therefore, cos c sec n = cos a sec m ; or, since n = b — m, cos c = cos a sec m cos (^ — m). It is evident that c may be computed, with the aid of logarithms, from the two equations tan m = tan a cos C, cose = cos a sec m cos (& — m). Example. Given a =- 97° 30 ' 20 ",& = 55° 12 '10", C=39°58'; find c. Fig. 46. log tan 6t =0.88025 (?i) log cos (7=9.88447 log tan 771 =0.76472(71) m= 99° 45' 14" 5-m = -44°33' 4" logcos 6^ = 9.11602(71) logcos (b — m) = 9. S52S6 log sec 7^^ = 0.77103(71 ) log cose = 9.73991 c = 56° 40' 20" Exercise XXXY. 1. Write formulas for finding, by ISTapier's Rules, the side a, when b, c, and A are given, and for finding the side b when a, c, and B are given. 158 SPHERICAL TRIGONOMETRY. 2. Given a = 88° 12' 20", ^> = 124° 7' 17", C=:50°2'l"; find A = 63° 15' 11", B = 132° 17' 59", c = 59° 4' 18". 3. Given a = 120° 55' 35", & = 88° 12' 20", C = 47°42'l"; find ^ = 129° 58' 3", ^ = 63° 15' 9", c = 55° 52' 40". 4. Given ^» = 63° 15' 12", c = 47° 42' 1", ^ =: 59° 4' 25"; find ^ = 88° 12' 24", C=: 55° 52' 42", a = 50°l'40". 5. Given ^» = 69° 25' 11", c = 109° 46' 19", ^ = 54° 54' 42"; find jg=: 56° 11' 57", C=rl23°21'12", a = 67° 13'. § 57. Case II. Criven the side c and the two adjacent angles A and B. The sides a and h may be found by the third and fourth of Napier's Analogies, tan ^{a-\-h) _Gosi(A — B) ~Qosi(A-\-B) tan 4- (a — Z») = -r— ftr ■ ..x tan -J c, tan ^ c, and then the angle C may be found by [44], by Napier's second Analogy, or by one of Gauss's equations, as, for instance, the second, which gives ^ cos 1 {a -I) ou.^.. Example. ^ = 107° 47' V .•.^(^-J5) = 34°24'20" B= 38° 58' 27''. n^ + J5) = 73°22'47" c= 51° 41' 14" ^c = 25° 50' 37" log cos |(^ - 5) = 9.91648 logsin^^- 5) = 9.75208 log sec H^ + ^) = 0.54359 logcsc 1(^ + 5) = 0.01854 logtanic =9.68517 logtanic =9.68517 logtanHa+ft) =0.14524 log tan } {a — h) = 9. 45579 logsini (-^ + -8) = 9.98146 ^(a+&) = 54°24'24.4" log sec i (a -6) =0.01703 i(a-&) = 15°56'25.6" log cos ic =9.95423 fa =70° 20' 50" log cos |C =9.95272 ]6 =38° 27' 59" i C = 26° 14' 52.5" C C = 52° 29' 45" THE OBLIQUE SPHERICAL TRIANGLE. 159 If the angle C alone is wanted, tlie best way is to decompose the triangle into two right triangles, and then apply Napier's Eules, as in Case I., when the side c alone is desired. Let (Fig. 47) ZABI) = x, Z.CBD = y, BD=p) then, Eule I., whence Eule II. whence whence Hence cos G = cot X cot Aj cot X = tan A cos c. cos ^=: cos ^ sin 0!!, cos 2^ = cos A CSC X. cos C = COS p sin y, cos p = cos C CSC y. cos C = cos ^ CSC cc sin 2/ = cos ^ CSC £c sin (5 — x). It is clear that C may be computed from the equations cot X = tan A cos c, cos C = cos ^ CSC X sin (B — x). Example. Given A = 35° 46' 15", B = 115° 9' 7", c = 51° 2 find a log tan ^ = 9.85760 log cose = 9.79856 log cot X =9.65616 X = 65° 37' 35" .•,B-x =49° 31' 32" log cos ^ =9.90992 logsin (5 — £c) = 9.88122 log CSC X = 0.04055 logcos (7= 9.83099 C=47°20'30" Exercise XXXVI. 1. What are the formulas for computing A when B, C, and a are given ; ^.nd for computing B when A, C, and b are given ? 2. Given A = 26° 58' 46", B = 39° 45' 10", e = 154° 46' 48"; find a = 37° 14' 10", h = 121° 28' 10", C= 161° 22' 11". 3. Given A = 128° 41' 49", B = 107° 33' 20", c = 124° 12' 31"; find a = 125° 41' 44", h = 82° 47' 34", C = 127° 22'. 160 SPHERICAL TRIGONOMETRY. 4. Given ^=153° 17' 6", (7= 78° 43' 36", a = 86° 15' 15"; find & = 152° 43' 51", c = 88° 12' 21", A = 78° 15' 48". 5. Given A = 125° 41' 44", C= 82° 47' 35", b = 52° 37' 57"; find a = 128° 41' 46", c = 107° 33' 20", B = 55° 47' 40". § 5S. Case III. Given two sides a and b, and the angle A o2yposite to a. The angle B is found from [44], whence we have sin B = sin A sin b esc a. When B has been found, C and c may be found from the fourth and the second of Napier's Analogies, from which we obtain ^"^^^^- sinH^-^^ ^"^(^-^)- eotiC = $4^^tanH^-^). ^ sin ^ (a — ^>) ^ ^ ^ The third and first of Napier's Analogies may also be used. Note 1. Since B is determined from its sine, the problem in general has two solutions ; and, moreover, in case sin B >► 1 , the problem is impossible. By geometric construction it may be shown, as in the corresponding case in Plane Trigonometry, under what conditions the problem really has two solutions, one solution, and no solution. But in practical applications a general knowledge of the shape of the triangle is known beforehand ; so that it is easy to see, without special investigation, which solution (if any) corresponds to the circumstances of the question. It can be shown that there are two solutions, when A and a are alike in kind and sin 6 > sin a > sin ^ sin 6 ; no solution when A and a are unlike in kind (including the case in which either ^ or a is 90°) and sin h is greater than or equal to sin a, or when sin a 90°, and a+6<180°; therefore, ^ + ^<180°; hence, -B< 90°, and only one solution. a + b ==88° 50' a-b =26°26' A + -B=140°51'53" A-B= 67° 59' 7" logsini(^ + J5) = 9.97416 logcsci(^-^)= 0.25252 log tan Ija — b) - 9. 37080 logtanic = 9.59748 ic = 21°35'38" c = 43° 11' 16" logsinJ. = 9.98609 log sin 6 =9.71435 log CSC g =0.07349 log sin I? = 9.77393 i? = 36°27'20" i(a + 6) =44° 25' ^{a-b) = 13° 13' ^(^ + 5) = 70° 26' 25' i (^ - ^) = 33° 59' 5' log sin 1 (a + b) = 9.84502 log CSC I {a - log tan I (A b) =0.64086 - B) = 9.82873 log cot 10= 0.31461 1C = 25°51'15' C= 51° 42' 30' Exercise XXXYII. 1. Given a = 73° 49' 38", b = 120° 53' 35", A = 88° 52' 42"; find B = 116° 42' 30", c = 120° 57' 27", C = 116° 47' 4". 2. . Given a = 150° 57' 5", b = 134° 15' 54", A = 144° 22' 42"; find ^1 = 120° 47' 45", Ci = 55°42'8", Ci = 97° 42' 55.4"; B^= 59° 12' 15", C2 = 23°57'17.4", C^ = 29° 8' 39". 3. Given a = 79° 0' 54.5", J = 82° 17' 4", ^ = 82° 9' 25.8"; find B = 90°; c = 45° 12' 19", C = 45° 44'. 4. Given a = 30° 52' 36.6", J = 31° 9' 16", ^ = 87° 34' 12"; show that the triangle is impossible. 162 SPHERICAL TRIGONOMETRY. § 59. Case IV. Given two angles A and By and the side a opposite to one of them. The side b is found from [44], whence sin b = sin a sin B esc A. The values of c and C may then be found by means of Napier's Analogies, the fourth and second of which give ^ . sm^(A + B)^ ,, sm ^(a — b) ^ ^ ^ Note 1. In this case the conditions for one, two, or no solutions can be deduced directly by the theory of polar triangles from the correspond- ing conditions of Case III. There are two solutions, when A and a are alike in kind and sin J5> sin ^ > sin a sin B ; no solution when A and a are unlike in kind (including the case in which either tI or a is 90°) and sin B is greater than or equal to sin A, or when sin J. < sin a sin B ; and one solution in every other case. Note 2. By proceeding as indicated in Case III., Note 2, formulas for computing c or C, independent of the side b, may be found ; viz. : tan m = tan a cos B, and sin (c — m) = cot A tan B sin m, cot X = cos a tan B, and sin (C — ic) = cos A sec B sin x. In these formulas m = BD, x— L BCD, D being the foot of the per- pendicular from the vertex C. Note 3. As in Case III. , only those values of b can be retained which are greater or less than a, according as B is greater or less than A. If log sin b is positive, the triangle is impossible. Exercise XXXVIII. 1. Given A = 110° 10', B = 133° 18', a = 147° 5' 32" ; find b = 155° 5' 18", c == 33° 1' 36", C = 70° 20' 40". THE OBLIQUE SPHERICAL TRIANGLE. 163 2. Given A = 113° 39' 21", B = 123° 40' 18", a = 65° 39' 46"; find b = 124° 7' 20", c = 159° 50' 14", C = 159° 43' 34". 3. Given A = 100° 2' 11.3", 5= 98° 30' 28", a =95° 20' 38.7"; find b = 90°, c = 147° 41' 43", C = 148° 5' 33". 4. Given A = 24° 33' 9", B = 38° 0' 12", a = 65° 20' 13" ; show that the triangle is impossible. § 60. Case V. Given the three sides, a, b, and c. The angles are computed by means of Formulas [47], and the corresponding formulas for the angles ^ and C. The formulas for the tangent are in general to be preferred. If we multiply the equation tan ^-4 = Vcsc s esc (s — a) sin (s — b) sin (5 — c) by the equation 1 = -. — \ ^> and put ^ ^ sin(s — a) ^ Vcsc s sin (s — a) sin (s — b) sin {s — c) = tan r, and also make analogous changes in the equations for tan-J^ and tan ^ C, we obtain tan ^A= tan r esc (s — a), tan -J- j5 = tan r esc (s — b), tan i C = tan r esc (5 — c), which are the most convenient formulas to employ when all three angles have to be computed. Example 1. a= 50° 54' 32' b= 37° 47' 18' c= 74° 51' 50' 2 8= 163° 33' 40' s= 81° 46' 50' s-a= 30° 52' 18' s-b= 43° 59' 32' s-c= 6° 55' 0' log CSC 5= 0.00448 log CSC (s — a) = 0.28978 log sin (s- 6)= 9.84171 log sin (s — c) = 9.08072 2 )19.21669 log tan i^ = 9.60835 1^ = 22° 5' 20" A = 44° 10' 40" 164 SPHERICAL TRIGONOMETRY. Example 2. a - 124° 12' 31" s -a= 13° 39' 5" b= 54° 18' 16" s -b= 83° 33' 20" c= 97° 12' 25" s - c= 40° 39' 11" 2 s = 275° 43' 12" log tan ^A = 0.30577 s = 137° 51' 36" log tan 15=9.68145 log sin {s — a) = 9.37293 log tan hC= 9.86480 logsin(s- 6) =9.99725 IA= 63° 41' 3.8" logsin(s- c) = 9.81390 ^B= 25° 39' 5.6" log CSC 5=0.17331 iC= 36° 13' 20.1" log tan2r= 9.35739 A = 127° 22' 7" logtanr = 9.67870 B= 61° 18' 11" C= 72° 26' 40" Exercise XXXIX. 1. Given a = 120° 55' 35", ^> = 59°4'25", c = 106° 10' 22"; find A = 116° 44' 50", B = 63° 15' 18", C = 91° 7' 22". 2. Given a = 50° 12' 4", ^» = 116° 44' 48", c = 129° 11' 42"; find A = 59° 4' 28", B = 94° 23' 12", C = 120° 4' 52". 3. Given a = 131° 35' 4", ^» = 108° 30' 14", c = 84° 46' 34"; find ^ = 132° 14' 21", ^ = 110° 10' 40", (7 = 99° 42' 24". 4. Given a = 20° 16' 38", ^' = 56° 19' 40", c = 66° 20' 44"; find A = 20° 9' 54", B = 55° 52' 31", C = 114° 20' 17". § 61. Case VI. Given the three angles, A, B, and C. The sides are computed by means of Formulas [48]. The formulas for the tangents are in general to be preferred. If we multiply the equation tan |- = 53° 49' 25", c = 97° 44' 19". 4. Given J = 4° 23' 35", ^ = 8° 28' 20", C= 172° 17' 56"; find a = 31° 9' 14", b = 84° 18' 2S", c = 115° 10'. § 62. Area of a Spherical Triangle. I. When the three angles, A, B, C, are given. Let R = radius of sphere, E = the spherical excess = A-\-B-\- C— 180°, i^=area of triangle. Three planes passed through the centre of a sphere, each perpendicular to the other two planes, divide the surface of the sphere into eight tri-rectangular triangles. It is convenient to divide each of these eight triangles into 90 equal parts, and to call these parts spherical degrees. The surface of every sphere, therefore, contains 720 spherical degrees. Since in spherical degrees, the AABC=E, and the entire surface of the sphere is equal to 720 spherical degrees, .-. A ABC: surface of the sphere = ^ : 720; or, since the surface of a sphere = AttB^, A ABC: 4.'7rR'' = E: 720; whence THE OBLIQUE SPHERICAL TRIANGLE. 167 II, When the thi^ee sides, a, b, c, are given. A formula for computing the area is deduced as follows : From the first of [49], cos ^{A-\- B) _ cos -J- (s , , / . T.. /, s —^^—^ = -t^u^^iA + B)Ur.U^-B), (b) in which for A and B we may substitute any other two angular magnitudes, as for example, ^(A-\-B) and (90°— -J- C), or ^((i-\-h) and J c. If we use in place of A and B the values ^{A-\-B) and (90° — ^ C), the first side of equation (b) becomes cos ^{A-\-B) —cos (90° — ^C) . cosi(^ + ^) + cos(90°-i(7)' and the second side becomes -tani(|^+i^+90°-iC)tani(i^+i^-90° + |C); or, -tani(^ + ^- (7+180°) tan J(^ + ^+C-180°). If we remember that £'=^ + ^+ C — 180°, and observe that tani(^+^-C+180°) = tani(360°-2(7+^+^+C-180°) = tani(360°-2(7+^) = tan[90°-i(2C-^)] = coti(2C-^), it will be evident that equation (b) may be written cosi(^+J?)-cos(90°-iC)^_ oosi(^ + 5)+cos(90°-jC) *"- '' * ^> 168 SPHERICAL TRIGONOMETRY. If we substitute, in equation (b), for A and B, the values ^{a-\-h) and \ c, and also substitute s ioi: ^ {a -\- h -\- c) and s — c for i- (a + ^ — c), equation (b) will become cos ^ (a -f ^) — cos ^ c ^ , ^ . ^ ,,. ^ — ^— f- f- = — tani5tani(s — c). (d) cos ^ (c)^ -}- ^) + cos I c ^ 2 ^ ^ ^ ^ Comparing (a), (c), and (d), we obtain cot i (2 C— ^) tan i ^= tan ^ s tan ^ {s — c). (e) By beginning with the second equation of [49], and treating it in the same way, we obtain as the result, tani (2 C—E) tanj JS'^tan^ (5 — a) tan i (s — h). (f) By taking the product of (e) and (f), we obtain the elegant formula, tan2iE=tan^staii^(s— a)tan|(s— b)tan^(s— c), [52] which is known as I'Huilier's Formula. By means of it E may be computed from the three sides, and then the area of the triangle may be found by [51]. III. In all other cases, the area may be found by first solving the triangle so far as to obtain the angles or the sides, whichever may be more convenient, and then applying [51] or [52]. Example 1. A = 102^ 14' \2" B= 54° 32' 24" C= 89° 5' 46'' 245° 52' 22' E= 65° 52' 22' = 237142" 180° = 648000" logE2 = logE2. log^ =5.37501 logF =0.06058 f logE2 F= 1.1497^2 If, therefore, we know the radius of the sphere, we can express the area of a spherical triangle in the ordinary units of area. * See Wentworth & HUl's Tables, page 20. THE OBLIQUE SPHERICAL TRIANGLE. 169 Example 2. a =133° 26^9'' b= 64° 50' 53'' c- 144° 13' 45" 2s = 342° 30' 57" s= 171° 15' 28. 5' -a= 37° 49' 9.5' - 6= 106° 24' 35. 5' - c= 27° 1'43.5' is=85°37'44' i(s-a) = 18°54'35' 1 (s - 6) = 53° 12' 18' 1(8 -c) = 13° 30' 52' log tan is = 1.11669 logtaiii(s- a) = 9.53474 logtani(s- 6) = 0.12612 log tan 1 (g — c) = 9.38083 logtan2^^ = 0.15838 0.07919 50° 11' ^ = 200° 46' 52' Exercise XLI. 1. Given ^ = 84° 20' 19", ^ = 27° 22' 40", (7 = 75° 33'; find ^=26159", F=0.126S2E^ 2. Given a = 69° 15' 6", b = 120° 42' 47", c = 159° 18' 33"; find ^=216° 40' 28". 3. Given a = 33° V 45", b = 155° 5' 18", C = 110° 10' ; find ^=133° 48' 53". 4. Find the area of a triangle on the earth's surface (regarded as spherical), if each side of the triangle is equal to 1°. (Eadius of earth = 3958 miles.) CHAPTER IX. APPLICATIONS OF SPHERICAL TRIGONOMETRY. § 63. Problem. To reduce an angle measured in space to the horizon. Let (Pig. 48) be the position of the observer on the ground, AOB = h,t\iQ angle measured in space, (for example, the angle between the tops of two church spires), OA^ and OB' the projec- tions of the sides of the angle upon the horizontal plane HR, AOA' = m and BOB' = n, the angles of inclination of OA and OB respectively to the horizon. Required the angle A'OB' = x made by the projections on the horizon. The planes of the angles of inclination AOA' and BOB' produced intersect in the line OC, which is perpendicular to the horizontal plane (Geom. § 520). From as a centre describe a sphere, and let its surface cut the edges of the trihedral angle 0-ABC in the points M, iV, and P. In the spherical triangle MNF the three sides MN= h, MP = 90° — m, NP = 90° — n, are known, and the spherical angle P is equal to the required angle x. From § 48 we obtain cos ^x= Vcos s COS (s — h) sec m sec n, where ^ (ni -\- n -{■ h) =^ s. Fig. 48. APPLICATIONS. 171 § 64. Problem. To find the distance between two places on the earth^s surface (regarded as spherical), given the latitudes of the places and the difference of their longitudes. Let M and iV (Fig. 49) be the places ; then their distance MN is an arc of the great circle passing through the places. Let F be the pole, AB the equator. The arcs MB and NS are the latitudes of the places, and the arc BS, or the angle MPJSf, is the difference of their longi- tudes. Let MB = bj JSfS = aj BS = I ; then in the spherical triangle MNF two sides, MF =90°— b, NF=90° — a, and the included angle MFN= I, are given, and we have (from § 56) tan m = cot a cos I, cos MN^ sin a sec m sin (b-\-rri). From these equations first find m, then the arc MN, and then reduce MN to geographical miles, of which there are 60 in each degree. Fig. 49. § Q)^. The Celestial Sphere. The Celestial Sphere is an imaginary sphere of indefinite radius, upon the concave surface of which all the heavenly- bodies appear to be situated. The Celestial Equator, or Equinoctial, is the great circle in which the plane of the earth's equator produced intersects the surface of the celestial sphere. The Poles of the equinoctial are the points where the earth's axis produced cuts the surface of the celestial sphere. 172 SPHERICAL TRIGONOMETRY. The Celestial Meridian of an observer is the great circle in which the plane of his terrestrial meridian produced meets the surface of the celestial sphere. Hour Circles, or Circles of Declination, are great circles passing through the poles, and perpendicular to the equinoctial. The Horizon of an observer is the great circle in which a plane tangent to the earth's surface, at the place where he is, meets the surface of the celestial sphere. The Zenith of an observer is that pole of his horizon which is exactly above his head. Vertical Circles are great circles passing through the zenith of an observer, and perpendicular to his horizon. The vertical circle passing through the east and west points of the horizon is called the Prime Vertical ; that passing through the north and south points coincides with the celestial meridian. The Ecliptic is a great circle of the celestial sphere, apparently traversed by the sun in one year from west to east, in consequence of the motion of the earth around the sun. The Equinoxes are the points where the ecliptic cuts the equinoctial. They are distinguished as the Vernal equinox and the Autumnal equinox ; the sun in his annual journey passes through the former on March 21, and through the latter on September 21. Circles of Latitude are great circles passing through the poles of the ecliptic, and perpendicular to the plane of the ecliptic. The angle which the ecliptic makes with the equinoctial is called the obliquity of the ecliptic ; it is equal to 23° 27', nearly, and is often denoted by the letter e. These definitions are illustrated in Figs. 50 and 51. In Fig. 50, AVBUi^ the equinoctial, P and P' its poles, NPZS the celestial meridian of an observer, NESW his horizon, Z his zenith, 31 a star, PMP' the hour circle passing through the star, ZMDZ' the vertical through the star. APPLICATIONS. 173 In rig. 51, AVBU represents the equinoctial, EVFU the ecliptic, P and Q their respective poles, F the vernal equinox, ?7 the autumnal equinox, M a star, PMB the hour circle through the star, QMT the circle of latitude through the star, andZ^r^^e. Fig. 51. The earth's diurnal motion causes all the heavenly bodies to appear to rotate from east to west at the uniform rate of 15° per hour. If in Fig. 50 we conceive the observer placed at the centre 0, and his zenith, horizon, and celestial meridian fixed in position, and all the heavenly bodies rotating around PP^ as an axis from east to west at the rate of 15° per hour, we form a correct idea of the apparent diurnal motions of these bodies. When the sun or a star in its diurnal motion crosses the meridian, it is said to make a transit across the meridian ; when it passes across the part NWS of the horizon, it is said to set; and when it passes across the part NES, it is said to rise (the effect of refraction being here neglected). Each star, as M, describes daily a small circle of the sphere parallel to the equinoctial, and called the Diurnal Circle of the star. The nearer the star is to the pole the smaller is the diurnal circle ; and if there were stars at the poles P and P\ they would have no diurnal motion. To an observer north of 174 SPHERICAL TRIGONOMETRY. the equator, the north pole P is elevated above the horizon (as shown in Fig. 50) ; to an observer south of the equator, the south pole P' is the elevated pole. § QQ. Spherical Co-ordinates. Several systems of fixing the position of a star on the sur- face of the celestial sphere at any instant are in use. In each system a great circle and its pole are taken as standards of reference, and the position of the star is determined by means of two quantities called its spherical co-ordinates. I. If the horizon and the zenith are chosen, the co-ordinates of the star are called its altitude and its azimuth. The Altitude of a star is its angular distance, measured on a vertical circle, above the horizon. The complement of the altitude is called the Zenith Distance. The Azimuth of a star is the angle at the zenith formed by the meridian of the observer and the vertical circle passing through the star, and is measured therefore by an arc of the horizon. It is usually reckoned from the north point of the horizon in north latitudes, and from the south point in south latitudes ; and east or west according as the star is east or west of the meridian. II. If the equinoctial and its pole are chosen, then the position of the star may be fixed by means of its declination and its hour angle. The Declination of a star is its angular distance from the equinoctial, measured on an hour circle. The angular distance of the star, measured on the hour circle, from the elevated pole, is called its Polar Distance. The declination of a star, like the latitude of a place on the earth's surface, may be either north or south ; but, in practical problems, while latitude is always to be considered positive, declination, if of a different name from the latitude, must be regarded as negative. APPLICATIONS. 175 If the declination is negative, the polar distance is equal numerically to 90° + the declination. The Hour Angle of a star is the angle at the pole formed by the meridian of the observer and the hour circle passing through the star. On account of the diurnal rotation, it is constantly changing at the rate of 15° per hour. Hour angles are reckoned from the celestial meridian, positive towards the west, and negative towards the east. III. The equinoctial and its pole being still retained, we may employ as the co-ordinates of the star its declination and its right ascension. The Eight Ascension of a star is the arc of the equinoctial included between the vernal equinox and the point where the hour circle of the star cuts the equinoctial. Right ascension is reckoned from the vernal equinox eastward from 0° to 360°. IV. The ecliptic and its pole may be taken as the standards of reference. The co-ordinates of the star are then called its latitude and its longitude. The Latitude of a star is its angular distance from the ecliptic measured on a circle of latitude. The Longitude of a star is the arc of the ecliptic included between the vernal equinox and the point where the circle of latitude through the star cuts the ecliptic. For the star M (Fig. 50), let ' 1 = the latitude of the observer, h = DM = the altitude of the star, z = ZM = the zenith distance of the star, a = /_PZM=^ the azimuth of the star, t=^ Z. ZPM^ the hour angle of the star, d = EM = the declination of the star, p = FM = the polar distance of the star, r = VR = the right ascension of the star, u = MT =the latitude of the star (Fig. 51), v=VT = the longitude of the star (Fig. 51). 176 SPHERICAL TRIGONOMETRY. In many problems, a simple way of representing the mag- nitudes involved, is to project the sphere on the plane of the horizon, as shown in Fig. 52. NESW is the horizon, Z the zenith, NZS the meridian, WZE the prime vertical, WAE the equinoctial projected on the plane of the horizon, P the elevated pole, M a star, DM its altitude, ZM its zenith dis- tance, Z FZM its azimuth, MR its declination, PM its polar distance, /_ ZPM its hour angle. § 67. The Astronomical Triangle. The triangle ZPM (Figs. 50 and 52) is often called the astronomical triangle, on account of its importance in problems in Nautical Astronomy. The side PZ is equal to the complement of the latitude of the observer. For (Fig. 50) the angle ZOB between the zenith of the observer and the celestial equator is obviously equal to his latitude, and the angle POZ is the complement of ZOB. The arc NP being the complement of PZ, it follows that the altitude of the elevated pole is equal to the latitude of the place of observation. The triangle ZPM then (however much it may vary in shape for different positions of the star M) always contains the following five magnitudes : PZ = co-latitude of observer = 90° — I, ZM= zenith distance of star = z, PZM= azimuth of star = a, PM=^ polar distance of star =p, ZPM= hour angle of star = t. APPLICATIONS. 177 A very simple relation exists between the hour angle of the sun and the local (apparent) time of day. Since the hourly rate at which the sun appears to move from east to west is 15°, and it is apparent noon when the sun is on the meridian of a place, it is evident that if hour angle = 0°, 15°, — 15°, etc., time of day is noon, 1 o'clock p.m., 11 o'clock a.m., etc. In general, if t denotes the absolute value of the hour angle, time of day = — p.m., or 12 — 3-3 a.m., according as the sun is west or east of the meridian. § 68. Problem. Given the latitude of the observer and the altitude and the azimuth of a star, to find its decimation and its hour angle. In the triangle ZPM (Fig. 52), given p^ == 90° — Z = co-latitude, ZM= 90° — h = co-altitude, Z. PZM =^a . = azimuth, to find FM= 90° — d = polar distance, Z ZFM =t = hour angle. Draw MQ ± to JSfS, and put ZQ = m, then, if a< 90°, FQ = 90° - (Z -f m), and if a> 90°, FQ = 90° - (Z- m) ; and, by Napier's Rules, cos a = =h tan m tan h, sin d = cos FQ cos MQ, sin h = cos m cos MQ ; whence, tanm= ± cot h cos a, sin d = sin h sin (I ± m) sec m, ^^ — in which the — sign is to be used if a > 90°. The hour angle may then be found by means of [44], whence we have sin t = sin a cos h sec d. -^- 178 SPHERICAL TRIGONOMETRY. § 69. Problem. To find the hour angle of a heavenly body when its declina- ^ tion, its altitude, and the lati- tude of the place are known. In the triangle ZPM (Fig. 53), given PZ = . , /l — COS ^ 18. tani^ = ±\/':7-^ ^ ^ 1 + cos ^ VI + cos ^ 1 19. cot^ cos 2; §30. 20. sin A + sin ^ == 2 sin i(A-\-B) cos i(A — B). 21 . sin ^ — sin ^ = 2 cos i(A-\-B) sin ^ (y1 — ^). 22. cos^ + cos5 = 2cosi(^ + 5)cosi(^— ^). 23. cos.4 — cos^ = — 2sin^(^+^)sin^(^— ^). sin ^ + sin B _ tan |- (^ + -^) . 31. 24. 25. sin ^ — sin ^ tan ^(A — B) a sin A b siwB 26. a^ = h'^-\-c^ — 2bcG0sA. a — h_ tan ^ (J — ^) ^' H^~'tani(^+^)* 28. sini^=V^^^¥^- §33. §34. § 35. §40. 188 FORMULAS. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. cosiA=\^-^ ^^. be ^ ^ s s — a ■c) s(s — a) hs-a)(s-b)(s-c) ^ ^. ^ s tSLU^A = F = F = i ac sin B. ahin B sin C 2sin(^+C) i^ = V^ (5 — a) (s — b)(s — c). Spherical Trigonometry cos c = cos a cos b. f sin «; = sine sin^. 1 sin ^ = sin c sin B. ( cos A = tan b cot c. ( cos B = tan a cot c. {cos ^ = cos a sin ^. cos^ = cos Jsin J. r sin & = tan a cot A. \ sin S^— ^)see(/S^— C) . §54. 49. 50. J cos ^ (^ + 5) cos ^ c = COS i(a-\-b) sin ^ C. sin K^ + ^) cos ^ c = cos i(a—b) cos | (7. COS i(A — B)smic = sin ^ (a + ^') sin | C. sin -J- (^ — ^) sin -J-c = sin ^ (a — J) cos -J C, ^ ^ COS -J (a -f ^>) ^ tan ^ (« + b) Gosi(A—B) cosi(A-\-B) tan-J-c. 1/ 7N sin4-M— ^) ^ 55. 51. F ■R^E 180 * 52. tan^|-^=tan-|-5 tan-J-(s — esc B = -' c c a a b 3. (i.) sin = I, cos= |, (ii.) sin = y\, etc. (v.) sin = ff, etc. tan = f, cot = I, (iii.) sin = yV» ^tc. (vi.) sin = i^f , etc. sec = f, CSC = f, (iv.) sin = j\, etc. 4. The required condition is that a^ + 6^ = c^. It is. 5. (1.) sin=-T— — 5, etc. (m.) sin = -, etc. (ii.) sm = „ , % ' etc. (iv.) sm = — > etc. 7. In (iii.) pV -j- ^232 = ^252 . in (iv.) m^n^s^ + m^H'^ = n^^r^. S. c= 145 ; whence, sin A = j\% =cosB; cos ^ = i-ff = sin B ; tan A = jW = cot 5 ; cot A = Vi^- = tan 5 ; sec A = f|| = esc ^ ; etc. TRIGONOMETKY. 9. b = 0.023 ; whence, tan A = cotB = ^H- ; cot ^ = tan B = ^%\, etc. 10. a= 16.8 ; whence, sin A = J || = cos B, etc. 11. c = ■» + g ; whence, sin ^ = ^ — ^ = cos 5 ; etc. 12. b='^q{p + q); whence, tan A = -l/- = cot E ; etc. P — Q 13. a = p — q : whence, sin^= — r-^ = cos B ; etc. ^ ^' p+ q 14. sin ^ = 2 V5 = 0.89443 ; etc. 15. sin vl = | ; etc. 16. sin J. = } (5 + V?) = 0.95572 ; etc. 17. cos^=^(V31- 1)^0.57097; sin^ = i (V31 + 1) = 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 ; 6 = 2 miles. 31. 400,000 miles. 30. a = 0.342, 6 = 0.940; a = 1.368, 6= 3.760. 32. 142.926 yards. Exercise III. 5. Through A (Fig. 3) draw a tangent, and take J. T = 3 ; the angle J. or is the required angle. 6. From ( Fig. 3) as a centre, with a radius = 2, describe an arc cut- ting at Tthe tangent drawn through B\ the angle ^OT is the required angle. 7. In Fig. 3, take OM = ^, and erect MP A_ OA and intersecting the circumference at P ; the angle POM is the required angle. 8. Since sin cc = cos x, OM = PM (Fig. 3), and x = 45° ; hence, con- struct 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 O erect a perpendicular to meet the circumference at some point P. Join OP ; the angle A OP is the required angle. 21. r sin x. 22. Leg adjacent to A = nc^ leg opposite to ^ = mc. 1. cos 60°. cot 1°. sin 45°. tan 75°. 2. cos 30°. cot 33°. sin 15°. tan 6°. 3. iV3 4. tan^ = cot^ = cot (90° 5. 30°. 7. 90°. 6. 30°. 8. 60°. ANSWERS. Exercise IV. sec 71° 50'. tan 7° 41^ sin 52° 36'. sec 35° 14' sec 20° 58'. tan 0° 1'. sin 4° 21'. sec 44° 59' A); hence, ^ = 90° - ^ and ^ = 45°. 9. 22° 30'. 11. 10°. 10. 18°. 90° 1^. n+1 Exercise VI. cot ^ = y\, sec A = -U, esc ^ = \ |. 2. cos^=0.6, tan J[ = 1.3333, cot^ = 0.75, sec^=1.6667, csc^=1.25. 3. sin ^= ^}, tan J.= l^i, cot^=ff, secJ.= fi, cscA = ^\. 4. sin J.=0.96, tan ^=3.4286, cot^ = 0.2917, sec J.=3.5714. 5. sin J. =0.8, cos J. =0.6, cot J.=0.75, sec J.=l. 6667, esc ^ = 1.25. 6. sin J.=W2, cos J.=i\^, tan J.=l, secJ.= V2, cscJ.= V2. 7. tan^=2, sin ^=0.90, cos ^=0.45, sec ^=2.22, esc ^ = 1.11. 8. cos7l=i, sin J.=iV3, tan J.= V3, cot^=iV3,. cscvl=fV3- 9. sin^=|A/2, cos J.=iV2, tan^=l, cot^ = l, secJ.= V2. 10. cos ^= Vl — nfi, tan^ = , ^ ^ Vl — m^, cot^=-Vl — m2. ' 1 — m2 m , 1 — m2 ^ , 2m ^ . 1 — m2 .1 + ^2 11. cosJ. = — -; -1 tan^=:; -, cot^=— , sec^ = 1 + m2 1 — m2 2m 1 — rri .^ . , m2 — n2 ^ ^ m2 — ri2 m^+rfi 12. sm ^=— T"^ — o- tan ^=-- ,sec^=— r 13. cot = 1, sin = 1 V2, cos = i V2, sec = V2, esc = \^. 14. cos = i V3, tan = i Vs, cot = V3, sec = f Vs, esc = 2. 15. sin = i V3, cos = i, tan = V3, cot = i V3, sec = 2. 16. sin = 1 V2 - V3, cos = 1 V2 + V3, cot = 2 + Vs. 17. sin = I V2^ ^, cos = 1 V2 + V2, tan = V2 - 1. 18. cos = 1, tan = 0, cot = 00, sec=l, esc = 00. 19. cos = 0, tan = oo, cot = 0, sec = oo, esc = 1. 20. sin = 1, cos = 0, cot = 0, sec = oo, esc = 1. TRIGONOMETRY. 21. cos^ = Vl-sin2^, tanA = ^r^£=, esc A = ~ 22. sin^ sec A 23. sin^- Vl — sin2 A ' """ "" sin A COS ^ V 1 — f^ne2 >1 1 COS^ ' tan J: csc^ = cos^ 1 Vl — cos^ J. 1 sec A 24. tan^ — / = ) cos ^ = VH- tan^^ Vl + tanM = Vl + tanM, cscA = ^^^^±^^ tSinA , cot J. = tan^ cos^ =: cot^ cot^ CSC ^ = Vl + cot2^, sin A sec A = Vl + cot2 A Vl + cot2^ Vl + cot2 A ' ^^ cot A iV5, cos^ = f^/^. 27. smA = ^\, cos^ = ff. 1 — .3cos2^ 4-8cos4^ 25. sin^ 26. sin A = ^Vl5, tan A = Vl5, 28. cos2 A — cos* A Exercise VII. 1. a; = 45°. 5. x = 60°. 2. x = 30°. 6. x = 45°. 3. a; = 0°, or 60°. 7. x = 45°. 4. x = 45°. S. x = 45°. 1. 6 - = cos vl ; .• b • C= 7- cos^ 2. - = sm ^ ; .• 6 -= cos J.; .• r- '^ sin A 3. . b = c cos ^. 4. 6. b - = cos A : .• c ' ^ = 90° - B, a = ccos B, 6 = c sin B. 6 • C= 7- COS J. 9. a; = 60°. 13. x = 0°, or 60°. 10. a; =60^ 14. x=30°. 11. a; = 30= 15. X = 30°, or 45° 12. a; = 45'^ 16. a; = 45°. 17. x = 60°. E VIII. 6. ^ = 90° - B, a = bcot Bf b sin B 7. ^ = 90° - B, 6 = a tan J5, cosB 8. cos ^ = - » c B=90°-A, a= ^c^- 62. ANSWKRS. fi Exercise IX. 31. c= 7.8112, A = 39° 48', B = 50° 12', F=15. 32. 6=69.997, A = 30' 12", B = 89° 29' 48", F= 21.525. 33. a =1.1886, A = 43° 20', B = 46° 40', ji^= 0.74876. 34. 6 = 21.249, c = 22.372, B = 71° 46', i^= 74.372. 35. a = 6.6882, c= 13.738, B=60°52', i?'= 40.129. 36. a = 63.859, 6=23.369, B = 20° 6', i?'= 746.15. 37. a =19.40, 6 = 18^778, A = 45° 56', -F= 182.15. 38. 6 = 53.719, c= 71.377, ^=41° 11', F = 1262.4. 39. a =12.981, c= 15.796, A = 55° 16', 2?^= 58.416. 40. a = 0.58046, 6=8.442, A= 3° 56', i?'= 2.4501. 41. F=i{c^ sin A cos A). 43. F=i(62tan^). 42. F=i{a'^ cot A). 44. F=i( ;aVc2 — a2). 45. 6=11.6, c = 15.315, A = 40° 45' 48", 5 = 49° 14' 12". 46. a = 7.2, c = 8.7658, B = 34° 46' 40" A = 55° 13' 20". 47. a = 3.6474, 6 = 6.58, c = 7.5233, B = 6] L°. 48. a =10.283, 6 = 19.449, A = 27° 52', B = 62° 8'. 49. 19° 28' 17" and 70° 31' 43". 50. 3 and 5.1961. 90° 51. a = c cos — -— ' n + 1 57. 59° 44' 35" 58. 95.34 feet. • 59. 1° 25' 56". • 6 = c sm — — r ' n+1 60. 7.0712 miles in each direction. 61. 20.88 feet. 52. 36° 52' 12" and 53° 7' 48". 62. 56.65 feet. 53. 212.1 feet. 63. 228.63 yards. 54. 732.22 feet. 64. 136.6 feet. 55. 3270 feet. 65. 140 feet. 56. 37.3 feet, 96 feet. 66. 84.74 feet. Exercise X. 1. C = 2 (90° — J.), c = 2acos^, h=asmA. 2. A = i{lSO°— C), c = 2acosA, h=asinA. 3. C = 2 (90° — ^), a = c -^2 008^, h=asmA. TRIGONOMETRY. 4. ^ = 1(180°- C), a = c -^ 2 cos A, h= a sin A. 5. C = 2 (90°-^), a = h-7-smA, c = 2 a cos ^. 6. ^ = i(180°-C'), a = y^-7-sin^, c = 2 a cos ^. 7. sin J. = h-^ a, C = 2(90°-^), c = 2acos^. 8. tan J. = h-7- ^c, (7 = 2(90°-^), a = /i-^sin^. 9. A = 67° 22' 50'^ C = 45° 14' 20", /i=13.2. 10. c = 0.21943, h = 0.27384, F= 0.03004. 11. a = 2.055, ;i= 1.6852, F= 1.9819. 12. a = 7.706, c = 3.6676, ' 2^^=13.725. 13. A = 79° 36' 30'', C = 20°47', c = 2.4206. 14. A = 77° 19' 11", C = 25° 21' 38", a = 20.5. 15. ^ = 25° 27' 47", 0=129° 4' 26", a = 81.41, h = S5. 16. ^ = 81° 12' 9", C= 17° 35' 42", a =17, c = 5.2. 17. F=icV4a2_c2. 22. 0.76537. 18. F= a2sin|Ocos^ C. 23. 94°20'. . 19. F= a2sin^cos^. 24. 2.7261. 20. F=A2tan^C. 25. 38° 56' 33". 21. 28.284 feet, 4525.44 sq. feet. 26. 37.699. Exercise XI. 1. r= 1.618, ;i= 1.5388, F= 7.694. 2. r= 11.271, h=10.SS6, F= 381.04. 3. h = 0.9848, p = 6.2514, F= 3.0782. 4. A =19.754, c = 6.2572, F=1236. 5. r= 1.0824, c = 0.82842, F= 3.3137. 6. r = 2.5933, A =2.4882, c= 1.4615. 7. r= 1.5994, 7^=1.441, p = 9.716. 8. 0.61803. 12. 0.2238. 17. 11.636. 9. 0.64984. 13. 0.31. 18. 99.64. 10. 0.51764. 14. 0.82842. 19. 1.0235. 11. 5- c 15. 94.63. 20. 16. 414.97. 0.635. o 90° ANSWERS. Exercise XII. 5. Two angles : one in Quadrant I. , the other in Quadrant II. 6. Four values : two in Quadrant I. , 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 cc = — f , ic is between 90° and 270° ; if cot cc = 4, x is between 0° and 90° or 180° and 270° ; if sec z = SO,x is between 0° and 90° or between 270° and 360° ; if esc x = — 3, cc is between 180° 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. sinx = — fV3, tanx = — 4V3, cot x = — y^ V3, esc x = — ^^ V3. 15. sinx = ± Jq VlO, cosx = qi j^VlO, tanx = — ^, secx = :piVlO. cscx = ± VlO. 16. 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 (860°+ 00°) = sin 90°= 1 ; tan 540° = tan 180° = ; cos 630° = cos 270° = ; cot 720° = cot 0° = 00 ; sin 810° = sin 90° = 1 ; esc 900° = esc 180° = 00. 21. 46°, 135°, 225°, 315°. 22. 0. 23. 0. 24. 0. 25. a2-62 + 4a&. Exercise XIII. 2. sin 172°= sin 8°. 8. sin 204° =— sin 24°. 3. cos 100° = — sin 10°. 9. cos 359° = cos 1°. 4. tan 125° = - cot 35°. 10. tan 300° = - cot 30°. 5. cot 91° = — tan 1°. 11. cot 264° = tan 6°. 6. sec 110° = — CSC 20°. 12. sec 244° = - esc 26°. 7. esc 157° = CSC 23°. 13. esc 271° = - sec 1°. 8 TRIGONOMETRY. 14. sin 163° 49' = sin 16° 11', 17. cot 139° 17' = - cot 40° 43'. 15. cos 195° 33' = - cos 15° 33'. 18. sec 299° 45' = esc 29° 45'. 16. tan 269° 15' = cot 0° 45'. 19. esc 92° 25' = sec 2° 25'. 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° = sin 20°, cos(— 200°) = cos 200° = — cos 20°, etc. 23. sin (- 345°) = - sin 345° = sin 15°, cos (— 345°) = cos 345° = cos 15°, etc. 24. 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' = sin 16° 54', cos (- 196° 54') = cos 196° 54' = - cos 16° 54', etc. . 26. sin 120° = i V3, cos 120° = - i, etc. 27. sin 135° = + i\^, cos 135° = - i V2, etc. 28. sin 150° = + h cos 150° = — -J V3, etc.' 29. sin 210° = - i, cos 210° = - | V3, etc. 30. sin 225° = — i ^, cos 225° = — i V2, etc. 31. sin 240° = - | Vs, cos 240° = - i, etc. 32. sin 300° = — i V3, cos 300° = + i, etc. 33. sin (- 30°) = -h cos (- 30°) = + ^ Vs, etc. 34. sin (- 225°) = + i V2, cos (- 225°) = - i^/^, etc. 35. cos a; = — i V2 or — V^, etc., x = 225°. 36. tan X = — V^, sin x = i, cos a; = — iV3, x = 150°. 37. sin 3540° = sin 300° = - sin 60° = - ^ V3, cos 3540° =+h etc. 38. 210° and 330°; 120° and 300°. 39. 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. 45. m sin x cos x. 44. (a — 6) sin x. 46. (a — b) cot x— (a + b) tan x. ANSWEES. 9 47. a2 + 62 + 2 a6 cos x. 49. cos xsiny — sin x cos y. 48. 0. . 50. tancc. 61. 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 (x — 90°) = sin x, etc. 54. sin (x — 180°) = — sin x, cos (x — 180°) = — cos x, etc. Exercise XIV. 1. sin (x + y) = ff, cos (x + y) = |f. 2. cosy, siny. 3. sin ( 90° + y)= cosy, cos ( 90° + y) = — sin y, etc. 4. sin (180° — y)= sin ?/, cos (180° — y) = — cosy, etc. 6. sin (180° + ?/) = — sin y, cos (180° + ?/) = — cosy, etc. ' 6. sin (270° — y) = — cosy, cos (270° — y) = — sin y, etc. 7. sin (270° + y) = — cosy, cos (270° + y) = sin y, etc. 8. sin (360° — y) = — sin y, cos (360° — y) = cos y, etc. 9. sin (360° + y) = siny, cos (360° + y) = cosy, etc. 10. sin (x — 90°) = — cos x, cos (x — 90°) = sin x, etc. 11. sin (x — 180°) = — sin X, cos (x — 180°) = — cos x, etc. 12. sin (x — 270°) = cos x, cos (x — 270°) = — sin x, etc. 13. sin ( — y) = — sin y, cos (— y) = cos y, etc. 14. sin(45°—y) = |^V2 (cosy— siny), cos(45°— y) = |V2 (cosy + siny), etc. 15. sin(45°+ y) = ^V2 (cos y+ siny), cos(45°+y) = | V2(cosy— siny), etc. 16. sin(30°+y) = i(cosy+ V3siny), cos(30° + y)=^(V3 cosy -siny), etc. 17. sin(60°—y) = i(V3 cosy— siny), cos(60°— ?/) = |(cosy+ V3siny), etc. 18. 3sinx-4sin3x. 19. 4 cos^x - 3 cos x. 20. 0. 21. iV3. ,, = ^i^^8 = 0.10051 ; cos I X = V^±f^ = 0.99494. 23. cos2x= — i, tan2x= — V3. 22. sini.. 10 TRIGONOMETRY. 24. sin 22i° = i V2 - V2 = 0.3827, cos 22i° = ^ V2 + V2 = 0.9239. tan22i°=V2-l =0.4142, cot22|° = V2+l =2.4142. 25. sin 15° = |- V2 — V3 = 0.2588, cos 15° = i V2 + VS = 0.9659. tan 15° =2-V3 =0.2679, cot 15° = 2 + Vs =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 2 X = 2 sin X cos x, and 2 tan x = 2 ' 1 + tan2x = sec^x = — w- • cos X cos-^x By making these substitutions in the given equation its truth will be evident. 34. sin J. + sin £ + sin C = sin J: + sin ^ + sin [180 - {A + B)] = sin A -\- sin B -i- sin {A + B) = 2 sin ^ (^ + JB) cos^ (^ - B) + 2 sin i (^ + B) cos 1{A + B) = 2 sin ^ (^ + B) [cos l{A-B)-{- cos i {A + B)] = 4sm^(^ + ^) cos ^ J. cos I ^, (see §§ 29 and 31). But cos^ C = cos [90° -i(^ + E)]= sin 1(^ + 5). Therefore, sin -4 + sin jK + sin C = 4 cos ^Acos^B cos | C. 35. Proof similar to that for 34. __ ^ . , ^ „ . , ^ si n ^cosg , cos J. sin B , sin C 36. tan A + tan 5 + tan C = :; -\ =: -\ cosJLcoSjB cos a cos 5 cosC sin G , sin C sin C cos C + cos A cos B sin C cos A cos 5 cos G cos A cos jB cos (7 _ (cos A cos B + cos G) sin C _ [cos AcosB— cos {A -h B)] sin G ~ cos J. cos JS cos C7 ~ cos A cos 5 cos C sin A sin B sin C ^ . , ^ , „ = — = tan A tan B tan C. cos A cos B cos G 37. Proof similar to that for 36. 38 2 42. tan2x. ^^ cos (x + y) sin2x cos (x - y) ' sinxsiny 39. 2 cot 2 X. ' cos X cosy 47. tan x tan y. ^^^ cos jx-y) ^^ cos (x + y) sin X cos 2/ ' cos X cosy cos (x + y) .- cos (x — y) sin X cosy ' sin x sin y ANSWERS. 11 Exercise XV. 1. sin-iiV3 =60° + 2n7t or 120° + 2 n tt. tan-i— = =S0° + 2n7i: or 210° + 2n7t. V3 vers-i i =±60° + 2n7t. cos-i /- ^\ = 135° + 2 n TT or 225° + 2n7t. csc-i V2 = 45° + 2 n TT or 135° + 2 w ;r. tan-i oo = 90° + 2 n ;r or 270° +2 nit. sec-12 =±60° + 2n^. cos-i (- ^ VS) = 150° + 2 w ^ or 210° + 2 n tt. 4. -^. 10. ±-|. 12. ±iV2. 2V2 13 8. 0°, 90°, 180°. 11. ± ^- 13. « = 0, or ± | V3. Exercise XVI. 1. If, for instance, B = 90°, [25] becomes - = sin A. 3. a2 = 62 + c2, a2 = 62 + c2-26c, a"^ = l^ + c^ + 2hc. ' 6. 90°. 7. (i.) = tan (J. — 45°), and a right triangle. (ii.) a + h= (a — &) (2 + V3), an isosceles triangle with the angles 30°, 30°, 120°. Exercise XVII. 9. 300 yards. 15. a = 5, c = 9.6592. 10. ^^=59.564 miles. 16. a = 7, 6=8.573. AC = 54.285 miles. 17. sides, 600 feet and 10-39.2 feet ; 11. 4.6064 miles, 4.4494 miles, altitude, 519.6 feet. 3.7733 miles. is. 855:1607. 12. 4.1501 and 8.67. 19. 5.438 and 6.857. 13. 6.1433 miles and 8.7918 miles. 20. 15.588. 14. 8 and 5.4723. 12 TRIGONOMETRY. 11. 420. Exercise XVIII. 12. 124.617. 11. 6. 12. 10.392. 14. 8.9212. Exercise XIX. 15. 25. 16. 3800 yards. 17. 729.67 yards. 18. 10.266 miles. 19. 6.0032 and 2.3385. 20. 26° 0' 10'' and 14° 5' 50' Exercise XX. 11. A = 36° 52' 12", B = 53° 7' 48", C = 90°. 16. 45°, 60°, 75°. 12. ^=5=.3.3°33'27", C = 112°53'6". 17. 4°23' W. of N., or W. of S. 13. A = B=C = 60''. 18. 60°. 14. Impossible. 20. 0.88877. 15. 45°, 120°, 15°. 21. 54.516 miles. Exercise XXI. 1. 4333600. 2. 365.68. 3. 13260. 4. 8160. 6. 240. 6. 26208. 7. 15540. 8. 29450 or 6982.8. 9. 17.3204. 10. 10.3919. 11. 0.19975. 12. db sin A. 13. \{a'^-b'^)t&nA. 14. 2421000. 16. 30°, 30°, 120°. 1. 21.166 miles 2. 6.3399 miles. 3. 119.29 feet. 4. 30°. Exercise XXII. 24.966 miles. 6. 20 feet. 6. 2.6247 or 21.4587 7. 276.14 yards. 8. 383.35 yards. ANSWERS. 13 MISCELLANEOUS EXAMPLES. 2. 106.70 feet; 21. 260.21 feet; 46. 294.69 feet. 142.86 feet. 3690.3 feet. 47. 12,492.6 feet. 3. 1023.9 feet. 22. 1.3438 miles. 48. 6.3.397 miles. 4. 37° 34' 5^'. 23. 235.80 yards. 49. 210.44 feet. 5. 238,400 miles. 27. 8 inches. 51. 757.50 feet. 6. 861,880 miles. 30. 460.46 feet. 52. 520.01 yards. 7. 2922.4 miles. 31. 88.936 feet. 53. 1366.4 feet. 8. 60°. 32. 13.657 miles. 54. 658.36 pounds; 9. 3.2068. 34. 56.564 feet. 22° 23' 47" with first 10. 0.6031. 35. 51.595 feet. force. 11. 199.56 feet. 36. 101.892 feet. 55. 88.326 pounds; 12. 43.107 feet. 38. N. 76° 56' E. ; 45° 37' 16" with 13. 45 feet. 13.938 miles an hr. known force. 14. 26° 34'. 39. 442.11 yards. 58. 500.16; 536.27. 15. 78.367 feet. 40. 255.78 feet. 59. 345.48 feet. 16. 75 feet. 41. 3121.1 feet ; 60. 345.46 yards. 17. 1.4446 miles. 3633.5 feet. 61. 61.23 feet. 18. 7912.4 miles. 42. 529.49 feet. 63. 307.77 yards. 19. 56.649 feet. 43. 41.411 feet. 64. 19.8; 35.7; 44.5. 20. 69.282 feet. 44. 234.51 feet. 65. ±45°, ±136°. 66. 45. cos A ■ 25.433 miles. — m± Vw2 + 4 ( 2 (n + E. 67. sm^ =\h, z . \ 1 — n^ cos B _ n / l — m' ~ m \ 1 - 7i2 68. 69. ±60°, ±120°. ^2_ r = h^c'''\ B: 0°, 180°, ± 60°. 2 n a 180° = 2^"* n 70. 0°, 30°, 180°, 210°. 73. i6csin^. 74. i c2 sin A sin B esc {A -\- B). 75. \/s{s-a){s-b){s-c). 14 TRIGONOMETRY. 77. 199 A. 3 R. 8. p. 94. 16,281. 114. S. 56° T 30'' E.; 78. 210 A. 3 R. 26 p. 95. 435.76 sq. ft. 202.6 miles. 79. 12 A. 3 R. 36 p. 96. 49,088 sq. ft. 115. N. 17° 25' W.; 80. 3 A. R. 6 p. 97. 749.95 sq. ft. 37° 46' N. 81. 12 a. 1r. 15 p. 98. 422.38 sq. ft. 116. S. 56° 11' E.; 244.3. 82. 4 A. 2 R. 26 p. 99. 1834.95 sq. ft. 117. 359.87 miles. 83. 14 A. 2 R. 9 p. 100. 26.87. 121. Long. 68° 55' W. 84. 61 A. 2 R. 103. 6. 122. 103.6 mUes. 85. 4 A. 2 R. 26 p. 108. 6. 124. 33° 18' K ; 86. 13.93, 23.21, 110. 6086.4 feet. 36° 24' W. 32.50 ch. HI. 5° 25' S. ; 125. N. 28° 47' E. ; 87. 9 A. R. 1 p. 457.5 miles. 1293 miles. 89. 876.34. 112. 460.8 miles; 126. S. 50°40'W.; 90. 1229.5. 383.1 miles. 250.8 ; 20° 9' W. 92. 1076.3. 113. 229 miles; 127. 38°21'N.; 93. 2660.4. lat. 11° 39' S. 55° 12' W. 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° 15' N. ; 20° 49' W. 132. N. 53° 20' E., 16° 7' W. ; or N. 53° 20' W., 25° 53' W. 133. N. 47° 42.5' 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. 134. Lat. 30°, 359.82 miles ; lat. 45°, 359.73 miles ; lat. 60°, 359.50 miles. 137. N. 72° 33' E. ; 45 miles ; 42° 15' N., 69° 5' W. 138. N. 72° 4' W., 287 miles ; 32° 54' S., 13° 2' E. ANSWERS. 15 PROBLEMS IN GONIOMETRY. [The solutions here given are for angles less than 360°.] 79 ±J_,±J^. 102. x=±i7t, ±^7e. V5 V5 • 103. x = 0°, ±60°. 80. ± V5 — 2. 104. X = tan-i Vi 81. ±iV3. 105. x=-15°, 105°. 82. ±1, ± f . 106. X = — 2 cot-i a. 83. ±W2-. ^,^^ x^cos-^ r'^^t^ ^^'V 84. h _ V 4 y or ' "V5-l V5 + 1 108. x = -45°, 135°, ^^- 4 ' 4 * isin-i(l-a). 86. x=i7t, f 7t. 109. x= ± 30°, ± 60°, ± 120°, 87. x = 90°, 270°. ±150°. , VS - 1 110. X = ± 60°, ± 90°, ± 120°. 88. X = sm-i — 2 111. x=±60°, ±90°, ±120°. 89. x = 0°, 90°. 112. x=120°. 90. x = 30°, sin-i(-i). ii3_ 3^ = 300, 150°, gin-ij. 91. X = 180°, cos-if. 114. X = ± 60°, ± 90°. 92. x = 0°, 120°, 180°, 240°. ng^ ^^0°, ±20°, ±100°, ±140°, 93. x = 45°, 225°, tan-i(-i). 180°. 94. x = 0°, ±60°, ±120°, 180°. 116. x = ± 45°, ±90°, ±1.35°. 95. x= - 45°, 135°, 117. x = ± 30°, ± 60°, ± 90°, i sin-i (2 V2 - 2). ±120°, ±150°. 96. X = 0°, 45°, 180°, 225°. 118. x = 0°, 45°, ± 90°, 225°. . I~Y\ 110. X = ± 30°, ± 60°, ± 120°, 97. x = coB-^i^±yj-^y ±1500. 96. x = 0°, 45°, 90°, 180°, 225°, 120. x = ± 30°, ±90°, ±150°. 270°. 121. x = 0°, 45°, 180°, 225°. 99. x = 0°, 180°, isin-if. 122. x = ± 45°, ±60°, ±120°, 100. x = 0°, ±90°, ±120°. ±135°. 101. x=0°, ±36°, ±72°, ±108°, 123. x = 0°, ±45°, ±135°. ± 144°, 180°. 124. X = ± 30°, ± 90°, ± 150°. 16 TRIGONOMETRY. 125. X = 8°, 168°. 139. x == ± |. 126. aj = tan-iVf 140. x = l. 127. x = ± 30°. 141. x = 0, 1, - 1. 128. x=± 60°, ± 120° 142. x = ± V|. 129. x=±30°, ±60°, ±120°, _ 1 ±150°. ^^^' ^~'^' 130. x=±sin-i|. 144^ (a^ + 65)1. 131. X = 30°, 150° - cos-i ^ • / 1 ± m\^ «/^ 145. ('-f^y(lT2m). 132. x^tan-i^^^, -tan-if. ^^g ^ 133. 2/ = — 90°, X indeterminate ; . .« , . /t ■ , ,/o x = 45°, y = 0°; ■^' =^^^'^- x=135°, y=:180°; ^^^' ^' ~t- x = 225°, y = 0°', 149 «+l , x = 315°, 2/ =180°. * V2a+1 134. X - tani ^6 ' 151. tan (x + y). y = t^n-i^^±:^/^^ZM, 152. '4^- 2 6 sm y 135. x = 45°, 225°. 153. -tanx. 136. x=±l, ±V3. ^^^ tan--^. 138. x = |V3. 156. cot2a:-tan2x. ENTRANCE EXAMINATION PAPERS. I. a . 90° 90° 6. rsm^j-q-j' '•cos^i^p^- 7. 475.27 feet. n. 4. sin = i V 2-V2, tan = V2 - 1, sec = V4-2V2, cos = iV2+V2, cot = V2+l, csc= V4 + 2V2. 6. (i.) one, (ii.) none, (iii.) none, (iv.) two. 7. 383.36 yards. ANSWERS. 17 2. (a)sin^=±i, tan^^rp / ' cot^=qiV3, III. 1 V3^ 2 sec ^ = — , CSC ^ = ± 2. Vs (6) 30°, 90°, 150°, 270°. 6. 161.41, 33° 34' 5'', 99° 4' 43''. 7. 69.812 yds. IV. 6. 230.03 feet. 7. ^ = 37° 24' 58", B = 51° 37' 52", C = 90° 57' 10' 1. 17| years. 4. 1. 2. siii2x=:±m, tan2x=±-p^=- ^' 1-7208. VI - m2 6^ j^^ 50O 18' E., 399 mUes. 3. X = 210°, 330°, 44° 25' 30", 135° 34' 30". VI. 1. 16. 4. 45°, 225°, 116° 33' 54", ' 3tana:-tan3x . 296° 33' 54". 1 — 3 tannic ' ^- ^irst ship, 223 miles; second 3. Third side, any value ; opposite ^^^P' ^^^ °^^®s- side, 13.766. 6. 0. VII. 1. 25. 4. ±90°, 180°. 2. 2. 5. S. 83°41' E.; 1907 miles. 3. 8.6816, -5^, 43° 43' 10", 106° 16' 50". VIII. 1. 27. 2. a = V2Ftan^, 6 = V2Fcot^. Z. a-± 45°, ± 135°; & = ± 30°, ± 150°. 4. Smallest value of opposite side, 1 ; 1.75, 53° 7' 48", 81° 52' 12" or 0.25, 126° 52' 12", 8° 7' 48". 5. 39° 29' N., 67° 14' W. 6. tan a - tan2 6 or - cot2&. 18 TRIGONOMETRY. IX. 1. 15.849. 2. a = 2(3 + V3), 6 = 2(V3 + 1), c = 4(V3 + l), B = 60°. 3. 155° 42' 20'^ 114° IT 42''. 4. 41° 24' 35", 82° 49' 10", 55° 46' 15". 6. N. 69°56'E.; 609 miles. 6. 1. X. 1. 1.23138. 4. 5,743^ 4 357. 2. a = 4, 6 = 3, c = 5, ^ = 53° 7' 48", 5. 14° 10' E. ; 342 miles. B = 36° 52' 12". 6. 2. 3. cos2^. XI. 1. logs 4 = 1-. 4. 114.92 feet. 3. 0.039345, 0.055226, 97° 45'. 5. 47°24'N.; 63° 43' W. XII. 7t V3 — 1 ^' 3 ^- — 2 ^- 462.34, 61° 37' 30", 56° 14' 30' XIII. 12- 1- ^- 7. 188,280. 8. 44° 35' 40". XIV. 1. 200° 32' 7". 5. 1, 7. a = 273.76, 6 = 272.94, c = 256.65, a = 62° 9' 42", ^ = 61° 50' 18", 7 = 56°. 8. 42° 49' 48". XV. 1. (a) 114° 35' 30", (6) |. 6. 205° 24' 47". 7. 461.94; 59° 11' 8". ANSWERS. 19 Exercise XXIII. 1. Iogio6 = 0.77815. logio 14 = 1.14613. logio 21 = 1.32222. logio4 = 0.60206. logio 12 = 1.07918. logio 5 = 0.69897. logioi = T.69897. logio i = 1.39794. logio 1 = 1.89086. logio U = 0.02119. 2. logs 10 = 3.3219. log2 5 = 2.3219. logs 5 = 1.4650. 10g7i = - 0.3562. 10g5 3l3 = - 2.2620. 3. l0ge2 = 0.69315. loge 3 = 1.09861. loge 5 = 1.60944. 10ge7 = 1.94591. loge 8 = 2.07944. loge 9 = 2.19722. logel- = - 0.40546. logef = - 0.22314. loge If = 0.25952. logeeV = -2.14843. 4. x= 1.54396. X = 0.83048. X = 0.42062. Exercise XXIV. 1. lOgeS = 1.09861. loge 5 = 1.60944. loge 7 = 1.94591. 2. loge 10 = 2.3025850930. 3. logio2 = 0.30103. logioe = 0.43429. logio 11 = 1.04139. 1. sin V = 0.00029088820. tan 1' = 0.000290888212. 2. sin 2' = 0.000581776. Exercise XXV. cos r = 0.99999995769. 3. sin 1° = 0.0175. 6. 0°40'9" Exercise XXVI. 1. sin 6' = 0.0017453 2. sin 2° = 0.034902 ; sin 3° = 0.052340 ; sin 4° =0.069762; cos G' = 0.9999995. cos 2° = 0.999392, cos 3° =0.998632. cos 4° = 0.997568. Exercise XXVII. 1. The 6 sixth roots of — 1 are : V3 + i . -\/3 + i -V3 2 ' 2 The 6 sixth roots of + 1 are : , 1 + V^ - 1 + V^ 1. 7, ' 7, -1, i, V3- - i 2 _ 1- -V- "3 v:^ 20 SPHERICAL TRIGONOMETRY. 2. -^— ' -^ , -I. 3. cos 67i° + i sin 67i°, cos 157i° + i sin 157^°, cos 24 7^° + i sin 247^'^ cos 337i° + i sin 337i°. 4. sin 4 ^ = 4 cos^ ^ sin ^ — 4 cos 6 sin^ ^. cos 4 ^ = cos* d — Q cos2 ^ sin2 d + sin* ^. Exercise XXVIII. 6. secx = l + ^%^ + ^V 2 ^ 24 ^ 720 6. xcotx=l-f-^-|^^- 3 45 945 7. sin 10° = 0.173648, cos 10° = 0. 984808. 8. tanl5° = 0.267958. SPHERICAL TRIGONOMETRY. Exercise XXIX. 1. 110°, 100°, 80° 2. 140°, 90°, 55°. 7. | 7t ft., 2 ;r ft., 2/;r ft. Exercise XXX. 3, (i.) Either a or 6 must be equal to 90°. (lii.) A = 90°, B = h. (ii.) A = 90° and B = b. (iv.) c = 90°, b = B = 90° Exercise XXXI. 2. I. The cosine of the middle part = the product of the cotangents of the adjacent parts. II. The cosine of the middle part = the product of the sines of oppo- site parts. ANSWERS. 21 Exercise XXXII. 24. A - 175° 57' 10'', B = 135° 42' 50", C = 135° 34' 7". 25. C= 104° 41' 39", a = 104° 53' 2", 6 = 133° 39' 48". 26. a = 90° ; b and B are indeterminate. 27. a = ^ = 60°, 6 = 90°, B = 90. 28. The triangle is impossible. 29. b = 130° 41' 42", c = 71° 27' 43", A = 112° 57' 2". 30. a = 26° 3' 61", A = 35°, B = 65° 46' 7". 31. Impossible. Exercise XXXIII. 1, cos ^ = cot a tan ^6, sin 1 5 = esc a sin I 6, cos ^ = cos a sec | &. 2, sin i J. = ^ sec ^ a. o • 1 . 1 180° . ^ . 1 180° 3, sm I A = sec 1 a cos , sm R = sm i a esc — » sm r = tan I a cot ^ n 4, Tetrahedron, 70° 31' 46"; octahedron, 109° 28' 14"; icosahedron, 138° 11' 36"; cube, 90°; dodecahedron, 116° 33' 44". 5, cot I A = Vcos a. Exercise XXXV. 1. (i.) tan m = tan 6 cos^, (ii.) tan m = tan c cos B, cos a = cos b sec m cos {c — m); cos b = cos c sec »w cos (a — m). Exercise XXXVI. 1. (i.) cot X = tan B esc a, (ii.) cot a; = tan C esc b, cos ^ = cos B CSC X sin (O — x) ; cos5 = cos C esc x sin (4 — x). Exercise XLI. 4. 2066.5 square miles. 22 SPHERICAL TRIGONOMETRY. Exercise XLII. 1. If X denotes the angle required, sin |x = cos 18° sec 9°, x — 148° 42'. 2. cos x—Q.0^ A cos B. 3. Let w = the inclination of the edge c to the plane of a and b. Then it is easily shown that V= abc sin I sin w. Now, conceive a sphere constructed having for centre the vertex of the trihedral angle whose edges are a, 6, c. The spherical triangle, whose vertices are the points where a, b, c meet the surface of this sphere, has for its sides I, m, n; and w is equal to the perpendicular arc from the side I to the opposite vertex. Let L, If, N denote the angles of this triangle. Then, by means of [39] and [48], we find that sin w = smmsin N=2 sin m sin i JV cos i N 2 / = -: — : Vsin s sin is — I) sin (s — m) sin (s — n), where s=- \{l + in-\- n) -, hence, ¥=2 abc Vsin s sin (s — I) sin (s — m) sin (s — n). 4. (i.) 9,976,500 square miles ; (ii.) 13,316,560 square miles. • 6. Let m = longitude of point where the ship crosses the equator, B = her course at the equator, d = distance sailed. Then tan m = sin I tan a, cos B = cos I sin a, cot d = cot I cos a. 6. Let k = arc of the parallel between the places, x = difference required ; then sin | fc = sin i d sec I. x — 90°( V2 — 1). 7. tan \{m — mf) — Vsec s sec (s — d) sin (s — Z) sin (s — T) ; where 2 s = Z + r + d, and 7m and m' are the longitudes of the places. 9. 44 min. past 12 o'clock. 10. 60°. 11. cos i = — tan d tan I ; time of sunrise = 12 — -- o'clock a.m. ; time t ^^ of sunset = —z o'clock p.m. ; cos a = sin d sec I. For longest day 15 at Boston : time of sunrise, 4 hrs. 26 min. 50 sec. a.m. ; time of sunset, 7 hrs. 33 min. 10 sec. p.m. Azimuth of sun at these times, 57° 25' 15'' ; length of day, 15 hrs. 6 min. 20 sec. ; for shortest day, times of sunrise and sunset are 7 hrs. 33 min. 10 sec. a.m. and 4 hrs. 26 min. 50 sec. p.m. ; azimuth of sun, 122° 34' 45"; length of day, 8 hrs. 53 min. 40 sec. 12. The problem is impossible when cot d l ; therefore this case is impossible. If Z = d, then cos < = 1, and t = 0° ; that is, the times both coincide with noon. The ex- planation of this result is, that for d = I the sun at noon is in the zenith, and south of the prime vertical at every other time. And if Z > d, the diurnal circle of the sun and the prime vertical of the place meet in two points which separate further and further as I increases. At the pole the prime vertical is indeterminate ; but near the pole, t = 90°, and the sun is always east at 6 a.m. 17. sin Z =: sin d esc /i. 18. 11° 50' 35". 19. The bearing of the wall, reckoned from the north point of the hori- zon, is given by the equation cot x = cosl tan d ; whence, for the given case, x = 75° 12' 38". 20. 55° 45' 6" N. 21. 63° 23' 41" N. or S. 22. (i.) cost — — tan I coip ; (ii.) t = z ; (iii.) the result is indeterminate. 23. cot a = cos Z tan (i. 28. sin d = sin e sin w, tan r = cos e tan w. 25. h = 65° 37' 20". 29. d = 32° 24' 12", r = 301° 48' 17". 26. h = 58° 25' 15", a = 152° 28'. 30. d = 20° 48' 12". 27. t = 45° 42', I = 67° 58' 54". 31. 3 hrs. 59 min. 27| sec. p.m. 32. cos I a = Vcos | (l + h + p)cosl{l + h — p) sec I sec h. FIVE -PLACE LOGAEITHMIC AND TRIGONOMETRIC TABLES ARRANGED BY G. A. WENTWORTH, A.M. G. A. HILL, A.M. Boston, U.S.A., and London PUBLISHED BY GINN & COMPANY 1897 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 Copyright, 1895, by G. A. Wkntworth and G. A. Hill. INTRODUCTION. 1. If the natural numbers are regarded as powers of ten, tlie exponents of the powers are the Common or Briggs Logarithms of the numbers. If A and B denote natural numbers, a and b their logarithms, then 10" = A, 10* = -5 ; or, written in logarithmic form, log^ = a, logB^b. . ' 2. The logarithm of a product is found by adding the logarithms of its factors. For, Ax B = 10« X 10» = 10« + ». Therefore, log{A x B) = a-\-b = \ogA-{- log B. 3. The logarithm of a quotient is found by subtracting the logarithm of the divisor from that of the dividend. Therefore, log — = a — 6 = log J. — log B. n 4. The logarithm of a power of a number is found by multiply- ing the logarithm of the number by the exponent of the power. For, ^« = (10«)'» = 10«». Therefore, log J." — an = n log A. 5. The logarithm of the root of a number is found by dividing the logarithm of the number by the index of the root. For, \C4 = -7To« = 10*. Therefore, log \(I = - = -^^— • 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 fractions. IV LOGARITHMS. For, 100 = 1, hence log 1 = 101 = 10, hence log 10 = 1 102 = 100, hence log 100 = 2 103 = 1000, hence log 1000 = 3 10-1 = 0. 1, hence log 0. 1 = — 1 ; 10-2 = 0.01, hence log 0.01 = — 2 ; 10-^ = 0.001, hence log 0.001 = — 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 t\iQ fractional part is dXwdt>j^ 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 + log 0.1 = 0.68124 - 1. (Page 1.) Again, 0.0007 = 7 X 0.0001. Therefore, log 0.0007 = log 7 + log 0.0001 = 0.84510 — 4. The logarithm 0.84510 — 4 is often written 4.84510. 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 0.78254 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 vnore than the number of zeros between the decimal point and the first significant figure of the given number. INTRODUCTION. V 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 correspond- ing 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 inte- gral part. If the characteristic is — 3, that is, 7.00000 — 10, the corresponding fraction has two zeros between the decimal point and the first significant figure. 11. The logarithms of numbers that can be derived one from another by 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 102) = log 4.6021 + log 10^ = 0.66296 + 2 = 2.66296. log 460210 = log (4.6021 X 10^) = log 4.6021 + log 10^ = 0.66296 + 5 = 6.66296. log 0.046021 = log (4.6021 -^ 102) = log 4.6021 - log 102 = 0.66296 - 2 = 8.66296 - 10. TABLE I. 12. In this table (pp. 1-19) the vertical columns headed N con- tain the numbers, and the other columns the logarithms. On page 1 both the characteristio and the mantissa are printed. On pages 2-19 the mantissa only is printed. The fractional part of a logarithm can be expressed only approx- imately, 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 4J, 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 6328751, it is written 63288. If it is 6328461 or 5328499, it is written in this table 53285. Again, if the mantissa is 5324981, it is written 63260 ; and if it is 4999967, it is written 60000. 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 by a zero, should be simply rejected, or whether the rejection should lead us to increase the preceding figure by one unit. Thus, the mantissa 1392^ 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 characteristic must be determined by § 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 145 = 2.16137, log 14500 = 4.16137. Page 14. log 716 = 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. Kequired 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 -f- (0.7 X 12) = 53449. Therefore, the required logarithm of 34237 is 4.53449. C^^ INTRODUCTION. Vll Eequired 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. Eequired 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-hundredths 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 Antilogarithm ; that is, the Number Corre- sponding 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 charac- teristic (§ 10). Find the number corresponding to the logarithm 0.92002. Page 16. The number for the mantissa 92002 is 8318. The characteristic is ; 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. The characteristic is 6 ; 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. The characteristic is — 3 ; 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 man- tissa 48762 lies, are found to be (page 6) 48756 and 48770. The corresponding numbers are 3073 and 3074. The smaller of these, 3073, contains the first four significant figures of the required number. 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 j'^j 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 6666 is f 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 man- tissas 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 denotes any number, then colog ^ = log — = log 1 — log ^ (§ 3) = — log 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 cologarifhin, it is customary to substitute for — log A its equivalent (10 - log ^) — 10. 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 A from 9 until we reach the last significant figure, which must be subtracted from 10. If log A 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 ^)- 20. So that, in this case, colog A is found by subtracting log A from 20, and then annexing — 20 to the remainder. Find the cologarithm of 4007. 10 -10 Page 8. log 4007 = 3.60282 colog 4007= 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 A. 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 log- arithm 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. LOGARITHMS. r Find the logarithms of Exercises. 1. 6170. 2. 0.617. 3. 2867. 4. 85.76. 5. 296.8. 6. 7004. 7. 0.8694. 8. 0.5908. 9. 73243. 10. 67.3208. 11. 18.5283. 12. 0.0042003. Find the cologarithms of : 13. 72433. 14. 802.376. 15. 15.7643. 16. 869.278. 17. 154000. 18. 70.0426. 19. 0.002403. 20. 0.000777. 21. 0.051828. Find the antilogarithms of : 22. 2.47246. 23. 7.89081. 24. '2.91221. 25. 1.26784. 26. 3.79029. 27. 6.18752. 28. 9.79029-10. 29. 7.62328-10. 30. 6.16465-10. Computation by Logarithms. 21. (1) Find the value of x, ii 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, ii x = 5250 -^ 23487. Page 10. log 5250 = 3.72010 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 Page 15. Page 9. Page 11. Page 17. Page 6. Page 4. Page 5. 899.1 X 0.00337 X 23435 log 7.56 = 0.87852 log 4667 = 3.66904 log 567 - 2.75358 colog 899.1 =7.04619-10 colog 0.00337 = 2.47237 colog 23435 = 5.63013-10 log a; = 2.44983 = log 281.73 .-. « =281.73. INTRODUCTION. XI (4) Find the cube of 376. Page 7. log 376 = 2.57519 Multiply by 3 (§ 4), 3 Page 10. log 3763 . = 7.72557 = log 53158600 .-. 3763 = 53158600. (5) Find the square of 0.003278. Page 6. log 0.003278 = 7.51561-10 2 Page 2. log 0.0032782 = 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 log V8322 = 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 divid- ing the logarithm by the index of the root, both the subtrahend and the number preceding the mantissa should be increased by 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. Pages. 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 .-. Vo.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 Vo. 076553 = 9.81400 - 10 = log 0.65163 .-. \/0.076553 = 0.65163. Exercises. Find by logarithms the value of : 45607 5.6123 2.567 31045' 0.01987* 0.05786 XU LOGARITHMS. 0.06547 4. 5. 7, 74.938 X 0.05938 4.657 X 0.03467 3.908 X 0.07189'. 0.0075389 X 0.0079 0.00907 X 009784' 312 X 7.18 X 31.82 519 X 8.27 X 6.132' 0.007X57.83X28.13 ^ ,00^ l io (> ^ 9.317 X 00.28 X 476.5 ' 5.55 X 0.0007632 X 0.87654 ^ Ol (^ i ^ 'T^^ 2.79X0.0009524X1.46785* ^ ^^ J ' ' / 0. 003457 X 43.387 X 99.2 X 0.00025 /~ ^ ^ /) ^ \ 0.005824 X 15.724 X 1.38 X 0.00089' ^ • / ^ ^ f 11 '/2' 23.815 X 29.36 X 0.007 X 0.62487 00072 X 9.236 X 5.924 X 3.0007 1 .<) i' 1 *• ■-4 / 3.1416 X 0.031416 X 0.0031416 '^'\7285 X 0.017285 X 0.0017285' 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 \ = 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 ^, = log a" =5.31443 log angle = 5.05429 = log 113316 .-. angle = 113316" = 31° 28' 36". INTRODUCTION. XIU 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 - arc angle = —-r^ — , 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. 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 equa- tion 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 in circular measure. C ^ C Since C = 2 irB, it follows that — = 2 tt, and ~- = it. Therefore, IC K If the arc = circumference, the angle = 2 tt. If the arc ^ semicircumference, the angle = tt. If the arc = quadrant, the angle = ^ tt. If the arc = radius, the angle =: 1. Therefore, tt = 180°, \ir = 90°, i tt = 60°, i tt = 45°, i tt = 30°, ■J TT = 22^°, and so on. Since 180° in common measure equals tt units in circular measure, 77" 1° in common measure = t^^: units in circular measure j 1 unit m circular measure = - — m common measure. TT 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 180° 1 unit in circular measure, and is equal to , or 57° 17' 45", TT very 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. XIV LOGARITHMS. But on pp. 30, 49, the characteristic changes one unit in valtie 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 8.24186 and 9.99993, and log tan or log cot lies between 8.24192 and 11.75808. 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' = 9.56631 — 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.. INTRODUCTIOK. XV 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 -^\ 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 q% 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. 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 16° 30'. The difference between 45334 and the given mantissa, 55359, is 25. The difference between 45334 and the next following mantissa, 45377, is 43, and If of 60" = 35". As the function is increasing, the 35" must be added to 16° 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 the 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". Exercises. Find ' 1. log sin 30° 8' 9". 9. log tan 25° 27' 47' 2. log sin 54° 54' 40". 10. log cos 56° 11' 57' 3. log cos 43° 32' 31". 11. log cot 62° 0' 4' 4. log cos 69° 25' 11". 12. log cos 75° 26' 58' 5. log tan 32° 9' 17". 13. log tan 33° 27' 13' 6. log tan 50° 2' 2". 14. log cot 81° 55' 24' 7. log cot 44° 33' 17". 15. log tan 89° 46' 35' 8. log cot 55° 9' 32". 16. log tan 1° 25' 56' XVI LOGARITHMS. i the angle A if 17. log sin ^= 9.70075. 25. log cos A = 9.40008. 18. log sin ^= 9.91289. 26. log cot A = 9.78815. 19. log cos ^= 9.86026. 27. log cos A = 9.34.301. 20. log cos ^= 9.54595. 28. log tan A = 10.52288. 21. log tan ^= 9.79840. 29. log cot A = 9 65349. 22. log tan^ = 10.07671. 30. log sin ^ = 8.39316. 23. log cot ^ = 10.00675. 31. log sin A = 8.06678. 24. log cot J. = 9.84266. 32. log tan A = 8.11148. 27. If log sec or log esc of an angle is desired, it may be found from the table by the formulas, sec A = 7 ; hence, log sec A = colog cos A. cos A CSC A = —. — 7 ; hence, log esc A == colog sin A. smA Page 31. log sec 8° 28' = colog cos 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 8.24186 and 9.99993 in the table; or, if a given log tan or log cot does not lie between the limits 8.24192 and 11.75808 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', pr 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° 5T 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. 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, minutes, and multiples of 10" are given or required. INTRODUCTION. XVll When the angle is between 0° and 2°, or 88° and 90°, and a greater degree of accuracy is desired than that given by the table, interpolation may be employed ; 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° 2M7'' = 6.90829 - 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 — 10. 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 colog cot = 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 formulas established in Trigonometry, that, between 90° and 180°, between 180° and 270°, log sin X = log sin (180° — x), log sin x = log sin (x ~ 180°)„, log cos X = log cos (180° — ic)„, log cos x = log cos (x — 180°)„, log tan X = log tan (180° — x)^, log tan x = log tan (x — 180°), log cot X = log cot (180° — x)^ ; log cot x = log cot (x — 180°) ; between 270° and 360°, log sin X = log sin (360° — x)^^ log cos £c = log cos (360° — ic), log tan X = log tan (360° — x)„, log cot ic = log cot (360° — x)„. XVIU LOGARITHMS. 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'' = 9.82766 - 10. log cos 137° 45' 22" = log„ cos 42° 14' 38" = 9.86940„ - 10. log tan 137° 46' 22" = log„ tan 42° 14' 38" = 9.95816n - 10. log cot 137° 45' 22" = log„ cot 42° 14' 38" = 0.04186„. log sin 209° 32' 60" = log„ sin 29° 32' 60" == 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 quadrant, and so related that if x denote the acute angle, the other three angles are 180° — ic, 180° -|- a;, and 360° — a;. If besides the given logaritlim 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 10° 3G' 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" m Quadrant II. and 343° 23' 4.3" 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. Q/juOX 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 i)age 50. The values of S and T are printed with the characteristic 10 too large, and in using them — 10 must always be annexed. Find log sin 0° 58' 17". 0° 58' 17" = 3497" log 3497 = 3.64370 S = 4.68565 - 10 log sin 0° 58' 17" = 8.22926 - 10 Find log cos 88° 26' 41.2". 90° - 88° 26' 41.2" = 1° .33' 18.8" = 5598.8" log 5598.8 = 3.74809 S = 4.68562 - 10 log cos 88° 20' 41.2" = 8.43361 - 10 INTRODUCTION. XIX Find log tan 0° 52' 47.5". 0° 62' 47.5" = 3167.5" log 3167.5 = 3.50072 T = 4.68561 - 10 log tan 0° 52' 47.5" = 8. 18633 - 10 Find log tan 89** 54' 37.362". 90° — 89° 54' 37.362" = 0° 5' 22.638" = 322.638" log 322.638 = 2.50871 T = 4.68558 — 10 log cot 89° 54' 37.362" = 7.19420 — 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. colog cot = 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. colog tan = 8.44593 — 10 T = 4.68569 - 10 Subtract, 3.76024 = log 5767.6 5757.6" = 1° 35' 67.6", and 90° - 1° 35' 57.6" = 88° 24' 2.4". Therefore, the angle reqmred 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. 82. Table VI. (pp. 52-69) contains the natural sines, cosines, tangents, and cotangents of angles from 0° to 90°, at inter- vals of 1'. 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 TABLE Vn. 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'. XX LOGARITHMS. If the bearing does not exceed 45° it is found in the left-h^md column, and the designations of the columns under "Distance" are taken from the toj) of the page; but if the bearing exceeds 45°, it is found in the right-hsind column, and the designations of the columns under ^'Distance" are taken from the botto7?i of the page. The method of using the table will be made plain by the follow- ing examples : — (1) Let it be required to find the latitude and departure of the course N. 35° 15' E. 6 chains. On p. 75, 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 = 3.463 chains. (2) Let it be required to find the latitude and departure of the course S. 87° W. 2 chains. As the hearing exceeds 45°, we look in the right-hand column of p. 70, 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 = 0.105 chains, and departure = 1.997 chains. Hence, latitude or southings 0.105 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. LAxrruDE. Departure. 20 = 2 X 10 7 0.3 =3^10 0.06 = 6 ^ 100 1.925 X 10 = 19.25 6.737 2.887 -MO =0.289 5.775 -r 100 = 0.058 0.543 X 10 = 5.43 1.90 0.814^10 =0.081 1.628 -i- 100 = 0.016 27.36 26.334 7.427 Hence, latitude = 26.334 chains, and departure = 7.427 chains. #• TABLE I. THE COMMON OE BRIGGS LOGAEITHMS OF THE NATUEAL NUMBEES From 1 to 10000. 1-100 N log N log N log N log N log 1 0.00 000 21 1.32 222 41 1. 61 278 61 1.78 533 81 1.90 849 2 0.30103 22 1.34 242 42 1.62 325 62 1. 79 239 82 1. 91 381 3 0.47 712 23 1. 36 173 43 1.63 347 63 1.79 934 83 1.91908 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. 41 497 46 1.66 276 66 1. 81 954 86 1.93 450 7 0.84 510 27 1. 43 136 47 1.67 210 67 1. 82 607 87 1.93 952 8 0.90 309 28 1. 44 716 48 1.68124 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 1.00 000 30 1.47 712 50 1.69 897 70 1.84 510 90 1. 95 424 11 1.04 139 31 1. 49 136 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. 51 851 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 1.17 609 35 1. 54 407 55 1.74 036 75 1. 87 506 95 1. 97 772 16 1.20 412 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 1.30103 40 1.60 206. 60 1. 77 815 80 1. 90 309 100 2.00 000 N log N log N log N log N log 1-100 100-150 N O 1 2 3 4 5 6 7 8 9 lOO 00 000 00 043 00 087 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 00 689 00 732 00 775 00 817 102 00 860 00 903 00 945 00 988 01030 01072 01115 01157 01199 01242 103 01284 01326 01368 01410 01452 01494 01536 01578 01620 01662 104 01703 01745 01787 01828 01870 01912 01953 01995 02 036 02 078 105 02 119 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 03 583 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 llO 04139 04 179 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 (H766 04 805 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 06 781 117 06 819 06 856 06 893 06 930 06 967 07 004 07 041 07 078 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 07 809 07 846 07 882 120 07 918 07 954 07 990 08 027 08 063 08 099 08135 08171 08 207 08 243 121 08 279 08 314 08 350 08 386 08 422 08 458 08 493 08 529 08 565 08 600 122 08 636 08 672 08 707 08 743 08 778 08 814 08 849 08 884 08 920 08 955 123 08 991 09 026 09 061 09 096 09132 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 09 968 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 10 585 10 619 10 653 10 687 128 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 •11 394 11428 11461 11494 11528 11561 11594 11628 11661 11694 131 11727 11760 11793 11826 11860 11893 11926 11959 11 992 12 024 132 12 057 12 090 12 123 12156 12189 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 12 678 134 12 710 12 743 12 775 12 808 12 840 12S72 12 905 12 937 12 969 13 001 135 13 033 13 066 13 098 13 130 13 162 13194 13 226 13 258 13 290 13 322 136 13 354 13 386 13 418 13 450 13 481 13 5J3 13 545 13 577 13 609 13 640 137 13 672 13 704 13 735 13 767 13 799 13 830 13 862 13 893 13 925 13 956 138 13 988 14 019 14 051 14 082 14114 14145 14176 14 208 14 239 14 270 139 14 301 14 333 14 364 14 395 14 426 14 457 14 489 14 520 14 551 14 582 140 14 613 14 644 14 675 14 706 14 737 14 768 14 799 14 829 14 860 14 891 141 14 922 14 953 14 983 15 014 15 045 15 076 15 106 15 137 15 168 15 198 142 15 229 15 259 15 290 15 320 15 351 15 381 15 412 15 442 15 473 15 503 143 15 534 15 564 15 594 15 625 15 655 15 685 15 715 15 746 15 776 15 806 144 15 836 15 866 15 897 15 927 15 957 15 987 16 017 16 047 16 077 16107 145 16137 16167 16 197 16 227 16 256 16 286 16 316 16 346 16 376 16 406 146 16 435 16 465 16 495 16 524 16 554 16 584 16 613 16 643 16 673 16 702 147 16 732 16 761 16 791 16 820 16 850 16 879 16 909 16 938 16 967 16 997 148 17 026 17 056 17 085 17114 17143 17173 17 202 17 231 17 260 17 289 149 17 319 17 348 17 377 17 406 17 435 17 464 17 493 17 522 17 551 17 580 150 17 609 17 638 17 667 17 696 17 725 17 754 17 782 17 811 17 840 17 869 IS^ O 1 2 3 4 5 6 7 8 . 9 100-160 150-200, 3 N O 1 2 -3 4" 5 6 7 8 9 150 17 609 17 638 17 667 17 696 17 725 17 754 17 782 17 811 17 840 17 869 151 17 898 17 926 17 955 17 984 18 013 18 041 18 070 18 099 18127 18156 152 18184 18 213 18 241 18 270 18 298 18 327 18 355 18 384 18 412 18 441 153 18 469 18 498 18 526 18 554 18 583 18 611 18 639 18 667 18 696 18 724 154 18 752 18 780 18 808 18 837 18 865 18 893 18 921 18 949 18 977 19 005 155 19 033 19 061 19 089 19117 19145 19173 19 201 19 229 19 257 19 285 156 19 312 19 340 19 368 19 396 19 424 19 451 19 479 19 507 19 535 19 562 157 19 590 19 618 19 645 19 673 19 700 19 728 19 756 19 783 19 811 19 838 158 19 866 19 893 19 921 19 948 19 976 20 003 20 030 20 058 20 085 20112 159 20140 20167 20194 20 222 20 249 20 276 20303 20 330 20 358 20 385 160 20 412 20 439 20 466 20 493 20 520 20 548 20 575 20 602 20 629 20 656 161 20 683 20 710 20 737 20 763 20 790 20 817 20 844 20 871 20 898 20 925 162 20 952 20 978 21005 21032 21059 21085 21112 21139 21165 21192 163 21219 21245 21272 21299 21325 21352 21378 21405 21431 21458 164 21484 21511 21537 21564 21590 21617 21643 21669 21696 21722 165 21748 21775 21 801 21827 21854 21880 21906 21932 21958 21985 166 22 011 22 037 22 063 22 089 22 115 22141 22 167 22 194 22 220 22 246 167 22 272 22 298 22 324 22 350 22 376 22 401 22 427 22 453 22 479 22 505 168 22 531 22 557 22 583 22 608 22 634 22 660 22 686 22 712 22 737 22 763 169 22 789 22 814 22 840 22 866 22 891 22 917 22 943 22 968 22 994 23 019 170 23 045 23 070 23 096 23 121 23147 23172 23198 23 223 23 249 23 274 171 23 300 23 325 23 350 23 376 23 401 23 426 23 452 23 477 23 502 23 528 172 23 553 23 578 23 603 23 629 23 654 23 679 23 704 23 729 23 754 23 779 173 23 805 23 830 23 855 23 880 23 905 23 930 23 955 23 980 24 005 24 030 174 24 055 24 080 24105 24130 24155 24180 24 204 24 229 24 254 24 279 175 24 304 24 329 24 353 24 378 24 403 24 428 24 452 24 477 24 502 24 527 176 24 551 24 576 24 601 24 625 24 650 24 674 24 699 24 724 24 748 24 773 177 24 797 24 822 24 846 24 871 24 895 24 920 24 944 24 969 24 993 25 018 178 25 042 25 066 25 091 25115 25139 25 164 25 188 25 212 25 237 25 261 179 25 285 25 310 25 334 25 358 25 382 25 406 25 431 25 455 25 479 25 503 * 180 25 527 25 551 25 575 25 600 25 624 25 648 25 672 25 696 25 720 25 744 181 25 768 25 792 25 816 25 840 25 864 25 888 25 912 25 935 25 959 25 983 182 26 007 26 031 26 055 26 079 26102 26126 26150 26174 26198 26 221 183 26 245 26 269 26 293 26 316 26 340 26 364 26 387 26 411 26 435 26 458 184 26 482 26 505 26 529 26 553 26 576 26 600 26 623 26 647 26 670 26 694 185 26 717 26 741 26 764 26 788 26 811 26 834 26 858 26 881 26 905 26 928 186 26 951 26 975 26 998 27 021 27 0+5 27 068 27 091 27 114 27138 27 161 187 27184 27 207 27 231 27 254 27 277 27 300 27 323 27 346 27 370 27 393 188 27 416 27 439 27 462 27 485 27 508 27 531 27 554 27 577 27 600 27 623 189 27 646 27 669 27 692 27 715 27 738 27 761 27 784 27 807 27 830 27 852 190 27 875 27 898 27 921 27 944 27 967 27 989 28 012 28 035 28 058 28 081 191 28 103 28126 28149 28171 28194 28 217 28 240 28 262 28 285 28 307 192 28 330 28 353 28 375 28 398 28 421 28 443 28 466 28 488 28 511 28 533 193 28 556 28 578 28 601 28 623 28 646 28 668 28 691 28 713 28 735 28 758 194 28 780 28 803 28 825 28 847 28 870 28 892 28 914 28 937 28 959 28 981 195 29 003 29 026 29 048 29 070 29 092 29115 29137 29159 29181 29 203 196 29 226 29 248 29 270 29 292 29 314 29 336 29 358 29 380 29 403 29 425 197 29 447 29 469 29 491 29 513 29 535 29 557 29 579 29 601 29 623 29 645 198 29 667 29 688 29 710 29 732 29 754 29 776 29 798 29 820 29 842 29 863 199 29 885 29 907 29 929 29 951 29 973 29 994 30 016 30 038 30 060 30 081 200 30103 30125 30146 30168 30190 30 211 30 233 30 255 30 276 30 298 N O 1 2 3 4 5 6 7 8 9 160 - 200 200-250 N O 1 2 3 4 5 6 7 8 9 200 30103 30125 30146 30168 30190 30 211 30 233 30 255 30 276 30 298 201 30 320 30 341 30 363 30 384 30 406 30 428 30 449 30 471 30 492 30 514 202 30 535 30 557 30 578 30 600 30 621 30 643 30 664 30 685 30 707 30 728 203 30 750 30 771 30 792 30 814 30 835 30 856 30 878 30 899 30 920 30 942 204 30 963 30 984 31006 31027 31048 31069 31091 31112 31133 31154 205 31175 31197 31218 31239 31260 31281 31302 31323 31345 31366 206 31387 31408 31429 31450 31471 31492 31513 31534 31555 31576 207 31597 31618 31639 31660 31681 31702 31 723 31 744 31 765 31 785 208 31806 31827 31848 31869 31890 31911 31931 31952 31973 31994 209 32 015 32 035 32 056 32 077 32 098 32118 32139 32160 32181 32 201 210 32 222 32 243 32 263 32 284 32 305 32 325 32 346 32 366 32 387 32 408 211 32 428 32 449 32 469 32 490 32 510 32 531 32 552 32 572 32 593 32 613 212 32 634 32 654 32 675 32 695 32 715 32 736 32 756 32 777 32 797 32 818 213 32 838 32 858 32 879 32 899 32 919 32 940 32 960 32 980 33 001 33 021 214 33 041 33 062 33 082 33102 33 122 33 143 33163 33183 33 203 33 224 215 33 244 33 264 33 284 33 304 33 325 33 345 33 365 33 385 33 405 33 425 216 33 445 33 465 33 486 33 506 33 526 33 546 33 566 33 586 33 606 33 626 217 33 646 33 666 33 686 33 706 33 726 33 746 33 766 33 786 33 806 33 826 218 33 846 33 866 33 885 33 905 33 925 33 945 33 965 33 985 34 005 34 025 219 34 044 34 064 34 084 34104 34 124 34143 34163 34183 34 203 34 223 220 34 242 34 262 34 282 34 301 34 321 34 341 34 361 34 380 34 400 34 420 221 34 439 34 459 34 479 34 498 34 518 34 537 34 557 34 577 34 596 34 616 222 34 635 34 655 34 674 34 694 34 713 34 733 34 753 34 772 34 792 34 811 223 34 830 34 850 34 869 34 889 34 908 34 928 34 947 34 967 34 986 35 005 224 35 025 35 044 35 064 35 083 35102 35 122 35141 35 160 35 180 35 199 225 35 218 35 238 35 257 35 276 35 295 35 315 35 334 35 353 35 372 35 392 226 35 411 35 430 35 449 35 468 35 488 35 507 35 526 35 545 35 564 35 583 227 35 603 35 622 35 641 35 660 35 679 35 698 35 717 35 736 35 755 35 774 228 35 793 35 813 35 832 35 851 35 870 35 889 35 908 35 927 35 946 35 96i 229 35 984 36 003 36 021 36 040 36 059 36 078 36 097 36116 36135 36154 230 36173 36192 36 211 36 229 36 248 36 267 36 286 36 305 36 324 36 342 231 36 361 36 380 36 399 36 418 36 436 36 455 36 474 36 493 36 511 36 530 232 36 549 36 568 36 586 36 605 36 624 36 642 36 661 36 680 36 698 36 717 233 36 736 36 754 36 773 36 791 36 810 36 829 36 847 36 866 36 884 36 903 234 36 922 36 940 36 959 36 977 36 996 37 014 37 033 37J251 37 070 37 088 37 218 37 236 37 254 37 273 235 37107 37125 37144 37162 37181 37199 236 37 291 37 310 37 328 37 346 37 365 37 383 37 401 37 420 37 438 37 457 237 37 475 37 493 37 511 37 530 37 548 37 566 37 585 37 603 37 621 37 639 | 238 37 658 37 676 37 694 37 712 37 731 37 749 37 767 37 785 37 803 37 822 239 37 840 37 858 37 876 37 894 37 912 37 931 37 949 37 967 37 985 38 003 240 38 021 38 039 38 057 38075 38 093 38112 38130 38148 38166 38184 241 38 202 38 220 38 238 38 256 38 274 38 292 38 310 38 328 38 346 38 364 242 38 382 38 399 38 417 38 435 38 453 38 471 38 489 38 507 38 525 38 543 243 38 561 38 578 38 596 38 614 38 632 38 650 38 668 38 686 38 703 38 721 244 38 739 38 757 38 775 38 792 38 810 38 828 38 846 38 863 38 881 38 899 245 38 917 38 934 38 952 38 970 38 987 39 005 39 023 39 041 39 058 39 076 246 39 094 39111 39129 39146 39164 39182 39199 39 217 39 235 39 252 247 39 270 39 287 39 305 39 322 39 340 39 358 39 375 39 393 39 410 39 428 248 39 445 39 463 39 480 39 498 39 515 39 533 39 550 39 568 39 585 39 602 .249 39 620 39 637 39 655 39 672 39 690 39 707 39 724 39 742 39 759 39 777 250 39 794 39 811 39 829 39 846 39 863 39 881 39 898 39 915 39 933 39 950 N O 1 2 3 - ■ 4 5 6 7 8 9 200-260 260-300 N 12 3 4 5 6 7 8 9 250 39 794 39 811 39 829 39 846 39 863 39 881 39 898 39 915 39 933 39 950 251 39 967 39 985 40 002 40 019 40 037 40 054 40 071 40088 40106 40123 252 40140 40 157 40 175 40192 40 209 40 226 40 243 40 261 40 278 40 295 253 40 312 40 329 40 346 40 364 40 381 40 398 40 415 40 432 40449 40466 254 40 483 40 500 40 518 40 535 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283 54 295 54 307 54 320 54 332 54 345 54 357 54 370 54 382 54 394 350 54 407 54 419 54 432 54 444 54 456 54 469 54 481 54 494 54 506 54 518 N O 1 2 3 4 5 6 7 8 9 300-360 35€>-400 m ; 7 N O 1 2 3 4 -T 6 7 8 9 350 54 407 54 419 54 432 54 444 54 456 54469 54 481 54 494 54 506 54 518 351 54 531 54 543 54 555 54 568 54 580 54 593 54 605 54 617 54 630 54 642 352 54 654 54 667 54 679 54 691 54 704 54 716 54 728 54 741 54 753 54 765 353 54 777 54 790 54 802 54 814 54 827 54 839 54 851 54 864 54 876 54 888 354 54 900 54 913 54 925 54 937 54 949 54 962 54 974 54 986 54 998 55 011 355 55 023 55 035 55 047 55 060 55 072 55 084 55 096 55 108 55 121 55133 356 55 145 55 157 55 169 55 182 55 194 55 206 55 218 55 230 55 242 55 255 357 55 267 55 279 55 291 55 303 55 315 55 328 55 340 55 352 55 364 55 376 358 55 388 55 400 55 413 55 425 55 437 55 449 55 461 55 473 55 485 55 497 359 55 509 55 522 55 534 55 546 55 558 55 570 55 582 55 594 55 606 55 618 360 55 630 55 642 55 654 55 666 55 678 55 691 55 703 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73 894 73 902 73 910 73 918 73 926 73 933 73 941 73 949 549 73 957 73 965 73 973 73 981 73 989 73 997 74 005 74 013 74 020 74 028 550 74 036 74 044 74 052 74 060 74 068 74 076 74 084 74 092 74 099 74107 N O 1 2 3 4 5 6 7 8 9 500-560 560 -^PO 11 N O 74 036 1 2 3 4 5 6 7 8 74 099 9 550 74 044 74 052 74 060 74 068 74 07^ 74 084 74 092 74107 551 74115 74 123 74 131 74 139 74 147 74 155 74162 74 170 74178 74186 552 74194 74 202 74 210 74 218 74 225 74 233 74 241 74 249 74 257 74 265 553 74 273 74 280 74 288 74 296 74 304 74 312 74 320 74 327 74 335 74 343 554 74 351 74 359 74 367 74 374 74 382 74 390 74 398 74 406 74 414 74 421 555 74 429 74 437 74 445 74 453 74 461 74 468 74 476 74 484 74 492 74^00 556 74 507 74 515 74 523 74 531 74 539 74 547 74 554 74 562 74 570 74 578 557 74 586 74 593 74 601 74 609 74 617 74 624 74 632 74 640 74 648 74 656 558 74 663 74 671 74 679 74 687 74 695 74 702 74 710 74 718 74 726 74 733 559 74 741 74 749 74 757 74 764 74 772 74 780 74 788 74 796 74 803 74 811 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790 88 795 88 801 88 807 88 812 773 88 818 88 824 88 829 88 835 88 840 88 846 88 852 88 857 88 863 88 868 774 88 874 88 880 88 885 88 891 88 897 88 902 88 908 88 913 88 919 88 925 775 88 930 88 936 88 941 88 947 88 953 88 958 88 964 88 969 88 975 88 981 776 88 986 88 992 88 997 89 003 89 009 89 014 89 020 89 025 89 031 89 037 777 89 042 89 048 89 053 89 059 89 064 89 070 89 076 89 081 89 087 89 092 778 89 098 89104 89109 89115 89120 89126 89131 89137 89143 89148 779 89154 89159 89165 89170 89176 89182 89187 89193 89198 89 204 780 89 209 89 215 89 221 89 226 89 232 89 237 89 243 89 248 89 254 89 260 781 89 265 89 271 89 276 89 282 89 287 89 293 89 298 89 304 89 310 89 315 782 89 321 89 326 89 332 89 337 89 343 89 348 89 354 89 360 89 365 89 371 783 89 376 89 382 89 387 89 393 89 398 89 404 89 409 89 415 89 421 89 426 784 89 432 89 437 89 443 89 448 89 454 89 459 89 465 89 470 89 476 89 481 785 89 487 89 492 89 498 89 504 89 509 89 515 89 520 89 526 89 531 89 537 786 89 542 89 548 89 553 89 559 89 564 89 570 89 575 89 581 89 586 89 592 787 89 597 89 603 89 609 89 614 89 620 89625 89 631 89 636 89 642 89 647 788 89 653 89 658 89 664 89 669 89 675 89 680 89 686 89 691 89 697 89 702 789 89 708 89 713 89 719 89 724 89 730 89 735 89 741 89 746 89 752 89 757 790 89 763 89 768 89 774 89 779 89 785 89 790 89 796 89 801 89 807 89 812 791 89 818 89 823 89 829 89 834 89 840 89 845 89 851 89 856 89 862 89 867 792 89 873 89 878 89 883 89 889 89 894 89 900 89 905 89 911 89 916 89 922 793 89 927 89 933 89 938 89 944 89 949 89 955 89 960 89 966 89 971 89 977 794 89 982 89 988 89 993 89 998 90 004 90 009 90 015 90 020 90 026 90 031 795 90 037 90 042 90 048 90 053 90 059 90 064 90 069 90 075 90 080 90 086 796 90 091 90 097 90102 90108 90113 90119 90124 90129 90135 90140 797 90146 90 151 90157 90162 90168 90173 90179 90184 90189 90195 798 90 200 90 206 90 211 90 217 90 222 90 227 90 233 90 238 90 244 90 249 799 90 255 90 260 90 266 90 271 90 276 90 282 90 287 90 293 90 298 90 304 800 90 309 90 314 90 320 90 325 90 331 90 336 90 342 90 347 90 352 90 358 N O 1 2 3 4 5 6 7 8 9 760-800 16 800-860 N O 1 2 90 320 3 4 5 6 90 342 7 8 9 90 358 800 90 309 90 314 90 325 90 331 90 336 90 347 90 352 801 90 363 90 369 90 374 90 380 90 385 90 390 90 396 90 401 90 407 90 412 802 90 417 90 423 90 428 90 434 90 439 90 445 90 450 90 455 90 461 90 466 803 90 472 90 477 90 482 90 488 90 493 90 499 90 504 90 509 90 515 90 520 ■ 804 90 526 90 531 90 536 90 542 90 547 90 553 90 558 90 563 90 569 90 574 805 90 580 90 585 90 590 90 596 90 601 90 607 90 612 90 617 90 623 90 628 806 90 634 90 639 90 644 90 650 90 655 90 660 90 666 90 671 90 677 90 682 807 90 687 90 693 90 698 90 703 90 709 90 714 90 720 90 725 90 730 90 736 808 90 741 90 747 90 752 90 757 90 763 90 768 90 773 90 779 90 784 90 789 809 90 79i 90 800 90 806 90 811 90 816 90 822 90 827 90 832 90 838 90 843 810 90 849 90 854 90 859 90 865 90 870 90 875 90 881 90 886 90 891 90 897 811 90 902 90 907 90 913 90 918 90 924 90 929 90 934 90 940 90 945 90 950 812 90 956 90 961 90 966 90 972 90 977 90 982 90 988 90 993 90 998 91004 813 91009 91014 91020 91025 91030 91036 91041 91046 91 052 91057 814 91062 91068 91 073 91078 91084 91089 91094 91100 91105 91110 815 91116 91121 91126 91132 91137 91142 91148 91153 91158 91164 816 91169 91174 91180 91185 91190 91196 91201 91206 91212 91217 817 91222 91228 91233 91238 91243 91249 91254 91259 91265 91270 818 91275 91281 91286 91291 91297 91302 91307 91312 91318 91323 819 91328 91334 91339 91344 91350 91355 91360 91365 91371 91376 820 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 821 91434 91440 91445 91450 91455 91461 91466 91471 91477 91482 822 91487 91492 91498 91503 91508 91514 91519 91524 91529 91535 | 823 91540 91545 91551 91556 91561 91566 91572 91577 91582 91587 824 91593 91598 91603 91609 91614 91619 91624 91630 91635 91640 825 91645 91651 91656 91661 91666 91672 91677 91682 91687 91693 826 91698 91703 91709 91714 91719 91724 91730 91735 91740 91745 827 91751 91756 91761 91766 91772 91777 91782 91787 91793 91798 828 91803 91808 91814 91819 91824 91829 91834 91840 91845 91850 829 91855 91861 91866 91871 91876 91882 91887 91892 91897 91903 830 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 | 831 91960 91965 91971 91976 91981 91986 91991 91997 92 002 92 007 832 92 012 92 018 92 023 92 028 92 033 92 038 92 044 92 049 92 054 92 059 833 92 065 92 070 92 075 92 080 92 085 92 091 92 096 92 101 92 106 92111 834 92 117 92 122 92127 92132 92137 92143 92 148 92 153 92 158 92163 835 92169 92 174 92179 92 184 92 189 92 195 92 '200 92 205 92 210 92 215 836 92 221 92 226 92 231 92 236 92 241 92 247 92 252 92 257 92 262 92 267 837 92 273 92 278 92 283 92 288 92 293 92 298 92 304 92 309 92 314 92 319 838 92 324 92 330 92 335 92 340 92 345 92 350 92 355 92 361 92 366 92 371 839 92 376 92 381 92 387 92 392 92 397 92 402 92 407 92 412 92 418 92 423 840 92 428 92 433 92 438 92 443 92 449 92 454 92 459 92 464 92 469 92 474 841 92 480 92 485 92 490 92 495 92 500 92 505 92 511 92 516 92 521 92 526 842 92 531 92 536 92 542 92 547 92 552 92 557 92 562 92 567 92 572 92 578 843 92 583 92 588 92 593 92 598 92 603 92 609 92 614 92 619 92 624 92 629 844 92 634 92 639 92 645 92 650 92 655 92 660 92 665 92 670 92 675 92 681 845 92 686 92 691 92 696 92 701 92 706 92 711 92 716 92 722 92 727 92 732 846 92 737 92 742 92 747 92 752 92 758 92 763 92 768 92 773 92 778 92 783 847 92 788 92 793 92 799 92 804 92 809 92 814 92 819 92 824 92 829 92 834 848 92 840 92 845 92 850 92 855 92 860 92 865 92 870 92 875 92 881 92 886 849 92 891 92 896 92 901 92 906 92 911 92 916 92 921 92 927 92 932 92 937 850 92 942 92 947 92952 92 957 92 962 92 967 92 973 92 978 7 92 983 92 988 N O 1 2 3 4 5 6 8 9 800-860 860-900 17 N O 1 2 92 952 3 4 5 6 7 8 9 850 92 942 92 947 92 957 92 962 92 967 92 973 92 978 92 983 92 988 851 92 993 92 998 93 003 93 008 93 013 93 018 93 024 93 029 93 034 93 039 852 93 044 93 049 93 054 93 059 93 064 93 069 93 075 93 080 93 085 93 090 853 93 095 93 100 93 105 93 110 93 115 93 120 93 125 93 131 93 136 93 141 854 93 146 93 151 93 156 93 161 93 166 93 171 93 176 93 181 93 186 93 192 855 93 197 93 202 93 207 93 212 93 217 93 222 93 227 93 232 93 237 93 242 856 93 247 93 252 93 258 93 263 93 268 93 273 93 278 93 283 93 288 93 293 857 93 298 93 303 93 308 93 313 93 318 93 323 93 328 93 334 93 339 93 344 858 93 349 93 354 93 359 93 364 93 369 93 374 93 379 93 384 93 389 93 394 859 93 399 93 404 93 409 93 414 93 420 93 425 93 430 93 435 93 440 93 445 860 93 450 93 455 93 460 93 465 93 470 93 475 93 480 93 485 93 490 93 495 861 93 500 93 505 93 510 93 515 93 520 93 526 93 531 93 536 93 541 93 546 862 93 551 93 556 93 561 93 566 93 571 93 576 93 581 93 586 93 591 93 596 863 93 601 93 606 93 611 93 616 93 621 93 626 93 631 93 636 93 641 93 646 864 93 651 93 656 93 661 93 666 93 671 93 676 93 682 93 687 93 692 93 697 865 93 702 93 707 93 712 93 717 93 722 93 727 93 732 93 737 93 742 93 747 866 93 752 93 757 93 762 93 767 93 772 93 777 93 782 93 787 93 792 93 797 867 93 802 93 807 93 812 93 817 93 822 93 827 93 832 93 837 93 842 93 847 868 93 852 93 857 93 862 93 867 93 872 93 877 93 882 93 887 93 892 93 897 869 93 902 93 907 93 912 93 917 93 922 93 927 93 932 93 937 93 942 93 947 870 93 952 93 957 93 962 93 967 93 972 93 977 93 982 93 987 93 992 93 997 871 94 002 94 007 94 012 94 017 94 022 94 027 94 032 94 037 94 042 94 047 872 94 052 94 057 94 062 94 067 94 072 94 077 94 082 94 086 94 091 94 096 873 94 101 94 106 94 111 94 116 94121 94 126 94131 94 136 94 141 94 146 874 94 151 94156 94 161 94 166 94 171 94176 94181 94 186 94 191 94196 875 94 201 94 206 94 211 94 216 94 221 94 226 94 231 94 236 94 240 94 245 876 94 250 94 255 94 260 94 265 94 270 94 275 94 280 94 285 94 290 94 295 877 94 300 94 305 94 310 94 315 94 320 94 325 94 330 94 335 94 340 94 345 | 878 94 349 94 354 94 359 94 364 94 369 94 374 94 379 94 384 94 389 94 394 879 94 399 94 404 94 409 94 414 94 419 94 424 94 429 94 433 94 438 94 443 880 94 448 94 453 94 458 94 463 94 468 94 473 94 478 94 483 94 488 94 493 881 94 498 94 503 94 507 94 512 94 517 94 522 94 527 94 532 94 537 94 542 882 94 547 94 552 94 557 94 562 94 567 94 571 94 576 94 581 94 586 94 591 883 94 596 94 601 94 606 94 611 94 616 94 621 94 626 94 630 94 635 94 640 884 94 645 94 650 94 655 94 660 94 665 94 670 94 675 94 680 94 685 94 689 885 94 694 94 699 94 704 94 709 94 714 94 719 94 724 94 729 94 734 94 738 886 94 743 94 748 94 753 94 758 94 763 94 768 94 773 94 778 94 783 94 787 887 94 792 94 797 94 802 94 807 94 812 94 817 94 822 94 827 94 832 94 836 888 94 841 94 846 94 851 94 856 94 861 94 866 94 871 94 876 94 880 94 885 889 94 890 94 895 94 900 94 905 94 910 94 915 94 919 94 924 94 929 94 934 890 94 939 94 944 94 949 94 954 94 959 94 963 94 968 . 94 973 94 978 94 983 891 94 988 94 993 94 998 95 002 95 007 95 012 95017 95 022 95 027 95 032 892 95 036 95 041 95 046 95 051 95 056 95 061 95 066 95 071 95 075 95 080 893 95 085 95 090 95 095 95 100 95 105 95 109 95 114 95 119 95 124 95 129 894 95 134 95 139 95 143 95 148 95 153 95 158 95 163 95 168 95 173 95 177 895 95 182 95 187 95 192 95 197 95 202 95 207 95 211 95 216 95 221 95 226 896 95 231 95 236 95 240 95 245 95 250 95 255 95 260 95 265 95 270 95 274 897 95 279 95 284 95 289 95 294 95 299 95 303 95 308 95 313 95 318 95 323 898 95 328 95 332 95 337 95 342 95 347 95 352 95 357 95 361 95 366 95 371 899 95 376 95 381 95 386 95 390 95 395 95 400 95 405 95 410 95 41i 95 419 900 95 424 95 429 95 434 95 439 95 444 95 448 95 453 95 458 95 463 95 468 N O 1 2 3 4 5 6 7 8 9 860-900 18 900-960 :n- 1 2 3 4 5 6 7 8 9 900 95 424 95 429 95 434 95 439 95 444 95 448 95 453 95 458 95 463 95 468 901 95 472 95 477 95 482 95 487 95 492 95 497 95 501 95 506 95 511 95 516 902 95 521 95 525 95 530 95 535 95 540 95 545 95 550 95 554 95 559 95 564 903 95 569 95 574 95 578 95 583 95 588 95 593 95 598 95 602 95 607 95 612 904 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 96 047 96 052 96 057 96 061 96 066 96 071 96 076 96 080 96 085 96 090 914 96 095 96 099 96104 96109 96114 96118 96123 96128 96133 96137 915 96142 96147 96] 52 96156 96161 96166 96 171 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 96 308 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 96 402 96 407 96 412 96 417 96 421 921 96 426 96 431 96 435 96 440 96 445 96 450 96 454 96 459 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 96 997 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 97 123 936 97128 97132 97137 97142 97 146 97151 97155 97 160 97 165 97169 937 97174 97179 97183 97188 97 192 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 97 414 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 97 497 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 9 900-960 950-1000 19 N o 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 98 182 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 98 421 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 98 664 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 98 816 98 820 98 825 98 829 98 834 98 838 98 843 98 847 98 851 974 98 856 98 860 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 99 034 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 99 180 99 185 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 99 436 99 441 99 445 99 449 99 454 99 458 99 463 99 467 99 471 988 99 476 99 480 99 484 99 489 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 9^800 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 99 913 99 917 99 922 99 926 99 930 99 935 99 939 99 944 99 948 99 952 999 99 957 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 4 00 022 00 026 00 030 00 03i 00039 9 N o 1 2 3 5 6 7 8 950-1000 20 TABLE IL-LOGAEITHMS OF CONSTANTS. Circomference of the Circle in degrees — 360 log 2.55 630 250 " 4.33 445 375 6.11260 500 0.49 714 9S7 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 X = 3. 14 159 265 358 979 323 846 26+ 338 3ZS Also: 2ir= 6.28318531 log 0.79817 987 x2 = 9. 86960440 log 0.99429975 4x= 12.56637 061 ^- 1.57 079633 1.09920986 0.19611988 1 = 0.10132118 9.00570025-10 1= 1.M719 755 3 ^= 4.18 879020 3 1= 0.78 539816 0. 02 002 862 V*-- 1.77 245 385 — = 0.56418958 0.24 857494 9. 75 142 506 - 10 0.62 208 861 V' 9. 89 508 988-10 1=0.97 720502 9.98998 569-10 1= 0.52359878 o 9.71899862-10 ^^ = 1.12 837 917 0. 05 245 506 1= 0.31830989 9.50285013-10 ^x = 1.46459189 0.16571662 ^= 0.15 915 494 9.20182013-10 -1=0.68 278406 9.83428338- 10 -= 0.95492966 9.97997138-10 ^t2 = 2. 14 502 940 0.33 143 32i ^= 1.27323954 0.10491012 i/'^ = 0.62 035 049 9. 79 263 713 - 10 ^= 0.23873241 9.37 791139-10 ^1 = 0.80599 598 9.90633 287-10 Aic a, whose length is equal to the radius r, is : log in degrees aP = 1^ = 57.29577951°. 1. 75 812 263 in minutes a' _ 10 800 - 3 437. 74 677' . 3.53 627 388 in seconds a" _ 648 000 - 2O6 264. 806" . . X 5.31442 513 Arc 2a, whose length is equal to twrice the radius, 2r, is : in degrees 2a9 .... = ^ = 114.59155 903° 2.05 915 263 in minutes 2 a' 21 600 — 6S75 49'??4' 3.83 730388 5.61545 513 w in seconds 2a" . . . . - ^ ^^^^ . . . = 412 529. 612" . . If the radius r = 1, the length of the arc is : for 1 degree \ - '' - 0. 01 745 329. . . 8. 24 1S7 737 - 10 for 1 minute . . . a" 180 ..^ — ^ — 000 029 089 6. 46 372 612 - 10 a' 10800 for 1 second 1 V ....-0.00000485... 4. 68 557 487 -10 a" 648000' for J degree ^■■'=^ = 0. 00 872 66i. . . 7. 94084 737-10 for 1 minute. . . .. — ' 000 014 ^44 6. 16 269 612 - 10 "2a' 2Um U.WUlloll... for J second . . . 1 X .... = 0.00000 242... 4. 38 454 487 - 10 2a" 1296000 Sin 1" in the unit cL role -0 00 000 485 4. 68 557 487 - 10 21 1 TA 'HE L( ELE r [I [THMS 3GAE] or TBDB trigoxo:metkic FU^CTIOXS: Prom O' to 0° 3', or 89^ 57' to 90^, for every second , Prom 0° to 2-, or 88*^ to 90°, foi every ten seconds; Prom 1= U Notk. T log: sin a 89^, for every minute 3. is to be appended. ]i«eM = 10.00000 o all the logarithms -1 0° o 0' 1' O r t f f f O' 1' o * ft 6. 46 373 6.76476 60 30 6.16270 6.63982 6 86167 30 1 4.68 557 6-47090 6.76 836 59 31 6.1769f 6.64462 6 86455 29 2 4.98660 6.47 797 6. 77 193 58 32 6.19072 6.64936 6 86 742 28 3 5. 16 270 6.48492 6.77 548 57 ZZ 6.20409 6.65406 6 87(^7 27 4 5.28 763 6.49175 6.77900 56 34 6.21705 6.65 870 6 87310 26 o 5.38454 6.49 849 6.78248 55 35 6.2296f 6.66330 6. 87 591 25 6 5.46373 6.50512 6.7859i 54 36 6.24188 6.66785 6 87870 24 ? 5. 53 067 6.51165 6.78938 53 37 6.25 378 6.67235 688147 23 S 5.58866 6.51808 6.79278 52 38 6.26536 6.67680 6.88423 22 9 5.63 982 6.52 442 6.79616 51 39 6.27664 6.68121 688697 21 lO 5. 68 557 6. 53 067 6.79952 50 40 6.28763 6.68557 688969 20 11 5.72 697 6.53 6S3 6.80285 49 41 6.29836 6.68990 689240 19 12 5.76476 6. 54 291 6.80615 48 42 6.30882 6.69418 689509 18 13 5.79952 6.54 890 6.80943 47 43 6.3190f 6.69841 6.89 776 17 14 5. 83 170 6. 55 481 6.81268 46 44 6.32903 6.70261 6 90042 16 15 5.86167 6.56064 6-81591 45 45 6.33879 6.70676 690306 15 16 5.88969 6.56639 6.81911 44 46 6.34833 6.710SS 690568 14 17 5.91602 6.57 207 6.82230 43 47 6.35767 6.71496 690829 13 18 5.94085 6-57 767 6.82545 42 48 6.36682 6 71900 691068 12 19 5.% 433 6-58320 6-82859 41 49 6.37577 6 72300 691346 11 20 5.98 660 6.58 866 6.83170 40 50 638454 6.72697 691602 lO — _ 6. 00 779 6.59406 6.83 479 39 51 6.39315 6 73090 6 91857 9 :: 6.02800 6.59939 6.83 786 38 52 640158 673479 6 92110 8 23 6.04 730 6-60465 6.84091 37 53 6.40985 6 73 865 692362 7 24 6.06 579 6.60985 6.84 394 36 54 641797 6 74248 692612 6 25 6. OS 351 6-61499 6.84694 35 55 642594 674627 692861 5 26 6.10055 6.62 007 6.84993 34 56 643376 6.75003 693109 4 27 6.1169f 6.62 509 6.85 289 33 57 6 44145 6 75 376 693 355 3 28 6.13 273 6.63 006 6.85 584 32 58 644900 6 75 746 693 599 2 29 6.14 797 6.63 496 6. 85 876 31 59 64S643 6 76112 693 843 1 30 6. 16 270 6.63982 6-86167 30 eo 6.46373 6.76476 694065 O 59' 58' 57' »» f f 59' 58' 57' ff lo|^oofc = ]ogcos kg sia^ 10. 00 000 89 log cos 22 0° t ff log sin log tan log COS ff f f ff log sin log tan log cos ff f O 10.00000 6O lOO 7.46 373 7.46 373 10.00000 5O 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 7.49175 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 59 110 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 7.51809 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 6.76 476 6.76 476 10.00000 58 12 7. 54 291 7. 54 291 10.00000 048 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 7.56064 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. 94 085 10.00000 057 13 7. 57 767 7. 57 767 10.00000 047 10 6. 96 433 6.96 433 10.00000 50 10 7. 58 320 7. 58 320 10.00000 50 20 6.98 660 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 7.59 406 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. 60 465 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 7.10 055 7.10 055 10.00000 40 20 7. 62 007 7. 62 008 10.00000 40 30 7. 11 694 7.11694 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. 19 072 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 7^4 188 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 7.26536 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 7.68121 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 7.30 882 10.00000 53 170 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 52 180 7. 71 900 7. 71 900 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. 40 158 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. 73 866 9. 99 999 10 9 7. 41 797 7. 41 797 10.00000 051 190 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. 75 745 7. 75 746 9. 99 999 20 50 7. 45 643 7. 45 643 10.00000 10 50 7. 76 112 7. 76 113 9. 99 999 10 lOO 7.46 373 7.46373 10.00000 5O 20 7.76 475 7. 76 476 9. 99 999 4O r ff log cos log cot log sin ff f f ft log cos log cot log sin ff f 89' 0° 23 r ff log sin 7.76 475 log tan log COS ff t f ff log sin log tan log cos ff r 20 7. 76 476 9.99 999 4O 30 7. 94 084 7.94 086 9. 99 998 3O 10 7. 76 836 7. 76 837 9. 99 999 50 10 7.94 32i 7. 94 326 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 9.99 998 10 210 7. 78 594 7.78 595 9.99 999 39 310 7.95 508 7.95 510 9.99 998 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. 96 432 7.96434 9. 99 998 20 50 7.80 284 7.80 285 9. 99 999 10 50 7. 96 660 7. 96 662 9.99 998 10 220 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. 97 114 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. 81 911 7. 81 912 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 10 230 7. 82 545 7. 82 546 9. 99 999 037 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. 99 998 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 10 240 7. 84 393 7. 84 394 9.99 999 36 34 7.99 520 7.99 522 9. 99 998 26 10 7.84 694 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 8.00154 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 10 250 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. 99 998 30 40 7.87 309 7.87 310 9. 99 999 20 40 8.01598 8.01600 9.99 998 20 50 7. 87 590 7. 87 591 9.99 999 10 50 8. 01 801 8. 01 803 9. 99 998 10 260 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 10 270 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 10 280 7. 91 088 7.91089 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 7.91857 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 10 290 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 8. 06 217 9. 99 997 20 50 7. 93 842 7. 93 844 9.99 998 10 50 8. 06 396 8. 06 399 9.99 997 10 30 t ft 7. 94 084 7.94 086 9.99 998 3O 4O0 8. 06 578 8.06 581 9.99 997 2O log cos log cot log sin tr r f ff log cos log cot log sin ff f 89' 24 0° f ff log sin 8.06 578 log tan log COS tf f f ff log sin log tan log cos ft f 40 8.06 581 9.99 997 2O 50 8. 16 268 8. 16 273 9. 99 995 OlO 10 8. 06 758 8. 06 761 9. 99 997 50 10 8. 16 413 8.16 417 9. 99 995 50 20 8. 06 938 8.06 941 9.99 997 40 20 8.16 557 8.16 561 9.99 995 40 30 8.07117 8. 07 120 9.99 997 30 30 8. 16 700 8. 16 705 9. 99 995 30 40 8. 07 295 8. 07 299 9. 99 997 20 40 8.16 843 8.16 848 9. 99 995 20 50 8. 07 473 8. 07 476 9. 99 997 10 50 8. 16 986 8. 16 991 9.99 995 10 410 8. 07 650 8. 07 653 9. 99 997 019 510 8. 17 128 8. 17 133 9.99 995 9 10 8.07 826 8.07 829 9.99 997 50 10 8. 17 270 8. 17 275 9. 99 995 50 20 8. 08 002 8. 08 005 9. 99 997 40 20 8.17 411 8.17 416 9. 99 995 40 30 8. 08 176 8. 08 180 9.99 997 30' 30 8.17 552 8.17 557 9.99 995 30 40 8.08 350 8.08 354 9.99 997 20 40 8. 17 692 8. 17 697 9. 99 995 20 50 8.08 524 8. 08 527 9. 99 997 10 50 8. 17 832 8. 17 837 9. 99 995 10 420 8.08 696 8.08 700 9. 99 997 018 52 8.17 971 8.17 976 9. 99 995 8 10 8. 08 868 8. 08 872 9. 99 997 50 10 8.18110 8.18115 9.99 995 50 20 8.09 040 8.09 043 9.99 997 40 20 8. 18 249 8. 18 254 9.99 995 40 30 8. 09 210 8. 09 214 9. 99 997 30 30 8.18 387 8.18 392 9. 99 995 30 40 8. 09 380 8. 09 384 9. 99 997 20 40 8. 18 524 8. 18 530 9. 99 995 20 50 8.09 550 8.09 553 9.99 997 10 50 8. 18 662 8. 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46 15 80120 91224 08 776 88 896 45 16 80136 91250 08 750 88 886 44 17 80151 91276 08 724 88 875 43 18 80166 91301 08 699 88 865 42 19 80182 91327 08 673 88 85i 41 20 80197 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 91482 08 518 88 793 35 26 80 290 91507 08 493 88 782 34 27 80 305 91533 08 467 88 772 33 28 80 320 91559 08 441 88 761 32 29 80 336 91585 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 80428 91739 08 261 88 688 25 36 80 443 9176i 08 235 88 678 24 37 80 458 91791 08 209 88 668 23 38 80 473 91816 08184 88 657 22 39 80 489 91842 08158 88 647 21 40 80 504 91868 08132 88 636 20 41 80 519 91893 08107 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 92125 07 875 88 531 10 51 80 671 92150 07 850 88 521 9 52 80 686 92176 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 O 9 log cos 9 log cot lO log tan 9 log sin r r 40° 47 r log sin log tan log cot log cos r 9 9 lO 9 O 80 807 92 381 07 619 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 88 394 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 07183 88 244 43 18 81076 92 843 07157 88 234 42 19 81091 92 868 07132 88 223 41 20 81106 92 894 07106 88 212 40 21 81121 92 920 07 080 88 201 39 22 81136 92 945 07 055 88191 38 23 81151 92 971 07 029 88180 37 24 81166 92 996 07 004 88169 36 25 81180 93 022 06 978 88158 35 26 81195 93 048 06 952 88148 34 27 81210 93 073 06927 88137 33 28 81225 93 099 06 901 88126 32 29 81240 93124 06 876 88115 31 30 81254 93 150 06 850 88105 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 87931 14 47 81505 93 584 06416 87 920 13 48 81519 93 610 06 390 87 909 12 49 81534 93 636 06 364 87 898 11 50 81549 93 661 06 339 87 887 lO 51 81563 93 687 06 313 87 877 9 52 81578 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 81622 93 789 06 211 87 833 5 56 81636 93 814 06186 87 822 4 57 81651 93 840 06160 87 811 3 58 81665 93 865 06135 87 800 2 59 81680 93 891 06109 87 789 1 60 81694 93 916 06 084 87 778 O 9 9 lO 9 r log cos log cot log tan log sin f 60' 49 48 4 1° f log sin log tan log cot log cos t 9 9 lO 9 O 81694 93 916 06 084 87 778 60 1 81709 93 942 06 058 87 767 59 2 81723 93 967 06 033 87 756 58 3 81738 93 993 06 007 87 745 57 4 81752 94 018 05 982 87 734 56 5 81767 94 044 05 956 87 723 55 6 81781 94 069 05 931 87 712 54 7 81796 94 095 05 905 87 701 53 8 81810 94 120 05 880 87 690 52 9 81825 94146 05 854 87 679 51 lO 81839 94171 05 829 87 668 50 11 81854 94197 05 803 87 657 49 12 81868 94 222 05 778 87 646 48 13 81882 94 248 05 752 87 635 47 14 81897 94 273 05 727 87 624 46 15 81911 94 299 05 701 87 613 45 16 81926 94 324 05 676 87 601 44 17 81940 94 350 05 650 87 590 43 18 81955 94 375 05 625 87 579 42 19 81969 94 401 05 599 87 568 41 20 81983 94 426 05 574 87 557 40 21 81998 94 452 05 548 87 546 39 22 82 012 94 477 05 523 87 535 38 23 82 026 94 503 05 497 87 524 37 24 82 041 94 528 05 472 87 513 36 25 82 055 94 554 05 446 87 501 35 26 82 069 94 579 05 421 87 490 34 27 82 084 94 604 05 396 87 479 33 28 82 098 94 630 05 370 87 468 32 29 82112 94 655 05 345 87 457 31 30 82 126 94 681 05 319 87 446 30 31 82 141 94 706 05 294 87 434 29 32 82 155 94 732 05 268 87 423 28 33 82 169 94 757 05 243 87 412 27 34 82 184 94 783 05 217 87 401 26 35 82198 94 808 05 192 87 390 25 36 82 212 94 834 05 166 87 378 24 37 82 226 94 859 05 141 87 367 23 38 82 240 94 884 05116 87 356 22 39 82 255 94 910 05 090 87 345 21 40 82 269 94 935 05 065 87 334 20 41 82 283 94 961 05 039 87 322 19 42 82 297 94 986 05 014 87 311 18 43 82 311 95 012 04 988 87 300 17 44 82 326 95 037 04 963 87 288 16 45 82 340 95 062 04 938 87 277 15 46 82 354 95 088 04 912 87 266 14 47 82 368 95 113 04 887 87 255 13 48 82 382 95 139 04 861 87 243 12 49 82 396 95 164 04 836 87 232 11 50 82 410 95 190 04 810 87 221 10 51 82 424 95 215 04 785 87 209 9 52 82 439 95 240 04 760 87198 8 53 82 453 95 266 04 734 87 187 7 54 82 467 95 291 04 709 87175 6 55 82 481 95 317 04 683 87164 5 56 82 495 95 342 04 658 87 153 4 57 82 509 95 368 04 632 87141 3 58 82 523 95 393 04 607 87 130 2 59 82 537 95 418 04 582 87119 1 60 82 551 95 444 04 556 87107 9 o 10 9 log sin r log cos log cot log tan f 42^ f log sin log tan log cot log cos / 9 9 lO 9 o 82 551 95 444 04 556 87107 60 1 82 565 95 469 04 531 87 096 59 2 82 579 95 495 04 505 87 085 58 3 82 593 95 520 04 480 87 073 57 4 82 607 95 545 04 455 87 062 56 5 82 621 95 571 04 429 87 050 55 6 82 635 95 596 04 404 87 039 54 7 82 649 95 622 04 378 87 028 53 8 82 663 95 647 04 353 87 016 52 9 82 677 95 672 04 328 87 005 51 10 82 691 95 698 04 302 86 993 50 11 82 705 95 723 04 277 86 982 49 12 82 719 95 748 04 252 86 970 48 13 82 733 95 774 04 226 86 959 47 14 82 747 95 799 04 201 86 947 46 15 82 761 95 825 04 175 86 936 45 16 82 775 95 850 04 150 86 924 44 17 82 788 95 875 04 125 86 913 43 18 82 802 95 901 04 099 86 902 42 19 82 816 95 926 04 074 86 890 41 20 82 830 95 952 04 048 86 879 40 21 82 844 95 977 04 023 86 867 39 22 82 858 96 002 03 998 86 855 38 23 82 872 96 028 03 972 86 844 37 24 82 885 96 053 03 947 86 832 36 25 82 899 96 078 03 922 86 821 35 26 82 913 96104 03 896 86 809 34 27 82 927 96129 03 871 86 798 33 28 82 941 96155 03 845 86 786 32 29 82 955 96180 03 820 86 775 31 30 82 968 96 205 03 795 86 763 30 31 82 982 96 231 03 769 86 752 29 32 82 996 96 256 03 744 86 740 28 33 83 010 96 281 03 719 86 728 27 34 83 023 96 307 03 693 86 717 26 35 83 037 96 332 03 668 86 705 25 36 83 051 96 357 03 643 86 694 24 37 83 065 96 383 03 617 86 682 23 38 83 078 96 408 03 592 86 670 22 39 83 092 96 433 03 567 86 659 21 40 83 106 96 459 03 541 86 647 20 41 83 120 96 484 03 516 86 635 19 42 83 133 96 510 03 490 86 624 18 43 83 147 96 535 03 465 86 612 17 44 83 161 96 560 03 440 86 600 16 45 83 174 96 586 03 414 86 589 15 46 83 188 96 611 03 389 86 577 14 47 83 202 96 636 03 364 86 565 13 48 83 215 96 662 03 338 86 554 12 49 83 229 96 687 03 313 86 542 11 50 83 242 96 712 03 288 86 530 lO 51 83 256 96 738 03 262 86 518 9 52 83 270 96 763 03 237 86 507 8 53 83 283 96 788 03 212 86 495 7 54 83 297 96 814 03 186 86 483 6 55 83 310 96 839 03 161 86 472 5 56 83 324 96 864 03 136 86 460 4 57 83 338 96 890 03 110 86 448 3 58 83 351 96 915 03 085 86 436 2 59 83 365 96 940 03 060 86 425 1 60 83 378 96 966 03 034 86 413 9 9 lO 9 9 log cos log cot log tan log sin f 48' 47' 43° f log sin log tan log cot log cos f 9 9 96 966 lO 03 034 9 86 413 o 83 378 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 83 446 97 092 02 908 86 354 55 6 83 459 97118 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 86188 41 20 83 648 97 472 02 528 86176 40 21 83 661 97 497 02 503 86164 39 22 83 674 97 523 02 477 86152 38 23 83 688 97 548 02 452 86140 37 24 83 701 97 573 02 427 86128 36 25 83 715 97 598 02 402 86116 35 26 83 728 97 624 02 376 86104 34 27 83 741 97 649 02 351 86 092 ZZ 28 83 755 97 674 02 326 86 080 32 29 83 768 97 700 02 300 86 068 31 30 83 781 97 72i 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 02123 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 0^7 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 98130 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 01794 85 827 11 50 84 046 98 231 01769 85 815 lO 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 84112 98 357 01643 85 754 5 56 84125 98 383 01617 85 742 4 57 84138 98 408 01592 85 730 3 58 84 151 98 433 01567 85 718 2 59 84 164 98 458 01542 85 706 1 60 84177 98 484 01516 85 693 9 log cos 9 log cot lO log tan 9 log sin f f 44° 49 f log sin log tan log cot log cos f 9 84 177 9 98 484 lO 01516 9 85 693 o 60 1 84190 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 01390 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 01213 85 547 48 13 84 347 98 812 01188 85 534 47 14 84 360 98 838 01162 85 522 46 15 84 373 98 863 01137 85 510 45 16 84 385 98 888 01112 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 99116 00 884 85 386 35 26 84 515 99141 00 859 85 374 34 27 84 528 99166 00 834 85 361 33 28 84 540 99191 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 ZZ 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 00177 85 037 7 54 84 873 99 848 00152 85 024 6 55 84 885 99 874 00126 85 012 5 56 84 898 99 899 00101 84 999 4 57 84 911 99 924 00 076 84 986 3 58 84 923 99 949 00 051 84 974 2 59 84 936 99 975 00 025 84 961 1 60 84 949 00 000 00 000 84 949 O 9 lO lO 9 f log cos log cot log tan log sin r 46' 46' 50 TABLE IV. ^ EoR Determining with Greater Accuracy than can be done by 1 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" -f S. log a" = log sin a - S, log tan a = log o" + 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° -ay = 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). Values of S aistd T. a" S log sin a a" T log tan a a T log tan a — — 5 146 8. 39 713 4. 68 557 4.68 557 4. 68 567 2409 8.06740 200 6.98 660 5 424 8.41999 4. 68 556 4. 68 558 4. 68 568 3 417 8. 21 920 1 726 7. 92 263 5 689 8. 44 072 4. 68 555 4.68 559 4. 68 569 3 823 8.26 795 2 432 8. 07 156 5 941 8. 45 955 4. 68 55i 4. 68 560 4. 68 570 4 190 8. 30 776 2 976 8. 15 924 6 184 8. 47 697 4. 68 554 4. 68 561 4. 68 571 4 840 8. 37 038 3 434 8. 22 142 6 417 8.49 305 4.68 553 4. 68 562 4. 68 572 5 414 8.41904 3 838 8.26 973 6 642 8.50 802 4. 68 552 4. 68 563 4. 68 573 5 932 8. 45 872 4 204 8.30 930 6 859 8. 52 200 4.68 551 4. 68 564 4. 68 574 6 408 8. 49 223 4 540 8. 34 270 7 070 8.53 516 4.68 550 4. 68 565 4.68 575 6 633 8.50 721 4 699 8. 35 766 7 173 8. 54 145 4. 68 550 4. 68 565 4. 68 575 6 851 8.52 125 4 853 8.37167 , 7 274 8. 54 753 4. 68 549 4. 68 566 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, ^i If ^ = the radius of the circle , the circumference = 2ir N. If iV = the radius of the circle, the area = wN\ If iV = the circumference of the circle, the radius = — 2 IT N. If iV — the circumference of the circle, the area — 4ir N'', N 2'ir^Y ir^2 27r 47r N 2 7r^ TTNi 2-ir 4Tr O 0.00 0.0 0.000 0.00 50 314. 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 6 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 1 661. 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 18146 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. 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 80S. 8 6.844 147. 14 93 584. 34 27 172 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 30172 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 2 7riVr 7 854.0 7.958 27r 198. 94 47r 100 628. 32 2iriV^ 31416 15.92 27r 795. 77 47r U N 52 TABLE VL - MTUEAL FUNCTIONS • f o° 1° 2° 3° 4° t sin cos sin cos sin cos sin cos sin COS O 0000 1.000 0175 9998 0349 9994 0523 9986 0698 9976 60 1 0003 1.000 0177 9998 0352 9994 0526 9986 0700 9975 59 2 0006 1.000 0180 9998 0355 9994 0529 9986 0703 9975 58 3 0009 1.000 0183 9998 0358 9994 0532 9986 0706 9975 57 4 0012 1.000 0186 9998 0361 9993 0535 9986 0709 9975 56 5 0015 1.000 0189 9998 0364 9993 0538 9986 0712 9975 55 6 0017 1.000 0192 9998 0366 9993 0541 9985 0715 9974 54 7 0020 1.000 0195 9998 0369 9993 0544 9985 0718 9974 53 8 0023 1.000 0198 9998 0372 9993 0547 9985 0721 9974 52 9 0026 ]..000 0201 9998 0375 9993 0550 9985 0724 9974 51 10 0029 1.000 0204 9998 0378 9993 0552 9985 0727 9974 50 11 0032 1.000 0207 9998 0381 9993 0555 9985 0729 9973 49 12 0035 1.000 0209 9998 0384 9993 0558 9984 0732 9973 48 13 0038 1.000 0212 9998 0387 9993 0561 9984 0735 9973 47 14 0041 1.000 0215 9998 0390 9992 0564 9984 0738 9973 46 15 0044 1.000 0218 9998 0393 9992 0567 9984 0741 9973 45 16 0047 1.000 0221 9998 0396 9992 0570 9984 0744 9972 44 17 0049 1.000 0224 9997 0398 9992 0573 9984 0747 9972 43 18 0052 1.000 0227 9997 0401 9992 0576 9983 0750 9972 42 19 0055 1.000 0230 9997 0404 9992 0579 9983 0753 9972 41 20 0058 1.000 0233 9997 (H07 9992 0581 9983 0756 9971 40 21 0061 1.000 0236 9997 0410 9992 0584 9983 0758 9971 39 22 0064 1.000 0239 9997 0413 9991 0587 9983 0761 9971 38 23 0067 1.000 0241 9997 0416 9991 0590 9983 0764 9971 37 24 0070 1.000 0244 9997 0419 9991 0593 9982 0767 9971 36 25 0073 1.000 0247 9997 0422 9991 0596 9982 0770 9970 35 26 0076 1.000 0250 9997 0425 9991 0599 9982 0773 9970 34 27 0079 1.000 0253 9997 0427 9991 0602 9982 0776 9970 33 28 0081 1.000 0256 9997 0430 9991 0605 9982 0779 9970 32 29 0084 1.000 0259 9997 0433 9991 0608 9982 0782 9969 31 30 0087 1.000 0262 9997 0436 9990 0610 9981 0785 9969 30 31 0090 1.000 0265 9996 0439 9990 0613 9981 0787 9969 29 32 0093 1.000 0268 9996 0442 9990 0616 9981 0790 9969 28 33 0096 1.000 0270 9996 0445 9990 0619 9981 0793 9968 27 34 0099 1.000 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 (H74 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 50 0145 9999 0320 9995 0494 9988 0669 9978 0843 9964 10 51 0148 9999 0323 9995 0497 9988 0671 9977 0845 9964 9 52 0151 9999 0326 9995 0500 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 cos sin 86° cos 8; sin >° r 87° f ' NATURAL SINES AND COSINES. 53 / 5° 6° 7° 8° 9° r sin cos 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 9961 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 10 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 16 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 0964 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 9953 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 lO 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 cos sin 84° cos sin 83° cos sin 82° cos sin 81° cos sin 80° f f 54 NATURAL SINES AND COSINES. r io° 11° 12° 13° 14° r 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 6 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 lO 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 2\\^ 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 980+ 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 2343 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 1840 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 1874 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 S3 1888 9820 2059 9786 2230 9748 2399 9708 2569 9665 7 54 ]891 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 2408 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 O cos sin 79° cos sin 78° cos sin 77° cos sin cos sin 75° r 76° f NATUKAL SINES AND COSINES. 55 f 15° sin cos 16° 17° 18° 19° r sin cos sin cos sin cos sin cos o 2588 9659 2756 9613 2924 9563 3090 9511 3256 9455 60 ] 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 2165 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 lO 2616 9652 2784 9605 2952 9555 3118 9502 3283 9446 50 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 ZZ 2681 9634 2849 9586 3015 9535 3181 9480 3346 9423 27 34 2684 9633 2851 9585 3018 9534 3184 9480 3349 9423 26 35 2686 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 lO 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 cos sin 74° cos sin 73° cos sin 72° cos sin 71° cos sin 70° r f 56 NATURAL SINES AND COSINES. f 20° 21° 22° sin cos 23° sin cos 24° sin cos f 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 lO 3448 9387 361 i 9325 3773 9261 3934 9194 4094 9124 50 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 9078 12 49 3554 9347 3716 9284 3878 9218 4038 9148 4197 9077 11 50 3557 9346 3719 9283 3881 9216 4041 9147 4200 9075 lO 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 9068 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 cos sin 69° cos sin 68° cos sin cos sin 66° cos sin f 67° 65° f NATURAL SINES AND COSINES. 57 / 25° 26° sin cos 27° 28° sin cos 29° f sin cos sin cos sin cos o 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 lO 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 10 51 4360 8999 4517 8922 4672 8842 4825 8759 4977 8673 9 52 4363 8998 4519 8921 4674 8840 4828 8757 498D 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 60 4384 8988 4540 8910 4695 8829 4848 8746 5000 8660 cos sin 64° cos sin 63° cos sin 62° cos sin cos sin 60° f 61° ' 58 NATURAL SINES AND COSINES. / 30° 31° 32° sin cos 33° sin cos 34° f 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 lO 5025 8646 5175 8557 5324 8465 5471 8371 5616 8274 50 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 8308 5710 8210 11 50 5125 8587 5275 8496 5422 8403 5568 8307 5712 8208 lO 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 58° cos sin 57° cos sin 56° cos sin 55° f f ) NATURAL SINES AND COSINES. 59 / 35° 36° 37° 38° 39° f 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 6298 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 lO 5760 8175 5901 8073 6041 7969 6180 7862 6316 7753 50 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 8168 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 7346 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 5800 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 5816 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 9376 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 5833 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 50 5854 8107 5995 8004 6134 7898 6271 7790 6406 7679 lO 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° f r 60 NATURAL SINES AND COSINES. / 40° 41° 42° 43° sin cos 44° r sin cos sin cos sin cos sin cos O 6428 7660 6561 7547 6691 7431 6820 7314 6947 7193 60 1 6430 7^j59 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 10 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 16 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 6611 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 31 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 7253 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 lO 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 cos sin 46° cos sin 45° f 47° f NAT UKAl ^ TANGENTS AND COTANGENTS. 61 f o° 1° 2° 3° 4° f 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 10 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 13.2480 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 ZZ 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 lO 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 O cot tan cot tan cot tan cot tan cot tan f 89° 88° 87° 86° 85° / 62 NATURAL TANGENTS AND COTANGENTS. r 5° 6^ 70 8° 9° f 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 9.2806 1254 7.9718 1432 6.9827 1611 6.2085 51 lO 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 10.8483 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 10 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 96220 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 cot tan cot tan cot tan cot tan cot tan r 84° 83° 82° 81° 80° / NATURAL TANGENTS AND COTANGENTS. 63 f lO^ 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 3950 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 6 1778 5.6234 1-959 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 lO 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 2941 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 3.8118 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 50 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 5.1^26 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 cot tan cot tan cot tan cot tan cot tan f 790 78° 770 76° 75° / 64 NATURAL TANGENTS AND COTANGENTS. f 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 10 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 3ZZZ 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 29916 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 lO 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 2.7600 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 cot tan cot tan cot tan cot tan cot tan f 74° 73° 72° 71° 70° f NATURAL TANGENTS AND COTANGENTS. 65 f 20^ 21° 22° 23° 24° f 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 22373 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 lO 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 2.5533 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 31 29 3736 2.6770 3936 2.5408 4139 2.4162 4345 2.3017 4554 2.1960 31 30 3739 2.6746 3939 2.5386 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 10 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 O f cot tan 69° cot tan 68° cot tan 67° cot tan 66° cot tan 65° f 66 NATURAL TANGENTS AND COTANGENTS. f 25^ 26° 27° 28° 29° f 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 lO 4699 2.1283 4913 2.0353 5132 1.9486 5354 1.8676 5581 1.7917 50 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 31 29 4766 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 10 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 cot tan cot tan cot tan cot tan f 64° 63° 62° 61° 60° f NATURAL TANGENTS AND COTANGENTS. 67 f 30^ 31° 32° 33° 34° r 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 STn 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 60H 1.6545 6285 1.5911 6531 1.5311 6783 1.4742 51 lO 5812 1.7205 6(H8 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 27 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 6933 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 50 5969 1.6753 6208 1.6107 6453 1.5497 6703 1.4919 6959 1.4370 lO 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 6?,?,8 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 r 59° 58° 57° 56° 55° t 68 KATURAL TANGENTS AND COTANGENTS. f 35^ 36° 37° 38° 39° f 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 1 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 lO 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 32 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 8248 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 lO 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 7S39 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 O cot tan cot tan cot tan cot tan cot tan f 54° 53° 52° 51° 50° f N. (^TUR. IL TANGENTS AND COTANGENTS. 69 r 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 16 8466 1.1812 8770 1.1403 9083 1.1009 9407 1.0630 9742 1.0265 46 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 25 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 ].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 lO 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 1.000 1.0000 cot tan cot tan cot tan cot tan cot tan f 49^ 48° 47° 46° 46° f 70 TABLE VII. -TRAVERSE TAELE. Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. O f Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o / 015 1.000 0.004 2.000 0.009 3.000 0.013 4.000 0.017 5.000 0.022 89 45 30 1.000 0.009 2.000 0.017 3.000 0.026 4.000 0.035 5.000 0.044 30 45 1.000 0.013 2.000 0.026 3.000 0.039 4.000 0.052 5.000 0.065 15 1 1.000 0.017 2.000 0.035 3.000 0.052 3.999 0.070 4.999 0.087 89 15 1.000 0.022 2.000 0.044 2.999 0.065 3.999 0.087 4.999 0.109 45 30 1.000 0.026 1.999 0.052 2.999 0.079 3.999 0.105 4.998 0.131 30 45 1.000 0.031 1.999 0.061 2.999 0.092 3.998 0.122 4.998 0.153 15 2 0.999 0.035 1.999 0.070 2.998 0.105 3.998 0.140 4.997 0.174 88 15 0.999 0.039 1.998 0.079 2.998 0.118 3.997 0.157 4.996 0.196 45 30 0.999 0.044 1.998 0.087 2.997 0.131 3.996 0.174 4.995 0.218 30 45 0.999 0.048 1.998 0.096 2.997 0.144 3.995 0.192 4.994 0.240 15 3 0.999 0.052 1.997 0.105 2.996 0.157 3.995 0.209 4.993 0.262 87 15 0.998 0.057 1.997 0.113 2.995 0.170 3.994 0.227 4.992 0.283 45 30 0.998 0.061 1.996 0.122 2.994 0.183 3.993 0.244 4.991 0.305 30 45 0.998 0.065 1.996 0.131 2.994 0.196 3.991 0.262 4.989 0.327 15 4 0.998 0.070 1.995 0.140 2.993 0.209 3.990 0.279 4.988 0.349 86 15 0.997 0.074 1.995 0.148 2.992 0.222 3.989 0.296 4.986 0.371 45 30 0.997 0.078 1.994 0.157 2.991 0.235 3.988 0.314 4.985 0.392 30 45 0.997 0.083 1.993 0.166 2.990 0.248 3.986 0.331 4.983 0.414 15 5 0.996 0.087 1.992 0.174 2.989 0.261 3.985 0.349 4.981 0.436 85 15 0.996 0.092 1.992 0.183 2.987 0.275 3.983 0.366 4.979 0.458 45 30 0.995 0.096 1.991 0.192 2.986 0.288 3.982 0.383 4.977 0.479 30 45 0.995 0.100 1.990 0.200 2.985 0.301 3.980 0.401 4.975 0.501 15 6 0.995 0.105 1.989 0.209 2.984 0.314 3.978 0.418 4.973 0.523 84 15 0.994 0.109 1.988 0.218 2.982 0.327 3.976 0.435 4.970 0.544 45 30 0.994 0.113 1.987 0.226 2.981 0.340 3.974 0.453 4.968 0.566 30 45 0.993 0.118 1.986 0.235 2.979 0.353 3.972 0.470 4.965 0.588 15 7 0.993 0.122 1.985 0.244 2.978 0.366 3.970 0.487 4.963 0.609 83 15 0.992 0.126 1.984 0.252 2.976 0.379 3.968 0.505 4.960 0.631 45 30 0.991 0.131 1.983 0.261 2.974 0.392 3.966 0.522 4.957 0.653 30 45 0.991 0.135 1.982 0.270 2.973 0.405 3.963 0.539 4.954 0.674 15 8 0.990 0.139 1.981 0.278 2.971 0.418 3.961 0.557 4.951 0.696 82 15 0.990 0.143 1.979 0.287 2.969 0.430 3.959 0.574 4.948 0.717 45 30 0.989 0.148 1.978 0.296 2.967 0.443 3.956 0.591 4.945 0.739 30 45 0.988 0.152 1.977 0.3(H 2.965 0.456 3.953 0.608 4.942 0.761 15 9 0.988 0.156 1.975 0.313 2.963 0.469 3.951 0.626 4.938 0.782 81 15 0.987 0.161 1.974 0.321 2.961 0.482 3.948 0.643 4.935 0.804 45 30 0.986 0.165 1.973 0.330 2.959 0.495 3.945 0.660 4.931 0.825 30 45 0.986 0.169 1.971 0.339 2.957 0.508 3.942 0.677 4.928 0.847 15 lO 0.985 0.174 1.970 0.347 2.954 0.521 3.939 0.695 4.924 0.868 80 15 0.984 0.178 1.968 0.356 2.952 0.534 3.936 0.712 4.920 0890 45 30 0.983 0.182 1.967 0.364 2.950 0.547 3.933 0.729 4.916 0.911 30 45 0.982 0.187 1.965 0.373 2.947 0.560 3.930 0.746 4.912 0.933 15 11 0.982 0.191 1.963 0.382 2.945 0.572 3.927 0.763 4.908 0.954 79 15 0.981 0.195 1.962 0.390 2.942 0.585 3.923 0.780 4.904 0.975 45 30 0.980 0.199 1.960 0.399 2.940 0.598 3.920 0.797 4.900 0.997 30 45 0.979 0.204 1.958 0.407 2.937 0611 3.916 0.815 4.895 1.018 15 12 0.978 0.208 1.956 0.416 2.934 0.624 3.913 0.832 4.891 1.040 78 15 0.977 0.212 1.954 0.424 2.932 0.637 3.909 0.849 4 886 1.061 45 30 0.976 0.216 1.953 0.433 2.929 0.649 3.905 0.866 4.881 1.082 30 45 0.975 0.221 1.951 0.441 2.926 662 3.901 0883 4.877 1.103 15 13 0.974 0.225 1.949 0.450 2.923 0.675 3.897 0.900 4.872 1.125 77 15 0.973 0.229 1.947 0.458 2.920 0.688 3.894 0.917 4.867 1.146 45 30 0.972 0.233 1.945 0.467 2.917 0.700 3.889 0.934 4.862 1.167 30 45 0.971 0.238 1.943 0.475 2.914 0.713 3.885 0.951 4.857 1.188 15 14 0.970 0.242 1.941 0.484 2.911 0.726 3.881 0.968 4.851 1.210 76 15 0.969 0.246 1.938 0.492 2.908 0.738 3.877 0.985 4.846 1.231 45 30 0.968 0.250 1.936 0.501 2.904 0.751 3.873 1.002 4.841 1.252 30 45 0.967 0.255 1.934 0.509 2.901 0.764 3.868 1.018 4.835 1.273 15 16 0.966 0.259 1.932 0.518 2.898 0.776 3.864 1.035 4.830 1.294 75 o r Dep. Lat. Dep. Lat. Distance 2. Dep. Lat. Dep. Lat. Dep. Lat. Distance 5. 9 ' Bearing. Distance 1. Distance 3. Distance 4. Bearing. 75°-90' 0°- -16° 71 Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. o f Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o r 15 6.000 0.026 7.000 0.031 8.000 0.035 9.000 0.039 10.000 0.044 89 45 30 6.000 0.052 7.000 0.061 8.000 0.070 9.000 0.079 10.000 0.087 30 45 5.999 0.079 6.999 0.092 7.999 0.105 8.999 0.118 9.999 0.131 15 1 5.999 0.105 6.999 0.122 7.999 0.140 8.999 0.157 9.999 0.175 89 15 5.999 0.131 6.998 0.153 7.998 0.175 8.998 0.196 9.998 0.218 45 30 5.998 0.157 6.998 0.183 7.997 0.209 8.997 0.236 9.997 0.262 30 45 5.997 0.183 6.997 0.214 7.996 0.244 8.996 0.275 9.995 0.305 15 2 5.996 0.209 6.996 0.244 7.995 0.279 8.995 0.314 9.994 0.349 88 15 5.995 0.236 6.995 0.275 7.994 0.314 8.993 0.353 9.992 0.393 45 30 5.994 0.262 6.993 0.305 7.992 0.349 8.991 0.393 9.991 0.436 30 45 5.993 0.288 6.992 0.336 7.991 0.384 8.990 0.432 9.989 0.480 15 3 5.992 0.314 6.990 0.366 7.989 0.419 8.988 0.471 9.986 0.523 87 15 5.990 0.340 6.989 0.397 7.987 0.454 8.986 0.510 9.984 0.567 45 30 5.989 0.366 6.987 0.427 7.985 0.488 8.983 0.549 9.981 0.611 30 45 5.987 0.392 6.985 0.458 7.983 0.523 8.981 0.589 9.979 0.654 15 4 5.985 0.419 6.983 0.488 7.981 0.558 8.978 0.628 9.976 0.698 86 15 5.984 0.445 6.981 0.519 7.978 0.593 8.975 0.667 9.973 0.741 45 30 5.982 0.471 6.978 0.549 7.975 0.628 8.972 0.706 9.969 0.785 30 45 5.979 0.497 6.976 0.580 7.973 0.662 8.969 0.745 9.966 0.828 15 5 5.977 0.523 6.973 0.610 7.970 0.697 8.966 0.784 9.962 0.872 85 15 5.975 0.549 6.971 0.641 7.966 0.732 8.962 0.824 9.958 0.915 45 30 5.972 0.575 6.968 0.671 7.963 0.767 8.959 0.863 9.954 0.959 30 45 5.970 0.601 6.965 0.701 7.960 0.802 8.955 0.902 9.950 1.002 15 6 5.967 0.627 6.962 0.732 7.956 0.836 8.951 0.941 9.945 1.045 84 15 5.964 0.653 6.958 0.762 7.952 0.871 8.947 0.980 9.941 1.089 45 30 5.961 0.679 6.955 0.792 7.949 0.906 8.942 1.019 9.936 1.132 30 45 5.958 0.705 6.951 0.823 7.945 0.940 8.938 1.058 9.931 1.175 15 7 5.955 0.731 6.948 0.853 7.940 0.975 8.933 1.097 9.926 1.219 83 15 5.952 0.757 6.944 0.883 7.936 1.010 8.928 1.136 9.920 1.262 45 30 5.949 0.783 6.940 0.914 7.932 1.044 8.923 1.175 9.914 1.305 30 45 5.945 0.809 6.936 0.944 7.927 1.079 8.918 1.214 9.909 1.349 15 8 5.942 0.835 6.932 0.974 7.922 1.113 8.912 1.253 9.903 1.392 82 15 5.938 0.861 6.928 1.004 7.917 1.148 8.907 1.291 9.897 1.435 45 30 5.934 0.887 6.923 1.035 7.912 1.182 8.901 1.330 9.890 1.478 30 45 5.930 0.913 6.919 1.065 7.907 1.217 8.895 1.369 9.884 1.521 15 9 5.926 0.939 6.914 1.095 7.902 1.251 8.889 1.408 9.877 1.564 81 15 5.922 0.964 6.909 1.125 7.896 1.286 8.883 1.447 9.870 1.607 45 30 5.918 0.990 6.904 1.155 7.890 1.320 8.877 1.485 9.863 1.651 30 45 5.913 1.016 6.899 1.185 7.884 1.355 8^70 1.524 9.856 1.694 15 10 5.909 1.042 6.894 1.216 7.878 1.389 8.863 1.563 9.848 1.737 80 15 5.904 1.068 6.888 1.246 7.872 1.424 8.856 1.601 9.840 1.779 45 30 5.900 1.093 6.883 1.276 7.866 1.458 8.849 1.640 9.833 1.822 30 45 5.895 1.119 6.877 1.306 7.860 1.492 8.842 1.679 9.825 1.865 15 11 5.890 1.145 6.871 1.336 7.853 1.526 8.835 1.717 9.816 1.908 79 15 5.885 1.171 6.866 1.366 7.846 1.561 8.827 1.756 9.808 1.951 45 30 5.880 1.196 6.859 1.396 7.839 1.595 8.819 1.794 9.799 1.994 30 45 5.874 1.222 6.853 1.425 7.832 1.629 8.811 1.833 9.791 2.036 15 12 5.869 1.247 6.847 1.455 7.825 1.663 8.803 1.871 9.782 2.079 78 15 5.863 1.273 6.841 1.485 7.818 1.697 8.795 1.910 9.772 2.122 45 30 5.858 1.299 6.834 1.515 7.810 1.732 8.787 1.948 9.763 2.164 30 45 5.852 1.324 6.827 1.545 7.803 1.766 8.778 1.986 9.753 2.207 15 13 5.846 1.350 6.821 1.575 7.795 1.800 8.769 2.025 9.744 2.250 77 15 5.840 1.375 6.814 1.604 7.787 1.834 8.760 2.063 9.734 2.292 45 30 5.834 1.401 6.807 1.634 7.779 1.868 8.751 2.101 9.724 2.335 30 45 5.828 1.426 6.799 1.664 7.771 1.902 8.742 2.139 9.713 2.377 15 14 5.822 1.452 6.792 1.693 7.762 1.935 8.733 2.177 9.703 2.419 76 15 5.815 1.477 6.785 1.723 7.754 1.969 8.723 2.215 9.692 2.462 45 30 5.809 1.502 6.777 1.753 7.745 2.003 8.713 2.253 9.682 2.504 30 45 5.802 1.528 6.769 1.782 7.736 2.037 8.703 2.291 9.671 2.546 15 15 5.796 1.553 6.761 1.812 7.727 2.071 8.693 2.329 9.659 2.588 75 o f Dep. Lat. Distance 6. Dep. Lat. Distance 7. Dep. Lat. Dep. Lat. Dep. Lat. o f Bearing. Distance 8. Distance 9. Distance 10. Bearing. 76° -90 72 16°- -30° Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. o f Lat. Dep, Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. O f 15 15 0.965 0.263 1.930 0.526 2.894 0.789 3.859 1.052 4.824 1.315 74 45 30 0.964 0.267 1.927 0.534 2.891 0.802 3.855 1.069 4.818 1.336 30 45 0.962 0.271 1.925 0.543 2.887 0.814 3.850 1.086 4.812 1.357 15 16 0.961 0.276 1.923 0.551 2.884 0.827 3.845 1.103 4.806 1.378 74 15 0.960 0.280 1.920 0.560 2.880 0.839 3.840 1.119 4.800 1.399 45 30 0.959 0.284 1.918 0.568 2.876 0.852 3.835 1.136 4.794 1.420 30 45 0.958 0.288 1.915 0.576 2.873 0.865 3.830 1.153 4.788 1.441 15 17 0.956 0.292 1.913 0.585 2.869 0.877 3.825 1.169 4.782 1.462 73 15 0.955 0.297 1.910 0.593 2.865 0.890 3.820 1.186 4.775 1.483 45 30 0.954 0.301 1.907 0.601 2.861 0.902 3.815 1.203 4.769 1.504 30 45 0.952 0.305 1.905 0.610 2.857 0.915 3.810 1.220 4.762 1.524 15 18 0.951 0.309 1.902 0.618 2.853 0.927 3.804 1.236 4.755 1.545 72 15 0.950 0.313 1.899 0.626 2.849 0.939 3.799 1.253 4.748 1.566 45 30 0.948 0.317 1.897 0.635 2.845 0.952 3.793 1.269 4.742 1.587 30 45 0.947 0.321 1.894 0.643 2.841 0.964 3.788 1.286 4.735 1.607 15 19 0.946 0.326 1.891 0.651 2.837 0.977 3.782 1.302 4.728 1.628 71 15 0.944 0.330 1.888 0.659 2.832 0.989 3.776 1.319 4.720 1.648 45 30 0.943 0.334 1.885 0.668 2.828 1.001 3.771 1.335 4.713 1.669 30 45 0.941 0.338 1.882 0.676 2.824 1.014 3.765 1.352 4.706 1.690 15 20 0.940 0.342 1.879 0.684 2.819 1.026 3.759 1.368 4.698 1.710 70 15 0.938 0.346 1.876 0.692 2.815 1.038 3.753 1.384 4.691 1.731 45 30 0.937 0.350 1.873 0.700 2.810 1.051 3.747 1.401 4.683 1.751 30 45 0.935 0.354 1.870 0.709 2.805 1.063 3.741 1.417 4.676 1.771 15 21 0.934 0.358 1.867 0.717 2.801 1.075 3.734 1.433 4.668 1.792 69 15 0.932 0.362 1.864 0.725 2.796 1.087 3.728 1.450 4.660 1.812 45 30 0.930 0.367 1.861 0.733 2.791 1.100 3.722 1.466 4.652 1.833 30 45 0.929 0.371 1.858 0.741 2.786 1.112 3.715 1.482 4.644 1.853 15 22 0.927 0.375 1.854 0.749 2.782 1.124 3.709 1.498 4.636 1.873 68 15 0.926 0.379 1.851 0.757 2.777 1.136 3.702 1.515 4.628 1.893 45 30 0.924 0.383 1.848 0.765 2.772 1.148 3.696 1.531 4.619 1.913 30 45 0.922 0.387 1.844 0.773 2.767 1.160 3.689 1.547 4.611 1.934 15 23 0.921 0.391 1.841 0.781 2.762 1.172 3.682 1.563 4.603 1.954 67 15 0.919 0.395 1.838 0.789 2.756 1.184 3.675 1.579 4.594 1.974 45 30 0.917 0.399 1.834 0.797 2.751 1.196 3.668 1.595 4.585 1.994 30 45 0.915 0.403 1.831 0.805 2.746 1.208 3.661 1.611 4.577 2.014 15 24 0.914 0.407 1.827 0.813 2.741 1.220 3.654 1.627 4.568 2.034 66 15 0.912 0.411 1.824 0.821 2.735 1.232 3.647 1.643 4.559 2.054 45 30 0.910 0.415 1.820 0.829 2.730 1.244 3.640 1.659 4.550 2.073 30 45 0.908 0.419 1.816 0.837 2.724 1.256 3.633 1.675 4.541 2.093 15 25 0.906 0.423 1.813 0.845 2.719 1.268 3.625 1.690 4.532 2.113 65 15 0.904 0.427 1.809 0.853 2.713 1.280 3.618 1.706 4.522 2.133 45 30 0.903 0.431 1.805 0.861 2.708 1.292 3.610 1.722 4.513 2.153 30 45 0.901 0.434 1.801 0.869 2.702 1.303 3.603 1.738 4.503 2.172 15 26 0.899 0.438 1.798 0.877 2.696 1.315 3.595 1.753 4.494 2.192 64 15 0.897 0.442 1.794 0.885 2.691 1.327 3.587 1.769 4.484 2.211 45 30 0.895 0.446 1.790 0892 2.685 1.339 3.580 1.785 4.475 2.231 30 45 0.893 0.450 1.786 0.900 2.679 1.350 3.572 1.800 4.465 2.250 15 27 0.891 0.454 1.782 0.908 2.673 1.362 3.564 1.816 4.455 2.270 63 15 0.889 0.458 1.778 0.916 2.667 1.374 3.556 1.831 4.445 2.289 45 30 0.887 0.462 1.774 0.923 2.661 1.385 3.548 1.847 4.435 2.309 30 45 0.885 0.466 1.770 0.931 2.655 1.397 3.540 1.862 4.425 2.328 15 28 0.883 0.469 1.766 0.939 2.649 1.408 3.532 1.878 4.415 2.347 62 15 0.881 0.473 1.762 0.947 2.643 1.420 3.524 1.893 4.404 2.367 45 30 0.879 0.477 1.758 0.954 2.636 1.431 3.515 1.909 4.394 2.386 30 45 0.877 0.481 1.753 0.962 2.630 1.443 3.507 1.924 4.384 2.405 15 29 0.875 0.485 1.749 0.970 2.624 1.454 3.498 1.939 4.373 2.424 61 15 0.872 0.489 1.745 0.977 2.617 1.466 3.490 1.954 4.362 2.443 45 30 0.870 0.492 1.741 0.985 2.611 1.477 3.481 1.970 4.352 2.462 30 45 0.868 0.496 1.736 0.992 2.605 1.489 3.473 1.985 4.341 2.481 15 30 0.866 0.500 1.732 1.000 2.598 1.500 3.464 2.000 4.330 2.500 60 o r Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. O f Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. 60° -76' 16°- -30 3 73 Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. o r Lat. Dep, Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o r 15 15 5.789 1.578 6.754 1.841 7.718 2.104 8.683 2.367 9.648 2.630 74 45 30 5.782 1.603 6.745 1.871 7.709 2.138 8.673 2.405 9.636 2.672 30 45 5.775 1.629 6.737 1.900 7.700 2.172 8.662 2.443 9.625 2.714 15 16 5.768 1.654 6.729 1.929 7.690 2.205 8.651 2.481 9.613 2.756 74 15 5.760 1.679 6.720 1.959 7.680 2.239 8.640 2.518 9.601 2.798 45 30 5.753 1.704 6.712 1.988 7.671 2.272 8.629 2.556 9.588 2.840 30 45 5.745 1.729 6.703 2.017 7.661 2.306 8.618 2.594 9.576 2.882 15 17 5.738 1.754 6.694 2.047 7.650 2.339 8.607 2.631 9.563 2.924 73 15 5.730 1.779 6.685 2.076 7.640 2.372 8.595 2.669 9.550 2.965 45 30 5.722 1.804 6.676 2.105 7.630 2.406 8.583 2.706 9.537 3.007 30 45 5.714 1.829 6.667 2.134 7.619 2.439 8.572 2.744 9.524 3.049 15 18 5.706 1.854 6.657 2.163 7.608 2.472 8.560 2.781 9.511 3.090 72 15 5.698 1.879 6.648 2.192 7.598 2.505 8.547 2.818 9.497 3.132 45 30 5.690 1.904 6.638 2.221 7.587 2.538 8.535 2.856 9.483 3.173 30 45 5.682 1.929 6.629 2.250 7.575 2.572 8.522 2.893 9.469 3.214 15 19 5.673 1.953 6.619 2.279 7.564 2.605 8.510 2.930 9.455 3.256 71 15 5.665 1.978 6.609 2.308 7.553 2.638 8.497 2.967 9.441 3.297 45 30 5.656 2.003 6.598 2.337 7.541 2.670 8.484 3.004 9.426 3.338 30 45 5.647 2.028 6.588 2.365 7.529 2.703 8.471 3.041 9.412 3.379 15 20 5.638 2.052 6.578 2.394 7.518 2.736 8.457 3.078 9.397 3.420 70 15 5.629 2.077 6.567 2.423 7.506 2.769 8.444 3.115 9.382 3.461 45 30 5.620 2.101 6.557 2.451 7.493 2.802 8.430 3.152 9.367 3.502 30 45 5.611 2.126 6.546 2.480 7.481 2.834 8.416 3.189 9.351 3.543 15 21 5.601 2.150 6.535 2.509 7.469 2.867 8.402 3.225 9.336 3.584 69 15 5.592 2.175 6.524 2.537 7.456 2.900 8.388 3.262 9.320 3.624 45 30 5.582 2.199 6.513 2.566 7.443 2.932 8.374 3.299 9.304 3.665 30 45 5.573 2.223 6.502 2.594 7.430 2.964 8.359 3.335 9.288 3.706 15 22 5.563 2.248 6.490 2.622 7.417 2.997 8.345 3.371 9.272 3.746 68 15 5.553 2.272 6.479 2.651 7.404 3.029 8.330 3.408 9.255 3.787 45 30 5.543 2.296 6.467 2.679 7.391 3.061 8.315 3.444 9.239 3.827 30 45 5.533 2.320 6.455 2.707 7.378 3.094 8.300 3.480 9.222 3.867 15 i 23 5.523 2.344 6.444 2.735 7.364 3.126 8.285 3.517 9.205 3.907 67 15 5.513 2.368 6.432 2.763 7.350 3.158 8.269 3.553 9.188 3.947 45 30 5.502 2.392 6.419 2.791 7.336 3.190 8.254 3.589 9.171 3.988 30 45 5.492 2.416 6.407 2.819 7.322 3.222 8.238 3.625 9.153 4.028 15 24 5.481 2.440 6.395 2.847 7.308 3.254 8.222 3.661 9.136 4.067 66 15 5.471 2.464 6.382 2.875 7.294 3.286 8.206 3.696 9.118 4.107 45 30 5.460 2.488 6.370 2.903 7.280 3.318 8.190 3.732 9.100 4.147 30 45 5.449 2.512 6.357 2.931 7.265 3.349 8.173 3.768 9.081 4.187 15 25 5.438 2.536 6.344 2.958 7.250 3.381 8.157 3.804 9.063 4.226 65 15 5.427 2.559 6.331 2.986 7.236 3.413 8.140 3.839 9.045 4.266 45 30 5.416 2.583 6.318 3.014 7.221 3.444 8.123 3.875 9.026 4.305 30 45 5.404 2.607 6.305 3.041 7.206 3.476 8.106 3.910 9.007 4.345 15 26 5.393 2.630 6.292 3.069 7.190 3.507 8.089 3.945 8.988 4.384 64 15 5.381 2.654 6.278 3.096 ^7.175 3.538 8.072 3.981 8.969 4.423 45 30 5.370 2.677 6.265 3.123 7.160 3.570 8.054 4.016 8.949 4.462 30 45 5.358 2.701 6.251 3.151 7.144 3.601 8.037 4.051 8.930 4.501 15 27 5.346 2.724 6.237 3.178 7.128 3.632 8.019 4.086 8.910 4.540 63 15 5.334 2.747 6.223 3.205 7.112 3.663 8.001 4.121 8.890 4.579 45 30 5.322 2.770 6.209 3.232 7.096 3.694 7.983 4.156 8.870 4.618 30 45 5.310 2.794 6.195 3.259 7.080 3.725 7.965 4.190 8.850 4.656 15 28 5.298 2.817 6.181 3.286 7.064 3.756 7.947 4.225 8.829 4.695 62 15 5.285 2.840 6.166 3.313 7.047 3.787 7.928 4.260 8.809 4.733 45 30 5.273 2.863 6.152 3.340 7.031 3.817 7.909 4.294 8.788 4.772 30 45 5.260 2.886 6.137 3.367 7.014 3.848 7.891 4.329 8.767 4.810 15 29 5.248 2.909 6.122 3.394 6.997 3.878 7.872 4.363 8.746 4.848 61 15 5.235 2.932 6.107 3.420 6.980 3.909 7.852 4.398 8.725 4.886 45 30 5.222 2.955 6.093 3.447 6.963 3.939 7.833 4.432 8.704 4.924 30 45 5.209 2.977 6.077 3.474 6.946 3.970 7.814 4.466 8.682 4.962 15 30 5.196 3.000 6.062 3.500 6 928 4.000 7.794 4.500 8.660 5.000 60 o f Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. o f Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. 60° -76^ T4 30°- -46 O Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing, O f Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o t 30 15 0.864 0.504 1.728 1.008 2.592 1.511 3.455 2.015 4.319 2.519 59 45 30 0.862 0.508 1.723 1.015 2.585 1.523 3.447 2.030 4.308 2.538 30 45 0.859 0.511 1.719 1.023 2.578 1.534 3.438 2.045 4.297 2.556 15 31 0.857 0.515 1.714 1.030 2.572 1.545 3.429 2.060 4.286 2.575 59 15 0.855 0.519 1.710 1.038 2.565 1.556 3.420 2.075 4.275 2.594 45 30 0.853 0.522 1.705 1.045 2.558 1.567 3.411 2.090 4.263 2.612 30 45 0.850 0.526 1.701 1.052 2.551 1.579 3.401 2.105 4.252 2.631 15 32 0.848 0.530 1.696 1.060 2.544 1.590 3.392 2.120 4.240 2.650 58 15 0.846 0.534 1.691 1.067 2.537 1.601 3.383 2.134 4.229 2.668 45 30 0.843 0.537 1.687 1.075 2.530 1.612 3.374 2.149 4.217 2.686 30 45 0.841 0.541 1.682 1.082 2.523 1.623 3.364 2.164 4.205 2.705 15 33 0.839 0.545 1.677 1.089 2.516 1.634 3.355 2.179 4.193 2.723 57 15 0.836 0.548 1.673 1.097 2.509 1.645 3.345 2.193 4.181 2.741 45 30 0.834 0.552 1.668 1.104 2.502 1.656 3.336 2.208 4.169 2.760 30 45 0.831 0.556 1.663 1.111 2.494 1.667 3.326 2.222 4.157 2.778 15 34 0.829 0.559 1.658 1.118 2.487 1.678 3.316 2.237 4.145 2.796 56 15 0.827 0.563 1.653 1.126 2.480 1.688 3.306 2.251 4.133 2.814 45 30 0.824 0.566 1.648 1.133 2.472 1.699 3.297 2.266 4.121 2.832 30 45 0.822 0.570 1.643 1.140 2.465 1.710 3.287 2.280 4.108 2.850 15 35 0.819 0.574 1.638 1.147 2.457 1.721 3.277 2.294 4.096 2.868 ^^ 15 0.817 0.577 1.633 1.154 2.450 1.731 3.267 2.309 4.083 2.886 45 30 0.814 0.581 1.628 1.161 2.442 1.742 3.257 2.323 4.071 2.904 30 45 0.812 0.584 1.623 1.168 2.435 1.753 3.246 2.337 4.058 2.921 15 36 0.809 0.588 1.618 1.176 2.427 1.763 3.236 2.351 4.045 2.939 54 15 0.806 0.591 1.613 1.183 2.419 1.774 3.226 2.365 4.032 2.957 45 30 0.804 0.595 1.608 1.190 2.412 1.784 3.215 2.379 4.019 2.974 30 45 0.801 0.598 1.603 1.197 2.404 1.795 3.205 2.393 4.006 2.992 15 37 0.799 0.602 1.597 1.204 2.396 1.805 3.195 2.407 3.993 3.009 53 15 0.796 0.605 1.592 1.211 2.388 1.816 3.184 2.421 3.980 3.026 45 30 0.793 0.609 1.587 1.218 2.380 1.826 3.173 2.435 3.967 3.044 30 45 0.791 0.612 1.581 1.224 2.372 1.837 3.163 2.449 3.953 3.061 15 38 0.788 0.616 1.576 1.231 2.364 1.847 3.152 2.463 3.940 3.078 52 15 0.785 0.619 1.571 1.238 2.356 1.857 3.141 2.476 3.927 3.095 45 30 0.783 0.623 1.565 1.245 2.348 1.868 3.130 2.490 3.913 3.113 30 45 0.780 0.626 1.560 1.252 2.340 1.878 3.120 2.504 3.899 3.130 15 39 0.777 0.629 1.554 1.259 2.331 1.888 3.109 2.517 3.886 3.147 51 15 0.774 0.633 1.549 1.265 2.323 1.898 3.098 2.531 3.872 3.164 45 30 0.772 0.636 1.543 1.272 2.315 1.908 3.086 2.544 3.858 3.180 30 45 0.769 0.639 1.538 1.279 2.307 1.918 3.075 2.558 3.844 3.197 15 40 0.766 0.643 1.532 1.286 2.298 1.928 3.064 2.571 3.830 3.214 50 15 0.763 0.646 1.526 1.292 2.290 1.938 3.053 2.584 3.816 3.231 45 30 0.760 0.649 1.521 1.299 2.281 1.948 3.042 2.598 3.802 3.247 30 45 0.758 0.653 1.515 1.306 2.273 1.958 3.030 2.611 3.788 3.264 15 41 0.755 0.656 1.509 1.312 2.264 1.968 3.019 2.624 3.774 3.280 49 15 0.752 0.659 1.504 1.319 2.256 1.978 3.007 2.637 3.759 3.297 45 30 0.749 0.663 1.498 1.325 2.247 1.988 2.996 2.650 3.745 3.313 30 45 0.746 0.666 1.492 1.332 2.238 1.998 2.984 2.664 3.730 3.329 15 42 0.743 0.669 1.486 1.338 2.229 2.007 2.973 2.677 3.716 3.346 48 15 0.740 0.672 1.480 1.345 2.221 2.017 2.961 2.689 3.701 3.362 45 30 0.737 0.676 1.475 1.351 2.212 2.027 2.949 2.702 3.686 3.378 30 45 0.734 0.679 1.469 1.358 2.203 2.036 2.937 2.715 3.672 3.394 15 43 0.731 0.682 1.463 1.364 2.194 2.046 2.925 2.728 3.657 3.410 47 15 0.728 0.685 1.457 1.370 2.185 2.056 2.913 2.741 3.642 3.426 45 30 0.725 0.688 1.451 1.377 2.176 2.065 2.901 2.753 3.627 3.442 30 45 0.722 0.692 1.445 1.383 2.167 2.075 2.889 2.766 3.612 3.458 15 44 0.719 0.695 1.439 1.389 2.158 2.084 2.877 2.779 3.597 3.473 46 15 0.716 0.698 1.433 1.396 2.149 2.093 2.865 2.791 3.582 3.489 45 30 0.713 0.701 1.427 1.402 2.140 2.103 2.853 2.804 3.566 3.505 30 45 0.710 0.704 1.420 1.408 2.131 2.112 2.841 2.816 3.551 3.520 15 45 0.707 0.707 1.414 1.414 2.121 2.121 2.828 2.828 3.536 3.536 45 O f Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. o f Bearing. Distance 1. Distance 2. Distance 3. Distance 4. Distance 5. Bearing. 46° -60^ 30°- -46 o 76 Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. o f Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. o t 3015 5.183 3.023 6.047 3.526 6.911 4.030 7.775 4.534 8.638 5.038 59 45 30 5.170 3.045 6.031 3.553 6.893 4.060 7.755 4.568 8.616 5.075 30 45 5.156 3.068 6.016 3.579 6.875 4.090 7.735 4.602 8.594 5.113 15 31 5.143 3.090 6.000 3.605 6.857 4.120 7.715 4.635 8.572 5.150 59 15 5.129 3.113 5.984 3.631 6.839 4.150 7.694 4.669 8.549 5.188 45 30 5.116 3.135 5.968 3.657 6.821 4.180 7.674 4.702 8.526 5.225 30 45 5.102 3.157 5.952 3.683 6.803 4.210 7.653 4.736 8.504 5.262 15 32 5.088 3.180 5.936 3.709 6.784 4.239 7.632 4.769 8.481 5.299 58 15 5.074 3.202 5.920 3.735 6.766 4.269 7.612 4.802 8.457 5.336 45 30 5.060 3.224 5.904 3.761 6.747 4.298 7.591 4.836 8.434 5.373 30 45 5.046 3.246 5.887 3.787 6.728 4.328 7.569 4.869 8.410 5.410 15 33 5.032 3.268 5.871 3.812 6.709 4.357 7.548 4.902 8.387 5.446 57 15 5.018 3.290 5.854 3.838 6.690 4.386 7.527 4.935 8.363 5.483 45 30 5.003 3.312 5.837 3.864 6.671 4.416 7.505 4.967 8.339 5.519 30 45 4.989 Z3ZZ 5.820 3.889 6.652 4.445 7.483 5.000 8.315 5.556 15 34 4.974 3.355 5.803 3.914 6.632 4.474 7.461 5.033 8.290 5.592^ 56 15 4.960 3.377 5.786 3.940 6.613 4.502 7.439 5.065 8.266 5.628 45 30 4.945 3.398 5.769 3.965 6.593 4.531 7.417 5.098 8.241 5.664 30 45 4.930 3.420 5.752 3.990 6.573 4.560 7.395 5.130 8.217 5.700 15 35 4.915 3.441 5.734 4.015 6.553 4.589 7.372 5.162 8.192 5.736 55 15 4.900 3.463 5.716 4.040 6.533 4.617 7.350 5.194 8.166 5.772 45 30 4.885 3.484 5.699 4.065 6.513 4.646 7.327 5.226 8.141 5.807 30 45 4.869 3.505 5.681 4.090 6.493 4.674 7.304 5.258 8.116 5.843 15 36 4.854 3.527 5.663 4.115 6.472 4.702 7.281 5.290 8.090 5.878 54 15 4.839 3.548 5.645 4.139 6.452 4.730 7.258 5.322 8.064 5.913 45 30 4.823 3.569 5.627 4.164 6.431 4.759 7.235 5.353 8.039 5.948 30 45 4.808 3.590 5.609 4.188 6.410 4.787 7.211 5.385 8.013 5.983 15 37 4.792 3.611 5.590 4.213 6.389 4.815 7.188 5.416 7.986 6.018 53 15 4.776 3.632 5.572 4.237 6.368 4.842 7.164 5.448 7.960 6.053 45 30 4.760 3.653 5.554 4.261 6.347 4.870 7.140 5.479 7.934 6.088 30 45 4.744 3.673 5.535 4.286 6.326 4.898 7.116 5.510 7.907 6.122 15 38 4.728 3.694 5.516 4.310 6.304 4.925 7.092 5.541 7.880 6.157 52 15 4.712 3.715 5.497 4.334 6.283 4.953 7.068 5.572 7.853 6.191 45 30 4.696 3.735 5.478 4.358 6.261 4.980 7.043 5.603 7.826 6.225 30 45 4.679 3.756 5.459 4.381 6.239 5.007 7.019 5.633 7.799 6.259 15 39 4.663 3.776 5.440 4.405 6.217 5.035 6.994 5.664 7.772 6.293 51 15 4.646 3.796 5.421 4.429 6.195 5.062 6.970 5.694 7.744 6.327 45 30 4.630 3.816 5.401 4.453 6.173 5.089 6.945 5.725 7.716 6.361 30 45 4.613 3.837 5.382 4.476 6.151 5.116 6.920 5.755 7.688 6.394 15 40 4.596 3.857 5.362 4.500 6.128 5.142 6.894 5.785 7.660 6.428 50 15 4.579 3.877 5.343 4.523 6.106 5.169 6.869 5.815 7.632 6.461 45 30 4.562 3.897 5.323 4.546 6.083 5.196 6.844 5.845 7.604 6.495 . 30 45 4.545 3.917 5.303 4.569 6.061 5.222 6.818 5.875 7.576 6.528 15 41 4.528 3.936 5.283 4.592 6.038 5.248 6.792 5.905 7.547 6.561 49 15 4.511 3.956 5.263 4.615 6.015 5.275 6.767 5.934 7.518 6.594 45 30 4.494 3.976 5.243 4 638 5.992 5.301 6.741 5.964 7.490 6.626 30 45 4.476 3.995 5.222 4.661 5.968 5.327 6.715 5.993 7.461 6.659 • 15 42 4.459 4.015 5.202 4.684 5.945 5.353 6.688 6.022 7.431 6.691 48 15 4.441 4.034 5.182 4.707 5.922 5.379 6.662 6.051 7.402 6.724 45 30 4.424 4.054 5.161 4.729 5.898 5.405 6.635 6.080 7.373 6.756 30 45 4.406 4.073 5.140 4.752 5.875 5.430 6.609 6.109 7.343 6.788 15 43 4.388 4.092 5.119 4.774 5.851 5.456 6.582 6.138 7.314 6.820 47 15 4.370 4.111 5.099 4.796 5.827 5.481 6.555 6.167 7.284 6.852 45 30 4.352 4.130 5.078 4.818 5.803 5.507 6.528 6.195 7.254 6.884 30 45 4.334 4.149 5.057 4.841 5.779 5.532 6.501 6.224 7.224 6.915 15 44 4.316 4.168 5.035 4.863 5.755 5.557 6.474 6.252 7.193 6.947 46 15 4.298 4.187 5.014 4.885 5.730 5.582 6.447 6.280 7.163 6.978 45 30 4.280 4.206 4.993 4.906 5.706 5.607 6.419 6.308 7.133 7.009 30 45 4.261 4.224 4.971 4.928 5.681 5.632 6.392 6.336 7.102 7.040 15 45 4.243 4.243 4.950 4.950 5.657 5.657 6.364 6.364 7.071 7.071 45 o r Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Q ' Bearing. Distance 6. Distance 7. Distance 8. Distance 9. Distance 10. Bearing. 46° -60' A TABLE OF THE ANGLES Which every Point and Quarter Point of the Compass makes with the Meridian. North. Points. ^ // 2 48 45 5 37 30 8 26 15 11 15 Points. 0-14 0-% 0-34 1 South. 1 N. by E. N. by W. S. by E. S. by -W. N.N.E. N.N.W. I-V4 14 3 45 16 52 30 19 41 15 22 30 2 * S.S.E. S.S.W. N.E, by N. N.-W. by N. 25 18 45 28 7 30 30 56 15 33 45 3 * S.E.byS. S.W. by S. N.E. N.^W. 3-V4 36 33 45 39 22 30 42 11 15 45 3-1/4 3-% .A S.E. S.'W. N.E. by E N.W.by-W. P 47 48 45 50 37 30 53 26 15 56 15 4-14 4-% 4-3/ 5 S.E.byE. S.W. by "W. E.N.E. W.N.W. P 59 3 45 61 52 30 64 41 15 67 30 5-1/4 6 * E.S.E. W.S.W. E. by N. ^W. by N. 6-y. 7 * 70 18 45 73 7 30 75 56 15 78 45 6 -1/4 7 E. by S. W. by S. East. "West. 7-V4 81 33 Ah 84 22 30 87 11 15 90 7-V4 East. West. 12 Oc»' '•^ S£p2 4l955LU, J^fct^^ j^P^i-io""*- ,4nA6T02a6)4T YC 2229D «Vj305100 WM-7 1595 THE UNIVERSITY OF CALIFORNIA LIBRARY ^a.