UNIVERSITY OF CALIFORNIA AT LOS ANGELES PHILLIPS-LOOMIS MATHEMATICAL SERIES ELEMENTS OF TRIGONOMETRY WITH TABLES PHILLIPS-LOOMIS MATHEMATICAL SERIES ELEMENTS OF TRIGONOMETRY PLANE AND SPHERICAL BY ANDREW W. PHILLIPS, PH.D. AND WENDELL M. STRONG, PH.D. YALE UNIVERSITY 45071 THE PHILLIPS-LOOMIS MATHEMATICAL SERIES. ELEMENTS OF TRIGONOMETRY, Plane and Spherical. By ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale University. Crown 8vo, Half Leather. ELEMENTS OF GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING FISHER, Ph.D., Professors in Yale University. Crown 8vo, Half Leather, $1 75. [By mail, $1 92.] ABRIDGED GEOMETRY. By ANDREW \V. PHILLIPS, Ph.D., and IRVING FISHER, Ph.D. Crown 8vo, Half Leather, $1 25. [Hy mail, $1 40.] PLANE GEOMETRY. By ANDREW W. PHILLIPS, Ph.D., and IRVING FISHEK, Ph.D. Crown 8vo, Cloth, 80 cents. [By mail, 90 cents.] LOGARITHMIC AND TRIGONOMETRIC TABLES. Five-Place and Four- Place. By ANDREW W. PHILLIPS, Ph.D., and WENDELL M. STRONG, Ph.D., Yale University. Crown 8vo. LOGARITHMS OF NUMBERS. Five-Figure Table to Accompany the "Elements of Geometry," by ANDREW W. PHILLIPS, Ph.D., and IRVING FISHER, Ph.D., Professors in Yale University. Crown 8vo, Cloth, 30 cents. [By mail, 35 cents.] NEW YORK AND LONDON : HARPER & BROTHERS, PUBLISHERS. Copyright, 1898, by HARPER & BROTHERS. All rights reserved. Mathematical Sciences _ Library O > IN this work the trigonometric functions are defined as I ratios, but their representation by lines is also introduced at % the beginning, because certain parts of the subject can be ^treated more simply by the line method, or by a combination \of the two methods, than by the ratio method alone. Attention is called to the following features of the book: The simplicity and directness of the treatment of both the Plane and Spherical Trigonometry. J The emphasis given to the formulas essential to the solu- j tion of triangles. v The large number of exercises. The graphical representation of the trigonometric, inverse trigonometric, and hyperbolic functions. The use of photo-engravings of models in the Spherical Trigonometry. The recognition of the rigorous ideas of modern math- \ ematics in dealing with the fundamental series of trigo- * nometry, ? The natural treatment of the complex number and the hyperbolic functions. The graphical solution of spherical triangles. Our grateful acknowledgments are due to our colleague, Professor James Pierpont, for valuable suggestions regard- ing the construction of Chapter VI. We are also indebted to Dr. George T. Sellew for making the collection of miscellaneous exercises. ANDREW W. PHILLIPS, WENDELL M. STRONG. YALE UNIVERSITY, December, 1898. TABLE OF CONTENTS PLANE TRIGONOMETRY CHAPTER I THE TRIGONOMETRIC FUNCTIONS PAGE Angles I Definitions of the Trigonometric Functions 4 Signs of the Trigonometric Functions 8 Relations of the Functions 10 Functions of an Acute Angle of a Right Triangle 13 Functions of Complementary Angles 14 Functions of o, 90, 1 80, 270, 360 15 Functions of the Supplement of an Angle . . 16 Functions of 45, 30, 60 17 Functions of ( x), (180 x), (i8o-f.r), (360 x) 18 Functions of (90 y), (90+^), (270^), (2"jo-\-y) 20 CHAPTER II THE RIGHT TRIANGLE Solution of Right Triangles 22 Solution of Oblique Triangles by the Aid of Right Triangles . . 28 CHAPTER III TRIGONOMETRIC ANALYSIS Proof of Fundamental Formulas (i i)- (14) 32 Tangent of the Sum and Difference of Two Angles 36 Functions of Twice an Angle 36 Functions of Half an Angle 36 Formulas for the Sums and Differences of Functions 37 The Inverse Trigonometric Functions 39 vi TABLE OF CONTENTS CHAPTER IV THE OBLIQUE TRIANGLE PAGE Derivation of Formulas 41 Formulas for the Area of a Triangle 44 The Ambiguous Case 45 The Solution of a Triangle : (i.) Given a Side and Two Angles 46 (2.) Given Two Sides and the Angle Opposite One of Them . 46 (3.) Given Two Sides and the Included Angle 48 (4.) Given the Three Sides 49 Exercises 50 CHAPTER V CIRCULAR MEASURE GRAPHICAL REPRESENTATION Circular Measure 55 Periodicity of the Trigonometric Functions 57 Graphical Representation 58 CHAPTER VI COMPUTATION OF LOGARITHMS AND OF THE TRIGONOMETRIC FUNC- TIONS DE MOIVRE'S THEOREM HYPERBOLIC FUNCTIONS Fundamental Series 63 Computation of Logarithms 64 Computation of Trigonometric Functions 68 De Moivre's Theorem 70 The Roots of Unity 72 The Hyperbolic Functions 73 CHAPTER VII MISCELLANEOUS EXERCISES Relations of Functions 78 Right Triangles 80 Isosceles Triangles and Regular Polygons 83 Trigonometric Identities and Equations 84 Oblique Triangles 88 TABLE OF CONTENTS vii SPHERICAL TRIGONOMETRY CHAPTER VIII RIGHT AND QUADRANTAL TRIANGLES PAGE Derivation of Formulas for Right Triangles 93 Napier's Rules i 94 Ambiguous Case 97 Quadrantal Triangles . 98 CHAPTER IX OBLIQUE-ANGLED TRIANGLES Derivation of Formulas 100 Formulas for Logarithmic Computation 101 The Six Cases and Examples 104 Ambiguous Cases 106 Area of the Spherical Triangle 108 CHAPTER X APPLICATIONS TO THE CELESTIAL AND TERRESTRIAL SPHERES Astronomical Problems no Geographical Problems 113 CHAPTER XI GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE 115 CHAPTER XII RECAPITULATION OF FORMULAS 119 APPENDIX RELATION OF THE PLANE, SPHERICAL, AND PSEUDO-SPHERICAL TRIGONOMETRIES 125 ANSWERS TO EXERCISES 129 PLANE TRIGONOMETRY CHAPTER I THE TRIGONOMETRIC FUNCTIONS ANGLES _/. In Trigonometry the size of an angle is measured by the amount one side of the angle has revolved from the position of the other side to reach its final position. Thus, if the hand of a clock makes one-fourth of a rev- olution, the angle through which it turns is one right angle; if it makes one-half a revolution, the angle is two right an- gles; if one revolution, the angle is four right angles; if one and one-half revolutions, the angle is six right angles, etc. O' B FIG. 2 FIG. 3 The amount the side OB has rotated from OA to reach its final position may or may not be equal to the inclination of the lines. In Fig. I it is equal to this inclination ; in Fig. 4 it is not. Two angles may have the same sides and yet be different. In Fig. 2 I PLANE TRIGONOMETRY and Fig. 4 the positions of the sides of the angles are the same ; yet in Fig. 2 the angle is two right angles, in Fig. 4 it is six right angles. The addition of any number of complete revolutions to an angle does not change the posi m of its sides. Qut^ton. Through how many right angles does the hour-hand of a clock revolve in 6 hours? the minute-hand ? Question. If the fly-wheel of an engine makes 100 revolutions per minute, through how many right angles does it revolve in i second ? Initial line \^J Initial line |J RIGHT ANGLES 5! RIGHT ANGLES Def. The first side of the angle that is, the side from which the revolution is measured is the initial line; the second side is the terminal line. Def. If the direction of the revolution is opposite to that of the hands of a clock, the angle is positive; if the same as that of the hands of a clock, the angle is negative. Initial line Initial line POSITIVE ANGLE NEGATIVE ANGLE The angles we have employed as illustrations those described by the hands of a clock are all negative angles. 2. Angles are usually measured in degrees, minutes, and seconds. A degree is one-ninetieth of a right angle, a min- ute is one-sixtieth of a degree, a second is one-sixtieth of a minute. THE TRIGONOMETRIC FUNCTIONS The symbols indicating degrees, minutes, and seconds are ' "; thus, twenty-six degrees, forty-three minutes, and ten seconds is written 26 43' 10". 3. The plane about the vertex of an angle is div. Jed into four quadrants, as shown in the figure; the first quadrant beeins at the initial line. ii in IV THE FOUR QUADRANTS III ANGLE IN 1ST QUADRANT II ANGLE IN 2D QUADRANT ANGLE IN 3D QUADRANT III ANGLE IN 4TH QUADRANT An angle is said to be in a certain quadrant if its terminal line is in that quadrant. EXERCISES 4. (i.) Express ^\ right angles in degrees, minutes, and seconds. In what quadrant is the angle? (2.) What angle less than 360 has the same initial and terminal lines as an angle of 745? (3.) What positive angles less than 720 have the same sides as an angle of 73 ? (4.) In what quadrant is an angle of 890? DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS 5. The trigonometric functions are numbers, and are de- fined as the ratios of lines. Let the angle AOP be so placed that the initial line is horizontal, and from P, any point of the terminal line, draw PS perpendicular to the initial line. s A ANGLE IN THE 1ST QUADRANT ANGLE IN THE 2D QUADRANT ANGI.K IN THH 3D QUADRANT Denote the angle A OP by x. SP ANGI.B IN THH 4TH QUADRANT -^=sine of x (written sin*). OS - = cosine of x (written cos*). THE TRIGONOMETRIC FUNCTIONS SP tangent of x (written OS > = cotangent of x (written cot^r). ^1 OP OS - = secant of x (written sec^r). - = cosecant of x To the above may be added the versed sine (written versin) and coversed sine (written coversin), svhich are defined as follows : versiii ic \ cos x\ coversiu x = i sin a/. The values of the sine, cosine, etc., do not depend upon what point of the terminal line is taken as P, but upon the angle. S S' S'S For the triangles OSP and OS'P' being similar, the ratio of any two sides of OS'P' is equal to the ratio of the corresponding sides of OSP. Def. The sine, cosine, tangent, cotangent, secant, and cosecant of an angle are the trigonometric functions of the angle, and depend for their value on the angle alone. 6. A line may by its length and direction represent a number; the magnitude of the number is expressed by the length of the line; the number is positive or negative ac- cording to the direction of the line. 6 PLANE TRIGONOMETRY 7. In 5, if the denominators of the several ratios be taken equal to unity, the trigonometric functions will be rep- resented by lines. SF SP Thus, sin.r=-y^= = SP the number represented by the line, that is, the ratio of the line to its unit of length. Hence SP may represent the sine of x. In a similar manner the other trigonometric functions may be represented by lines. In the following figures a circle of unit radius is described about the vertex O of the angle A OP, this angle being de- noted by x. Then from 5 it follows that C cot C Cot B FIG 4 THE TRIGONOMETRIC FUNCTIONS 7 SP represents the ine of x. OS represents the coine of x, A T represents the tangent of x. BC represents the cotangent of x. O T represents the secant of x. OC represents the coecant of x. For the sake of brevity, the lines SP, OS, etc., of the preceding figures are often spoken of as the sine, cosine, etc. Hence, we may also define the trigonometric functions in general terms as follows: If a circle of unit radius is described about the vertex of an angle, (i.) The sine of the angle is represented by the perpendicular upon the initial line from the intersection of the terminal line with the circumference, (2.) The cosine of the angle is represented by the segment of the initial line extending from the vertex to the sine. (3.) The tangent of the angle is represented by a line tangent to the circle at the beginning of the first quadrant, and extending from the point of tangency to the terminal line. (4.) The cotangent of the angle is represented by a line tangent to the circle at the beginning of the second quadrant, and extending from the point of tangency to the terminal line. (5.) The secant of the angle is represented by the segment of the terminal line extending from the vertex to the tangent. (6.) The cosecant of the angle is represented by the segment of the terminal line extending from the vertex to the cotangent. The definitions in 5 are called the ratio definitions of the trigonometric functions, and those in 7 the line definitions. The introduction of two definitions for the same thing should not embarrass the student. We have shown that they are equivalent. In some cases it is convenient to use the first definition, and in other cases the second, as the student will observe in the course of this study. It is therefore important that he should be- come familiar with the use of both. 8 PLANE TRIGONOMETRY SIGNS OF THE TRIGONOMETRIC FUNCTIONS 8. Lines are regarded as positive or negative according to their directions. Thus, in the figures of 5, OS is posi- tive if it extends to the rig/it of O along the initial line, negative if it extends to the left ; SP is positive if it extends upward from OA, negative if it extends doivnward. OP, the terminal line, is always positive. The above determines, from 5, the signs of the trigono- metric functions, since it shows the signs of the two terms of each ratio. By the line definitions the signs may be determined di- rectly. The sine and tangent are positive if measured up- ward from OA, and negative if measured doivnward, The cosine and cotangent are positive if measured to the right from OB, and negative if measured to the left. B Cot-f- Cot- B FIG. 3 THE TRIGONOMETRIC FUNCTIONS The secant and cosecant are positive if measured in the same direction as the terminal line, OP; negative if measured in the opposite direction. The signs of the functions of angles in the different quadrants are as follows : Quadrant I II Ill IV Sine and cosecant + + - - Cosine and secant + - - + Tangent and cotangent + - + - 0. It is evident that the values of the functions of an angle depend only upon the position of the sides of the angle. If two angles differ by 360, or any multiple of 360, the position of the sides is the same, hence the values of the functions are the same. Thus in Fig. i the angle is 120, in Fig. 2 the angle is 840, yet the lines which represent the functions are the same for both angles. EXERCISE Determine, by drawing the necessary figures, the sign of tan 1000; cos 810; sin 760; cot 70; cos 550; tan 560; sec 300; cot 1560; sin 130; cos 260; tan 310. 10 PLANE TRIGONOMETRY RELATIONS OF THE FUNCTIONS 10. By 5, whatever may be the length of OP, we have SP 9JL x- - -2- OP ' ~d~p ~ cos * ' os ~ an * ' SP~ ' * ' os OP B Cot C We have, then, from Figs. 2 and 3, SP sinac -rr. = tan ac = ; OS -o* / os_ SP~ Multiplying (i) by (2), C09iC or tan x = laii./ cot 07=1, i _ i cot x ' tan x Again, from Figs. 2 and 3, OP 1 -= = ec x = ; OS cos x ' From Figs. 2 and 3, or and sin*jr= I COS'JT ; cos*j:= I sin s jr. Also, OA*+AT*=Or, and or 1 + tan 8 a? 1 + cot'oe = csc'ac. FIG. 3 (I) (2) (3) (4) (5) (6) (7) (8) THE TRIGONOMETRIC FUNCTIONS \\ The angle x has been taken in the first quadrant ; the results are, however, true for any angle. The proof is the same for angles in other quadrants, except that SP be- comes negative in the third and fourth quadrants, and OS in the second and third. EXERCISES 11. (i.) Prove cos-r sec.r= i. (2.) Prove sin-r CSC.T I. (3.) Prove tan .r cos x sin x. (4.) Prove sin x \/i cos'' x i cos*x. (5.) Prove tan x + cot x = . sm.r cos-r (6.) Prove sin 4 x cos 4 x = i 2 cos 2 .r. (7.) Prove = sin x. cotx secx (8,) Prove tan x sin x -+- cos x = sec jr. 12. The formulas (i)-(8) of 10 are algebraic equations connecting the different functions of the same angle. If the value of one of the functions of an angle is given, we can substitute this value in one of the equations and solve to find another of the functions. Repeating the process, we find a third function, etc. In solving equation (6), (7), or (8) a square root is extracted ; unless something is given which determines whether to choose the positive or negative square root, we get two values for some of the functions. The reason for this is that there are two angles less than 360 for which a function has a given value. EXERCISES 13. (i.) Given x less than 90 and sin.r = ^; find all the other functions of x. Solution. COSJT V i 4= A/3. Since x is less than 90, we know that COSJT is positive. 12 PLANE TRIGONOMETRY Hence cos x = ,- = --- =y 3 ; 2 iv/3 i CSCJT=- = 2. 2 (2.) Given tan* = and * in quadrant IV; find sin* and cos jr. Solution, hence sin x COSJT 3 sin.r = -*: COS X, hence 10 sin 2 jr = !; cos.r= T 3 1 jV' 10. (3.) Given sin( 30) = ; find the other functions of 30. (4.) Given x in quadrant III and sin.r = ; find all the other functions of x. (5.) GiveVi y in quadrant IV and sin/= $, find all the other functions oiy. (6.) Given cos 6o = J; find all the other functions of 60. (7.) Given sin o = o; find coso and tano. (8.) Given tan^r = |and z in quadrant I; find the other functions of z. (9.) Given cot45= I ; find all the other functions of 45. (10.) Given tanj=i\/5 and cos^ negative; find all the other functions of y. (11.) Given cot 30= \/3; fid tne other functions of 30. (12.) Given 2 sinjr=i cos* and x in quadrant II; find sin* and cos JT. (13.) Given tan x-*r cot* = 3 and x in quadrant I ; find sin*. THE TRIGONOMETRIC FUNCTIONS FUNCTIONS OF AN ACUTE ANGLE OF A RIGHT TRIANGLE 14. The functions of an acute angle of a right triangle can be expressed as ratios of the sides of the triangle. Remark. Triangles are usually lettered, as in Fig. 2, the capital letters denoting the angles, the corresponding small letters the sides opposite. In the right triangle ABC, by 5, RC a cos A = r- = - = sin B ; AB c =!=! = an B. 15. From 14, for an acute angle of a right triangle, we have side opposite angle - ; hypotenuse side adjacent to angle cosine = - T-=! - - ; hypotenuse side opposite angle tangent = -^3 - -~-^- , ' ' side adjacent to angle side adjacent to angle v cotangent = 73 - - : . side opposite angle (9) 1 4 PLANE TRIGONOMETRY FUNCTIONS OF COMPLEMENTARY ANGLES 16. From 14, we have sin A=cos=co*(9QA); co* A = s'm= sin (9O 4); tan A = cot B cot (9O A) 5 cot A = tan B = tan (9O A). Because of this relation the sine and cosine are called co-func- tions of each other, and the tangent and cotangent are called co- functions of each other. The results of this article may be stated thus: A function of an acute angle is equal to the co-function of its complementary angle. Tha values of the functions of the different angles are given in " Trigo- nometric Tables." By the use of the principle just proved, each function of an angle 'between 45 and 90 can be found as a function of an angle less than 45. Consequently, the tables need to be constructed for angles up to 45 only. The tables are so arranged that a number in them can be read either as a function of an angle less than 45 or as the co-function of the complement of this angle. EXERCISES J7. (i.) Express as functions of an angle less than 45: sin 70 ; cos 89 30' ; tan 63 ; cos66; cot47; sin 72 39'. (2.) cos;r = sin2.r; find*. (3.) tan x cot 3* ; find A: - (4.) sin2jr = cos3-r ; find x. (5.) cot(30 jr) = tan(3o + 3jr); find*. (6.) A, B, and C are the angles of a triangle; prove that Hint. A + B + C= 1 80. THE TRIGONOMETRIC FUNCTIONS 18. As the angle x decreases towards o (Fig. i), sinx de- creases and cos.* increases. When OP comes into coincidence with OA, SP becomes o, and OS becomes OA( \). Hence sino = o. coso = i. FIG. 3 As the angle x increases towards 90 (Fig. 2), sin.* increases and cos:r decreases. When OP comes into coincidence with OB, SP becomes OB{~\) and OS becomes o. Hence 8^90 = !, cos 90 ^o. As the angle x decreases towards o (Fig. 3), tan* decreases and cot* increases. When OP comes into coincidence with OA, A T becomes o and B C has increased without limit. Hence tano = o, coto = oo. As the angle x increases towards 90 (Fig. 4), tan.* increases and cot* decreases. When OP comes into coincidence with OB, ^47'has increased without limit, and BCQ. Hence tan 90 = 00, cot9o=:o. Remark. By coto=oo we mean that as the angle approaches indefinitely near to o its cotangent increases so as to become greater than any finite quan- tity we may choose. The symbol oo does not denote a definite number, but simply that the number is indefinitely great. i6 PLANE TRIGONOMETRY In every case where a trigonometric function becomes indefinitely great it is in a positive sense if the angle approaches the limiting value from one side, in a negative sense if the angle approaches the limiting value from the other side. Thus cot o = -j- oo if the angle decreases to o, but cot o= oo if the angle increases from a nega- tive angle to o. We shall not often need to distinguish between + 00 and oo, and shall in general denote either by the symbol oo. By a similar method the functions of 180, 270, and 360 may be deduced. The results of this article are shown in the following table : Angle 90 1 80 270 300 sin o I I O cos I -I I tan o CO o CO o cot oo 00 O 00 19. It may now be stated that, as an angle varies, its sine and cosine can take on 'values from / to + / only, its tangent and cotangent all values from oo t<> -\- oo, its secant and cosecant all values from oo to -(- . ex. j)t those between / and -f- 1. FUNCTIONS OF THE SUPPLEMENT OF AN ANGLE 20. Suppose the triangle OPS (Fig. i) equal to the tri- angle OP'S' (Fig. 2), then SP=S'P' and OS=OS f , and the angle A OP' (Fig. 2) is equal to the supplement of AOP (Fig. i). Also, in the triangle AOP' (Fig. 3), angle AOP' = angle AOP' (Fig. 2). V r o FIG. a FIG. 3 THE TRIGONOMETRIC FUNCTIONS It follows from 5 and 8 that sin (1O as) = sin x ; co (1O x) = cos a? ; tan (1O x) = tan a? ; ' cot (1O a?) = cot x. The results of this article may be stated thus : The sine of an angle is equal to the sine of its supplement, and the cosine, tangent, and cotangent are each equal to minus the same functions of its supplement. The principle just proved is of great importance in the solution of tri- angles which contain an obtuse angle. FUNCTIONS OF 45, 30, AND 6b 21. In the right triangle OSP (Fig. i) angle = angle /> = 45. and OP = i. Hence OS = SP = i -y/2. Therefore sin 45 = 00545 =^-\/2; 14,16 tan 45 = cot 45= i. P \J \.rj- O v _L Q n v * ^^ O 2 FIG. I FIG. 2 In equilateral triangle 0/^4 (Fig. 2) the sides are oi unit length /* bisects angle OP A, is perpendicular to OA, and bisects 0,4. Hence, in the right triangle OPS, OS = %, SP = %-\/'$. Therefore sin 30 := cos 60 = ^ ; 14 tan 30 = cot 60 = i v X 3 : cot 30 = tan 60 \/3- 1 8 PLANE TRIGONOMETRY 22. The following values should be remembered : Angle 30 45 60 90 sin o i iv/2 iv/3 i cos i *V1 ii/5 i o EXERCISES Prove that if x = 30, (i.) sin 2jr = 2 sin^r cos^r; (2.) cos 3_r = 4 cos 3 .r 3 cos x ; (3.) cos 2jr = cos s jr sin 2 A-; (4.) sin 3* = 3 sin.r cos 2 ;r sin s .r; 2 tan JT (5.) tan 2;r = 3. i tan- x (6.) Prove that the equations of exercises i and 3 are cor- rect if ^r = 45. (7) Prove that the equations of exercises (2) and (4) are cor- rect if JT= 120. The following three articles, 23-25, are inserted for completeness. They include the functions of (90 x] and (180 x), which, on account of their great importance, were treated separately in 16 and 20. FUNCTIONS OF ( x), (l8o X\ (l8o+*), (360 *) 23. The line representing any function as sine, cosine, etc. of each of these angles has the same length as the line repre- senting the same function of x. Thus in Figs. 2 and 3, triangle OS'P'= triangle OSP, hence SP=S'/ y> , and OS=OS'. THE TRIGONOMETRIC FUNCTIONS B C C' B FIG. 3 In Figs, i and 4, triangle OSP'=tria.ngle OSP. hence SP'=SP. In Figs, i, 2, and 4, triangle OA 7"=triangle OA T, hence A T' - A 7. In Figs, i, 2, and 4, triangle OC' = triangle OBC. hence C'=BC. Therefore any function of each of the angles ( x}. (180 x), ^), (360 x\ is equal in numerical value to the same function of x. Its sign, however, depends on the direction of the line repre- senting it. Putting in the correct sign, we obtain the following table: sin ( x) = sin x cos( *) = cos* tan ( x) = tan x cot ( x) = cot x sin (180 + *) = sin* cos ( 1 80 + x) = cos x tan(i8o-r-*) = tan* cot (180 + *) = cot* sin (180 x) = sin* cos (180 x) cos* tan(i8o *)= tan* cot(i8o- x) -cot* sin (360 *)=: sin * cos (360 *) = cos* tan (360 *) = tan * cot (360 *) = - cot* 2O PLANE TRIGONOMETRY FUNCTIONS OF (90 }'), (90 +j), (270 -y\ (270 -fj) 24. The line representing the sine of each of these angles is of the same length as the line representing the cosine of j; the cosine, tangent, or cotangent, respectively, are of the same length as the sine, cotangent, and tangent of y. For Triangle OS'P' = triangle OSP, hence S'P'=OS, and OX = S Triangle OA T ' triangle OBC, hence A T' = RC. Triangle OBC = triangle OA T, hencf BC = AT. Therefore any function of each of the angles (90 }'), (90 (270 y), (2 70 +y), is equal in numerical value to the co-function THE TRIGONOMETRIC FUNCTIONS 21 of y. Its sign, however, depends on the direction of the line repre- senting it. Putting in the correct sign, we obtain the following table : sin (90 y) = cos^ sin (90 + y) = cos_y cos (90 v) = sinj' cos (90 + y) = sinjv tan (90 y) coty tan (90 + y) = cot v cot (90 y) = tany cot (90 +y) = tany sin (270 y) = cos_v sin (270 +y) = cosy cos (270 y) smy cos (270 +jy) = sinj tan (270 y) = coty tan (270 + y) = cotjv cot (270 y) = tany cot (270 + r) = tan v 25. Either of the two preceding articles enables us directly to express the functions of any angle, positive or negative, in terms of the functions of a positive angle less than 90. Thus, sin 212 =sin(i8o+ 32)= sin 32; cos 260 = cos (270 10) = sin 10. EXERCISES (i.) What angles less than 360 have the sine equal to \-\/2 ? the tangent equal to\/3? (2.) For what values of x less than 720 is sin;r = ^^/3? (3.) Find the sine and cosine of 30; 765; 120; 210. (4.) Find the functions of 405; 600; 1125; 45; 225. (5.) Find the functions of 120; 225; 420; 3270. (6.) Express as functions of an angle less than 45 the functions of 233: -197: 894. (7.) Express as functions of an angle between 45 and 90, sin 267; tan ( 254) ; cos 950. (8.) Given cos 164 = .96, find sin 196. (9.) Simplify cos(9o + jr)cos(27o .r) sin(i8o x)s'\n(^6o .r). 0-> Simplify;!; / ( ^I^tan(90 + ^) + , ;M . ' _ rt - sin 5 (270 (H.) Express the functions of (x 90) in terms of functions of x. CHAPTER II THE RIGHT TRIANGLE 2V. To solve a triangle is to find the parts not given. A triangle can be solved if three parts, at least one of which is a side, are given. A right triangle has one angle, the right angle, always given ; hence a right triangle can be solved if two sides, or one side and an acute angle, are also given. The parts of the right triangle not given are found by the use of the following formulas: opposite side adjacent side (i) sine =r s - ; (2) cosine =-^- ; 14 (3) tangent = hypotenuse opposite side adjacent side (4) cotangent = hypotenuse adjacent side (5) c'=a* 16 opposite side (6) B = (go A). To solve, select a formula in which two given parts enter; substituting in this the given values, a third part is found. Continue this method till all the parts are found. In a given problem there are several ways of solving the triangle ; choose the shortest. EXAMPLE The hypotenuse of a right triangle is 47.653, a side is 21.34; find the remaining parts and the area. THE RIGHT TRIANGLE SOLUTION WITHOUT LOGARITHMS The functions of angles are given in the table of " Natural Functions." a 21.34 sm/J =-= 7 c 47.653 190612 227880 190612 372680 333571 391090 381224 9866 sin ^ = .4478 ^=26 36' &=<: cos A =47-653 x -8942 47-653 .8942 95306 190612 428877 381224 42.6113126 =42.61 f B =(90 -26 36)^63= = x 21.34 X42. 21-34 42.61 2134 12804 4268 8536 SOLUTION EMPLOYING LOGARITHMS It is usually better to solve triangles by the use of logarithms. The logarithms of the functions are given in the tables of " Logarithms of Functions." * sin A log sin A =log a log c Iog2i.34 =1.32919 log 47.653 = I- 67809 sub. log sin ^4=9.6511010 ^=26 36' 14" cos A=- c log b = log c + log cos A log 47. 653 = 1.67809 log cos 26 36' 14" =9.95140 10 log =1.62949 =42.608 =(90 -26 36' I4")=63 23' 46" area = %ab log area = log + log a + log b log =9.69897 10 Iog2i. 34=1. 32919 log 42. 608 = 1.629-19 log area=2. 65765 2)909.2974 454.6487 area=454.6 * In this solution the five-place table of the " Logarithms of Functions" is used. f No more decimal places are retained, because the figures in them are not accurate ; this is due to the fnct that the table of " Natural Functions" is only iour-place. 24 PLANE TRIGONOMETRY CHECK ON THE CORRECTNESS OK THE WORK = 90.263 x 5.043 90.263 5-043 270789 361052 4513150 ' = 455- Extracting the square root, a = 21.34, which proves the solution cor- rect. = 90.261 x 5.045 log 90.261 1.95550 log 5.045 = 0.70286 2)2.65836 Iog2i.34 = 1.32918 a = 21.34, which proves the solu- tion correct. Remark. The results obtained in the solution of the preceding exercise without logarithms are less accurate than those obtained in the solution by the use of logarithms; the cause of this is that four- place tables have been used in the former method, five place in the latter. EXERCISES 28. (i.) In a right triangle = 96.42, c= 114.81 ; find a and A. (2.) The hypotenuse of a right triangle is 28.453,3 side is 18.197; find the remaining parts. (3.) Given the hypotenuse of a right triangle = 747-24. an acute angle =23 45' ; find the remaining parts. (4.) Given a side of a right triangle = 37.234, the angle opposite = 54 27' ; find the remaining parts and the area. (5.) Given a side of a right triangle = 1.1293, the angle adjacent = 74 13' 27" ; find the remaining parts and the area. (6.) In a right triangle A 1 5 22' 11 ", ^ = . 01793; find b. (7.) In a right triangle B = f\ 34' 53", = 896.33; find a. (8.) In a right triangle <: = 3729.4. = 2869.1 ; find A. (9.) In a right triangle a= 1247, b= 1988 ; find c. Co.) In a right triangle a = 8.6432, = 4.7815 ; find B. The angle of elevation or depression of an object is the angle a line from the point of observation to the object makes with the horizontal. THE RIGHT TRIANGLE *"Yfrus angle x (Fig. i) is the angle of elevation of P if O is the point of observation ; angle y (Fig. 2) is the angle of depressipn of P if O is the point of observation. (11.) At a horizontal distance of 253 ft. from the base of a tower the angle of elevation^of the top is 60 20' ; find the height of the tower. (12.) Fronft|gr top of a vertical cliff 85 ft. high the angle of depres- sion of a buoy is 24 31' 22"; find the distance of the buoy from the foot of the cliff. (13.) A vertical pole 31 ft. high casts a horizontal shadow 45 ft. long ; ' find the angle of elevation of the sun above the horizon. (14.) From the top of a tower 115 ft. high the angle of depression of an object on a level road leading away from the tower is 22 13' 44" ; find the distance of the object from the top of the tower. (15.) A rope 324 ft. long is attached to the top of a building, and the inclination of the rope to the horizontal, when taut, is observed to be 47 r 21' 17"; find the height of the building. (16.) A light- house is 150 ft. high. How far is an object on the surface of the water visible from the top? [Take the radius of the earth as 3960 miles.] (17.) Three buoys are at the vertices of a right triangle; one side of the triangle is 17,894 ft., the angle adjacent to it is 57 23' 46". Find the length of a course around the three buoys. ('Y\aX\W<\><* : ^ (i 8.) The angle of elevation of the top of a tower observed from a point at a horizontal distaitce of 897.3 ft. from the base is 10 27' 42"; find, the height of the tower. (19.) A ladder 42^ ft. long leans against the side of a building; its foot is 25^ ft. from the building. What angle does it make with the ground ? (20.) Two buildings are on opposite sides of a street 120 ft. broad. 26 PLANE TRIGONOMETRY The height of the first is 55 ft. ; the angle of elevation of the top of the second, observed from the edge of the roof of the first, is 26 37'. Find the height of the second building. (21.) A mark on a flag-pole is known to be 53 ft. 7 in. above the ground. This mark is observed from a certain point, and its angle of elevation is found to be 25 34'. The angle of elevation of the top of the pole is then measured, and found to be 34 17'. Find the height of the pole. (22.) The equal sides of-an isosceles triangle are each 7 in. long; the base is 9 in. long. Find the angles of the triangle. Hint. Drnw the perpendicular BD. BD bisects the base, and also the angle ABC. In the right triangle ABD, AB-J in., AD^,\ in., hence ABD can be solved. Angle C=angle A, angle ABC 2 angle ABD (23.) Given the equal sides of an isosceles triangle each 13.44 in., and the equal angles are each 63 21' 42"; find the remaining parts and the area. (24.) The equal sides of an isosceles triangle are each 377.22 in., the angle between them is 19 55' 32". Find the base and the area of the triangle. (25.) If a chord of a circle is 18 ft. long, and it subtends at the centre an angle of 45 31' 10", find the radius of the circle. (26.) The base of a wedge is 3.92 in., and its sides are each 13.25 in. long; find the angle at its vertex. THE RIGHT TRIANGLE 27 (27.) The angle between the legs of a pair of dividers is 64 45', the legs are 5 in. long; find the distance between the points. (28.) A field is in the form of an isosceles triangle, the base of the triangle is 1793.2 ft. ; the angles adjacent to the base are each 53 27' 49". Find the area of the field. (29.) A house has a gable roof. The width of the house is 30 ft., the height to the eaves 25^ ft., the height to the ridge-pole 33^ ft. Find the length of the rafters and the area of an end of the house. (30.) The length of one side of a regular pentagon is 29.25 in. ; find the radius, the apothem, and the area of the pentagon. Hint. The pentagon is divided into 5 equal isosceles triangles by its radii. Let A OB be one of these triangles. ^#=29.25 in. ; angle AOB=^ .of 36o = 72. Find, by the methods previously given, OA, OD, and the area of the triangle A OB. These are the radius of the pentagon, the apothem of the pentagon, and \ the area of the pentagon respectively. (31.) The apothem of a regular dodecagon is 2 ; find the perimeter. (32.) A tower is octagonal ; the perimeter of the octagon is 153.7 ft. Find the area of the base of the tower. (33.) A fence extends about a field which is in the form of a regular polygon of 7 sides; the radius of the polygon is 6283.4 ft. Find the length of the fence. (34.) The length of a side of a regular hexagon inscribed in a circle is 3.27 ft. ; find the perimeter of a regular decagon inscribed in the same circle. (35.) The area of a field in the form of a regular polygon of 9 sides is 483930 sq. ft. ; find the length of the fence about it. 28 PLANE TRIGONOME TR Y SOLUTION OF OBLIQUE TRIANGLES BY THE AID OF RIGHT TRIANGLES 29. Oblique triangles can always be solved by the aid of right triangles without the use of special formulas ; the method is frequently, however, quite awkward ; hence, in a later chapter, formulas are deduced which render the solu- tion more simple. The following exercises illustrate the solution by means of right triangles : (i.) In an oblique triangle a = 3.72, ^ = 47 52', C= 109 10'; find the remaining parts. The given parts are a side and two angles. C Hint. /t=[i8o-(S+ C)]. Draw the perpendicular CD. Solve the right triangle BCD. Having thus found CD, solve the right triangle A CD. (2.) In an oblique triangle a = 89.7, c 125.3, &= 39 8'; find the remaining parts. The given parts are two sides and the included angle. = 125.3 THE RIGHT TRIANGLE 29 Siint.Dra.v/ the perpendicular CD. Solve the right triangle CBD. Having thus found CD and AD(=c DB), solve the right triangle ACD. (3.) In an oblique triangle a = 3.67, = 5.81, A = 27 23'; find the remaining parts. The given parts are two sides and an anglt opposite one of them. C A B' D B Either of the triangles ACB, ACB' contains the given parts, and is a solution. There are two solutions when the side opposite the given angle is less than the other given side and greater than the perpendicular, CD, from the extremity of that side to the base.* , Hint. Solve the right triangle ACD. Having thus found CD, solve the right triangle CDB (or CDB'). (4.) The sides of an oblique triangle are a= 34.2, = 38.6, - = 55.12; find the angles. The given parts are the three sides. C =66.12 * A discussion of this case is contained in a later chapter on the solution of oblique triangles. Hint. PLANE TRIGONOMETRY Let DB=x, Hence In each of the right triangles ACD and BCD the hypotenuse and a side are now known ; hence these triangles can be solved. (5.) Two trees, A and B, are on opposite sides of a pond. The distance of A from a point-C is 297.6 ft., the distance of B from C is 864.4 ft., the angle ACB is 87 43' 12". Find the distance AB. (6.) To determine the distance of a ship A from a point B on shore, a line, BC, $00 ft. long, is measured on shore ; the angles, ABC and ACB, are found to be 67 43' and 74 21' 16" respectively. What is the distance of the ship from the point Bt (7.) A light-house 92 ft. high stands on top of a hill; the distance from its base to a point at the water's edge is 297.25 ft. ; observed from this point the angle of elevation of the top is 46 33' 15". Find the length of a line from the top of the light-house to the point. (8.) The sides of a triangular field are 534 ft., 679.47 ft., 474.5 ft. What are the angles and the area of the field ? (9.) A certain point is at a horizontal distance of 117^ ft. from a river, and is li ft. above the river; observed from this point the angle of depression of the farther ban k is i 1 2'. What is the width of the river? (10.) In a quadrilateral ABCD, AB= 1.41, BC 1.05, CD= 1.76. DA = 1.93, angle ^=75 2i J ; find the other angles of the quadrilateral. THE RIGHT TRIANGLE 31 Hint. Draw the diagonal DB. In the triangle A BD two sides and an included angle are given, hence the triangle can be solved. The solution of triangle ABD gives DB. Having found DB, there are three sides of the triangle DBC known, hence the triangle can be solved. (n.) In a quadrilateral ABCD, AB= 12.1, AD = 9.7, angle A 17 1 8', angle # = 64 49', angle Z)= 100; find the remaining sides, Hint. Solve triangle ABD to find BD. CHAPTER III TRIGONOMETRIC ANALYSIS 30. In this chapter we shall prove the following funda- mental formulas, and shall derive other important formulas from them : sin (05 + j/) = sin x cos y + cos x sin /, -in / //sin./- cost/ -./ sin?/, cos(a? + ?/) = cos a? cosy sin aj slniy, cos(x j/) = cos x cosy + sin aj sinj/; PROOF OF FORMULAS (l l)-(l4) 31. Let angle y2<9>=;tr, angle QOP=y\ then angle A OP -(*+?)- The angles x and j are each acute and positive, and in Fig. i (x-\-y) is less than 90, in Fig. 2 (x+y) is greater than 90. Q In both figures the circle is a unit circle, and SP is perpendicular to OA ; hence $/>= sin (JT +j), C>6'= cos (jc +. TRIGONOMETRIC ANALYSIS 33 Draw DP perpendicular to OQ ; then DP=s\ny, OD = cosy, angle SPD = angle AOQ = x. (Their sides being perpendicular.) Draw DE perpendicular to OA, DH perpendicular to SP. (s,'mx} x OD = s\nx cosy. ED (For OED being a right triangle, = sin jr.) =(cosx] x DP=cosx sinj. HP (For HPD being a right triangle, = cos x.~) Therefore, sin(x + j/) = sinic cos?/ + cosa? sin?/. (ii) Cos O + j) = O5 = OE - HD* OE = (cos x] x 0Z? cos x OE (For OED being a right triangle, - = cos x.) snx sn/. Ff D (For PHD being a right triangle, -~- =sin x.) Therefore, cos (x + y] = cosx co?/ sin a? sin?/. (13) 3. The preceding formulas have been proved only for the case when x and y are each acute and positive. The proof can, however, readily be extended to include all values of x and y. Let y be acute, and let x be an angle in the second quad- rant ; then ;r (90 + ^') where x' is acute. sin (x +y] sin (90 + x' +y) = cos(V-f-j) 24 = cos x' cosj sin x' sin y = sin (90 + x'} cos/ + cos (90 + x') sin _y 24 cosj-|-cos;tr sinj/. If (jc + r) is greater than 90, OS is negative. 34 PLANE TRIGONOMETRY Thus the formula has been extended to the case where one of the angles is obtuse and less than 180. In a similar way the formula for cos(x+y] is extended to this case. By continuing this method both formulas are proved to be true for all positive values of x and y. Any negative angle y is equal to a positive angle y, minus some multiple of 360. The functions of y are equal to those of y' , and the functions of (x-\-y) are equal to those of (x+y 1 ). 9 Therefore, the formulas being true for (x+y'\ are true for A repetition of this reasoning shows that the formulas are true when both angles, x and y, are negative. 33. Substituting the angle y for y in formula (11), it becomes sin (x y) = s\v\x cos( y) + cosx sin {y). But cos( y) = cosjy, and sin ( y)= sin/. 23 Therefore, sin (as y) = s\nx cosy cosx sin y. (12) Substituting (y) for y in formula (13), it becomes cos(x y)=- cosx cos( y) sin^r sin ( j/), cosx cosy + sinx sinjj/. Therefore, co*(x y) = cosx cosy + sinx slny.* (14) EXERCISES 34. (r.) Prove geometrically where x and y are acute and positive : sin(.r ^/) = sin x cos/ COSJT siny, cos(;r _y) = cos^r cosy -f sin^- sin/. * Formulas (12) and (14) are proved geometrically in 34. The geometric proof is complicated by the fact that OD and DP are functions of /, while the functions of y are what we use. TRIGONOMETRIC ANAL YSTS 35 Hint. Angle AOQx, angle POQ=y, and angle AOP-(x-y). Draw PD perpendicular to OQ. Then Z?/'=sin( ;(')=: sin v; but DP is negative, therefore PD taken as positive is equal to sin y : OD = cos ( y) = cos y, Angle HPD=a.ng\e AOQ=x. their sides being perpendicular. Draw DH perpendicular v.o SP, DE perpendicular to OA. s(x-y)=SP=ED-PH. P'rom right triangle OED, ED.= (s'm x)x 0Z>=sin x cos/. From right triangle DHP, PS/=(cosx)xPZ)=cosx siny. Therefore, sin (x j)=sin x cosy cos x siny. Cos (.r - v) = O -S = OE + DH. From right triangle OED, OE=(cosx)x OD=cosx cosy. From right triangle DHP, Z>//=(sin x) x / > Z> = sin x siny. Therefore, cos (xy)=cos x cos j + sin x sin_y. (2.) Find the sine and cosine of (45+^), (30 .r), (6o+.r), in terms of sin x and cos.r. (3.) Given sin.r=f, slt\}>=.^, .r and y acute; find sin(-tr+j) and sin(.r y). (4.) Find the sine and cosine of 75 from the functions of 30 and 45. Hint. 75=(45 + 30). (5.) Find the sine and cosine of 15 from the functions of 30 and 45. (6.) Given x and y, each in the second quadrant, sin x = $, siny = find sin(^r+j) and cos(jr y). (7.) By means of the above formulas express the sine and cosine of (i 80 x), (i8o+jr), (270 ,r), (270-}- x), in terms of sin x and cosx. (8.) Prove sin (6o+ 45) + cos (60 + 45) = cos 45. (9.) Given sin 45 = -^\/ 2 > COS 45 i "V/ 2 ! find sin 90 and cos 90. (10.) Prove that sin (60 -f- x) sin (60 .r):=sin.r. 36 PLANE TRIGONOMETRY TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES ?- Tan/'*- ,A _ sin (x+y)_ sin* cosj+cos^r sinjy r>. 1 clll i;t -p y I cos(;r+j) cos* cosj Dividing each term of both numerator and denominator of the right-hand side of this equation by cos* cosj, and sin remembering that = tan, we have cos tan x + tan y *"(+*)=!_ tang tBn V (15) In a similar way, dividing formula (12) by formula (14), we obtain tan x tan // FUNCTIONS OF TWICE AN ANGLE 36. An important special case of formulas (11), (13), and (15) is when y = x', we then obtain the functions of 2x in terms of the functions of x. From (11), s\n(x-\-x} s\\\x C.OSX + CQSX sin^r. Hence sin2ic = 2 sinoc cosx. (17) From (13), cos2a; = co8 2 iK-8ln 2 a;. (18) Since cos s ^=: I sin 2 ;r, and sin"^r= I cos";r, we derive from equation (18), cos2x= i 2sinV, (19) and cos2^r=2 cosV I. (20) From (i 5), t** = c ' (21) FUNCTIONS OF HALF AN ANGLE ,77. Equations (19) and (20) are true for any angle; there- fore for the angle \x. From(i9), cosx=i 2 sin 9 ^; TRIGONOMETRIC ANALYSIS I cos;tr or sin $x /I cos a? , N therefore, sm-x = \/ -- - (22) From (20), cos.* = 2 cos^x i ; 1 + cosx or cos \x -- 5 /I + cos a? therefore, cosfx = y gj Dividing (22) by (23), we obtain /I cos a; / x tan \x = \ / (24) V 1 + cosx FORMULAS FOR SUMS AND DIFFERENCES OF FUNCTIONS 38, From formulas (n)-(i4), we obtain sin (x + j) + sin (x y) = 2sinx cosj ; sin (x-\-y) sin (x y} = 2cosx siny; cos (x -\-y) + cos (xy) = 2 cosx cosy ; cos(^4-jv) cos(^r y) 2sin^r siny. Let u (x +y) and v (x y) ; Substituting in the above equations, we obtain siiiw + sin v = 2 sin-J(? +v)cos-|(u v); (25) sin n - sin v = 2 cos-J (i + v) sin (le - v) ; (26) cos M 4- cos v= 2 cofrj (e 4- v) cos^- (ie v) ; (27) cos? cos^= 2sin-^-(w + v) sin^(u v). (28) Dividing (25) by (26), slnu+slnv _tan^(u+v)^ , . EXERCISES 39. Express in terms of functions of x, by means of the formulas of this chapter, 45071 38 PLANE TRIGONOMETRY (i.) Tan(i8o .r); tan (i8o + .r). (2.) The functions of (x 180). (3.) Sin (x 90) and cos (x 90). (4.) Sin(jr 270), and cos (^ 270). (5.) The sine and cosine of (45*); of (45 -f- x). (6.) Given tan 45= i, tan 30 = 1/3; find tan 75; tan 15. cot a? cot?/ 1 (7.) Provecot (x + y) = . (30) cot y -i cot x Hint. Divide formula (13) by formula (n). COtiC C0t?/ + l (8. Prove cot,x-y ) = -. (31) cot //-cot x (9.) Prove cos (3o+j) cos (30 y) = sin/. (10.) Prove sin 3^ = 3 sin x 4 sin'jr. Hint. Sin 3jr = sin (x+2x). (n.) Prove cos ^x = 4 cos s 3 cos x. (12.) If x and / are acute and tan .* = , tan/ = J, prove that (*+/) = 45. (13.) Prove that tan (.*-}- 45) = i tan^r (14.) Given sin/ = f and/ acute; find sin /, cos^y, and tan^/. (15.) Given cos;r= | and ^r in quadrant II; find sin 2x and cos 2.r. (i 6.) Given cos 45 = \ \/2 ; find the functions of 22^. (17.) Given tan x = 2 and jr acute ; find tan \x. (i 8.) Given cos 30 = -y/3 : find the functions of 15. (19.) Given cos9o = o; find the functions of 45. (20.) Find sin^-r in terms of sin^r. (21.) Find cos5,r in terms of cos x. (22.) Prove sin (x -\-y -j- ?) = sin x cos/ cos .z+cos x sin y cos s+cos x cosy sinz sin x slny sinz. Hint. Sin (x+y+z)- sin (x+y) c (23.) Given tan 2.*- = 3 tan x; find jr. (24.) Prove sin 32 + sin 28 = cos 2. (25.) Prove tan x -f cot x = 2 esc 2x. (26.) Prove (sin^-r + cosfr)^ i -4-sin x. (27.) Prove (sin %x cos %x)* \ sin x. TRIGONOMETRIC ANALYSIS 39 (28.) Prove cos 2x = cos 4 .r sin 4 .r. (29.) Prove tan (45 -+- x) + tan (45 x) = 2 sec 2x. 2 tan x (30.) Prove sin2jr = (31.) Prove cos2.r = i+tan 2 .*-' i tanlr i -f- tan a .f ' I 4- sin 2.r /tan x-\- iV (32.; Prove- =(- -) i sin 2.r \tanjr i/ (33-) Prove (34.) Prove I +COS X sin x i cos x . , cos x cosy (35.) Express as a product -.* cos x -j- cosy COSJT cos r _ 2 sini^+r) sin^(jr y) COSJT + COSJ 2 co tan x 4- tan_y (36.) Express as a product . cot x + cot_y cos (x -\-y) (37.) Prove i tan-r tan y= -^-. cosj THE INVERSE TRIGONOMETRIC FUNCTIONS 40. Dcf. The expressions sin-'tf, cos-'tf, tan-', etc., de- note respectively an angle whose sine is a, an angle whose cosine is a, an angle whose tangent is a, etc. They are called the inverse sine of a, the inverse cosine of a, the inverse tangent of a, etc., and are the inverse trigono- metric functions. Sin l a is an angle whose sine is equal to a, and hence de- notes, not a single definite angle, but each and every angle whose sine is a. * Since quantities cannot be added or subtracted by the ordinary operations with logarithms, an expression must be reduced to a form in which no addition or subtraction is required, to be convenient for logarithmic computation. 40 PLANE TRIGONOMETRY Thus, if sin*=i, *=3O, 150, (30 + 360), etc., and sin~ '^=30, 150, (3O + 36o ), etc. Remark. The sine or cosine of an angle cannot be less than i or greater than -f- 1; hence sin- 1 ** and cos~'a have no meaning unless a is between i and + i. In a similar manner we see that sec~'a and csc-'tf have no meaning if a is between i and -j- i. EXERCISES 41. (i.) Find the following angles in degrees: s\n- l %\/2, tan-'( i), sin-'( i). cos-'^, cos" 1 1, (2.) If x cot~4, find tan x. (3.) If A- = sin- I |, find cos^r and tan x. (4.) Find sin(tan-'^ -v/3)- (5.) Find sin (cos -I |). (6.) Find cot (tan- 1 ^f). (7.) Given sin~'a = 2 cos-'rt, and both angles acute; find a. (8.) Given sin-'a = cos~ I ; find the values of sin-'a less than 360. (9.) Given tan~'i = \ tan~'o, and both angles less than 360; find the angles. (10.) Given sin-'rt r^cos-'rt and sin-'a + cos-'a =450; find sin-'rt. (11.) Prove sin (cos -I rt):=: V 1 <** Hint. Let JT=COS '<7 ; then a = cbsjc, sin x= \/i cos' 2 * = \/i a 9 . (12.) Prove tan^an-'a-l-tan- 1 ^)^ _ , a b ( H-) Prove tan (tan-'rt tan~'^) = - i -\- ao (14.) Prove cos (2 cos- 1 ^ ) = 2 rt s i. (15.) Prove sin(2cos-'rt)=2rt y/i a*. (i 6.) Prove tan(2tan- r rt)= : - (17.) Prove cos (2tan~'a)= j- (18.) Prove sin(sin~ 'a + cos~"'^) = rt V CHAPTER IV THE OBLIQUE TRIANGLE DERIVATION OF FORMULAS 42. The formulas derived in this and the succeeding articles reduce the solution of the oblique triangle to its simplest form. r. C C FIG. 3 Draw the perpendicular CD. Let CD=h, h Then T : (In Fig. 2 -=sin(i8o-//)=sin/0 and . - = sm B. /T T-' tl (In Fig. 3 -=rs a By division we obtain, sin A b sin i; Remark. This formula expresses the fact that the ratio of two sides of an oblique triangle is equal to the ratio of the sines of the angles opposite, and does not in any respect depend upon which side has been taken as the base. Hence if the letters are advanced one step, as shown in the figure, we obtain, as another form of the same formula, PLANE TRIGONOMETRY b _ sin B c sinC Repeating the process" we obtain c sin C - = . rt sin ^4 The same procedure may be applied to all the formulas for the solution of oblique triangles. Henceforth only one expression of each formula will be given. Formula (32) is used for the solution of triangles in which a side and two angles, or two sides and an angle, opposite one of them are given. 43. We obtain from formula (32) by division and compo- sition, a b sin/i sin .5 a + b ~ sin A + sin B ' By formula (29), denoting the angles by A and B, in- stead of u and v, Therefore, a b tan |(/2 -{-/?)' ^(^l B) (33) This formula is used for Hie solution of triangles in which two sides and the included angle are given. 44. Whether A is acute or obtuse, we have C C (If A is acute (Fig. i),AD bcosA,DB = AB AD c bcosA, CD . If A is obtuse (Fig. 2), AD = cos (iBoP-A) = -b cos A, DB-AB , CD b sin(i8o- A}-= THE OBLIQUE TRIANGLE 43 a y = (e-bcosA}' > + (dsmA)\ =e*-2 be cos A + 2 (cosM +sinM). Therefore, a 9 6 2 -f tf-zbc cos A. (34) Tliis formula is used in deriving formula (37). It is also used in the solution without logarithms of tri- angles of which tivo sides and the included angle or three sides are given. 2L2 | 2 2 5. From formula (34), cos A = ; 2bc From formula (22), 37, P^- 1 2 sin A = i cos A = i 2 be Hence 2 sin 2 ^A= ; , 2 be 2bc 2 be Let s = , then (a b + c} = 2(s b], and 2 (S-C). ,_2 ( S -b}(s-c] be Substituting, 2 sin 8 i Hence sin^ = x /^ y- C J" (35) From formula (23), 37, 2 cos 2 1 A = 2 be 2s(s- -a} ~~bc~ In extracting the root the plus sign is chosen because it is known that A is positive. 44 PLANE TRIGONOMETRY Hence cos^A =. Dividing (35) by (36), we obtain (s a) (s-a)(s-b](s-c) " sa (36) (37) K Let js s a Formulas (37) and (38) are used to find the angles of a tri- angle when the three sides are given. FORMULAS FOR THE AREA OF A TRIANGLE 46. Denote the area by 5. (In Fig. I, CD=asinB; in Fig. 2, CD = a sin (180-^) = a sin Z?.) In Figs, i and 2, S=\c.CD. Hence S = $acsinB. (39) From formula (17), sin B2 sin^.5 cos^Z?. THE OBLIQUE TRIANGLE 45 Substituting for s'm^H and cos^-^ the values found in formulas (35) and (36), we obtain sin> = \/s(s a)(s b)(s c}. ac* Therefore, S=*/s(sa)(sb)(sc). (40) This formula may also be written, S=sK. (41) Formula (39) is used to find the area of a triangle when two sides and the included angle are known; formula (40) or formula (41), when the three sides are known. THE AMBIGUOUS CASE 47- The given parts are two sides, and the angle opposite one of them. Let these parts be denoted by a, b, A. C If a is less than b and greater than the perpendicular CD (Fig. i), there are the two triangles ACB and ACB', which contain the given parts, or, in other words, there are two solutions. If a is greater than b (Fig. 2), there is one solution. If a is equal to the perpendicular CD, there is one solu- tion, the right triangle A CD. 46 PLANE TRIGONOMETRY If the given value of a is less than CD, evidently there can be no triangle containing the given parts. Since CD=bs\i\A t there is no solution when a < l>s\nA ; there is one solution, the right triangle A CD when a=bs\nA; there are two solutions when a < l> and 48. CASE I. Given a side and two angles. EXAMPLE Given a = 36.738, A = 36 55' 54", B 72 5' 56", C=i8o (A-\-B)=. 180 109 r 5o"m7o58' 10". To find c. To find b. I > _sm a sin A log fl=l. 56512 log sin ^=9.97845 cologsin .,4 =0.22 1 23 log =1.76480 ^ = 58.184 sn a sin A =r. 56512 logsinC=9-97559-io colog sin A =o. 22123 log=i. 76194 ^=57.80 Determine from c, C, and .5 by the formula This check is long, but is quite certain to reveal an error. A check which is shorter, but less sure, is b _ sin B c sin C Solve the following triangles : (i .) Given a = 567.25, A = n 15', B = 47 12'. (2.) Given a = 783.29, A = 81 52', B = 42 27'. (3.) Given c= 1125.2, A = 79 15', -5=55 n'. (4.) Given =15.346, ^=15 51', C=58 10'. (5.) Given a = 5301. 5, ^=69 44', =41 18'. (6.) Given =1002.1, ^ = 48 59', = 76 3'. 4,9. CASE II. Given two sides of a triangle and tJie angle opposite one of them. THE OBLIQUE TRIANGLE 47 EXAMPLE Given a 23.203, b 35.121, A = 36 8' 10". C To find B and B' . sin A ~ a log =1.54556 log sin ,4=9. 77064 10 colog=8. 63445 10 log sin /?=g. 95065 10 ^=63 12' C and C'. C =i8o-(A +)=8o 39' 50' 6" ' = 180 -(/* + ')= 27 3' 50" 7't> find c and c . c sin C a sin ^4 log 0=1. 36555 log sin C= 9. 99421 10 colog sin /f =0.22936 log f=i. 58912 ^=38.825 log a = i. 36555 log sin C' =9.6580010 colog sin A =0.22936 log f'=i. 25291 ^' = 17.902 Check. Determine b from c, C, and j? by the formula This check is long, but is quite certain to reveal an error. A check which is shorter, but less sure, is c sin C (i.) How many solutions are there in each of the following? (i.) ^ = 30, a= 15, b 20; (2.) A = 30, a = 10, b = 20 ; (4.) # = 37 23', a = g.i, 6 = 7.5. 48 PLANE TRIGONOMETRY Solve the following triangles, finding all possible solutions: (2.) Given A = 147 12', a = 0.63735, = 0.34312. (3.) Given A = 24 31', a = 1.7424, = 0.96245. (4.) Given A = 21 21', a = 45.693, b= 56.723. (5.)Given^4= 61 16', a = 9.5124, = 12.752. (6.) Given C= 22 32', # =0.78727, <: = 0.473 1 i. ?0. CASE III. Given two sides and the included angle. EXAMPLE Given a = 41. 003, = 48.718, C 68 33' 58" ; find the remaining parts and the area. To find A and B. n(# - A) _b - a b-a 7.715 b + a = 89.721 log (b a) =0.88734 colog (b + a) = 8.04710 10 log tan %(B + A) = o.i 6639 log ia.n^(B A) = 9.10083 10 l(B - A)= 7 11' 20" ^ = 62 54' 2l" ^=48 31' 41" To find c. c __ sin C a sin/4 logrt = 1.61281 log sin C= 9.96888 -10 colog sin// =0.12535 log * = 1.70704 <= 50.938 To find the area. S=\ab sin C log ^ = 9.69897 10 logrt = 1.61281 log<= 1.68769 log sin (7=9.96888 10 log 6'= 2.96835 5= 929.72 sin C Check. c ~b sin B log sin 2? = 9.94951 10 log c = 1 . 70704 colog b 8.31231 10 log sin C 9.96886 10 THE OBLIQUE TRIANGLE 49 Solve the following triangles, and also find their areas : (i.) Given A= 41 15', =0.14726, c=. 0.10971. (2.) Given C= 58 47', =i 1.726, ^=16.147. (3.) Given B 49 50', ^ = 103 74, ^-=99.975. (4.) Given A= 33 31', =0.32041, ^=0.9203. (5.) Given C=i28 7', =17.738, 0=60.571. TJf. CASE IV. Given the three sides. EXAMPLE Given = 32.456, = 41.724, c =53.987 ; find the angles and area. ^ = 64.084 (s a) 31.628 (s b} = 22.360 (s c) 10.097 / (s - a)(s - />)(s - f) A \ / v s log (s a) = 1.50007 log (s-t) =1.34947 log (s r)=i.oo4ig colog .r = 8. 19325 10 2)2.O46q8 log K 1.02349 To find A. K tanl// = s a log A"= i. 02 349 log (s-a) = i. 50007 sub. log tan|^=g. 52342 10 |^ = i8 27' 23" ^=36 54' 46" To find B. K" tan | .#= . j = 11.698, ^=33.328. (5.) Given a = ii3.oS, =131.17, ^=114.29. (6.) Given a= .9763, =1.2489, c= 1.6543. EXERCISES 52. (i.) A tree, A, is observed from two points, B and C, 1863 ft. apart on a straight road. The angle BCA is 36 43', and the angle CBA is 57 21'. Find the distance of the tree from the nearer point. (2.) Two houses, A and B, are 3876 yards apart. How far is a third house, C, from A, if the angles ABC and AC are 49 17' and 58 18' respectively ? (3.) A triangular lot has one side 285.4 ft. long. The angles adja- cent to this side are 41 22' and 31 19'. Find the length of a fence around it, and its area. (4.) The two diagonals of a parallelogram are 8 and 10, and the angle between them is 53 8' ; find the sides of the parallelogram. (5.) Two mountains, A and B, are 9 and 13 miles from a town, C; the angle ACB is 71 36' 37". Find the distance between the moun- tains. (6.) Two buoys are 2789 ft. apart, and a boat is 4325 ft. from the nearer buoy. The angle between the lines from the buoys to the boat is 16 13'. How far is the boat from the farther buoy? Are there two solutions? (7.) Given a =64.256, r= 19.278, C= 16 19' u"; find the differ- ence in the areas of the two triangles which have these parts. (8.) A prop 13 ft. long is placed 6 ft. from the base of an embank- ment, and reaches 8 ft. up its face; find the slope of the embank- ment. (9.) The bounding lines of a township form a triangle of which the sides are 8.943 miles, 7.2415 miles, and 10.817 miles; find the area of the township. (10.) Prove that the diameter of a circle circumscribed about a triangle is equal to any side of the triangle divided by the sine of the angle opposite. THE OBLIQUE TRIANGLE 5! Hint. By Geometry, angle A OJ3= 2 C. Draw OD perpendicular to AB. Angle DOB-\AOB=C. DBr sin DOB=r sin C. Hence c=2rsmC, ** x-i - smC (ii.) The distances AB, BC, and AC, between three cities, A, B, and Care 12 miles, 14 miles, and 17 miles respectively. Straight rail- roads run from A to B and C. What angle do they make ? (12.) A balloon is directly over a straight road, and between two points on the road from which it is observed. The points are 15847 ft. apart, and the angles of elevation are found to be 49 12' and 53 29' respectively. Find the distance of the balloon from each of the points. (13.) To find the distance from a point A to a point B on the op- posite side of a river, a line, AC, and the angles CAB and ACB were measured and found to be 315.32 ft., 58 43', and 57 13' respectively. Find the distance AB. (14.) A building 50 ft. high is situated on the slope of a hill. From a point 200 ft. away the building subtends an angle of 12 13'. Find the distance from this point to the top of the building. (15.) Prove that the area of a quadrilateral is equal to one-half the product of the diagonals by the sine of the angle between them. (16.) From points A and B, at the bow and stern of a ship respec- tively, the foremast, C, of another ship is observed. The points A and B are 300 ft. apart ; the angles ABC and BAC are found to be 52 PLANE TRIGONOMETRY 65 31' and 1 10 46' respectively. What is the distance between the points A and C of the two ships ? (17.) TAVO steamers leave the same port at the same time ; one sails, directly northwest, 12 miles an hour; the other 17 miles an hour, in a direction 67 south of west. How far apart will they be at the end of three hours ? (18.) Two stakes, yf and ft, are on opposite sides of a stream; a third stake, C, is set 62 ft. from A ; the angles ACB and CAB are found to be 50 3' 5" and 61 18' 20" respectively. How long is a rope connecting A and Z?? (19.) To find the distance between two inaccessible mountain-tops, A and B, of practically the same height, two points, C and D, are taken one mile apart. The angle CDA is found to be 88 34', the angle DC A is 63 8', the angle CDS is 64 27', the angle DCB is 87 9'. What is the distance? (20.) Two islands, B and C, are distant 5 and 3 miles respectively from a light-house, A, and the angle BAC is 33 11'; find the dis- tance between the islands. (21.) Two points, A and B, are visible from a third point C, but not from each other; the distances AC, BC, and the angle ACB were measured, and found to be 1321 ft., 1287 ft., and 61 22' respectively. Find the distance AB. (22.) Of three mountains, A, B, and C, B is directly north of C 5 miles, A is 8 miles from C and 11 from B. How far is A south of /?? (23.) From a position 215.75 ft. from one end of a building and 198.25 ft. from the other end, the building subtends an angle of 53 37' 28"; find its length. (24.) If the sides of a triangle are 372.15, 427.82, and 404.17 ; find the cosine of the smallest angle. (25.) From a point 3 miles from one end of an island and 7 miles from the other end, the island subtends an angle of 33 55' 15"; find the length of the island. (26.) A point is 13581 in. from one end of a wall 12342 in. long, and 10025 in. from the other end. What angle does the wall subtend at this point? (27.) A straight road ascends a hill a distance of 213.2 ft., and is in- THE OBLIQUE TRIANGLE 53 clined 12 2' to the horizontal; a tree at the bottom of the hill subtends at the top an angle of 10 5' 16". Find the height of the tree. (28.) Two straight roads cross at an angle of 37 50' at the point A ; 3 miles distant on one road is the town B, and 5 miles distant on the other is the town C. How far are B and C apart ? (29.) Two stations, A and B, on opposite sides of a mountain, are both visible from a third station, C; AC =11.5 miles, BC = 9.4 miles, and the angle ACB = 59 31'. Find the distance from A to B, (30.) To obtain the distance of a battery, A, from a point, B, of the enemy's lines, a point, C, 372.7 yards distant from A is taken ; the an- gles ACB and CAB are measured and found to be 79 53' and 74 35' respectively. What is the distance ABt (31.) A town, B, is 14 miles due west of another town, A. A third town, C, is 19 miles from A and 17 miles from B. How far is C west of A} (32.) Two towns, A and B, are on opposite sides of a lake. A is 18 miles from a third town, C, and B is 13 miles from C; the angle ACB is 13 17'. Find the distance between the towns A and B. (33.) At a point in a level plane the angle of elevation of the top of a hill is 39 51', and at a point in the same direct line from the hill, but 217.2 feet farther away, the angle of elevation is 26 53'. Find the height of the hill above the plane. (34.) It is required to find the distance between two inaccessi- ble points, A and B. Two stations, C and D, 2547 ft. apart, are chosen and the angles are measured ; they are ACB^=2j 21', BCD =33 14', BDA = \% 17', and ADC$\ 23'. Find the distance from AloB. (35.) Two trains leave the same station at the same time on straight tracks inclined to each other 21 12'. If their average speeds are 40 and 50 miles an hour, how far apart will they be at the end of the first fifteen minutes ? (36.) A ship, A, is seen from a light-house, B; to determine its dis- tance a point, C, 300 ft. from the light-house is taken and the angles BCA and CBA measured. If BCA = 108 34' and CBA =65 27', what is the distance of the ship from the light-house? 54 PLANE TRIGONOMETRY (37.) Prove that the radius of the inscribed circle of a triangle is equal to a sin^J5 sln^C sec^A. Hint. Draw OB, OC, and the perpendicular OD. OB and 0C bisect the angles B and C respectively, and ODr. cosj^l Hence sin-^ Li sin \ C sin -J- H sin -J C sin ^ j9 sin ^ C = a sin A^ sin A Csec -J/4. CHAPTER V CIRCULAR MEASURE GRAPHICAL REPRESENTATION CIRCULAR MEASURE 53. The length of the semicircumference of a circle is TrR (^ = 3.14159-!-); the angle the semicircumference sub- tends at the centre of the circle is 180. Hence an arc whose length is equal to the radius will subtend the angle T 80 ; this angle is the unit angle of circular measure, and is called a radian. 7T R If the radius of the circle is unity, an arc of unit length subtends a radian ; hence in the unit circle the length of an arc represents the circular measure of the angle it subtends. 7T 7T Thus, if the length of an arc is , it subtends the angle - radians. C' A' bmce one radian = I8 7T V, , we have 90 radians, i8o = 7r radians, 56 PLANE TRIGONOMETRY 270= -- radians, 360 27r radians, etc. The value of a radian in degrees and of a degree in radians are : i radian = 57.29578, = 57 i 7' 45". i .0174533 radian. In the use of the circular measure it is customary to omit the word radian ; thus we write - , it. etc., denoting - radians, ir radians, etc. On the other 2 to 2 hand, the symbols ' " are always printed if an angle is measured in degrees, minutes, and seconds ; hence there is no confusion between the systems. EXERCISES (i.) Express in circular measure 30, 45, 60, 120, 135, 720, 990. (Take 77=3.1416.) (2.) Express in degrees, minutes, and seconds the angles , ,-,-. (3.) What is the circular measure of the angle subtended by an arc of length 2.7 in., if the radius of the circle is 2 in.? if the radius is 5 in. ? 54. The following important relations exist between the circular measure x of an angle and the sine and tangent of the angle. 7T (i .) If x is less than -, sin x < x < tan x. O s Draw a circle of unit radius. By Geometry, SP >- . As x approaches the limit o, cos^r approaches the length of the radius, that is, I, as a limit. Therefore, approaches the limit I. x sinje Dividing i > > cos^r by cos^r, we obtain X i tan;ir "^ -^. T COS X X As x approaches the limit o, cos^r approaches the limit I ; hence approaches the limit I. cos* tcLtl X Therefore, - approaches the limit I. PERIODICITY OF THE TRIGONOMETRIC FUNCTIONS 55. The sine of an angle x is the same as the sine of (^+360), (>-|-720 ), etc. that is, of (x+2mr\ where n is any integer. The sine is therefore said to be a periodic* function, hav- ing the period 360, or 2?r. The same is true of the cosine, secant, and cosecant. * If a function, denoted by/(^), of a variable x, is such that f(x + k)=f(x) for every value of x, k being a constant, the function f(x) is periodic; if k is the least constant which possesses this property, k is the period of /(*). 58 PLANE TRIGONOMETRY The tangent of an angle x is the same as the tangent of (,r-f 1 80), (^-+360), etc. that is, of (x+mr\ where n is any integer. The tangent is therefore a periodic function, having the period 180, or TT. The same is true of the cotangent. GRAPHICAL REPRESENTATION 56. On the line OX lay off the distance OA(=x) to rep- resent the circular measure of the angle x. At the point A erect a perpendicular equal to sin x. If perpendiculars are thus erected for each value of x, the curve passing through their extremities is called the sine curve. If sin* is negative, the perpendicular is drawn downward. In a similar manner the cosine, tangent, cotangent, secant, and cosecant curves can be constructed. Sine Curve Cosine Curve GRAPHICAL REPRESENTATION 59 1 I f I ' Tangent Curve f Cotangent Curve 6o PLANE TRIGONOMETRY V.JT SECANT CURVE If the distances on OX are measured from O' instead of (9, we obtain from the secant curve the cosecant curve. In the construction of the inverse curves the number is represented by the distance to the right or left from O\ the circular measure of the angle by the length of the per- pendicular erected. All of the preceding curves, except the tangent and co- tangent curves, have a period of 2?r along the line OX\ that is, the curve extended in either direction is of the same form in each case between 2ir and 477-, 477 and 677, 2?r and o, etc., as between o and 2?r, while the corresponding inverse curves repeat along the vertical line in the same period. The period of the tangent and cotangent curves is TT. GRAPHICAL REPRESENTA TION 61 -1 +1 INVERSE SINE CUKVE INVERSE COSINE CUKVH J I -3 -2 -1 +2 +3 INVERSE TANGENT CURVE 62 PLANE TRIGONOMETRY -3 -1 t-1 +2 + 3 INVERSE SECANT CHAPTER VI COMPUTATION OF LOGARITHMS AND OF THE TRIG- ONOMETRIC FUNCTIONS-DE MOIVRE'S THEOREM HYPERBOLIC FUNCTIONS 57. A convenient method of calculating logarithms and the trigonometric functions is to use infinite series. In works on the Differential Calculus it is shown that /-f2 /y3 /y4 (1 +)=*-+-- + . . . (I) - - a? 2 , a? 1 x 6 Another development which we shall use later is e * = l + Ti + & + . + *< + - <4> where ^=2.7182818 ... is the base of the Naperian system of logarithms. 58. The series (i) converges only for values of x which satisfy the inequality i<;r^i. The series (2), (3), and (4) converge for all finite values of x. It is to be noted that the logarithm in (i) is the Naperian, and the angle x in (2) and (3) is expressed in circular measure. * 31 denotes 1x2x3; 41 denotes I X 2x3x4, etc. 64 PLANE TRIGONOMETRY COMPUTATION OF LOGARITHMS 59. We first recall from Algebra the definition and some of the principal theorems of logarithms. The logarithm to the base a of the number m is the number x which satisfies the equation, a* = m. This is written x = \og a m. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Thus \og a mn = log a m + ]og a n. The logarithm of the quotient of two numbers is equal to the log- arithm of the dividend minus the logarithm of the divisor. fflf Thus log a - = log a m log a . The logarithm of the power of a number is equal to the logarithm of the number multiplied by the exponent. Thus log a m^p log a m. To obtain the logarithm of a number to any base a from its Na- perian logarithm, we have log, m log a m = - = M a log, m, log, a where M. = , - ; M is called the modulus of the system. log,tf 60. We proceed now to the computation of logarithms. The series (i) enables us to compute directly the Naperian logarithms of positive numbers not greater than 2. Example. To compute log*- to five places of decimals. 2 Substitute - for x in (i): 2 2/22 2* 3 2 4 2 If the result is to be correct to five places of decimals, we nyist take enough terms so that the remainder shall not affect the fifth decimal place. Now we COMPUTATION OF LOGARITHMS know by Algebra that in a series of which the terms are each less in numerical value than the preceding, and are also alternately positive and negative, the re- mainder is less in numerical value than its first term. Hence we need to take enough terms to know that the first term neglected would not affect the fifth place. Positive terms i .5OOOOOO 2 I j .0416667 3 2 i I .0062500 5 2 I , , .OOIIl6l 7 2 1 i I .0002170 9 2 i I 77 .OOOO444 i I ~T = cos_y sin (30 /), cos (yP-\-y) = cos (30 y) sin/. 08. To employ series (2) and (3) in computing the sine and cosine we must first convert the angle into circular measure. To do. this we recall that i = . 017453293, i ' = .0002908882, i " = .000004848 1 37. Example. To compute the sine and cosine of 12 15' 39". 12 = .209439516 15' =.004363323 39" = .000189076 12 15' 39" = .213991915 in circular measure. PLANE TRIGONOMETRY * i * sm;r = x 1 - 3! 5! x=. 2139919 x* =.0000037 .2139956 subtract =.0016332 sin jr=. 2 1 23624 Correct to five decimal places. COS*=I --- --- 2 ! 4 ! 1 = 1.0000000 X* == .0000874 4! _ _ 1.0000874 jt 2 subtract -= .0228963 COSJT= .9771911 Correct to five decimal places. DE MOIVRE'S THEOREM 09. In Algebra we learn that the complex number (8) -may be represented graphically thus : Y Take two lines, OX and OY, at right angles to each other. To the number a will correspond the point A, whose dis- tances from the two lines of reference are (3 and a re- spectively. This geometrical representation shows at once that we can also write a in the form = r(cos + t sin 5). (9) 70. From Algebra we recall the definition of the sum of the complex numbers a = a + ifi and b = y + fi>\ namely Subtraction is defined as the inverse of addition, so that DE MO IV RE'S THEOREM 71 Multiplication is most conveniently defined when a and b are written in form (9). If a r (cos^+/ sin ?) and bs (cos^+/ sin0), their product is defined by the equation ab rs [cos (& + ^>) -f- * sin ($-\- 0)]. (i o) Division is defined as the inverse of multiplication, so that - = - [cos (3 - 0) 4- * sin (> - f)]. Finally, we recall that in an equation between complex numbers, a + //3 = y + a, we have = y, 13 = $. (n) 7 1. Consider the different powers of the complex number *:=cos $ + / sin. By (10) we have x* = (cos -f * sin ^) (cos 3+t sin -&), cos 2^ + 1 'sin 2$. x* = x* . jc = (cos2$+/ sin 2$) (cos &-!-/' sin ^), = cos3^ + / sin 3.?. And, in general, for any integer n, ^ = (cos ^+/ sin -&)* = cos n$-\-i sin n. From this equation we have De Moivre's Theorem, which is expressed by the formula nw-&). (12) 72. An interesting application of De Moivre's Theorem is the expansion of sin nx and cosnx in terms of sin x and cos^r. Expanding the left-hand side of (12) by the bino- mial theorem, and substituting x for S, we have "~ 2 cosnx+t sin ^ = cos** x-\-n cos""" 1 x (/sin x) -{- 3 ! )' -f . 72 PLANE TRIGONOMETRY or cosnx + i sinw-x^fcos*.* -- - - - cos*" 2 * sin 8 *-}- . . .) \ 2 I ) n ( i) (n 2) 1 + t \n cos"" 1 x sin* -- -i cos*" 3 * sin'^-f .... Equating real and imaginary parts, as in (u), we have cos n ~ 2 x sin' ft (fi I ) (fi cosnx=cos"x -- - j : cos n ~ 2 x sin'^-j- . . . (13) 3 , : + ...<-4> Example. n = 5. cos 5 x = cos 6 x 10 cos s .r sin".r-|-5 COSJT sin* jr. sin 5^-= 5 COS*JT sin.r 10 cos 2 x sin 3 .r + sin* x. THE ROOTS OF UNITY 75. We find another application of De Moivre's Theorem in obtaining the roots of unity. The w th roots of unity are by definition the roots of the equation Every equation has n roots and no more ; hence, if we can find n distinct numbers which satisfy this equation we shall have all the th roots of unity. Consider the )i numbers 2-rrr . . 2irf x r = cos (-; sin , n n r=o, i, 2, ... n i. Geometrically these numbers are represented by the n vertices of a regular polygon. They are, therefore, all dif- ferent. We shall see now that they are precisely the th roots of unity. In fact, we have by (12), / 2irr 2irr\ n -*":=( cos f-z sin - ] . \ H n I THE ROOTS OF UNITY 73 / 27rr\ . . / 2irr\ =cos I n . -|4-< Sin ( . ), V / V / = cos 27rr+/sin 2:rr, = 1 4- z . o = i . Therefore ,t' r is one of the roots of unity. Thus the cube roots of unity are represented by the points A, P, and Q of the following figure. In the figure OA = i, angle AOP = that is, the circumference is di- vided into three equal parts by the points A, P, and Q. Then OD = , and DP = DQ = $\/3. Hence we see from the method of represent- ing a complex number given above that A represents -\-i,P represents i + z 'lA/3. Q represents / ^-y/3- P^ = 120, angle AOQ = = 240 3 3 EXERCISES 74. (i.) Express sin 4* and cos 4* in terms of sin* and cos*. (2.) Express sin 6* and cos 6* in terms of sin* and cos*. (3.) Find the six 6 th roots of unity. (4.) Find the five 5 th roots of unity. THE HYPERBOLIC FUNCTIONS 75. The hyperbolic functions are defined by the equations e*-e-* siiili a? = cosh x (16) in which sinh^r and cosh;r denote the hyperbolic sine and 74 PLANE TRIGONOMETRY hyperbolic cosine of x respectively. These functions are called the hyperbolic sine and cosine on account of their relation to the hyperbola analogous to the relation of the sine and cosirte to the circle. A natural and convenient way to arrive at the hyperbolic functions and to study their properties is by using complex numbers in the following manner. The series (2), (3), and (4) give the value of sin *, cos*, and e* for every real value of x. These series also serve to define sin*, cos*, and ^*for complex values of x. In the more advanced parts of Algebra it is shown that the following fundamental formulas which we have proved only for a real variable, sin (x+y) = s\n x cos^ + cos* sin_y, (17) cos (x+y) = cosx cosy s'\nx siny, (18) e*+'=f*e 3 ', (19) hold unchanged when the variable is complex. This fact enables us to calculate with ease sin*, cos*, and e x for any complex value of the variable. In so doing we are led directly to the hyperbolic func- tions. At the same time a relation between the trigono- metric and hyperbolic functions is established by means of which the formulas of Chapter III. can be converted into corresponding formulas for the hyperbolic functions. Taking x and y real and replacing y in (17), (18), and (19) by iy, we get sin (x+iy) = s'\nx costy+cosx sin iy, cos (x+iy) = cos x cos iy sin x. sin iy, Thus the calculation of these functions when the variable is complex is made to depend upon the case where the vari- able is a pure imaginary. HYPERBOLIC FUNCTIONS 75 If we replace x by ix in series (4) we obtain Y (ix) 3 () 3 ! 5 ! 7 ! A comparison with series (2) and (3) shows that these two series are cos^r and sin x respectively; hence the important formula due to Euler e'*=cosx-\-i siii.r. (20) This enables us to calculated from sin.r and cos^r when ix is a pure imaginary ; that is, when x is real. To find sin ix and cosix replace x in (20) by ix\ we obtain e~*=cos ix-\-i sin ix. (21) Again replacing x by ix in (20), we obtain e*=cos ix i sin ix. (22) The sum and difference of (21) and (22) give - = cosh x, (23) 2 = i si nh x. (24) J If we compute the value of e* by the aid of series (4) for a succession of values of x, we find that sinh-ar and cosh A* are represented by the curves on page 76. The system of formulas belonging to the hyperbolic func- tions is obtained from those of the trigonometric functions by using (23) and (24). This shows that for every formula in analytic trigonometry there exists a corresponding for- mula in hyperbolic trigonometry which we get by this sub- 7 6 PLANE TRIGONOMETRY stitution. In the examples which follow, this method is used to obtain important formulas in hyperbolic trigonome- try. Replacing x by ix in (23) and (24), we get e ix +e~ ix COS JC = sill x 2 e ix_ e -ix (25) (26) which are formulas frequently used. Example. sinh (.r -\-y) = / sin i(x -\-y), = i [sin ix cos iy -\- cos ix sin z'y], = i \i sinh x cosh y-\-i cosh x sinh j], = sinh x cosh y -\- cosh x sinh y. Example. sinh x-\- sinh^ = /(sin tx 4- sin iy), =. i 2 sin i(x-\-y) cos /(.r y\ = 2 sinh ^ (x -\-y) cosh ^ (-r y). sinh HYPERBOLIC FUNCTIONS 77 EXERCISES 76. (i.) Prove sinh 0=0, cosho=i. (2.) Prove sinh ^7J7=/, cosh ^TT/=O. (3.) Prove sinh TTI' O, coshyrz^ i. Prove that (4.) sin ( ix) = sin ix. (5.) cos ( ix) = cos ix. (6.) sinh( x) = sinh .r. (7.) cosh( .r) = cosher. Remark. The hyperbolic tangent, cotangent, secant, and cosecant are defined by sinh.jtr cosher tanh Y > coth x > cosher sinh.r i i seen x ^^ : > csch x ^^ . ; coshjf smh^r Prove that (8.) tan (ix) = i tanh x. (9-) coth ( x) = coth x. (10.) sech ( x) = sech x. (ii.) cosh ! .r sinh a ^r=:j. (12.) sech 2 ^- + tanh z .r = i. (13-) cot hlr csch 2 ^ = I . (14.) sinh (x y) = sinh x coshj/ cosh x sinhj/. (15-) cosh (,r y) cosh x cosh^j/ sinh^r sinhj/. (16.) coshj^r= t / 1-f-cosh.r (17.) sinh s'\nhv = 2 cosh| (u -\- v) sinh ^ (u v). (i 8.) cosh u-\- cosh?/ = 2 cosh %(u-{-v) cosh \(uv). (19.) cosh u cosh v = 2 sinh^(-fz/)sinh|( v). CHAPTER VII MISCELLANEOUS EXERCISES RELATION OF FUNCTIONS 77> Prove the following: (i.) cos.r = sin.r cot.r. (2.) CSC.T tan x = sec x. (3.) (tan x -}- cot x) sin x cos x = I . (4.) (sec/ tan/) (sec/ 4- tan/) = I. (5.) (esc 2 cot s) (esc z 4- cot z) = i . (6.) cos 2 / 4- (tan/ cot/) sin/ cosy = sin 2 ^ (7.) cos'-r 5!^^+ i =2 cos" a-. (8.) (sin_y cos/) 4 = 1 2 sin/ cosj. (9.) sin 3 jr-f-cos 8 jr = (sin jr-j-C 08 -*") ( l s ' n x . y (10.) - - =cot.rtany. tan x-\-co\.y (n.) cos*/ sin"/ = 2 cos"/ i. (12.) i tan 4 jr = 2 sec a -r sec*.*. COS.T (13.) - - j = tan jr. v J/ sm.r cot a .r (14.) sec"/ esc*/ = tan 2 / -(-cot 5 / +2. (15.) cot/ esc/ sec/ (i 2 sin 2 /) = tan/. N / I Y I COS.? (i 6.) I . cot2J =- -- \sin 2 / i -f- cosr sec/ tan/ sin/ (I7-) i-j-cos/ sin 3 / (i8.) i+?-^lf=(sin.r4-cos.r) 2 . SGC X (i 9-) ; sin 3 jr = (cos.r sin x] (i 4- sin x COS.T). y t sec 8 * (20.) (sin^r cos/ 4- cos x si n/) 2 4- (cos x cos y sin. v sin /)" = ! MISCELLANEOUS EXERCISES 79 (21.) (a cos.r b sin xf -\-(a sin x-\-b cosxf a"-\-^. i 4 tan 2 / ' ' /33^2 7, ^T^Ta"7.\s ~r~ 7T~ (cos 2 / sin 2 /) 2 (i tan 2 /) 2 Find an angle not greater than 90 which satisfies each of the fol- lowing equations: (23.) 4 cos x = 3 sec .r. (24.) sin/ = csc/ f. (25.) -\/2sin.r tan.i' = o. (26.) 2 cos A- "V/3 cot.r = o. (27.) tanj + cotj" 2 = - (28.) 2 sin 2 / 2 = \/2 cos_y. (29.) 3 tan 2 ^ i =4 sin 2 .r. (30.) cos 2 x-{- 2 sin 2 .r |sin^r = o. (31.) csc.r = tan^-. (32.) sec a- -f- tan x ^/y. (33.) tan x -\- 2 -y/3 cos x = o. (34.) 3sin.r 2cos 2 .t'=o. Express the following in terms of the functions of angles less than 45: (35.) sin 92. (36.) cos 127. (37.) tan 320. (38.) cot 350. (39.) sin 265. (40.) tan 171. (41.) Given sin x = and x in quadrant II; find all the other functions of x. (42.) Given cos.r = | and x in quadrant III; find all the other functions of x. (43.) Given tan^" = f and x in quadrant III; find all the other functions of x. (44.) Given cot.r = | and x in quadrant IV; find all the other functions of x. 8o PLANE TRIGONOMETRY In what quadrants must the angles lie which satisfy each of the following equations : (45.) sin x cos x = \ \/3. (46.) sec-r tan .r = 2 -y/3. (47.) ta.ny-{- -y/2o cosj= o. (48.) cos x cot x = $ . Find all the values of y less than 360 which will satisfy the fol- lowing equations: (49.) tan^ + 2 sin^ = o. (50.) (i + tan*) (i 2 sin .*):= o. (51.) sin x cos x (i -\- 2 cos x) = o. Prove the following: (52.) cos78o = . (53.) sin i48s = ^\/2. (54,) cos 2 5 50 = (39.) sin6o-f- sin 20 = 2 sin 40 cos 20. (40.) sin 40 sin io = 2cos25 sin 15. (41.) COS2J: cos4-r = 2 sin3~vsin.r. (42.) tan 1 5 = 2 y/3~. (43.) (A/I -f-sin^r \/i sin^) 2 = 4 sin ! ;r. (44.) \/i -f si . v (45.) - -^ ^-' 2 cos (x +y) = -r sm.r , ., . (46.) -: ^- = 2 COS 2X. sm2jr (47.) sin 50 sin 70 -4- sin io = o. , o \ IT 1T fyt IT (40.) cos -- cos - = 2 sin sin 32 12 12 cos 7 5 -f cos 1 5 (Si.) tan'^ = ft V (52.) (53.) sin3J--f sin 5^ = 2 sin4JT cos^r. (54.) cos 5-r + cos gx = 2 cos jx cos 2x. (55.) sinis^ 2V 2 / (56.) - - = tan jr. cos 3_r + cos JT (57.) sin $y= 5 sinj 20 sin 3 /+ 16 si (58.) cos 57 = 5 cos_y 20 cos 3 j-f- 16 4 tan _r(i tan 2 .r) (60.) (6 1.) cos 3^ -\- cos 5 JT -J- cos jx + cos 1 5 x = 4 cos 4^ cos 5^ cos 6^. MISCELLANEOUS EXERCISES 87 (62.) sin 2 ^r(cotf.r i) 2 = i sin.r. cos 3.r 4- 3 cos x ~ (64.) sin x(\. 4- tan .r) 4- cos .r( i 4-cot.r) = csc. (65.) cosV- S in^ = 24-sin2.r < cosx smjr 2 (66.) cos j + cos (i 20 y) + cos (i 20 -\-y) = o. . sin ~\x (67.) . =2 COS 2^T 4- I. sin JT (cosj/ cos 3/)(sin 8_y + sin 2y) _ ^ (sin y/ sin_y)(cos4^/ cos6/) (69.) cos.r . sin3 (70.) -r-2 sm.r COSJT ' COS.T ^ - ' - (73-) = tan 4- r - cos .i- + cos 3-tr -|- cos 5_r -|- cos "jx If ^f, B, and C are the angles of a triangle, prove the following : (74.) sin iA 4- sin 7.B -\- sin 2C = 4 sin A sin^ sin C. (75.) sin 2^4 + sin 2B sin 2(7 = 4 cos .4 cos B sin C. (76.) sinM-fsin2# + sin 2 C= 2 _^_ 2 cos ^ cos 5 cos C (77.) tan ^4 + tan B + tan C = tan A tan ^ tan C. Solve the following equations for values of x less than 360. (78.) cos2^r+ cos - ;1: ' = i. (79.) sin x-\-?>\njx = sin4_r. (80.) cos.r sin 2x co&^x o. '(81.) COS.T sin3;r cos2.r = o. (82.) sin4_r 2sin2.r:=:o. (83.) sin 2x cos2.r sin jr + cos.r = o. (84.) sin (60 x) si n (60 + x) = + \ y^. (85.) sin (30 + x) cos (60 + x) = \ (86.) csc-r = i -4-cot-r. (87.) cos *x cos *x. (88.) 2 siny=s\n2y. (89.) sin3/-|-sin2y-|-sin (90.) sin"jr-f 5 cosV = 3. (91.) tan(45 OBLIQUE TRIANGLES 81. (i.) It is required to find the distance between two points, A and B, on opposite sides of a river. A line, AC, and the angles BAC and ACB are measured and found to be 2483 ft., 61 25', and 52 17' respectively. (2.) A straight road leads from a town ^4 to a town B, 12 miles distant ; another road, making an angle of 77 with the first, goes from A to a town C, 7 miles distant. How far are the towns B and C apart ? (3.) In order to determine the distance of a fort, A, from a battery, /?, a line, /?C, one-half mile long, is measured, and the angles ABC and ACB are observed to be 75 18' and 78 21' respectively. Find the distanced/?. (4.) Two houses, A and B, are 1728 ft. apart. Find the distance of a third house, C, from A \i BACM Q 51' and ABC 57 23'. (5.) In order to determine the distance of a bluff, A, from a house, B, in a plane, a line, BC, was measured and found to be 1281 yards, also the angles ABC and BCA 65 31' and 70 2' respectively. Find the distance AB. (6.) Two towns, 3 miles apart, are on opposite sides of a balloon. The angles of elevation of the balloon are found to be 13 19' and 20 3'. Find the distance of the balloon from the nearer town. (7.) It is required to find the distance between two posts, A and B, which are separated by a swamp. A point C is 1272.5 ft. from A, and 2012.4 ft. from B, The angle ACB is 41 9' u". (8.) Two stakes, A and B, are on opposite sides of a stream ; a third point, C, is so situated that the distances AC and BC can be found, and are 431.27 yards and 601.72 yards respectively. The angle ACB is 39 53' 13". Find the distance between the stakes A and B. MISCELLANEOUS EXERCISES 89 (9.) Two light-houses, ,4 and B, are u miles apart. A ship, C, is observed from them to make the angles J5AC=^i 13' 31" and ABC = 21 46' 8". Find the distance of the ship from A. (10.) Two islands, A and B, are 6103 ft. apart. Find the distance from A to a ship, C, if the angle ABC is 37 25' and BAC is 40 32'. (u.) In ascending a cliff towards a light-house at its summit, the light-house subtends at one point an angle of 21 22'. At a point 55 ft. farther up it subtends an angle of 40 27'. If the light-house is 58 ft. high, how far is this last point from its foot? (12.) The distances of two islands from a buoy are 3 and 4 miles respectively. The islands are 2 miles apart. Find the angle sub- tended by the islands at the buoy. (13.) The sides of a triangle are 151.45, 191.32, and 250.91. Find the length of the perpendicular from the largest angle upon the opposite side. (14.) A tree stands on a hill, and the angle between the slope of the hill and the tree js 110 23'. At a point 85.6 ft. down the hill the tree subtends an angle of 22 22'. Find the height of the tree. (15.; A light-house 54 ft. high is built upon a rock. From the top of the light-house the angle of depression of a boat is 19 10', and from its base the angle of depression of the boat is 12 22'. Find the height of the rock on which the light-house stands. (16.) Three towns, A, B, and C, are connected by straight roads. AB At miles, BC=$ miles, and AC '=7 miles. Find the angle made by the roads AB and BC. (17.) Two buoys, A and B, are one-half mile apart. Find the dis- tance from A to a point C on the shore if the angles ABC and BAC are 77 7' and 67 17' respectively. (i 8.) The top of a tower is 175 ft. above the level of a bay. From its top the angles of depression of the shores of the bay in a certain direction are 57 16' and 15 2'. Find the distance across the bay. (19.) The lengths of two sides of a triangle are \/2 and \/^. The angle between them is 45. Find the remaining side. (20.) The sides of a parallelogram are 172.43 and 101.31, and the angle included by them is 61 16'. Find the two diagonals. (21.) A tree 41 ft. high stands at the top of a hill which slopes 90 PLANE TRIGONOMETRY 10 12' to the horizontal. At a certain point down the hill the tree subtends an angle of 28 29'. Find the distance from this point to the foot of the tree. (22.) A plane is inclined to the horizontal at an angle of 7 33'. At a certain point on the plane a flag-pole subtends an angle 20 3', and at a point 50 ft. nearer the pole an angle of 40 35'. Find the height of the pole. (23.) The angle of elevation of an inaccessible tower, situated in a plane, is 53 19'. At a point 227 ft. farther from the tower the angle of elevation is 22 41'. Find the height of the tower. (24.) A house stands on a hill which slopes 12 18' to the horizontal. 75 ft. from the house down the hill the house subtends an angle of 32 5'. Find the height of the house. (25.) From one bank of a river the angle of elevation of a tree on the opposite bank is 28 31'. From a point 139.4 ft. farther away in a direct line its angle of elevation is 19 10'. Find the width of the river. (26.) From the foot of a hill in a plane the angle of elevation of the top of the hill is 21 7'. After going directly away 21 1 ft. farther, the angle of elevation is 18 37'. Find the height of the hill. (27.) A monument at the top of a hill is 153.2 ft. high. At a point 321.4 ft. down the hill the monument subtends an angle of 11 13'. Find the distance from this point to the top of the monument. (28.) A building is situated on the top of a hill which is inclined 10 12' to the horizontal. At a certain distance up the hill the angle of elevation of the top of the building is 20 55', and 115.3 ft. farther down the hill the angle of elevation is 15 10'. Find the height of the building. (29.) A cloud, C, is observed from two points, A and B, 2874 ft. apart, the line AB being directly beneath the cloud. At A, the angle of elevation of the cloud is 77 19', and the angle CAB is 51 18'. The angle ABC is found to be 60 45'. Find the height of the cloud above A. (30.) Two observers, A and B, are on a straight road, 675.4 ft. apart, directly beneath a balloon, C. The angles ABC and BAC are 34 42' and 41 15' respectively. Find the distance of the balloon from the first observer. MISCELLANEOUS EXERCISES 91 (31.) A man on the opposite side of a river from two objects, A and B, wishes to obtain their distance apart. He measures the dis- tance CD 357 ft., and the angles ACB=2<) 33', BCD = 38 52', ADB = 54 10', and ADC =34 n'. Find the distance AB. (32.) A cliff is 327 ft. above the sea-level. From the top of the cliff the angles of depression of two ships are 15 u' and 13 13'. From the bottom of the cliff the angle subtended by the ships are 122 39'. How far are the ships apart ? (33.) A man standing on an inclined plane 112 ft. from the bottom observed the angle subtended by a building at the bottom to be 33 52'. The inclination of the plane to the horizontal is 18 51'. Find the height of the building. (34) Two boats, A and B, are 451.35 ft. apart. The angle of ele- vation of the top of a light-house, as observed from A, is 33 if. The base of the light-house, C, is level with the water; the angles ABC and CAB are 12 31' and 137 22' respectively. Find the height of the light-house. (35.) From a window directly opposite the bottom of a steeple the angle of elevation of the top of the steeple is 29 21'. From another window, 20 ft. vertically below the first, the angle of elevation is 39 3'. Find the height of the steeple. (36.) A dock is i mile from one end of a breakwater, and i miles from the other end. At the dock the breakwater subtends an angle of 31 n'. Find the length of the breakwater in feet. (37.) A straight road ascending a hill is 1022 ft. long. The hill rises i ft. in every 4. A tower at the top of the hill subtends an angle of 7 19' at the bottom. Find the height of the tower. (38.) A tower, 192 ft. high, rises vertically from one corner of a triangular yard. From its top the angles of depression of the other corners are 58 4' and 17 49'. The side opposite the tower subtends from the top of the tower an angle of 75 15'. Find the length of this side. (39.) There are two columns left standing upright in a certain ruins ; the one is 66 ft. above the plain, and the other 48. In a straight line between them stands an ancient statue, the head of which is 100 ft. from the summit of the higher, and 84 ft. from the top of the lower 92 PLANE TRIGONOMETRY column, the base of which measures just 74 ft. to the centre of the figure's base. Required the distance between the tops of the two columns. (40.) Two sides of a triangle are in the ratio of 1 1 to 9, and the opposite angles have the ratio of 3 to i. What are these angles? (41.) The diagonals of a parallelogram are 12432 and 8413, and the angle between them is 78 44'; find its area. (42.) One side of a triangle is 1012.6 and two angles are 52 21' and 57 32' ; find its area. (43.) Two sides of a triangle are 218.12 and 123.72, and the included angle is 59 10' ; find its area. (44.) Two angles of a triangle are 35 15' and 47 18', and one side is 2104.7 ; find its area. (45.) The three sides of a triangle are 1.2371, 1.4713, and 2.0721 ; find the area. (46.) Two sides of a triangle are 168.12 and 179.21, and the included angle is 41 14' ; find its area. (47.) The three sides of a triangle are 51 ft., 48.12 ft., and 32.2 ft. ; find the area. (48.) Two sides of a triangle are m.iSand 12 1.21, and the included angle is 27 50' ; find its area. (49.) The diagonals of a parallelogram are 37 and 51, and they form an angle of 65 ; find its area. (50.) If the diagonals of a quadrilateral are 34 and 56, and if they intersect at an angle of 67, what is the area? SPHERICAL TRIGONOMETRY CHAPTER VIII RIGHT AND QUADRANTAL TRIANGLES RIGHT TRIANGLES 82. Let O be the centre of a sphere of unit radius, and ABC a right spherical triangle, right angled at A, formed by the intersection of the three planes A OC, A OB, and BOC with the surface of the sphere. Suppose the planes DAC" and EEC passed through the points A and B respectively, and perpendicular to the line OC. The plane angles DC" A and BC'E each measure the angle C of the spherical tri- angle, and the sides of the spherical triangle #, b, c have the same numerical measure as BOC, AOC, and AOB respec- 94 SPHERICAL TRIGONOMETRY tively, then, AD = ta.nc, BE sine, C' = sina, OC' = cosa, cos&, OE cosc, AC"= sin b. In the two similar triangles OEC' and OAC", cos c cos c cos a OA ' i cosb In the triangle BC'E, , or cos a = cost? cose. (i) ~ BE . ~ sin c sine = -757^, or sin 6= sin In the triangle DAC" , DA r tan c tan C -.,, . , or tan 6 = - (3) Combining formulas (2) and (3) with (i), -_ tan b , v Again, if AB were made the base of the right spherical triangle ABC, we should have z? sin ^ ^\ Sm slrTrt' ^ 5 ^ .-. Ltlll //r\ tan^^-^-^. cos^= (7) tatirt From the foregoing equations we may also obtain by combinations, cos^=sin C cosb. (8) cos7 sin# cos^r. (9) cotC (10) NAPIER'S RULES OF CIRCULAR PARTS 83. The above ten formulas are sufficient to solve all cases of right spherical triangles. They may, however, be RIGHT AND QUADRANTAL TRIANGLES 95 expressed as two simple rules, called, after their inventor, Napier's rules. The two sides adjacent to the right angle, the complement of the hypotenuse, and the complements of the oblique an- gles are called the circular parts. The right angle is not one of the circular parts. comp a comp G< Thus there are five circular parts namely, />, c, comprt, comp B, compC. Any one of the five parts may be called the middle part, then the two parts next to it are called adjacent parts, and the remaining two parts are called the oppo- site parts. Thus if c is taken for the middle part, comp B and b are adjacent parts, and comprt and comp C are opposite parts. The ten formulas may be written and grouped as follows : 1st Group. sin comp C = tan comprt tan b. sin comp ,5=tan comp a tan c. sin comp a =tan comp.Z? tan comp C. sin c =tan comp B la.nl>. sin b =tan comp C tanr. "id Group. sin comp a= cos/' cos c. sin <=cos comp a cos comp B. sin f=cos comprt cos comp C. sin comp j9=cos comp C cos b. sin comp C =cos comp B cos c. Napier's rules may be stated : I. The sine of the middle part is equal to tlie product of tJie tangents of the adjacent parts. II. The sine of the middle part is equal to the product of the cosines of the opposite parts. 96 SPHERICAL TRIGONOMETRY 84, In the right spherical triangles considered in this work, each side is taken less than a semicircumference, and each angle less than two right angles. In the solution of the triangles, it is to be observed, (i.) If the two sides about the right angle are both less or both greater than 90, the hypotenuse is less than 90; if one side is less and the other greater than 90, the hypotenuse is greater than 90. (2.) An angle and the side opposite are either both less or both greater than 90. EXAMPLE 85. Given a = 63 56', 6 = 40 o', to find c, B, and C. To find c. comp a is the middle part. c and b are the opposite parts. sin comprt=cos cos=cos comprt cos comp.Z?, or sin /;=sin a sin B. sin b sin B-- smrt log sin />=9.8o8o7 colog sin rt.=o. 04659 log sin ,#=9.85466 ^ = 45 41' 28" Check. Use the three parts originally required. comp C is the middle part, comp .5 and c are opposite parts. sin comp C=cos^ cos comp B, or cos C=cos c sin B. log cos c =9. 75863 log sin #=9.85466 log cos "=9.61329 C=6 5 45' 54" RIGHT AND QUADRANTAL TRIANGLES 97 AMBIGUOUS CASE 86. When a side about the right angle and the angle opposite this side are given, there are two solutions, as illustrated by the fol- lowing figure. Since the solution gives the values of each part in terms of the sine, the results are not only the values of a, b, B, but 180 a, 1 8o , 180 B. Given c = 26 4'. C=36o'. To find a, a', b, b' and B, B' , using Napier's rules. To find B and B '. sin comp C= cos comp B cose, 3r cos C=sin B cos c, cos C cos c log cos =9.90796 colog cos c =0.04659 sin = log sin = 9.95455 B= 64 14' 30" ' = i8o-=iis 45' 30" To find b and b'. sin =tan c tan comp (7, sin =tan c cot C. log tan c= 9. 68946 log cot (7=0.13874 log sin =9.82820 b= 42 19' 17" '=180 =137 40' 43" To find a and a'. sin c =cos comp a cos Comp C, >r sin c =sin a sin C, sin c ir sin = - - . sin C log sin c=<) 64288 colog sin (7=0.23078 log sin 0=9.87366 a= 48 22' 55" a' = i8o 0=131 37' 5"+ (Discrepancy due to omitted decimals.) Check. sin =cos comp , r sin =sina log sin a or rt'=g. 87366 log sinZ? or '=9.95455 log sin =9.82821 = 42 19' 21" '=180 =137 40' 39" 98 SPHERICAL TRIGONOMETRY QUADRANTAL TRIANGLES 87. Def. A quadrantal triangle is a spherical triangle one side of which is a quadrant. A quadrantal triangle may be solved by Napier's rules for right spherical triangles as follows : By making use of the polar triangle where we see that the polar triangle of the quadrantal triangle is a right triangle which can be solved by Napier's rules. Whence we may at once derive the required parts of the quadrantal triangle. EXAMPLE Given A = 1 36 4'. B = 140 o'. a = 90 o'. The corresponding parts of the polar triangle are rt' = 6356', ' = 4oo', ^' = 90. By Napier's rules we find #' = 45 41 '28", C' = 6s45' 58", c = 54 59' 47"; whence, by applying to these parts the rule of polar triangles, we obtain b 134 18' 32". c 114 14' 2", C=i25o' 13". EXERCISES 88. (i.) In the right-angled spherical triangle ABC, the side a= 63 56', and the side = 40. Required the other side, c, and the angles B and C. (2.) In a right-angled triangle ABC, the hypotenuse a = 91 42', and the angle B = <)5 6'. Required the remaining parts. (3.) In the right-angled triangle ABC, the side = 26 4', and the angle = 36. Required the remaining parts. (4.) In the right-angled spherical triangle ABC, the side c=S4 30', and the angle .5 = 44 50'. Required the remaining parts. Why is not the result ambiguous in this case? RIGHT AND QUADRANTAL TRIANGLES 99 (5.) In the right-angled spherical triangle ABC, the side = 55 28', and the side ^ = 63 15'. Required the remaining parts. (6.) In the right-angled spherical triangle ABC, the angle B = 6g 20', and the angle C = 58 16'. Required the remaining parts. (7.) In the spherical triangle ABC, the side a = 90, the angle C= 42 10', and the angle A -=.115 20'. Required the remaining parts. Hint. The angled of the polar triangle is a right angle. (8.) In the spherical triangle ABC, the side = 90, the angle C= 69 13' 46", and the angle A = 72 12' 4". Required the remaining parts. (9.) In the right-angled spherical triangle ABC, the angle (7=23 27' 42", and the side b 10 39' 40". Required the angle B and the sides a and c. (10.) In the right spherical triangle ABC, the angle = 47 54' 20", and the angle C=6i 50' 29". Required the sides. CHAPTER IX OBLIQUE-ANGLED TRIANGLES 89, Let O be the centre of a sphere of unit radius, and ABC an oblique-angled spherical triangle formed by the three planes AOB, BOC, and AOC. Suppose the plane AED passed through the point A perpendicular to AO, in- tersecting the planes A OB, BOC, and AOC, in A, ED, and AD respectively. Then AD=tan b, AE tan c, OD sec b, OE=secc. In the triangle ROD, ED 1 = sec 8 ^ -f sec V 2 sec b sec c cos a. In the triangle AED, ED 1 tan 8 ^ -f- tan V 2 tan (5 tan <: cos A. Subtracting these two equations and remembering that sec a tan"=i, we have = 2 2 sec secf cosrt + 2 tan tan^r cos A. Reducing, we have cosa=cos& co8c+sin6 sine cos .4. (i) OBLIQUE-ANGLED TRIANGLES 101 If we make b and c in turn the base of the triangle, we obtain in a similar way, cos b = cos c cos a -\- sin c sin a cos B, Remark. In this group of formulas the second may be obtained from the first, and the third from the second, by advancing one letter in the cycle as shown in the figure; thus, writing b for a, c for b, a for c, B for A, C for B, and A for C. The same principle will apply in all the formulas of Oblique- Angled Spherical Triangles, and only the first one of each group will be given in the text. 00. By making use of the polar triangle where c=i%o-C C=i8o-c' we may obtain a second group of formulas. Substituting these values of a, b, c, and A in (i), and remembering that cos(i8o A) = cos A and sin (180 A) = sin A, we have cos^4' = cos/?' cosC'-f-sin-Z?' sin C' cosa'. Since this is true for any triangle, we may omit the accents and write, cos 4= cosU cosC+sinU sin C cos a. (2) FORMULAS FOR LOGARITHMIC COMPUTATION 91 Formula (i), cos a cos b cose + sin b sin c cos A, cosa cos$ cos*: gives cos A = : r : . sm# sin c By 36, cos^4:=i 2 sin 2 ^ cos# cos^ cose Whence i 2s\rr$A= . . , sini8o, then (a-J-)>i8oP, and if (A + B)< , then ( 104 SPHERICAL TRIGONOMETRY 94. All cases of oblique-angled triangles may be solved by applying one or more of the formulas I, II, III, IV, V, VI, VII, as shown in the following cases. CASES (i.) Given three sides, to find the angles. Apply formula I. Check: apply V or VI. (2.) Given three angles, to find the sides. Apply formula II. Check : apply III or I V. (3.) Given two sides and the included angle. Apply V and VI, and VI 7. Check : apply III or I V. (4.) Given two angles and included side. Apply III and I V, and VII. Check : apply V or VI. (5.) Given two angles and an opposite side. Apply VII, V, and III. Check : apply I V. (6.) Given two sides and an opposite angle. Apply VII, V, and I V. Check : apply III. EXAMPLE CASE (l) 95. Given a = 8i 10' = 6o2o' r=ii225' To find A, B, and C. a= 81 10' f =112 25 j = i26 57' 30" s-a=4S 47' 30" s 6=66 37' 30" j c = i4 32' 30" 2)19 60464 log sin ,r=9. 90259 log tan \A= 9.80232 log sin (s-a)=g. 85540 log sin(.-*)=n ofi^r i//=32 2 3 ' 19" log sin (.?-<)= To find A. sin s sin (s a) log sin (s ^=9.96281 log sin (s ^=9.39982 colog sin s=o. 14460 colog OBLIQUE-ANGLED TRIANGLES 10: To find B. sins sm(j b) log sin (.r rt) = g. 85540 log sin (sf) = g. 39982 colog sin.r =0.09741 colog sin (5 /') =0.03719 2)19.36982 log tan 5= 9.69491 Jfl=262l' 6" .5=52 42' 12" tan i C= , /si"('-)si('-l V sill j sin(j f) log sin (j )=9 97501 1 45' colog sin (rt - b) =0.74279 25' cot (7=9.74210 C= 61 5' 32' C=I22 II' 4" EXAMPLE CASE (3) 96. Given a = 78 1 5' = 56 20' To find (0 + ^=67 17' 30" (*-*)= 10 57' 30" ^ C=6o Formula V may be written cosA( /^) cot A 6" 2 2 cos ( + /;) log cos(rt )= 9.99201 log cot (7= 9.76144 colog cos ^ (a + />)=. 0.41337 log tan (A + B)= io. 16682 ) = 5544' 36"- f-^?)= 6 47' 4" ^4=62 31' 40" ^=48 57' 32"- ?, and c. log sin C=I20 (rt +^=9.96498 log cos %(a + ^=9.58663 log sin J(rt ^=9.27897 lg cos J()= 9. 28696 colog sin %(Ah)= 0.92762 log sin c =9.98029 log tan c= 10. 13183 \f= 53 33' 5"- ^=107 i 51" (Discrepancy due to omitted decimals ) AMBIGUOUS CASES 97. (i.) Two sides and an angle opposite one of them are the given parts. If the side opposite the given angle differs from po more than the other given side, the given angle and the side opposite being either both less or both greater than 90, there are two solutions. (2.) Two angles and a side opposite one of them are the given parts. If the angle opposite the given side differs from 90 more than the other given angle, the given side and the angle opposite being either both less or both greater than 90, there are two solutions. Remark. There is no solution if, in either of the formulas, sin A sin b sin b sin A 1 0*1 M- . n sm a sin B the numerator of the fraction is greater than the denominator. OBLIQUE-ANGLED TRIANGLES 107 EXAMPLE 98. Given 0=40 16' d= To find B, B', To find B and B'. Formula VII may be written sin B= sin A sin b sin a log sin A=g. 89947 log sin =9. 86924 colog sin =o. 18953 log sin ^=9.95824 = 65 1 6' 30" ^' = 114 43' 30" To find c. Formula IV may be written _ cosl(A+) tanjfri + J) cosi(^-2?) log cos $(A + fi)=(). 71326 log tan(-f ^=9.98484 COlog COS (.4 .5) = O.O02 70 log tan ^f=g. 70080 ^=26 39' 42" '=53 19' 24" log c log tan $( colog cos^(^4 )=<). 98484 log tan ^'=9.09860 \c'= 7 9' 9" = 50 10' 30", and the angle ^ = 34 15' 3". Required the remaining parts. AREA OF THE SPHERICAL TRIANGLE 100. It is proved in geometry that the area of a spherical triangle is equal to its spherical excess, that is, area = (A-\--{-C2rt. angles) X area of the tri-rectangular triangle, where A, B, and C are the angles of the spherical triangle. Hence area __ A+-\-CiSo surface of sphere ~ 720 The surface of the sphere is 47rft*, therefore A + B+C 18OA The following formula, called Lhuilier's theorem, simpli- fies the derivation of (A + B+C 180) where the three OBLIQUE-ANGLED TRIANGLES 109 sides of the spherical triangle are given ; in it a, b t and c denote the sides of the triangle, and 2s = a + & + c. tan /d+g+C-180\ _ ytmni s tan i(s-a) tan i(s-6) tan i (s-c). EXERCISES (i.) The angles of a spherical triangle are, ^ = 63, 5=84 21', (7=79; the radius of the sphere is 10 in. What is the area of the triangle ? (2.) The sides of a spherical triangle are, a = 6.47 in., = 8.39 in., = 9.43 in. ; the radius of the sphere is 25 in. What is the area of the triangle ? (3.) In a spherical triangle, ^ = 75 16', B = yf 20', c = 26 in. ; the radius of the sphere is 14 in. Find the area of the triangle. (4.) In a spherical triangle, a = 441 miles, ^ = 287 miles, 7 = 38 21'; the radius of the sphere is 3960 miles. Find the area of the triangle, CHAPTER X APPLICATIONS TO THE CELESTIAL AND TERRES- TRIAL SPHERES ASTRONOMICAL PROBLEMS 101, An observer at any place on the earth's surface finds himself seemingly at the centre of a sphere, one-half of which is the sky above him. This sphere is called the celestial sphere, and upon its surface appear all the heavenly bodies. The entire sphere seems to turn completely around once in 23 hours and 56 minutes, as on an axis. The im- aginary axis is the axis of the earth indefinitely produced. The points in which it pierces the celestial sphere appear stationary, and are called the north and south poles of the heavens. The North Star (Polaris) marks very nearly (with- in i 16') the position of the north pole. As the observer travels towards the north he finds that the north pole of the heavens appears higher and higher up in the sky, and that its height above the horizon, measured in degrees, corre- sponds to the latitude of the place of observation. The fixed stars and nebulae preserve the same relative positions to each other. The sun, moon, planets, and com- ets change their positions with respect to the fixed stars continually, the sun appearing to move eastward among the stars about a degree a day, and the moon about thir- teen times as far. AP PLICA TIONS 1 1 1 The zenith is the point on the celestial sphere directly overhead. The horizon is the great circle everywhere 90 from the zenith. The celestial equator is the great circle in which the plane of the earth's equator if extended would cut the ce- lestial sphere. The ecliptic is the path on the celestial sphere described by the sun in its apparent eastward motion among the stars. The ecliptic is a great circle inclined to the plane of the equator at an angle of approximately 23^-. The poles of the equator are the points where the axis of the earth if produced would pierce the celestial sphere, and are each 90 from the equator. The poles of the ecliptic are each 90 from the ecliptic. The equinoxes are the points where the celestial equa- tor and ecliptic intersect ; that which the sun crosses when coming north being called the vernal equinox, and that which it crosses when going south the autumnal equinox. The declination of a heavenly body is its distance, meas- ured in degrees, north or south of the celestial equator. The right ascension of a heavenly body is the distance, measured in degrees eastward on the celestial equator, from the vernal equinox to the great circle passing through the poles of the equator and this body. The celestial latitude of a heavenly body is the dis- tance from the ecliptic measured in degrees on the great circle passing through the pole of the ecliptic and the body. The celestial longitude of a heavenly body is the dis- tance, measured in degrees eastward on the ecliptic, from 112 SPHERICAL TRIGONOMETRY the vernal equinox to the great circle passing through the pole of the ecliptic and the body. EXERCISES (i.) The right ascension of a given star is 25 35', and its declina- tion is 4-(no rtn ) 63 26'. Assuming the angle between the celestial equator and the ecliptic to be 23 27', find the celestial latitude and celestial longitude. In this figure AB is the celestial equator, AC the ecliptic, P the pole of the equator, P' the pole of the ecliptic. S is the position of the star, and the lines SB and SC are drawn through P and P' perpendicular to AB and AC. AB is the right ascension and BS the declination of the star, while AC is the longitude and SC the latitude of the star. In the spherical triangle P' PS, it will be seen that P' S is the comple- ment of the celestial latitude, PS the complement of the declination, and P' PS is 90 plus the right ascension. It is to be noted that A is the ver- nal equinox. (2.) The declination of the sun on December 2ist is (south) 23 27'. At what time will the sun rise as seen from a place whose latitude is 41 18' north ? The arc ZS which is the distance from the zenith to the centre of the sun when the sun's upper rim is on the horizon is 90 50'. The 50' is made up of the sun's semi-diameter of 16', plus the correction for refraction of 34'. AP PLICA TIONS 1 1 3 (3.) The declination of the sun on December 2ist is (south) 23 27'. At what time would the sun set as seen from a place in lati- tude 50 35' north ? In these figures P is the pole of the equator, Z the zenith, Q the celes- tial equator. AS is the declination of the sun, ZS=go 5Q 1 , PS= 9O + dec- lination, PZ=go latitude. The problem is to find the angle SPZ. An angle of 15 at the pole corresponds to I hour of time. GEOGRAPHICAL PROBLEMS 102. The meridian of a place is the great circle passing through the place and the poles of the earth. The latitude of a place is the arc of the meridian of the place extending from the equator to the place. Latitude is measured north and south of the equator from o to 90. The longitude of a place is the arc of the equator extend- ing from the zero meridian to the meridian of the place. The meridian of the Greenwich Observatory is usually taken as the zero meridian. Longitude is measured east or west from o to 180. The longitude of a place is also the angle between the zero meridian and the meridian of the place. 114 SPHERICAL TRIGONOMETRY In the following problems one minute is taken equal to one geo- graphical mile. (I.) Required the distance in geographical miles between two places, D and E, on the earth's surface. The longitude of D is 60 15' E., and the latitude io 10' N. The longitude of E is 115 20' E., and the latitude 37 20 N. In this figure A C represents the equator of the earth, P the north pole, and A the intersection of the meridian of Greenwich with the equator. PB and PC represent meridians drawn through D and E respectively. Then AB is the longitude and BD the latitude of D ; AC the longitude and CE the latitude of E. (2.) Required the distance from New York, latitude 40 43' N., longitude 74 o' W., to San Francisco, latitude 37 48' N., longitude 1 22 28' W., on the shortest route. (3.) Required the distance from Sandy Hook, latitude 40 28' N., longitude 74 i' W., to Madeira, in latitude 32 28' N., longitude 16 55, \V., on the shortest route. (4.) Required the distance from San Francisco, latitude 37 48' N., longitude 122 28' W., to Batavia in Java, latitude 6 9' S., longi- tude 106 53' E., on the shortest route. (5.) Required the distance from San Francisco, latitude 37 48' N., longitude 122 28' W., to Valparaiso, latitude 33 2' S., longitude 71 41' W., on the shortest route. CHAPTER XI GRAPHICAL SOLUTION OF A SPHERICAL TRIANGLE 103. The given parts of a spherical triangle may be laid off, and then the required parts may be measured, by making use of a globe fitted to a hemispherical cup. The sides of the spherical triangle are arcs of great circles, and may be drawn on the globe with a pencil, using the rim of the cup, which is a great circle, as a ruler. The rim of the cup is graduated from o to 180 in both directions. The angle of a spherical triangle may be measured on a great circle drawn on the sphere at a distance of 90 from the vertex of the angle.* CASE I. Given the sides a, b, and c of a spJierical triangle \ to determine the angles A , B, and C. Place the globe in the cup, and draw upon it a line equal to the number of degrees in the side c, using the rim of the cup as a ruler. Mark the extremities of this line A and B. With A and B as centres, and b and a respectively as radii, draw with the dividers two arcs intersecting at C (Fig. i). Then, placing the globe in the cup so that the points^ and C shall rest on the rim, draw the line AC=b, and in the same way draw BCa. To measure the angle A place the arc AB in coincidence * Slated globes, three inches in diameter, made of papier-mache, and held in metal hemispherical cups, are manufactured for the use of students of spherical trigonometry at a small cost. with the rim of the cup, and make AE equal to 90. Also make AF in AC produced equal to 90. Then place the globe in the cup so that E and F shall be in the rim, and note the measure of the arc EF. This is the measure of the angle A. In the same way the angles B and C can be de- termined. CASE II. Given the angles A, B, and C, to find the sides a, b, and c. Subtract A, B, and C each from 180, to obtain the sides a', b', and c' of the polar triangle. Construct this polar tri- angle according to the method employed in Case I. Mark its vertices A', B', and C'. With each of these vertices as a centre, and a radius equal to 90, describe arcs with the di- viders. The points of intersection of these arcs will be the vertices A, B, and C of the given triangle. The sides of this triangle a, b, and c can then be measured on the rim of the cup. GRAPHICAL SOLUTION 117 CASE III. Given two sides, b and c, and the included angle A, to find B, C, and a. Lay off (Fig. 3) the line AB equal to c, and mark the point D in AB produced, so that AD equals 90. With the dividers mark another point, F, at a distance of 90 from A. Turn the globe in the cup till D and Fare both in the rim, and make DE equal to the number of degrees in the angle A. With A and E in the rim of the cup, draw the line AC equal to the number of degrees in the side b. Join C and B. The required parts of the triangle can then be measured. FIG. 3 CASE IV. Given the angles A and B and the included side c, to find a, b, and C. Lay off the line AB equal to c. Then construct the given angles at A and B, as in Case III., and extend their sides to intersect at C. CASE V. Given the sides b, a, and the angle A opposite one of these sides, to find c, B, and C. (Ambiguous case.) Il8 SPHERICAL TRIGONOMETRY Lay off (Fig. 4) AC equal to b, and construct the angle A as in Case III. Take c in the dividers as a radius, and with C as a centre describe arcs cutting the other side of the tri- angle in B and B' y and measure the remaining parts of the two triangles. If the arc described with C as a centre does not cut the other side of the triangle, there is no solution'. If tangent, there is one solution. CASE VI. Given the angles A, B, and the side a opposite one of the angles. Construct the polar triangle of the given triangle by Case V.; then construct the original triangle as in Case II., and measure the parts required. The constructions given above include all cases of right and quadrantal triangles. CHAPTER XII RECAPITULATION OF FORMULAS ELEMENTARY RELATIONS ( IO) sin* cos* tan x = - , cot x = . , cos* sin* r i sec * = - , esc * = . cos* sin* tan * cot * = i , sin 2 * + cos 2 * = i, i + tan 2 * = sec' 1 *, i + cot 2 * = csc a *. RIGHT TRIANGLES ( 14 AND 2j} b sm .? = -, c b a cos A = - , cos B = - , c c a b tan A=-r, tan B = -, b a -b a cot A = - , cot B = i, a b where c = hypotenuse, a and b sides about the right angle; A and B the acute angles opposite a and b. FUNCTIONS OF TWO ANGLES ( 30-34) sin (*+/) = sin* cosj + cos* sin^, sin (* j)r=sin* cos y cos * sin y, cos (* -\-y) = cos * COS^K sin * sin^, cos (* /) = cos* cos/ + sin* siny. 120 RECAPITULATION OF FORMULAS tan^r+tan y tan (x-V-y) i tan x tany tan.r tan y tan (x y) = i+tan.r cot x cot^ i cot (x y) = cotj + cot^r ' cot x cotj-f- 1 cotj cot^r FUNCTIONS OF TWICE AN ANGLE ( 36) sin 2^r = 2 sm^r cosx, cos 2 JT = cos" x sin" JT =. i 2 sin a ^r, = 2 cos 2 jr i, 2 tan x tan 2.r=: cot 2.r = i tan a jr cot a .r i 2 cot JT FUNCTIONS OF HALF AN ANGLE ( 37) tan = \/- cot *jc=* I COS X SUMS AND DIFFERENCES OF FUNCTIONS ( 38) sin u + sin z> = 2 sin (w + -v) cos ^ ( v), sin a sinw=:2 cosj (u-\-v) sin ^( f), cos w -f- cos v = 2 cos i ( -f- 7 ') cos i ( f), cos cost/ = 2 sin \(u-\-v) sin ^( ^). sin u + sin z> tan ^ ( -|- v) sin sin v~ tan ( z*)' RECAPITULATION OF FORMULAS 121 OBLIQUE TRIANGLES ( 42-45) a sin A a sin A b sin B b sin B ' c sin C ' c sin C ' = r 2 -\- a? 2ca cos B, where s= tan \A = 2 AT. -AT .AT 5-^*1 , Ldll IF'-* 7 t .$ a .y b ,. ^ l(sa) (sb) (sc) where K \ * s AREA OF A TRIANGLE ( 46) S~\ac sin B. S=ba sin C. S=cb sin ^4. LOGARITHMIC, COSINE, SINE, AND EXPONENTIAL SERIES (58) X X 3C^ =JT J + -J- + etc - 122 RECAPITULATION OF FORMULAS X X* X 7 sm;r = .r 1 ~i ~"~ 71 ~~ T~ """' etc< X X" X -! + 3 - + 4 -l+- etc - DE MOIVRE'S THEOREM ( 71) (cos;tr-f- \/~ * 8ittjr)*=cos*Jr-f*V ' sin.r. ( l)( 2) sm #.r = n cos" '.r sm.r --- : - - cos"~ 3 .r sin 3 .r-f, etc. n (n i ) HYPERBOLIC FUNCTIONS ( 75) e* e~ x sinh x = , sn JT = 2 ' = cos x-\-i sinx e** e~ tx 2 /(** *-*) " sin ix = - L t sinh ^r, cos t'x = =cosh x. 2 SPHERICAL TRIANGLES RIGHT AND QUADRANTAL TRIANGLES _( 83, 87) Use Napier's rules. OBLIQUE TRIANGLES ( 89-93) cosrt = cosl> cos +sin b sinr cos A. cos A cos B cos C-\- sin B sin C cos a. . sin s sin (j a) RECAPITULATION OF FORMULAS 123 / ~V -cos S cos (S A) cos (S) cos(S-C)' sin (A cos *,(A ff) tan $ sin ( + ) cot j C sin i ( ^)~tan | (A B) cos ^ (# + <5) _ cot | C cos (a b) ~tan ' sin a sin b sin A sin^' AREA OF SPHERICAL TRIANGLES ( lOl) tan \ _ ' _^ tan j _ APPENDIX RELATIONS OF THE PLANE, SPHERICAL, AND PSEUDO- SPHERICAL TRIGONOMETRIES We have up to the present considered the trigonometries which deal with figures on a plane or spherical surface. A characteristic feature of these two surfaces is that the curv- ature of the plane is zero, while that of the sphere is a posi- tive constant p. If the radius of the sphere is increased in- definitely, its surface approaches the plane as a limit while its curvature p approaches o. In works on absolute geometry it is shown that there ex- ists a surface which has a constant negative curvature: it is called a pseudo-sphere, and the trigonometry upon it pseudo- spherical trigonometry. We observe that as p passes continuously from positive to negative values, we pass from the sphere through the plane to the pseudo-sphere. Thus the formulas of plane trigonometry are the limiting cases of those of either of the two other trigonometries. In the treatment of spherical trigonometry the radius of the sphere has been taken as unity. If, however, the radius of tlie sphere is r, and a, b, and c denote the lengths of the sides of the spherical triangle, the formulas are changed, in that a is replaced by -, b by -, and c by - ; thus, 126 APPENDIX . ~ sine sin C= sin a . c sin- ^- becomes sinC= . a sm- r The formulas for pseudo-spherical trigonometry are the same as the formulas of spherical trigonometry, except that the hyperbolic functions of -, -, and - are substituted for r r r the trigonometric. Thus, corresponding to the above formula of spherical trigonometry, is the formula sinh- sinC= u a smh- r of pseudo-spherical trigonometry. PSEUDO-SPHERE The pseudo-sphere is generated by revolving the curve whose equation is r-\- -J^^7 l . '. y=r log * yV-.r' about its y axis. The radius of the base of the pseudo-sphere is r. APPENDIX 127 Hence the formulas of plane trigonometry can be derived from the formulas of either spherical or pseudo- spherical trigonometry by expressing the functions in series and al- lowing r to increase without limit. Example. Show that if r is increased indefinitely the following corresponding formulas for the spherical and pseudo-spherical right triangle a b c cos = cos- cos-' (i) r r r cosh - = cosh - cosh - , (2) r r r reduce to the corresponding formula for a plane right triangle; that is, to . a*=y+c\ (3) Substituting the series cos -, etc., in equation (i), we obtain ( r (\i ^ ( i W-i. \ ( r (<\. \ ( I -- ,(- ) +...) = ( I -- -(-) + . . . ) I I -- -(-) +...), \ 2 !W / \ 2 \\rj J \ 2\\rJ J 1 2 , I rt 4 , I 6* i c 2 . I l> 4 . Or I -- ; -. H --- 3 -h ... =5 I -- : -s -- , -H --- 2"<"''' (4' 2 ! r 2 4 i r 4 2 ! r* 2 ! r 2 4 1 r 4 Substituting in equation (2) the series for cosh - , etc. , which we obtain from x ~ x , we have . i ,i , f or i + :-n + -j+... = i+-7 -; + :T + -- 1+... (5) 2 ! ;-* 4 ! r 4 2 ! r* 2 ! ; J 4 i r* Cancelling I in equations (4) and (5), multiplying by r 2 , and, finally, allowing r to increase without limit, we get from either equation EXERCISES Derive each of the following formulas of plane trigonometry from the corresponding formula of spherical trigonometry, and also from the corresponding formula of pseudo-spherical trigonometry. 123 APPENDIX Right triangles ; A Bright angle. (I.) Plane, sin C=-- c . . ^ sin c Spherical, sin C=- -- sin a Pseudo-spherical, sin C=-r-r Oblique Triangles. (2.) Plane, a 7 b 1 + ? 2 be cos A Spherical, cos a = cos cos c-\- sin 3 sin cos A Pseudo-spherical, cosh a = cosh b cosh r + sinh b sinhc cos A. (3.) Plane, Spherical, ---r- , , t i 1 :an v - -^ -? = tani-tani - tan $ -- tan i Pseudo-spherical, (iSoO-^-H^+C 1 ) */ T^~~ TU-") , (*-*) . tan s -=Y tanh ^ - tanh ^ ' tanh J S ' tanh | ANSWERS TO EXERCISES 4 (page 3). (5.)cos/ = 4, tan/ = f, (i.) 192 51' 25^". cot y = J, sec/ = f , Quadrant III. csc/ = . (2.) 2 5 . (6.) sin 60 = ^ \/3, (3.) 2870, 647. tan 60 = \/3, (4.) Quadrant III. cot 60 = ^ \/3> sec 60 = 2, 9 (page 9). esc 60 = f -v/3. tan 1000 is negative. (7.) cos o = i, tan o = o. cos 810 is o. (8.) sin^ = f, cos 2 = |, sin 760 is positive. cot 2 = |, sec .2 = f , cot 70 is negative. CSC2- = |. cos 550 is negative, tan 560 is negative. (9.) sin 45 = cos 45 = i -v/2, tan 45= i, sec 300 is positive, cot 1 560 is negative. sec 45 = esc 45 = ^'2. . ( T Q \ Clrji/ __ 1 -t/ ^ f*dz 1/ sin 130 is positive. ^ 5^y 3, -ub_y cos 260 is negative. cot/ = f v/5, sec/ = |, tan 310 is negative. csc/ = ! y'5. (11.) sin 30 = ^ cos3o = |-v/ 13 (page n). tan 30 = | -v/3, (3.) cos 30 = -v/3. sec ^ O o _ a ^~ tan 3 o = v/3, CSC 30 = 2. cot 30 = -v/3, (12.) sin J r = f, cos^ = --|. sec 30 = | -v/3, (13.) J\.\ Vs- CSC 30 = 2. v (4.) cos.r= \/2, 17 (page 14). tan .r = ^ \/2, (i.) sin 70 = cos 20, cot .r = 2 \/2, cos 60 = sin 30, sec .r | -v/2, cos 89 31'= sin 29', CSC X ~ 3. cot 47= tan 43, 9 ANSWERS TO EXERCISES (2.) (3.) (4.) (5.) (i.) (2.) (3.) tan 63= cot 27, sin 72 39'= cos 17 21' .r = 3 o. -r = 22 30'. .r=i8. (4.) (5. 25 (page 21). 225 and 315, 60 and 240. 60, 120, 420, 480. sin 30= -|, cos 30=^ -v/3. sin 765= cos 765 = \ -\/2, sin 1 20= \ i/3- cos 120 = , sin 210= \, cos 210= \ \/3. The functions of 405 are equal to the functions of 45. sin 6oo= \ \/$, cos 600 = , tan 600= -y/3, cot 600 = \/3, sec 600 = 2, esc 600 = | -y/3. The functions of 1125 are equal to the functions of 45. i sin 45 = -/I, cos 45= | "v/2. tan 45= cot 45= i , sec 45=-v/2, CSC 45= -y/2. sin 225= cos 225= -v/2, tan 225= cot 225= i, sec 225= esc 225= v/ 2 - The functions of 120 are the same as those of 600 given in (4). sin 225 = ^ \/2, cos 225 = 1\/ 2 ' tan 225= cot 225= i , sec 225= -\/2, CSC 225= -\/2, sin 420 = ^ v/3, cos 420 ^, tan 420 = y'T cot 420 = i \/3, sec 420 = 2, esc 420 == | \/3_ The functions of 3270 are equal to the functions of 30. (6.) sin 233 = cos 37, cos 233 = sin 37, tan 233 = cot 37, cot 233 = tan 37, sec 233 = esc 37, esc 233 = sec 37. sin 1 97 = sin 17, cos 197 = cos 17, tan 197 = tan 17, cot 1 97 = cot 1 7, sec 1 97 = sec 1 7, esc 1 97 = esc 17. sin 894 = sin 6, cos 894 = cos 6, tan 894 = tan 6, cot 894 = cot 6, sec 894 = sec 6, esc 894 = esc 6. (7.) sin 267 = sin 87, tan 254 = tan 74, cos 950 = cos 50. (8.) 0.28. ANSWERS TO EXERCISES (9.) 2 sin 5 x. (10.) i -{-sec 2 x. (11.) sin (x 90)= cos. r, cos (x 90) = sin .r, tan (x 90) = cot x, cot (x 90) = tan x, sec (x 90) = esc x, esc (.r 90) = sec x. 28 (page 24). (I.) a =62.324, y4 = 32 52' 40". (2.) =21.874, yj = 39 45' 28". 5 = 50 14' 32". (3.) a = 300.95, = 683.96, # = 66 15'. (4.) = 26.608, c = 45- 763, # = 35 33'- area = 495-34- (5-) ^ = 3-9973. ^ = 4.1537, .4 = 15 46' 33", area = 2. 257. (6.) = 0.01729. (7.) = 298.5. (8.) ^ = 39 42' 24". (9.) ^- = 2346.7. (10.) # = 28 57' 8". dr.) 444.16 ft. (12.) 186.32 ft. (I3-) 34 33' 44"- % (14.) 303.99 ft. (15.) 238.33 ft. (16.) 15 miles (about). (17.) 79-079 ft. (18.) 165.68 ft. (I9-) 53 33'- (20.) 115.136 ft. (21.) 76.355 ft. (22.) = 8o 32", ,4 = C = 49 59' 44". (23.) =53 i6' 3 6", = 12.0518 in., area = 72. 392 sq. in. (24.) = 130.52 in., area = 24246 sq. in. (25.) 23.263 ft. (26.) 1 7 48". (27.) 5.3546 in. (28.) 1084950 sq. ft. (29.) 17 ft., 885 sq. ft. (30.) radius = 24.882 in., apothem = 20.13 ' in - area= 1472 sq. in. (31.) 12.861. (32.) 1782.3 sq. ft. (33.) 38168 ft. (34.) 20.21 ft. (35.) 2518.2 ft. 29 (page 28). (I.) ^ = 22 58', = 7-07. c = 9.0046. (2.) = 79-435. ^=45 27' 14", C = 95 24' 46". (3.) A3 = 7.674$, ^L' = 2.6435, B = 46 43 '50', ^' = 133 16' 10", io$ 53' 10", ' = \g 20' 50". (4.) A = 37 53'. # = 43 52' 25", 132 ANSWERS TO EXERCISES C = 9 8? 14' 35" (5-) 902.94- (6.) 1253.2 ft. (7.) 357-224 ft. (8.) ^ = 44 2' 9", = $i 28' u", C = 84 29' 40", area = 126100 sq. ft. (9.) 407.89 ft. (io.) B= \2\ 7' 16", C = 92 20' 38", D = 7\ u' 6". (u.) BC- 6.6885, DC 1.9915. 34 (page 34). (2.) sin (454- -i^) = i-y/ 2 (cos.r-f sin x), cos (45+ f) = 4 V 2 < cos x sin ;r), sin (yfx) = ^ (cos* -y/3 sin JT), cos (30^ = i ( V / 3 cos -*" + sin ^), sin (60+*) = ^ (-V/3 cos A- 4- sin x), cos (60+-*-) = i(cos.r -y/3 sin.r)- (3.) sin (.r+_v) = f. sin (.r .y)=. U-) sin 75 = cos 75 = (j.) sin 15= cos 15= 4 V6 y/2 4 -y/6 \/2 39 (Page 37). (5.) sin(45-.r) = 5 \/2 (cos .1 sin .V), cos (45 .r) = | \/ 2 (cos a- -f- sin x), sin(45+.r) = - ^ -y/2 (cos .i- + sin .v), cos(45+x) = \-\/2 (cos x sin.r). (6.) tan 7 5 =2 4- V3. tan 1 5 = 2 v/3- (14.) ini7= cos \y (15.) sin 2.r = ff, cos 2x = $ 5 . (16.) sin 22^ = -K/2 -y/2, CSC 22^ c = -y/4 + 2 \/ 2 - 07-) (i 8.) sin I5 = K/ 2 ""V3. ANSWERS TO EXERCISES 133 esc 15 = 2 \J 2 (20.) sin 5_r = 5 sin x 20 sin 3 x + 16 sin 5 .f. (21.) cos 5-r = 5 cos x 20 cos 3 x -f- 1 6 cos 5 x. (23.) The values of jr<36o are, o, 30, 1 50, 1 80, 210, 330. (36.) tan.r tanj. 41 (page 40). (i.) sin ' | -v/ 2 =45, 135. 45+ 360, etc., cos ' \ 60, 300, etc., tan-' ( i)= 1 3 5, 3 1 5, etc., cos 1 i =0, 360, etc., sin 1 ( 1) = 210, 330, etc. (2.) (3-) c (4.) si (5.) sin(cos-'J) = f (6.) cot (tan > j\) = 1 7 (8.) 45, 225. (9.) JT=:45 ,J =180. (10.) sin"" 1 ^ = 225. 48 (page 46). (i.) C=m33'. ^ = 2133.5, c = 2477.8. (2.) C=554i'- ^ = 534.05, ^ = 653.52. (3.) C'=45 34', a 1548.1, ^=1293.7. (4.) ^ = 105=59', CZ =: 54.OI8, <: =47.738. (5.) v9 = 6858' ( ^ = 5274.9, ^ = 3730- (6.) ^=54 58', a = 923.4, c-= 1 187.7. 49 (page 47). (i.) (i.) Two solutions. (2.) One solution, a right tri- angle. (3.) One solution.. (4.) Two solutions. (2.) ff=i6 57' 21", C- 1 5 5' 39". ^ = 0.32122. (3.) r= 2.5719, ^=13 15' i", C= 142 13' 59"- (4.) c = 93-59. ^' = 54-069, ^ = 26 52' 7", #'=133 7' 53". C = i3i46 / 53",C'=253i'7". (5.) No solution. (6.) <= 1.0916, '=0.36276, ,4 =z 3937 ' 1 6", ,4 ' =: 1 4o2 2 '44", ^ .ii7 5o'44",^'=i7 5' 1 6". 50 (page 48). (i.) a = 0.097 1, # = 90 35' 36", C=489'34", 5 = 0.0053261. 134 ANSIVERS TO EXERCISES (2.) C \\.1\\, 7>=48 D 44' 32", A = 76 20' 5", C = 95 15' 56", # = 44 52 55". 5 = 0.60709. 5 = 80.962. (3.) = 85.892, 52 (page 50). A =67 21' 42", (i.) 1116.6 ft. C = 6248' 1 8", (2.) 3081.8 yards. 5=3962.8. (3.) 638.34 ft., (4.) a =0.6767, 14653 sq. ft. />' = 1 5 9' 2 1 ", (4.) 4.1 and 8.1. C= 131 19' 39". (5.) 13.27 miles. 5 = 0.08141. (6.) 6667 ft. One solution. (5.) r = 72.87, (7-) 121.97. ^ = 40 50' 32". (8.) 44 2' 56". =ll 2' 28". (9.) 32.151 sq. miles. 5 = 422.65. (11.) 54 29' 12". (12.) a 12296 ft., 51 (page 49). ^=13055 ft. (13.) 294.77 ft. (i.) ,4 = 55 20' 42", (14.) 222.1 ft. ^=106 35' 36", C=i8 3' 42", 5 = 267.92. (2.) A = 34 24' 26", (i 6.) 4202.1 ft. (17.) 72.613 miles. (18.) 50-977 ft- (19.) 0.85872 miles. # = 73 14' 56", C = 72 20' 36", 5=3.6143. (20.) 2.98 miles. (21.) 1 393.9 ft. (22.) 8.2 miles. (3.) A = 52 20 24", =107 19' 14", (23.) 1 87.39 ft. (24.) 0.60 1 1. C=20 20' 24", (25.) 4.8112 miles. 5=1437.5. (4.) A = 97 48', (26.) 60 51' 8". B= 18 21 48", (27-) 37.365 ft- C=63 50' 12", (28.) 3.2103 miles. 5=193.13. (29.) 10.532 miles. (5.) A = 54 20' 1 6". (30.) 851.22 yards. ^ = 70 27' 46". (3 1 -) 9-5722 miles. C= 54 72', (32.) 6.1271 miles. 5 = 6090. (33.) 280.47 ft. (6.) A = 35 59' 30". (34-) 1 23.33 ft. ANSWERS TO EXERCISES 135 (35-) 4-8ii2 miles. (4.) .(-=!, .1^ = 0.3090 +/ 0.95 n. (36.) 2666.1 ft. .r 2 = 0.8090 + i o. 5878, .r 3 = 0.8090 / 0.5878. 53 (page 56). .1-^ = 0.3090 i 0.951 1. (i.) 30 = 0.5236, 45 = 0.7854. 60 = 1.0472, I 20 2.0944, 135= 2.3562, J20 12.5664, 77 (page 78). (23.) ^ = 30. (24.) y = 30. (25.) x o or 45. (26.) -r = 6o. 990= 17.2788. (28.) / = 45- (2.) -JT = 22 30', (29.) jr = 45. -=18, 10 i = 28 38' 53", l~ = 1 00 1 6' 4". (3-) I-35.0.54. (30.) .r = 30. (31.) ,r = 6o. (32.) x = yP. (33.) No angle < 90. (34.)* =30. (35.) sin 92 = cos 2. 74 (page 73). (36.) cos 127=: sin 37. (37.) tan 320 = tan 40. (i.) sin 4jr = 4 cos 3 ;r sin^r (38.) cot 350 = cot 10. 4 cos x sin 3 x. (39.) sin 265 = cos 5. cos 4_r = cos 4 .r 6 cos 2 x sin 2 x -}- sin 4 x. (2.) sin 6;r = 6 cos 5 x sin .r (40.) tan 171= tan 9. (41.) cos.r = 1^/33' 20 cos 3 x sin 3 .r ssV33. + 6 cos x si n 5 JT, cot^ = i-v/33, cos 6x = cos 6 or sec x g'g -v/3~3~, y' 1 5 cos* .* sin 2 x CSC X = |. + 1 5 cos 2 x sin 4 .r sin 6 x. (42.) sin^ = -i-/55. (3.) * =i, ^ | = j + /^3, tan^r^i-v/55- /- cot x = fg \/55, -r 2 = | + /, .r a = i, sec x = |, ,- csc.r=r /?; \/SS- x \ ? Z ^~' (43.)^n.r = -^V^ jr 5 = \ i . cot.r^f, sec.i- = %\/i3, 136 ANSWERS TO EXERCISES cscx = ^13. (21.) 71 33' 54"- (44.) sin x = J f \/74. (22.) 858,160 miles. cos x =. , 7 r \/74. (23.) 238,850 miles. < T / T^' tan .r = f , sec .r = \ -\/74' i / (24.) 2163.4 miles. (25.) 90,824,000 miles. esc -t = i y 74. (26.) 432.08 ft. (45.) Quadrant II or IV. (27.) 60.191 ft. (46.) Quadrant I or II. (28.) 0.32149 mile. (47.) Quadrant III or IV. (29.) 193.77 ft. (48.) Quadrant I or II. (49.) .r = o, 120, 1 80, 240. 79 (page 83). (50.) .r = 30, 135, 150, 315. (i.) 3.416 ft. (51.) .r = o, 90, 120, 180, 240, (2.) 3.7865 ft. 270. (3.) 20.45 ft- (57-) o. (4.) 36.024^. (58.) a. (5.) 8.6058 sq.ft. (59.) 2(a-b\ (6.) 181.23 in. (60.) i(fl 2 b). (7-) 2.9943 ft- 78 (page 80). (8.) 5.1311 in. (9.) 25.92 ft. (i.) 306.32 ft. (10.) 92 i' 24". (2.) 831.06 ft. (11.) 1.2491. (3.) 53 28' 14". (12.) 33 1 2' 4". (4-) 49-39 ft. (13.) 11248 ft. (5.) 0.43498 mile. (14.) 0.60965 miles. (6.) 209.53 ft. (15.) 1.3764. (7.) 7.3188 ft. (16.) 1.9755- (80 37 36' 30". (17.) 19.882. (9.) 109.28 ft. (i 8.) 0.9397. (10.) 502.46 ft. (19.) 6.4984. (u.) 6799.8 ft. (20.) 3.4641- (12.) 219.05 ft. (21.) 6.1981. (13.) 49i.76ft. (22.) 6.9978. (14.) 50 32' 44". (23.) 15.25. (15.) 49 44' 38". 80 (page 84). (i 6.) 34.063 ft. (78.) x 90, 1 20. 240, 270. (17.) 32.326 ft., 29 6' 35". (79.) .r = o, 20, 45, 90, ioo c (18.) 5.6569 miles an hour. 135, 140, 1 80, 220 C (19.) 56.295 ft. 225, 260, 270, 3i5 c (20.) 103.09 ft. 340. ANSWERS TO EXERCISES 137 (80.) ,r = o, 30, 90, 150, 1 80, 270. (8 1.) x = o, 45, 120, 240, 225, 270. (82.) x = o, 90, 1 80, 270. (83.) x = cP, 90, 210, 330. (84.) x = 240, 300. (85.) .r = 2io, 330. (86.) x = o, 90. (87.) ,r = o, 1 80. (88.) .r=zo, 1 80. (89.) x = cP, 90, 120, 1 80, 240 270. (90.) x = 4S, 135- 22 5. 3'5 (91.) .r = 30 , 150, 210, 330. 81 (page 88). (i.) 2145.1 ft. (2.) 12.458 miles. (3.) 1.1033 miles. (4.) 1 508.4 ft. (5-) i7i93Y ards - (6.) 1.2564 miles. (7.) 1346.3^. (8.) 387.1 yards. (9.) 5.1083 miles. (10.) 3791-8 ft. (n.) 4-4152 ft- (12.) 28 57' 20". (13.) 115.27. (14.) 44.358 ft. (15.) 92.258 ft. (16.) 101 32' 16". (17.) 0.83732 mile. (18.) 539.1 ft. (19.) 1.239. (20.) 152.31 and 238.3. (21.) 68.673 ft. (22.) 32.071 ft. (23.) 137.78 ft. (24.) (25-) (26.) (27-) (28.) (29-) (30.) (31-) (32.) (33.) (34.) (35.) (36.) (37-) (38.) (39-) (40.) (4I-) (42.) (43-) (44-) (45-) (46.) (47-) (48.) (49-) (50-) 55-74 ft. 247.52 ft. 556.34 ft. 465.72 ft. 109.22 ft. 2639.4 ft. 396.54 ft. 287.75 ft- 2280.6 ft. 64.62 ft. 127.98 ft. 45-183 ft- 4365.2 ft. 140.17 ft. 610.45 ft. i56.66ft. 41 48' 39" and 125 25 57' 51,288,000. 366680. i i 586. 947460. 9929-3- 751-62 sq. ft. 3I45.9- 855.1. 876.34. 88 (page 98). (i.) ^=54 59' 47". = 45 41 '28", C=6 5 4 5' 58", (2.) C= 7 i 36' 47". b = 95 22', ' = 96 13' 23". 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I 9 .2 18.4 17.6 8 16.8 16.0 15.2 9 24.3 23.4 22.5 9 21 .6 20.7 19.8 o 18.9 18.0 17.1 19O 23O N O 1 2 3 4 5 6 7 8 9 190 27 876 898 921 944 967 989 *OI2 *o35 *o58 *o8i 191 28 io3 126 149 171 ic;4 21 7 240 262 285 307 192 33o 353 3 7 5 3 9 8 421 443 466 488 5n 533 I 9 3 556 5 7 8 601 623 646 668 691 7 i3 7 35 758 ig4 780 8o3 825 847 870 892 914 9 3 7 9 5 9 981 igS 29 oo3 026 o48 070 092 11 K. i3 7 i5g 181 203 196 226 248 270 292 3i4 336 358 38o 4o3 425 197 447 469 491 5i3 535 55 7 579 601 623 645 198 66 7 688 710 732 754 776 798 820 842 863 199 885 907 929 9 5i 973 994 *oi6 *o38 *o6o *o8i 200 3o io3 125 i46 168 190 21 I 233 2 55 276 298 2OI 320 34 1 363 384 4o6 428 449 471 492 5i4 202 535 55 7 5 7 8 600 621 643 664 685 707 728 203 75 771 792 8i4 835 856 878 899 920 942 204 963 984 *oo6 *O27 *o48 *o6g *Q9I *I 12 *i33 *i54 205 3i 175 197 218 23g 260 281 302 323 345 366 2O6 38 7 4o8 429 450 4 7 i 492 5i3 534 555 576 207 697 618 63 9 660 681 702 723 744 765 7 85 208 806 827 848 869 890 911 g3i 952 9 7 3 994 209 32 015 o35 o56 077 098 118 i3g 1 60 181 201 210 222 243 263 284 305 325 346 366 387 4o8 21 I 4'28 44g 46 9 490 5io 53i 552 5 7 2 5 9 3 6i3 212 634 654 675 6 95 7 i5 ?36 7 56 777 797 818 2l3 838 858 879 899 919 940 960 980 *OOI *02I 2l4 33 o4r 062 082 102 122 i43 1 63 i83 203 224 2l5 244 264 284 3o4 325 34s 365 385 4o5 425 216 445 465 486 5o6 526 546 566 586 606 626 217 646 666 686 706 726 ?46 766 786 806 826 218 846 866 885 goS 925 945 9 65 985 *oo5 *025 219 34o44 o64 o84 io4 124 i43 i63 i83 203 223 220 242 262 282 3oi 321 34i 36i 38o 4oo 420 221 43 9 45 9 4?9 4 9 8 5i8 53 7 55 7 5 77 5 9 6 616 222 635 655 6 7 4 6 9 4 7 i3 7 33 7 53 772 792 811 223 83o 850 869 889 908 928 947 967 986 *oo5 224 35 025 o44 o64 o83 1 02 122 i4i 160 180 199 225 218 2 38 257 276 2g5 3i5 334 353 3 7 2 3 9 2 226 4n 43o 44g 468 488 507 526 545 564 583 227 6o3 622 64 1 660 679 698 717 7 36 7 55 774 228 79 3 8i3 832 85i 870 889 908 927 946 9 6 5 229 984 *oo3 *O2I *o4o *o59 *078 *097 *n6 *i35 *i54 230 36 i 7 3 192 21 I 229 2^8 267 286 305 324 342 N 1 2 3 4 5 6 7 8 9 23O 26O N O 1 2 3 4 5 6 7 8 9 230 36 173 I 9 2 21 I 22 9 248 267 286 305 324 342 831 36 1 38o 3 99 4i8 436 455 4 7 4 4c >3 Sii 53o 232 549 568 586 605 624 64a 66 1 680 6 9 8 717 233 7 36 7 54 773 79 I 810 829 84? 866 884 9 o3 234 922 9 4o 9 5 9 977 996 *oi4 *o33 *o5i ^070 *o88 235 3 7 io 7 125 i44 162 181 199 218 236 254 273 a36 291 3io 3a8 346 365 383 4oi 420 438 45 7 23 7 475 4g3 5n 53o 548 566 585 6o3 621 63 9 238 658 676 6 9 4 712 7 3i 749 767 7 85 8o3 822 23 9 84 Q 858 876 8 9 4 912 93i 949 9 6 7 985 *oo3 240 38 021 039 057 o 7 5 o 9 3 I 12 i3o 1 48 1 66 1 84 s4i 202 220 238 256 274 292 3io 328 346 364 242 382 399 417 435 453 471 48 9 5o 7 525 543 243 56i 578 596 6i4 632 650 668 686 7 o3 721 244 7 3 9 7 5 7 775 792 810 828 846 863 881 899 245 917 934 g52 97<> 987 *oo5 *023 *o4i *o58 *o 7 6 246 39 094 i ii 129 i46 1 64 182 199 . 2I 7 235 252 247 270 287 305 322 34o 358 3 7 5 3 9 3 4io 428 248 445 463 48o 4 9 8 5i5 533 55o -568 585 602 a4 9 620 63 7 655 672 690 707 7 24 7^ (2 7 5 9 777 250 794 811 82 9 846 863 88 1 898 9 i5 9 33 9 5o 5i 967 985 *002 *oi 9 *o37 *o54 *o7i *o88 *io6 *I23 252 4o i4 i5 7 175 I 9 2 209 226 243 261' 2 7 8 295 253 3l2 32 9 346 364 38i 398 4 15 432 44 9 466 a54 483 5oo 5i8 535 552 56 9 586 6o3 620 63 7 255 654 671 688 705 722 7 3 9 756 773 79 807 256 824 84 1 858 875 892 909 926 943 9 6o 976 257 99 3 *OIO *027 *o44 *o6i *o 7 8 ^ ( ) 95 *in "128 *i45 258 4i 162 79 196 212 229 246 263 280 296 3i3 25 9 33o 34 7 363 38o 3 97 4i4 43o 447 464 48 1 260 497 5i4 53i 547 564 58i 5 97 6i4 63i 64? N 1 2 3 4 5 7 8 9 PP 19 18 17 16 15 14 i 1.9 1.8 1.7 i 1.6 i.5 ~i4 2 3.8 3.6 3.4 a 3.2 3.o 2.8 3 5. 7 5.4 5.i 3 4.8 4.5 4-2 4 7.6 7.2 6.8 4 6.4 6.0 5.6 5 9 .5 9 .o 8.5 5 8.0 7-5 7.0 6 ii.4 10.8 10.2 6 9.6 9.0 8.4 7 i3.3 12.6 1 1.9 7 i 1 1.2 io.5 9-8 8 i5.2 i4.4 i3.6 8 12.8 12.0 I 1.2 9 17.1 16.2 i5.3 9 j i4.4 i3.5 12.6 26O-3OO N 1 2 3 4 5 6 7 8 9 2GO 4i 497 5i4 53i 547 564 58i 5 97 614 63i 64 7 261 664 68 1 697 714 7 3i ?4? 764 780 797 8i4 262 83o 84? 863 880 896 913 929 946 963 979 263 996 *OI2 *O2g *o45 *o62 *c>78 *o 9 5 *ui *I27 *i44 a64 42 160 I 77 193 210 226 243 25g 275 292 3o8 265 3^5 34 1 35? 3 7 4 390 4o6 423 43 9 455 4 7 2 266 488 5o4 621 53 7 553 670 586 602 619 635 267 65i 667 684 700 716 732 749 765 781 797 268 8i3 83o 846 862 878 8 9 4 911 927 943 9 5 9 269 97 5 991 *oo8 *O24 *o4o *o56 *072 *o88 *io4 *I2O 270 43 i36 l52 169 185 2OI 217 233 249 265 28l 271 297 3i3 329 345 36i 3?7 3 9 3 409 425 44 1 272 45 7 4?3 48 9 505 521 53? 553 56 9 584 600 2 7 3 616 632 648 664 680 696 712 727 743 7^9 274 77 5 791 807 823 838 854 870 886 902 917 2 7 5 9 33 94g 9 6 5 981 996 *OI2 *028 *o44 *o5 9 *o 7 5 276 44091 107 122 1 38 i54 I7O i85 2OI 217 232 277 248 264 279 2 95 3u 326 342 358 3 7 3 38 9 278 4o4 420 436 45i 46 7 483 498 5i4 529 545 279 56o 5 7 6 5 9 2 607 623 638 654 669 685 7OO 280 716 7 3i 747 762 778 79 3 809 824 84o 855 281 871 886 902 917 932 948 9 63 979 994 *OIO 282 45 025 o4o o56 071 086 102 117 i33 i48 i63 283 179 ig4 209 225 24o 255 271 286 3oi 3i 7 284 332 347 362 3 7 8 3 9 3 4o8 423 43 9 454 46 9 285 484 500 5'5 53o 545 56i 5 7 6 5gi 606 621 286 637 652 667 682 697 712 728 743 7 58 77 3 287 788 8o3 818 834 849 864 879 8 9 4 909 924 288 9 3 9 954 969 984 *ooo *oi5 *o3o "045 *o6o *75 289 46 090 105 120 135 150 165 1 80 J 95 210 225 290 24o 255 270 285 3oo 315 33o 345 35 9 3?4 291 38 9 4o4 4l9 434 44g 464 479 494 5og 523 292 538 553 568 583 5g8 6i3 627 642 657 672 298 687 702 716 7 3i 746 761 776 79 8o5 820 294 8X5 850 864 879 894 909 923 g38 9 53 967 2g5 982 997 *OI2 *O26 *o4i *o56 *O7O *o85 *IOO *u4 296 4? 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'9 42 8.47 226 8.4? 245 i.52 755 9.99981 18 43 8.47 650 424 8.4? 669 424 i.52 33i 9.99981 '7 419 44 8.48069 8.48089 416 i .5i 911 9.99980 16 45 8.48485 8.48 5o5 i .5i 495 9.99980 i5 46 8.48896 411 8.48 917 412 i .5 i o83 9-99979 i4 47 8.49 3o4 408 8.49 325 i 50675 9-99979 i3 48 8.49 708 8.49 729 i .5o 271 9-99979 12 49 8.5o 1 08 8.5o i3o 401 i .4g 870 9.99978 I I 50 8.5o 5o4 396 8.5o 527 397 1.49473 9.99978 10 5! 8.5o 897 393 8.5o 920 393 i .49 080 9-99 977 9 r 8.5i 287 8.5i 3io i .48 690 9-99977 8 53 8.5i 673 382 8.5i 696 300 383 i.48 3o4 9-99977 7 54 8.52o55 8 .52 079 180 i .47 921 9.99976 6 55 8.52434 8.52 45g i .4? 54i 9.99976 5 56 8.52 810 37 6 8.52 835 37 6 i .47 i65 9.99975 4 373 373 5 7 8.53 i83 8.53 208 i .46 792 9-99975 3 58 8.53 552 8.53 578 i .46 422 9.99974 2 5 9 8.53 919 367 8.53 945 3 6 7 i.46o55 9.99974 I 60 8.54282 8.54 3o8 i .45 692 9.99974 L.Cos. d. | L. Cotg. d. L. Tang. L. Sin. 88. PP 470 455 441 424 4 o8 396 386 376 367 .1 47.0 45-5 44.1 .1 42.4 40.8 39-6 .1 38.6 37.6 36.7 .2 94.0 91 o 88.2 .2 84.8 81.6 79.2 .2 77.2 75.2 73-4- 3 141.0 '3 6 -5 132.3 3 127.2 122.4 118.8 3 115-8 II2.8 10. 1 4 188.0 182.0 176.4 4 169.6 163.2 158.4 4 '544 150.4 46.8 5 235.0 227.5 220.5 5 212. 204.0 198.0 5 193.0 188.0 183-5 6 282.0 2 73- 264.6 .6 254-4 244-8 237.6 .6 231.6 225.6 220.2 7 329.0 318-5 308.7 7 206.8 285.6 277-2 7 270.2 263.2 856.9 .8 376.0 364.0 352-8 .8 339-2 326 4 316.8 .8 308.8 300.8 293.6 .9 423.0 409. 5 396. g 9 38i.fi ^67.2 356-4 9 347-4 338.4 330.3 33 2. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 8.54282 8.54 3o8 i .45 692 9.99974 60 I 8.54642 357 8.54669 M 1 1.45 33i 9.99973 5 9 2 8.54 999 8.55 02 7 i .44 9?3 9.99973 58 3 8.55 354 355 8.55382 355 1.44618 9.99972 5 7 4 8.55 7 o5 8.55 734 352 i .44 266 9.99972 56 5 8.56 o54 8.56o83 i.43 917 9.99971 55 6 8.56 4oo 346 8.56429 34 6 i.43 571 9.99971 54 7 8.56 743 343 8.66773 344 i.43 227 9.99970 53 8 8.57 o84 8.5 7 n4 1.42 886 9.99970 62 9 8.57 421 337 8.57452 338 1.42 548 9.99969 5i 10 8.5 77 5 7 33 6 8.57 788 33 6 I .42 212 9.99969 50 1 1 8.58 089 8.58 121 333 33 I ./ ii8 79 9.99968 49 12 8.58 419 8.5845i I .1 ii 54g 9.99968 48 1 3 8.58 7 47 8.58 779 3 2 I .4l 221 9.99967 47 i4 i5 S.Sg 072 8.5 9 3 9 5 325 323 8.59 io5 8.59428 326 323 I .40 895 i .4o 572 9.99967 9.99967 46 45 16 8.5g 715 8.59 749 321 i .40 25i 9.99966 44 i? 8.60 o33 316 8.60068 319 Jyfi i.3g 982 9.99966 43 18 8.60 349 8.60 384 i.Sg 616 9.99965 42 19 8.60 662 3'3 8.60698 3M i .39 3o2 9.99964 4i 20 8.60 973 8. 6 1 009 3" i .33 991 9.99964 40 21 8.61 282 307 8.61 3ig 310 i. 3868i 9.99963 3 9 22 8.61 58g 8.61 626 i.38 374 9.99963 38 23 8.61 894 8.61 9 3i 305 1.38069 9.99962 3 7 302 3 1 24 8.62 196 301 8.62 234 1.37 766 9.99962 36 25 8.62 497 8.62 535 1.37465 9.99961 35 26 8.62 795 8.62834 299 1.37 166 9.99961 34 296 29 r 27 28 8.63 091 8.63 385 294 8.63 i3i 8.63426 295 1.36 869 i.36 574 9.99960 9.99960 33 32 2 9 8.63 678 293 8.63 718 i.36 282 9-99 9 5 9 3i 30 8.63 968 8.64 009 291 1.35 991 9-99959 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. 87 3O'. PP 360 350 340 33<> 330 310 300 290 38 5 .1 ,6 35 34 .! 33 32 3' .1 3 29 28.5 .2 72 70 68 2 66 64 62 .2 60 s 57-0 3 1 08 105 IO2 3 99 96 93 3 90 8 7 85.5 4 144 140 itf 4 132 128 124 4 120 116 114.0 5 1 80 175 170 165 1 60 155 5 ISO '45 142 5 .6 216 210 204 6 198 192 1 86 .6 1 80 '74 171.0 7 252 245 2 3 8 7 231 224 217 7 3IO 203 199-5 .8 288 280 272 8 264 256 248 .8 240 232 228.0 9 3 2 4 SJS 306 2Q7 288 279 9 270 256-5 34 2 3O . , L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 30 8.63 968 8.64 009 1 i .35 991 9-99 9 5 9 30 3i 8.64 256 287 8.64 298 287 i .35 702 9.99958 2 9 32 8.64 543 8.64585 i.35 415 9.99958 28 33 8.64827 8.64 870 285 i .35 i3o 9-99 U 5 7 27 283 28, 34 8.65 no 8.65 i54 281 1.34846 9.99956 26 35 8.65 3 9 i 8.65435 1.34565 9.99956 25 36 8.65 670 279 8. 65 715 1.34285 9.99 955 24 277 2? 3? 8.65 9 4? 8.65 99 3 i .34 007 9.99955 23 38 8.66223 8.66 269 i.33 7 3i 9.99954 22 39 8.66497 274 8.66 543 274 1.33457 9.99 9 5 4 21 40 8.66 769 8.66816 i.33 1 84 9.99953 20 4i 8 .67 o3g 8.67 087 271 i .32 gi3 9.99952 19 42 8.67 3o8 2 9 8.6 7 356 i.32 644 9.99962 18 43 8.6 7 5 7 5 266 8.6 7 624 266 1.32 3 7 6 9.99951 17 44 8.67841 8.6 7 890 vfit I .32 I IO 9.99951 16 45 8.68 io4 2 3 8.68 1 54 i ."i !i 846 9.99950 i5 46 8.68367 263 260 8.68 417 263 26l i.3i 583 9.99949 i4 47 8.68 627 8.68678 260 i .r !l 322 9.99949 i3 48 8.68 886 8.68 g38 i .3i 062 9-99948 12 49 8.69 144 258 8.69 196 258 i.3o 8o4 9-99948 I I 50 8 .69 4oo 8.69453 257 i.3o54 7 9-9994? 10 5i 8.69 654 254 8.69 708 255 i . 3o 292 9.99946 9 52 8.69 907 8.69 962 i.3oo38 9.99946 8 53 8.70 i5g 252 8.70 214 252 i .29 7 86 9.99945 7 250 25 54 8.70 409 8.70465 1.29 535 9-99944 6 55 8.70 658 8.70 714 i .29 286 9.99944 5 56 8.70905 247 8.70 962 2 4 a i .29 o38 9.99943 4 240 5? 8.71 i5i 8.71 208 i .28 7 g2 9.99942 3 58 8.71 3g5 8.71 453 1.28 54 7 9.99942 2 5 9 8.71 638 243 8. 7 i 6g 7 244 1.28 3o3 9-99 94 i I 60 8.71 880 8.71 g4o i .: 28 060 9.99940 L. Cos. d. L. Cotg. d. L. Tang. | L. Sin. f 87. PP 280 275 270 265 260 255 250 45 240 .1 28.0 27-5 27.0 .1 26.5 26.0 25-5 .1 25.0 24-5 24.0 .2 56.0 SS-o 54-o .2 53.0 52.0 51.0 .2 50.0 49.0 48.0 3 84.0 82.5 81.0 3 79-5 78.0 76.5 3 75-o 73-5 72.0 4 112. IIO.O 108.0 .4 Td6.0 104.0 IO2.O 4 100.0 98.0 96.0 5 140.0 '37-5 135-0 5 132-5 130.0 127.5 5 125.0 122.5 I2O.O .6 168.0 165.0 162.0 .6 159.0 156.0 '53-0 .6 150.0 147.0 144.0 7 196.0 192-5 189.0 7 '85.5 182.0 178.5 7 175-0 171.5 1 68.0 .8 224.0 220. o 216.0 .8 212. 208 o 204.0 .8 2OO.O 196.0 192.0 < 252.0 247-5 2 43- 9 238.5 234.0 229.5 .p 225.0 220.5 216.0 35 3. ' L. Sin. d. L.Tang. d. L. Cotg. L. Cos. 8.71 880 340 239 238 237 235 234 232 232 230 229 228 226 226 224 223 222 22O 22O 2ig 2I 7 216 216 214 213 212 211 2IO 209 208 208 8.71 g4o 241 239 239 237 236 234 234 232 231 329 229 227 226 225 224 222 222 220 219 2IQ 217 216 215 214 213 211 211 210 209 208 I .28 060 9.99 y 4o 60 I 2 3 4 5 6 7 8 9 8.72 120 8.72 359 8.72 697 8. 72 834 8.73 069 8. 7 3 3o3 8.73535 8.73 767 8. 7 3 997 8.72 181 8.72 420 8.72 65g 8.72 896 8.73 i32 8.73 366 8.73 600 8.73832 8. 7 4o63 I .27 819 i .27 58o i .27 34i i .27 104 1.26868 1.26634 i .26 4oo 1.26 168 i .25 937 9.99940 9.99 9 3 9 9.99 938 9.99938 9.99 9 3 7 9.99936 9.99936 9.99935 9.99 9 34 5 9 58 5 7 56 55 54 53 52 5i 10 8.74 226 8.74 292 i .25 708 9.99934 50 1 1 12 i3 i4 i5 16 i? 18 '9 8.74454 8. 7 468o 8.74 906 8. 7 5 i3o 8.75 353 8.75575 8. 7 5 79 5 8.76 oi5 8.76234 8.74 52i 8.74748 8.74 974 8.75 199 8.75423 8. 7 5645 8. 7 586 7 8.76 087 8.76 3o6 i .25 479 I .25 252 I .25 O26 i .24 801 1.24 577 1.24355 1.24 i33 i.23 gi3 i .23 694 9.99933 9.99932 9.99932 9.99 9 3i 9.99930 9.99929 9.99929 9.99928 9.99 927 49 48 47 46 45 44 43 42 4i 20 8.76451 8.76 525 i .23 4?5 9.99926 40 21 22 23 24 25 26 27 28 29 8.76667 8.76883 8.77097 8.77 3io 8.77 522 8.7 7 733 8.77943 8.78 i52 8.78 36o 8.76 742 8.76958 8.77173 8.77 387 8.77 600 8.77 811 8.78 022 8.78 232 8. 7 844i i.23 258 I .23 O42 I .22 827 I .22 6l3 I .22 4<"> I .22 189 I .21 978 I .21 768 I .21 55g 9.99926 9.99925 9.99924 9.99923" 9.99923 9.99922 9.99921 9.99920 9.99920 3 9 38 3? 36 35 34 33 32 3i 30 8.78 568 8.78 64g I .21 35l 9.99919 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 86 3O . PP 238 34 23.4 46.8 70.2 93.6 117.0 140.4 163.8 187.2 210.6 329 .i .2 3 4 '.6 9 335 33O 316 313 308 304 .1 .2 3 4 '.6 .8 9 23.8 47 .6 71.4 5 95.2 119.0 142.8 166.6 190.4 214.2 22.9 45.8 68.7 91.6 "45 37-4 160.3 183.2 206.1 22.5 45- o 67.5 90.0 112.5 i35-o '57-5 180.0 52- 5 22.0 &0 88.0 IIO.O 132.0 154.0 176.0 198.0 21.6 .1 432 .2 64.8 .3 86.4 .4 108.0 .5 129.6 .6 151.2 .7 172.8 104.4 -9 21.2 42.4 63.6 84.8 I o6.0 127.2 148.4 169.6 90.8 20.8 4 ,.6 62.4 832 104.0 124.8 145.6 166.4 187.2 20.4 40.8 61.2 81.6 IO2.O 122-4 142.8 163.2 183.6 36 3 30 . / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 30 8.78 568 8.78 649 i .21 35i 9.99 919 30 3i 8.78774 8.78 855 .21 145 9.99918 29 32 8.78 979 .79 061 .20 989 9.99917 28 33 8.79 i83 8 . 79 266 .20 ?34 9-99 9 1 ? 27 203 204 34 8.79 386 8.79-470 .20 53o 9.99916 26 35 8.79 588 8.79.673 .20 32? 9.99915 25 36 8.79 789 8.79 875 .20 125 9.99914 24 2OI 201 ?7 8.79990 8.8c ) 076 . 19 924 9.99913 23 38 8.80 189 8.8( ) 277 201 . 19 ?23 9.99913 22 3 9 8.80 388 199 8.8c ' 4?6 I 99 . 19 524 9.99912 21 40 8.80 585 '97 8.80 6 7 4 I.19 326 9.99 911 20 4i 8.80 782 '97 8.8< 5872 1 06 I . I 9 128 9-99 9 10 19 42 8.8 3978 8.81 068 I. l8 932 9.99909 18 43 8.81 i 7 3 8.81 264 '9 i . 18 ?36 9.99909 17 194 '95 44 8.81 367 8.81 409 1. 18 54 1 9.99908 16 45 8.81 56o 8.81 653 1.18 34? 9.99907 i5 46 8.81 752 192 8.81 846 '93 1. 1 8 1 54 9.99906 i4 192 192 4 7 8. 8 1 944 8.82 o38 1.17 962 9.99905 i3 48 8.82 1 34 8.82 23o 1.17 770 9.99904 12 49 8.82 324 190 8.82 420 190 1.17 58o 9.99 904 I I 50 8.82 5i3 8.82 610 1.17 3go 9.99 903 10 5i 8.82 701 187 8.8 2 799 1.17 201 9.99902 9 52 8.8 2888 8.82 987 1.17 oi3 9.99901 8 53 8.83 075 1 86 8.83 175 186 1.16825 9.99900 7 54 8.83 261 18 8.83 36i i . 16 63g 9.99899 6 55 8.83 446 8.8354? ifl i.i6453 9.99898 5 56 8.83 63o 8.83 7 32 1.16 268 9.99898 4 183 - 57 8.83 8i3 8.83 916 1.16084 9.99 897 3 58 8.83 996 8.84 ioo 182 i . 1 5 900 9.99896 2 5 9 8.84 177 181 8.8 4 282 182 i . 5 718 9.99 895 I 60 8.84 358 8.84464 i.i5 536 9.99 894 L. Cos. d. L. Cotg. d L. Tang. L. Sin. ' 86. PP 2OI 198 195 192 189 187 185 183 181 .1 2O. I 19.8 '9-5 .1 19.2 18.0 18.7 .1 i8. S .8.3 18.1 .2 40.2 39- 6 39.0 .2 38-4 37-8 37-4 .2 37-o 36.6 36.2 3 60.3 59-4 58-5 3 57-6 56.7 56.1 3 55-5 54-9 54-3 4 80.4 79.2 78.0 4 76.8 75-6 74.8 4 74.0 73-2 72.4 5 100.5 99.0 97-5 5 96.0 Q4-.S 93-5 5 9 2 -5 9'-5 9-5 .6 120.6 118.8 117.0 .6 115.2 "3-4 1 1 2. 2 .6 III.O 109.8 108.6 7 140.7 138.6 136-5 7 134-4 I32-3 130.9 7 129.5 128.1 126.7 .8 160.8 158.4 150.0 .8 151.2 149.6 .8 148.0 146.4 144.8 .9 iSo.q 172.8 170.1 168.3 .9 166.5 164.7 162.9 4. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 8.84358 8.84464 182 I . i5 536 9.99894 60 I 8. 8453 9 179 8.84646 180 i . 5 354 9.99893 5y 2 8.84 718 8.84826 i . i 5 174 9.99 892 58 3 8.84897 179 8. 85 006 J -i4 994 9.99891 57 4 8.85 075 178 8.85 185 179 i.i48i5 9.99891 56 5 8.85 252 8.85 363 i . 4 637 9.99890 55 6 8.85 429 177 8.8554o 177 i . i4 46o 9.99889 54 176 '7i 7 8.85 605 8.85 717 176 i . 4283 9.99 888 53 8 8.85 780 8.858 9 3 i . i4 107 9.99887 52 9 8.85 955 '75 8.86 069 176 i . 3 g3i 9.99 886 01 10 8.86 128 173 8.86243 '74 i.i3 757 9.99 885 50 1 1 8.86 3oi 8.86417 '74 I . 3 583 9.99884 49 12 8.864?4 8.86 5 9 i i . 1 3 4og 9.99883 48 i3 8.86645 171 8.86763 172 i . 3 237 9.99 882 47 171 "7 ! i4 8.86816 8.86935 i . 1 3 065 9.99 881 46 i5 8.86 987 169 8.87 106 i . 12 894 9.99 880 45 16 8.87 i56 169 8.87 277 171 170 1.12 723 9.99879 44 17 8.87 325 169 8. 87 447 160 i. 12 553 9.99879 43 18 8.87494 8.87 616 i. 12 384 9.99878 42 '9 8.87661 107 1 68 8.87785 169 1 . 12 2l5 9.99 877 4i 20 8.87 829 166 8.87 953 I . 12 047 9.99 876 40 21 8.87995 166 8.88 120 167 I .11 880 9.99875 3 9 22 8.88 161 16 8.88 287 i . 1 1 7i3 9.99874 38 23 8.8 8 326 164 8.8 8453 16s i . ii 547 9.99873 ^7 24 8.88 490 164 8.8 8618 1. 1 1 382 9.99872 36 25 8.8 8654 8.88 7 83 i . 1 1 217 9.99871 35 26 8.8 8817 103 8.8 8 9 48 105 I . I I 052 9.99870 34 '63 16 * 27 O t 8 980 8.8 9111 I. 10 889 9.99869 33 28 8.8 9 142 8.8 9 274 i . 10 726 9.99868 32 29 8.8 9 3o4 8.8 9437 163 I . 10 563 9.99 867 3i 30 8. 89 464 8.89 598 I . 10 4oa 9.99 866 30 L. Cos. i d. L. Cotg. d. L. Tang. L. Sin. t 85 3O . PP 181 179 177 175 173 .7> 168 1 66 164 .1 18.1 17.9 17-7 .1 '7-5 17-3 17.1 .1 16.8 16.6 16.4 .2 36-2 35-8 35-4 .2 35-o 34-6 34-2 .2 33-6 33-2 32.8 3 54-3 53-7 53- 3 52.5 5'-9 5' 3 3 50.4 49.8 49.2 4 724 71-6 70.8 4 70.0 69.2 68.4 4 67.2 66.4 6 5 6 5 9-5 88.5 5 8 7 -5 86. s 85.5 5 84.0 83.0 82.0 .6 108.6 107.4 106.3 .6 105.0 103.8 1 02. 6 .6 100.8 99.6 98.4 .7 126.7 125-3 123-0 7 122.5 121. 1 119.7 7 117.6 116. 2 114.8 .8 144.8 "43-2 41.6 .8 140.0 138.4 136.8 .8 '34-4 132.8 131.2 9 162.9 161.1 159-3 9 "57-5 '55-7 '53-9 151.2 141.4 147.6 38 4 3D . L. Sin. d. L. Tang. d. L. Cotg. L Cos. 30 8. 89 464 8.89 598 I . 10 402 9.99 866 30 3i 8.6 9 625 8.8 9 760 160 I . IO 240 9.99 865 29 32 8. 9?84 159 8.8 9 920 i . 10 080 9.99 864 28 33 M 9 943 '59 8.90 080 I .09 920 9.99 863 27 '59 j 34 8.90 1 02 TC8 8.90 24o I .09 760 9.99 862 26 35 8.90 260 8 .90 399 i .09 601 9.99 861 25 36 8.90 417 157 8.90 557 158 I .09 443 9.99 860 24 157 15 -, 37 8.90 574 8.9 o? 1 ! i .09 285 9.99859 23 38 8.90 730 8.90 872 i .09 128 9.99 858 22 39 8. 90 885 155 8.91 029 i .08 971 9.99 857 21 40 8.91 o4o 155 8.91 185 150 i. 08 8i5 9.99 856 20 4i 8.91 195 155 8.91 34o !55 i .08 660 9.99855 i 42 8.91 34g 8.91 4g5 i .08 Sog 9.99 854 18 43 8.91 5o2 153 8.91 650 D i. 08 35o 9.99 853 i? 153 15 J 44 8.91 655 8.91 8o3 i .08 197 9.99 852 16 45 8.91 807 8.91 957 i. 08 o43 9.99 85i i5 46 8.91 959 152 8.92 no "S3 i .07 890 9.99850 i4 I e j 15 1 47 8.92 no 8.92 262 1.07 738 9.99848 i3 48 8.92 261 8.92 4i4 1.07 586 9.99847 12 49 8.92 4i i ISO 8.92 565 15 1.07 435 9.99 846 I I 50 8.92 56i ISO 8.92 716 i .07 284 9.99 845 10 5i 8.92 710 149 8.92 866 i .07 1 34 9.99 844 9 52 8.92 85g 8.93 016 i .06 984 9.99843 8 53 8.93 007 8.93 165 i.o6835 9 . 99 842 7 147 '4 54 8.93 i54 8.93 3i3 1.06687 9.99 84i 6 55 8.93 3oi 8.93462 i. 06 538 9.99 84o 5 56 8.93448 J 47 146 8.93 609 '47 i .06 391 9 . 99 83o 4 5? 8 . 93 5g4 ii6 8.93 756 i .06 244 9.99 838 3 58 8.93 740 8 .93 go3 i .06 097 9.99837 2 5 9 8.93 885 J 45 8 . 94 049 I4& **6 i ,o5 g5i 9.99 836 I 60 8.94 o3o 8.94 igS i .o5 805 9.99 834 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. < 85. PP 162 160 159 157 155 'S3 151 149 M7 .1 16.2 16.0 15-9 .1 '5-7 15-5 IS-3 -I IS-I 14.9 14.7 .2 32 4 32.0 3 i.8 .2 31-4 31.0 30.6 .2 30.2 29.8 29-4 3 48.6 48.0 47-7 3 47-' 40-5 45-9 -3 45-3 44-7 44.1 4 64.8 64.0 63.6 4 62.8 62.0 61.2 .4 60.4 59- 6 58.8 5 81.0 80.0 79-5 5 78.5 77-5 76-5 -5 75-5 74-5 73-5 .6 97.2 96.0 95-4 .6 94.2 93.0 91.8 .6 90.6 89.4 88.2 7 "3-4 112. "1-3 . 7 109.9 08.5 107.1 .7 ioS-7 104.3 102.9 .8 129.6 128.0 127.2 .8 125.6 24 o 122.4 .8 120.8 119.2 117.6 .9 145.8 144.0 '43- 1 9 J4i-3 '39-5 '37-7 -9 '35-9 134.1 132. 3 9 5. ' L. Sin. d. L. Tang. d. L. Cotg. L Cos. 8.94 o3o 8.94 195 i .o5 805 9.99 834 60 I 8.94 i?4 144 8.94 34o MS i ,o5 660 9.99 833 5 9 2 8.94317 8. 9 4485 i .o5 515 9.99 832 58 3 8.94461 4 63o MS i .o5 370 9.99 83i 57 142 *4 i 4 8.g46o3 8.94773 i .o5 227 9.99 83o 56 5 8-94746 8.94 917 i.o5 o83 9.99 829 55 6 8.94887 141 8.95 060 M3 i .04 940 9.99 828 54 142 i j 2 7 8.g5 029 8.g5 202 i .04 798 9.99827 53 8 8.g5 170 8.95 344 i.o4656 9.99 825 52 9 8.95 3io 140 8. 9 5486 142 i.o4 5i4 9.99 824 5i 10 8.95 450 8.g5 627 141 i.o4 373 9.99 823 50 ii 8.95 58 9 139 8.95 767 140 i.o4 233 9.99 822 49 12 8.g5 728 8.95 908 i .04 092 9.99 821 48 i3 8.95 867 139 .38 8.96 047 139 140 i.o3 953 9.99 820 47 i4 8.c )6 oo5 138 8.96 187 | i.o3 8i3 9.99819 46 i5 8.96 i43 8.96 325 i .o3 675 9.99817 45 16 8.96 280 8.96464 '3V i.o3 536 9.99 816 44 '37 i- '7 8.96 417 116 8.96 602 i.o3 398 9.99 8i5 43 18 8.96 553 8.96 739 i .o3 261 9-99 8i4 42 [9 8.96689 136 8.96 877 i.o3 123 9.99 8i3 4i 20 8.96 825 8.97 oi3 I .02 987 9.99812 40 21 8.96 960 135 8.97 150 135 i .02 85o 9.99 810 39 22 8.97 095 8.97285 1.02 715 9.99809 38 23 8.97 229 8.97 421 1 .02 579 9.99 808 3? '34 3 s 24 8.97 363 133 8.97 556 135 I .02 444 9.99807 36 25 8 .97 496 8.97 691 i .02 3og 9.99 806 35 26 8.97629 8.97 825 13 4 i .02 175 9.99 8o4 34 '33 M ^ 27 8.97 762 8.97959 I .02 O4l 9.99 8o3 33 28 8.97894 8.9 8 092 I .OI 908 9.99 802 32 2 9 8.c >8 026 8.9 8 225 i.oi 775 9.99 801 3i 30 8.98 157 8.98 358 I .Ol 642 9.99 800 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 84 3O . PP MS M3 141 139 '38 136 135 133 z .1 .2 M-5 29.0 M-3 28.6 14.1 28.2 .1 .2 13.9 13.8 27.8 27.6 13.6 .1 27.2 .2 '3-5 27.0 III 'I' 1 26.2 3 43-5 42.9 42.3 3 4'-7 4'-4 40.8 . 3 4-5 39-9 39-3 4 58.0 57-2 56.4 4 55-6 55-2 54-4 -4 54-0 532 52-4 5 72-5 70-5 5 69.5 69.0 68.0 .5 67- S 66.5 65-5 .6 87.0 85.8 84.6 .6 83.4 82.8 81.6 .6 81.0 79.8 78.6 is 101.5 116.0 100. 1 114.4 98.7 112. 8 :l 97.3 96.6 III. 2 IIO.4 95.2 .7 108.8 .8 94-5 08.0 93-' 106.4 91.7 104.8 9 <3"-5 128. 126.9 I25.I 124.2 i??.4 .0 121.5 119.7 "7-9 4o 5 3O . / L. Sin. d. L, Tang. d. L. Cotg. L. Cos. 30 8.98 167 8.98 358 I .01 642 9.99 800 30 3i 8.9 3 288 131 8.9 3 490 132 i .01 5io 9.99798 29 32 8.98 419 8.9 3 622 i .01 378 9-99797 28 33 8.9 3549 130 8.9 3 7 53 131 i .01 247 9-99 79 6 27 130 131 34 8.9 3 670 8.9 3884 i .01 1 16 9-99795 26 35 8.98 808 8.99015 i .00 985 9.99793 25 36 8.9 S 9 3 7 8.99 i45 130 i .00 855 9-99 79 2 24 129 130 3? 8.9 9 066 128 8.99 275 i .00 725 9.99791 23 38 8.99 194 8.99 405 J i .00 5g5 9-99 79 22 3 9 8.99 322 8.99 534 i .00 466 9.99 788 21 40 8.99 450 8.99 662 i. oo 338 9.99 787 20 4i 8.99 577 127 8.99791 128 i .00 209 9.99 786 *9 42 8.99 704 8.99919 I .00 08 I 9-99 7 8 5 18 43 8.99 83o 126 9.00 046 128 0.99 954 9.99788 17 44 8.99 956 126 9.00 174 0.99 826 9.99 782 16 45 9.00 082 9.00 3oi 0.99699 9-99 7 Sl i5 46 9.00 207 125 125 9.00 427 126 0.99 573 9.99 780 i4 4? 9.00 332 9.00 553 0.99 447 9.99 778 i3 48 9.00 456 9.00 679 0.99 32i 9-99777 12 49 9.00 58 1 125 9.00 805 0.99 195 9.99 776 I I 50 9 .00 704 9.00 930 0.99 070 9-99 775 10 5i 9.00 828 124 9.01 o55 124 O.( )8 9 45 9.99 773 9 52 9.00 g5i 9.01 179 O.( )8 821 9.99 772 8 53 9.01 074 9.01 3o3 O.( >86 97 9-99 77 1 7 . 54 9.01 196 9.01 427 O.( )8 573 9.99769 6 55 9.01 3i8 9.01 55o o.c 18 450 9-99 7 68 5 56 9.01 44o 9.01 673 0.98 327 9-99 7 6 7 4 121 < 5? 9.01 56i 9.01 796 0.( j8 204 9.9 9 765 3 58 9.01 682 9.01 918 O.( j8 082 9-99 7 6 4 2 5 9 9.01 8o3 9.02 o4o 0.97 960 9-99 7 63 I 60 9.01 923 9.02 162 0.97838 9.99 761 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. / 84. PP 13 129 128 126 "5 123 122 121 I2O .1 13.0 12.9 12.8 .1 12.6 12.5 12.3 .1 12.2 12. 1 12.0 .2 26.0 25.8 25.6 .2 25.2 25.0 24.6 .2 24.4 24.2 24.0 3 39-o 38-7 38.4 3 37.8 37- S 36.9 3 36.6 36.3 36.0 4 52.0 51-6 51-2 4 50-4 50.0 49.2 4 48.8 48.4 48.0 5 65.0 64-5 64.0 5 63.0 62.5 61.5 5 61.0 60.5 60. a .6 78.0 77-4 76.8 .6 75-6 75-o 73-8 .6 73-2 72.6 72.0 7 91.0 90.3 89.6 . 7 88.2 87- S 86.1 . 7 85 7 o3 9.9973? 42 '9 9.04 1 49 9 . o4 4 1 3 0. ?5 58 7 9.99 7 36 4i 20 9.04 262 9.04 528 "5 o . g5 4?2 9-99 7 3 4 40 21 g.o4 3?6 114 9.04 643 o j5 35? 9.99 ?33 3 9 22 9.04 4go 9.04 758 O }5 242 9.99731 38 23 9.04 6o3 9.04873 "5 o. ? 5 127 9.99730 3? ii i 24 9 . o4 7 1 5 "3 9-o4 987 o . j5 oi3 9.99728 36 25 9.04828 9.o5 101 0.94 899 9.99727 3b 26 9.04 94o 9-o5 2i4 0.94 786 9-99 726 34 ii L 27 9.0 >5 o52 9-o5 328 0.94 672 9-99724 33 28 9-o5 1 64 9.o5 44i 0.94 559 9-99723 32 2 9 9.o5 275 9.o5 553 0.94 44? 9.99 721 3i 30 g.o5 386 9.o5 666 "3 0.94 334 9.99 720 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 83 3O . PP 131 120 119 118 117 116 "5 114 "3 .1 12. 1 I2.O II.9 .1 ii. 8 11.7 ii. 6 .1 "5 11.4 "3 .2 24.2 24.0 23.8 2 23.6 23.4 23.2 .2 23.0 22.8 22.6 3 30-3 36.0 35-7 3 35-4 35-i 34-8 3 34-5 34-2 33-9 4 48.4 48.0 47.6 4 47.2 46.8 46.4 4 46.0 45 6 45 -5 60. S OO.O 59- 5 5 59- 58.5 58.0 5 57-5 57-o 56.5 .6 72.6 72.0 71.4 .6 70.8 70.2 69.6 .6 69.0 68.4 67.8 7 84-7 84.0 83-3 7 82.6 81.9 81.2 <7 80.5 79.8 79.1 8 96.8 96.0 95.2 .8 94-4 93.6 92.8 .8 92.0 91.2 90.4 .9 108.9 I08.0 107.1 9 106. 2 105. 3 104.4 IO2.6 101.7 6 3O . / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 30 9-o5 36 9 .o5 666 o. 9 4 334 9-99 7 20 30 3i 9.0 5497 9.0 5 778 0.94 222 9-99 7 l8 29 32 9.0 5 607 9.o5 890 0.94 no 9.99 717 28 33 g.oS 717 9.06 002 O.C ;3 998 9.99 716 27 no in 34 g.o5 827 9.06 1 13 O.C )3 887 9.99 714 26 35 9.06 937 9 .06 224 O.g3 776 9.99 713 25 36 9 .06 046 9.06 335 0.93 665 9.99 711 24 109 no 3? 9.06 1 55 9.06 445 O.C >3 55,5 9.99710 23 38 9.06 264 9.06 556 0.93 444 9.99 708 22 3 9 9.06 372 9.06 666 0.93 334 9-99 77 21 40 9.06 48i 9.06 775 109 0.93 225 9-99 7 5 20 4i 9.06 589 9.06 885 o.g3 n5 9.99 704 '9 42 9.06 696 9.06 994 0.93 006 9.99702 18 43 9.06 8o4 9.07 io3 109 0.92 897 9.99 701 17 107 1 08 44 9.06 91 1 9.07 21 I 0.92 789 9.99699 16 45 9.07 018 9.07 820 109 0.92 680 9.99698 i5 46 9.07 124 9.07 428 0.92 572 9.99696 i4 107 4 7 9.07 281 9.07 536 0.92 464 9.99695 i3 48 9.07 337 9.07 643 0.92 357 9.99693 12 49 9 07 442 i5 9.07 751 0.92 249 9.99692 I I 50 9.07 548 9.07 858 107 0.92 142 9.99690 10 5i 9.07 653 9.07 964 0.92 o36 9.99689 9 62 9.07 758 9.0 8071 0.91 929 9.99687 8 53 9.07863 105 9.08 177 106 0.91 828 9.99 686 7 54 9.07 968 9.08 283 106 0.91 717 9.99 684 6 55 9.08 072 9.0 8 389 0.91 611 9.99 683 5 56 9.08 176 9.0 84 9 5 0.91 5o5 9.99 681 4 104 105 5? 9.0 8 280 9.08 600 0.91 4oo 9.99 680 3 58 9.0 8 383 9.08 705 0.91 295 9.99678 2 5 9 9.08 486 103 9.08 810 105 0.91 190 9.99 677 I 60 9.08 58g 9.0 8 914 0.91 086 9.99675 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 83. PP 112 in no 109 108 107 1 06 105 104 .1 II. 2 ii. i II. O .1 10.9 10.8 10.7 ,! 10.6 10.5 10.4 .2 22-4 22.2 22. .2 21.8 21.6 21.4 .2 21.2 21. 20.8 3 33-6 33-3 33- 3 32-7 32-4 3 2 -i 3 31.8 3'-5 31.2 4 44.8 44-4 44.0 -4 436 43.2 42.8 4 42.4 42.0 41.6 5 56.0 55-5 SS- 5 54-5 54- 53-5 5 53-o 52-5 52.0 .6 67.2 66.6 66.0 .6 65.4 64.8 64.2 .6 63.6 63.0 62.4 7 78.4 77-7 77.0 . 7 7 6 -3 75-6 74-9 7 74-2 73-5 72.8 8 89.6 oo. 8 88.0 .8 87.2 86 4 85.6 .8 84.8 84.0 83-2 9 TOO. 8 99.9 99.0 98.1 97.2 96.3 9 95-4 94-5 93.6 43 r L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 9.08 589 9.0 8 9 i4 0.91 086 9 . 99 675 60 1 9 .c >8 692 103 9.09 019 105 0.90 981 9 . 99 674 5 9 2 9 .c >8 795 9.09 123 0.90 877 9-99 6 7 2 58 3 9.08 897 9.09 227 104 0.90 773 9.99670 57 IO 1 4 9.08 999 9.09 33o 0.90 670 9.99669 56 5 9 .09 101 9.09 434 0.90 566 9.99 667 55 6 9.09 202 9.09 537 10 i 0.90 463 9.99 666 54 IO 1 7 9.09 3o4 9.09 64o 0.90 36o 9.99 664 53 8 9.09405 9.09 742 0.90 258 9.99 663 52 9 9.09 5o6 9.09 845 103 0.90 i55 9.99 661 1 10 9.09 606 9.09947 0.90 o53 9.99659 50 1 1 9.09 707 9.10 049 o.f 3 9 9 5i 9.99 658 49 12 9 .09 807 9.10 i5o o.i 39 850 9.99 656 48 i3 9.09907 9. IO 252 o.i 39748 9.99655 47 99 i i4 9. to 006 9.10 353 o.f 3 9 647 9.99 653 46 ID 9.10 106 9. 10 454 0.89 546 9.99 65i 45 16 9.10 205 99 9. 10 555 0.89445 9.99650 44 99 IO : 17 9.10 3o4 08 9.10 656 o.f 39344 9.99 648 43 18 9. i O 4O2 9.10 756 o.f 3g 244 9 . 99 647 42 '9 9.10 5oi 99 nR 9.10 856 o.i 3 9 1 44 9.99 645 4i 20 9 . 10 599 9.10 956 o.i 3 9 o44 9 . 99 643 40 21 9.10 697 98 9.11 o56 99 o.f 38 9 44 9.99 642 3 9 22 9. 10 795 08 9.11 i55 o.f 38 845 9.99 64o 38 23 9. 10 8 9 3 97 9.11 254 99 o.f 38 746 9.99 638 37 24 9. 10 990 97 9.11 353 99 o.f 38647 9.99637 36 25 9.11 087 9.11 452 o.f 38 548 9.99 635 35 26 9 . 1 1 184 97 9.11 55i 98 o.f 3844 9 9.99 633 34 27 9 . 1 1 281 06 9.11 64 9 | o.f 3835i 9.99 632 33 28 9.11 3 7 7 9. ii 74? 1 0.88 253 9.99 63o 32 29 9- 1 1 4?4 06 9.11 845 n 1 o.f 58 155 9.99629 3i 30 9.11 570 9.11 943 o.f 38o5 7 9.99627 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 82 30 . PP 105 104 103 103 101 1 00 99 98 97 .1 10.5 10.4 10.3 .i IO. 2 10. I 10. .1 9. 9 9.8 9.7 .2 21. 20.8 20 6 .2 20.4 20.2 20. o .2 19 8 19.6 19.4 3 3i 5 31-2 3-9 3 30.6 30.3 30.0 3 29.7 29.4 29.1 4 42.0 41.6 41.2 4 40.8 40.4 40.0 4 19-6 392 3 8.8 5 52.5 52.0 S'-5 5 51.0 5-5 50.0 5 49-5 49.0 48.5 .6 63.0 62.4 61.8 .6 6l.2 60.6 60.0 .6 594 58.8 58.2 7 73-5 72.8 72.1 7 71.4 70.7 70.0 -7 69.3 68.6 67.9 8 84.0 8,. 2 82.4 .8 8l.6 80.8 80.0 .8 79.2 78.4 77.6 9 94- > 91.8 OO.g 90.0 89.1 88.2 87.3 44 7 3O . , L. Sin. d. L. Tang. d. L. Cotg. Cos. 30 9.11 570 9 6 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 92 92 9 1 92 9 1 9 1 90 9 1 go 9 1 90 9.11 943 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 9 1 92 d. 0.88 067 9-99 62 7 30 3i 32 33 34 35 36 3? 38 3 9 9.11 666 9.11 761 9.11 85y 9.11 g52 9.12 o4? 9.12 142 9.12 236 9.12 33i 9.12 425 9.12 o4o 9.12 i38 9.12 235 9.12 332 9.12 428 9.12 525 9.12 621 9.12 717 9.12 8i3 0.87 960 0.87 862 0.87 765 0.87 668 0.87 572 0.87475 0.87 379 0.87 283 0.87 187 9.99 625 9.99 624 9 .99 622 9.99 620 9.99 618 9.99617 9.99 6i5 9.99 6i3 9 . 99 612 29 28 2 7 26 25 24 23 22 2 I 40 9.12 5ig 9.12 909 0.87 091 9 .99 610 20 4i 42 43 44 45 46 4? 48 49 9.12 612 9.12 706 9.12 799 9.12 892 9.12 985 9. 1 3 078 9. i3 171 g.i3 263 9. i3 355 9. 1 3 oo4 9. 1 3 099 9 . 1 3 1 94 9. i3 289 9. 1 3 384 9.13478 g.i3 573 9-i3 667 g.iS 761 0.86 996 0.86 901 0.86 806 0.86 711 0.86 616 0.86 522 0.86 427 0.86 333 0.86 239 9 .99 608 9.99607 9.99605 9.99 6o3 9 .99 601 9.99 600 9-99 5 9 8 9.99 5 9 6 9.99595 '9 1 8 I? 16 i5 i4 i3 12 I I 50 9. i3 447 9.i3 854 0.86 i46 9.99 5 9 3 10 5i 52 53 54 55 56 5? 58 5 9 9. i3 53g 9. i3 63o 9. i3 722 9.i3 8i3 9. 1 3 904 9-i3 994 9. i4 o85 9.14 i?5 9. i4 266 9. 1 3 948 9 . 1 4 o4 1 9 .i4 1 34 9. i4 227 9. l4 32O 9.14412 9. 14 5o4 9. i4 597 9.14 688 0.86 o52 o.85 g5g o.85 866 o.85 773 o.85 680 o.85 588 o.85 496 o.854o3 o.85 3i2 9.99 5 9 i 9.99589 9.99 588 9.99 586 9.99 584 9.99 582 9.99 58i 9.99579 9-99 5 77 9 8 7 6 5 4 3 2 I 60 9. i4 356 9 . i4 780 o.85 220 9.99 5 7 5 L. Cos. d. L. Cotg. L. Tang. L. t 82. PP. .1 .2 3 4 .6 .8 9 97 96 95 .1 2 3 4 .6 8 9 94 93 92 .1 .2 3 4 .6 ',8 9 91 90 9-7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.3 9 .6 19.2 28.8 38.4 48.0 57-6 67.2 76.8 86.4 9-5 19.0 28.5 38.0 47-5 57-o 66.5 76.0 85.5 9-4 9-3 18.8 18.6 28.2 27.9 37-6 37-2 47.0 46.5 56.4 55-8 63.8 65.1 75-2 74-4 84.6 81.7 9.2 i8. 4 27.6 3 6.8 46.0 55-2 64.4 73- 6 9 .i 18.2 - 2f -3.. 36-4 45-5 54- 6 63-7 72.8 81.9 9.0 18.0 ..27-0 36.0 45-o 54-o 63.0 72.0 81.0 45 8. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 9.14 356 9. i4 780 o.85 220 9 99 5 7 5 60 I 9.14445 9.14 872 92 o.85 128 9 99 574 5o 2 9. i4 535 9.14 963 o.85 037 9 .99 572 5* 3 9.14 624 Cy 9. 1 5 o54 9 1 0.84946 9 .99 570 3 7 4 9-i4 7' 4 90 80 9. i5 i45 9' 0.8485s 9 99 568 56 5 9. i4 8o3 9. 1 5 236 9 1 o.84 764 9 .99 566 55 6 9. i4 891 9. i5 327 9' 0.84673 9 99 56 5 54 7 9. i4 98 f(n 9. i 54i7 9 0.84583 9 .99 563 53 8 9. 1 5 069 9. 1 5 5o8 o.84 492 9 .99 56i 52 9 9. i5 i5j 9. i5 598 90 o.84 i [O2 9 .99 55g 5i 10 9. 1 5 245 88 9.i5 688 90 o.84 3i2 9 .99 557 50 1 1 g.i5 333 88 9.1 5 777 9 0.84 223 9 .99 556 49 12 9. 1 5 421 9-i5 867 o.84 i33 9 .99 554 48 i3 9-i5 5o8 87 9. i5 956 9 o.84o44 9 .99 552 4? i4 9. i5 596 88 8? 9.16 o46 90 0.83954 9 .99 55o 46 i5 9.i5 683 87 9.16135 0.83865 9 . 99 548 45 16 9 .i5 77 o 87 9. 16 224 89 88 o.83 776 9 .99 546 44 17 9. i5 857 87 9. 16 3i2 80 0.83688 9 .99 545 43 18 9. 1 5 94 4 86 9.16 4oi o.83 599 9 .99 543 42 '9 9.16 o3o 86 9.16 48g o.83 5n 9 .99 54r 4i 20 9.16 1 16 87 9. 16 577 0.83423 9 .99 SSg 40 21 9. 16 2o3 86 9.16665 88 o.83 335 9 .99537 3 9 22 9. 16 289 Be 9.16 753 o.83 247 9 .99 535 38 23 9.16 37 4- 86 9.16841 87 o.83 [5 9 9 .99 533 3? 24 9.16 46o 85 9. 16 928 88 o.83 072 9 .99 532 36 25 9.16 545 86 9.17 016 0.82 984 9 .99 53o 35 26 9.16 63i 9.17 io3 7 0.82 897 9 .99 528 34 85 87 27 9.16 716 85 9.17 190 87 0.82 810 9 .99 526 33 28 9.16 801 9.17277 0.82 723 9 .99 524 32 2 9 9.16 886 85 9.17 363 0.82 637 9 .99 522 3i 30 9. 16 970 84 9.17 450 87 0.82 55o 9 .99 52O 30 L. Cos. d. L. Cotg. d. L. Tang. | L. Sin. ' 813O. PP 9* 9* 90 89 88 87 86 .1 9.2 9.1 9.0 .1 8-9 8.8 .1 8.7 8.6 .2 18.4 18.2 1 8.0 .2 17.8 .7.6 .2 17.4 17.2 3 =7-6 27-3 27.0 3 26.7 26.4 3 26.1 25.8 .4 36.8 36.4 36.0 4 35-6 35 2 4 34.8 34.4 5 46.0 45-5 45-o 5 44-5 44-o S 43-5 43-o .6 55.2 54.6 54-o .6 53-4 52.8 .6 52.2 51.6 7 64.4 63.7 63.0 7 62.3 61.6 7 60.9 60.2 .8 73-6 72.8 72.0 .8 71.2 70.4 .8 69.6 68.8 .9 828 -i. i 81.0 .9 80. i 9 78.3 77.4 46 8 3O . ' L. Sin. d. L. Tang. d. L. Cotg. B L. Cos. 30 9. 16 970 9.17 450 0.82 55o 9 99 52O 30 3i 9.17 o5 = 85 8i 9.17 536 86 0.82 464 9 99 5i8 29 32 9.17 i3 9 9.17 622 0.82 378 9 99 5l 7 28 33 9.17 223 84 9.17 708 86 0.82 292 9 99515 27 34 9.17 307 9 . 1 7 794 86 0.82 206 9 99 5i3 26 35 9.17 3 9 i 4 9.17 880 O.82 120 9 99 5i i 25 36 9.174?' i 8 3 9.17 965 85 86 0.82 c 3 5 9 99 Sog 24 37 9.17 558 9. 18 o5i 0.81 9 49 9 99 So? 23 38 9.17 64i 9.18 i36 0.81 864 9 99 5o5 22 3 9 9.17 724 83 9.18 221 85 0.81 779 9 99 5o3 21 40 9.17 807 83 9.18 3o6 85 0.81 694 9 .99 5oi 20 4i 9.17 89 3 83 83 9.18 391 85 84 0.81 609 9 99499 '9 42 9.17973 9.i84?5 0.81 525 9 99497 18 43 9.18 o55 9.18 56o 0.81 44o 9 99 4 9 5 17 84 44 9.18 i3 1 9.18 644 0.81 356 9 99494 16 45 9.18 220 3 9.18 728 0.81 272 9 .99 492 i5 46 9.18 3o2 9.18 812 84 0.81 188 9 99 490 i4 8l 84 4 7 9.18 383 9.18 896 R-3 0.81 io4 9 .99488 i3 48 9-i8465 9.18979 0.81 021 9 .99486 12 4 9 9.18 547 9.19 o63 84 0.80 987 9 99484 I I 50 9.18 628 9. 19 i46 0.80 854 9 .99 482 10 5i 9.18 709 81 9. 19 229 03 83 0.80 771 9 .99 48o 9 52 9.18 790 9.19 3i2 81 0.80 e 88 9 .99478 8 53 9.18871 9.19 3g5 83 0.80 605 9 99 476 7 54 9. 18 952 81 9.19 4?8 83 O.8o 522 9 .99 4?4 6 55 9.19 o33 9. 19 56i O.8o 439 9 . 99 472 5 56 9.19 1 13 9.19 643 O.8o 35? 9 .99470 4 5? 9.19 193 80 9.19 725 82 O.8o 275 9 .99 468 3 58 9. 19 273 9.19 807 0.80 1 93 9 .99 466 2 5 9 9.19 353 g o 9. 19 889 82 O.8o III 9 99 464 I 60 9.19433 9.19 971 O.8o 029 9 .99 462 L. Cos. d. L. Cotg. d. L. Tang. \ L. Sin. > 81. PP 86 85 84 83 82 81 80 .1 8.6 8.5 8.4 .1 8.3 8.2 .1 8.1 8.0 .2 17.2 17.0 16.8 .2 16.6 16.4 .2 16.2 16.0 3 25.8 25.5 25.2 3 24.9 24.6 3 24-3 24.0 4 34-4 34-o 33- 6 4 33-2 32.8 4 32-4 32.0 5 43-o 42-5 42-0 5 4'-5 41.0 5 40.5 40.0 .6 S1 .6 51.0 50.4 .6 49-8 49.2 .6 48.6 48.0 .7 60.2 59-5 58.8 -7 58.1 57-4 . 7 567 56 o .8 68.8 68.0 67.2 .8 66.4 656 .8 64.8 64.0 9 74-4 76.5 75.6 9 74-7 73- 8 9 72 9 72.0 47 9. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 9.19 433 9.19971 0.80 029 9 .99 462 60 I 9.19 5i3 9.20 o53 81 0.79947 9 .99 460 5 9 2 9.19 5g2 80 9.20 1 34 0.79 866 9 . 99 458 58 3 9. 19 672 9.20 216 0.79 784 9 .99 456 57 79 4 9.19 761 9.20 297 81 0.79 703 9 99 454 56 5 9. 19 83o 9.20 378 0.79 622 9 .99 452 b5 6 9.19909 79 9.20 45g 0.79 54i .99450 54 7S 81 7 9.19 988 9.20 54o 81 0.79 46o 9 99448 53 8 9.20 067 9.20 621 0.79 379 9 .99 446 52 9 9.20 i45 78 9.20 701 0.79 299 9 99 444 5 i 10 9 . 2O 223 9.20 782 0.79 218 9 99 442 50 1 1 9.20 3o2 79 78 9.20 862 80 0.79 i38 9 99 44o 49 12 9.20 38o 9.20 942 0.79 o58 9 .99 438 48 i3 9.20458 9.21 022 0.78 978 9 .99 436 47 77 i4 9.20 535 78 9.2110-2 80 0.78 f $98 9 .99434 46 i5 9 .20 6i3 9.21 182 0.78 818 9 .99 432 45 16 9 .20 691 77 9.21 261 79 80 0.78 739 9 99 429 44 17 9.20 768 9.21 34i o. 78 65g 9 99 427 43 18 9.20845 9.21 420 0.78 58o 9 .99 425 42 '9 9.20 922 77 9.21 499 79 0.78 5oi 9 .99 423 4i 20 9.20 999 9.21 578 79 0.78 422 9 .99 421 40 21 9.21 076 77 9.21 657 79 7O 0.78 343 9 .99419 3 9 22 9.21 i53 76 9.21 736 0.78 264 9 .99417 38 23 9.21 229 9.21 8i4 7 0.78 186 9 .99415 37 77 79 24 9.21 3o6 76 9.21 893 78 0.78 107 <; . 99 4i3 36 25 9.21 382 -7 ft 9.21 971 o. 78 029 9 99 4n 35 26 9.21 458 7 6 9.22 049 78 o. 779 5i 9 99 409 34 2 7 9.21 534 76 9.22 127 78 0.77 873 9 .99407 33 28 9,21 610 9.22 2o5 78 0-77795 9 994o^ 32 2 9 9.21 685 ?6 9.22 283 78 0.77 717 9 .99 4O2 3 1 30 9.21 761 9 .22 36i 0.77 63g 9 .99 4oo 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. ' 8O 3O . PP 83 81 so 79 78 77 76 .1 8.2 8.1 8.0 ., 7-9 7 .8 .1 7-7 7.6 .2 16.4 16.2 16.0 .2 15.8 .5.6 .2 '5-4 15-2 3 24.6 2+- 3 24.0 3 23-7 23.4 3 23-' 22.8 .4 32.8 32.4 32.0 4 31.6 31-2 4 30 8 3-4 5 4'-o 40.5 40.0 5 39-5 39- 5 38.5 38.0 .6 49.2 48.6 48.0 ,6 47-4 46.8 .6 46.2 45-6 7 57-4 56.7 56.0 . 7 55-3 54-6 7 53-9 53-2 .8 65.6 64.8 64.0 .8 63.2 62.4 .8 61.6 60.8 9 73-8 72 9 72.0 9 /'' 7" 2 9 69.3 68.4 48 9 3O . 1 L. Sin. d. L. Tang. d. L. Cotg. L. Cos. 30 9.21 761 75 76 75 75 75 74 75 75 74 74 74 74 74 74 73 74 73 73 73 73 73 73 72 73 72 72 73 7i 72 72 9 .22 36i 77 78 77 77 77 77 77 76 77 76 76 77 76 76 75 76 75 76 75 75 75 75 75 74 75 74 75 74 74 74 0.77 689 9 99 4oo 30 3i 32 33 34 35 36 3? 38 3 9 9.21 836 9.21 912 9.21 987 9 .22 062 9.22 187 9.22 211 g.22 286 9.22 36i 9.22 435 9.22 438 9.22 5i6 9.22 5g3 9 .22 670 9.22 747 9.22 824 9.22 901 9.22 977 9.28 o54 o. 77 562 0.77 484 0.77 407 0.77 33o 0.77 253 0.77 176 0.77 099 0.77 028 0.76 946 9- 9- 9- 9- 9- 9- 9 9 9 99 898 99 896 99 894 99 892 99 890 99 388 99 385 99 383 99 38i 29 28 2 7 26 25 24 23 22 21 40 9 .22 5og 9.23 i3o o . 76 870 9 99 3 79 20 4i 42 43 44 45 46 4 7 48 49 9.22 583 9.22 657 9.22 781 9.22 805 9.22 878 9.22 g52 9-23 025 9. 23 098 9.23 171 9.23 206 9.23 283 9.23 SSg 9.28 435 9.23 5io 9.23 586 9.28 661 9.23 787 9.28 812 0.76 794 0.76 717 0.76 64i 0.76 565 0.76 490 0.76 4i4 0.76 889 0.76 268 0.76 188 9 9 9 9 9 9 9 9 9 99 3 77 99 3 75 99 872 99370 99 368 99 366 99 364 99 862 99 35 9 '9 18 17 16 i5 i4 i3 12 I I 50 g .23 244 9.28 887 0.76 1 13 9 99 35 7 10 5i 52 53 54 55 56 5y 58 5 9 9.23 317 9 .23 890 9.28 462 9.28 535 9.28 607 9.28 679 9-23 752 9.23 823 g.23 895 9.28 962 9.24 087 9 .24 112 9.24 186 9.24 261 9.24 335 9.24 4io 9.24484 9.24 558 0.76 o38 0.75 968 0.75888 0.75 8i4 0.75 789 0.75 665 0.75 Sgo 0.75 5i6 0.75 442 9.99 355 9.99 353 9.99 35i 9.99 348 9.99 346 9.99 344 9". 9 9 342 9.99 34o 9-99 33 7 9 8 7 6 5 4 3 2 I 60 9.28 967 9.24 682 0.75 368 9 .99 335 L. Cos. cl. L. Cotg. d. L. Tang. L. Sin. / 80. PP 77 7 6 75 .1 .2 3 4 .6 .8 74 73 .1 .2 3 4 J 9 72 71 .1 7.7 .2 15.4 3 23.1 4 3-8 5 38.S .6 46.2 7 53-9 ,8 61.6 9 6 9-3 7 .6 15-2 22.8 30.4 38.0 45-6 53-2 60.8 68.4 7-5 15.0 22.5 30.0 37-5 45.0 52-5 60.0 ii 32.2 29.6 37- 44-4 51.8 59.2 7-3 14.6 21.9 29.2 36.5 43-8 5'-i 58.4 6s-7 7.2 14.4 21.6 28.8 36.0 43.2 50.4 57- 6 6 4 .3 7-i 14.2 21-3 28.4 35-5 42.6 497 568 63.9^ 4 9 1O C ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. g .28 967 72 71 71 72 71 71 71 70 71 7 7' 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 9 24 632 74 73 74 73 74 73 73 73 73 73 72 73 72 73 72 72 72 72 72 7i 72 7 72 7' 7' 7' 7 1 70 7' 7' 0.75 368 9-99 335 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 60 I 2 3 4 5 6 7 8 9 9.24 oSg 9.24 no 9.24 181 9.24 253 9.24 324 9.24 395 9.24 466 9.24 536 9.24 607 9 9 9 9 9 9 9 9 9 24 706 24 779 24853 24 926 25 ooo 25 073 25 i46 25 219 25 292 o. 75 294 O.75 221 0.75 147 O.75 O74 0.75 ooo 0.74 927 0.74854 0.74 781 0.74 708 9-99 9.99 9-99 9-99 9.99 9.99 9-99 9.99 9.99 333 33i 328 326 324 322 319 3i 7 3,5 5 9 58 57 56 55 54 53 52 5i 10 9.24 6 77 9 25365 o. 7 4635 9-99 3i3 50 1 1 12 i3 i4 i5 16 i? 18 '9 9.24 748 9.24 818 9.24 888 9.24 g58 9.25 028 9.25 098 9-25 168 9.25 237 9.25 307 9 9 9 9 9 9 9 9 9 25437 25 5io 25 582 25 655 25 727 25 799 25 871 25 943 26 015 0.74 563 0.74 4go 0.74418 0.74345 0.74 273 O.74 2OI O.74 129 0.74 057 o. 7 3 985 9.99 3 to 9.99 3o8 9.99 3o6 9.99 3o4 9 .99 3oi 9.99299 .9.99297 9.99294 9.99 292 49 48 4? 46 45 44 43 42 4i 20 9.25 376 9 26086 0.73 914 9-99 290 40 21 22 24 25 26 27 28 2 9 9.25 445 9.25 5i4 9-25 583 9.25 652 9.25 721 9 .25 790 9.25 858 9.25 927 9.25 995 9 9 9 9 9 9- 9 9 9 26 i58 26 229 26 3oi 26 372 26443 26 5i4 26 585 26655 26 726 0.73 842 0.73 771 0.73 699 0.73 628 0.73 557 0.73486 0.73415 0.73345 0.73 274 9.99 288 9.99 285 9.99 283 9.99 281 9.99278 9.99276 9.99274 9.99271 9.99269 39 38 37 36 35 34 33 3i 30 9.26 o63 9 26 797 0.73 2o3 9.99 267 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 79 3O'. PP .1 .2 3 4 .6 !s '? 74 73 7* .1 .2 3 4 .6 :l 71 70 69 6.9 13.8 20.7 27.6 34-5 41.4 48-3 55-2 62.1 .1 .2 3 4 .6 .8 68 3 7-4 14.8 22.2 29.6 37-o 44-4 5i.8 59- 2 7-3 7-2 14.6 14-4 21.9 21.6 29.2 28.8 36.5 36.0 43.8 43.2 51-1 504 58.4 57.6 6=1.7 64.8 7-' 14.2 21.3 28.4 35- 5 42.6 4 2i 56.8 7.0 14.0 2I.O 28.0 35-o 42.0 49.0 56.0 63.0 6.8 13.6 20.4 27.2 34-o 40.8 47.6 54-4 61.2 0.3 0.6 0.9 1.2 i-5 1.8 2.1 2.4 2 -7 5o 1O 30 . , L. Sin. d. L. Tang-, d. L. Cotg. L. Cos. d. 30 9 .26 o63 9.26 797 0.73 2o3 9.99 267 30 3i 9 .26 i3 68 9.26 867 0.73 1 33 9.99 264 2 29 32 9 .26 199 9.26 9 3 7 0.73 o63 9.99 262 28 33 9 .26 267 9.27 008 7 1 0.72 992 9.99 260 27 68 7 3 34 9 .26335 68 9.27 078 0.72 922 9.99257 2 26 35 9 .26 4o3 9.27 i48 0.72 852 9.99255 25 36 9 .26 470 67 9.27 218 7 0.72 782 9.99 252 24 68 7 3 7 9 .26 538 f,-, 9.27 288 60 0.72 712 9.99 25o 2 23 38 9 .26 6o5 i 9.27 35 7 o. 72 643 9.99 248 22 3 9 9 . 26 672 7 9.27 427 70 0.72 573 9.99 245 21 40 9 .26 7 3g 9.27 4 9 6 9 0.72 5o4 9.99 243 20 4i 9 .26 806 VJ 67 9.27 566 7 60 0.72 434 9.99 241 '9 42 9 .26873 9.27 635 O. 7 2 365 9.99 238 18 43 9 .26 940 67 9.27 704 69 O. 7 2 296 9.99 236 '7 67 69 3 44 9 .27 007 66 9.27 77 3 fin 0.72 227 9.99 233 2 16 45 9 .27073 9.27 842 0.72 i58 9.99 23l i5 46 9 .27 i4< > 67 66 9.27 911 69 69 0^72 089 9.99229 3 i4 47 9.27 206 67 9.27 980 fin 0.72 020 9.99 226 2 i3 48 9.27 273 9.28 049 0.71 g5i 9.99 224 12 49 9 .27 33 9 9.28 117 0.71 883 9-99 221 I I 50 g .27 405 9.28 186 o. 71 8i4 9.99 219 10 5i 9 .27471 66 9.28 254 60 0.71 746 9.99 2I 7 9 52 9.27537 9.28 323 0.71 677 9.99 2l4 8 53 9.27 602 65 66 9.28 391 68 0.71 609 9.99 212 3 7 54 9.27 668 66 9.28 459 68 o. 71 54i 9.99 209 2 6 55 9.27 734 9.28 527 o. 71 4?3 9-99 2 7 5 56 9.27 799 65 9.28 ^95 o. 7 i 4o5 9.99 204 4 65 67 57 9.27 864 66 9.28 662 68 0. 71 338 9.99 2O2 2 3 58 9. 27 g3o 9.28 73o o. 7 I 2 7 O 9.99 2OO 2 5 9 9.27 995 65 9.28 798 fi-T 0. 7 I 202 9-99 '97 I 60 9.28 060 9.28 865 0. 7 I iSg 9.99 195 " L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 79. PP 70 69 68 67 66 65 3 .1 7.0 6.9 6.8 .1 6. 7 6.6 .1 6-S 0.3 .2 140 13-8 13-6 .2 13.4 13.2 .2 13.0 0.6 3 21 O 20.7 20.4 . 3 20. i 19.8 3 J9-5 0.9 4 28 o 27.6 27.2 .4 26.8 26.4 4 26.0 1.2 5 350 34-5 34-o 5 33-5 33-o 5 32-5 "5 .6 42 o 41.4 40.8 .6 40.2 39- 6 .6 39- 1.8 7 49.0 48.3 47-6 7 46.9 46.2 7 45-5 2.1 .8 56.0 55-2 54-4 .8 53.6 52.8 .8 52.0 2.4 _ 63.0 62.1 61.2 .9 60.3 59-4 9 58.5 2.7 5i 11. ' L. Sin. d. L Tang. d. L. Cotg. L. Cos. d. 9.28 060 65 6 4 6s 65 64 6 4 65 6 4 6 4 6 4 63 6 4 6 4 - 63 6 3 6 3 9 28 865 68 67 67 67 67 67 66 66 67 66 66 66 66 66 66 66 65 66 65 66 65 65 65 64 0.71 1 35 9-99 '95 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 2 3 2 3 2 3 2 3 2 3 3 2 60 I 2 3 4 5 6 7 8 9 9.28 125 9.28 190 9.28 254 9.28 3ig 9.28 384 9.28448 9.28 5i2 9.28 577 9.28 64 1 9 y 9 9 9 9- 9 9 9 28 933 29 ooo 29 067 29 i34 29 201 29 268 29 335 29 4O2 29 468 0.71 067 0.71 ooo 0.70 933 0.70 866 0.70 799 0.70 732 0.70 665 0.70 5g8 0.70 532 9-99 9.99 9-99 9-99 9-99 9-99 9-99 9-99 9.99 192 190 187 185 182 1 80 177 175 172 5 9 58 57 56 55 54 53 52 5i 10 9.28 705 9 29 535 o. 70 465 9.99 170 50 1 1 12 i3 i4 i5 16 17 18 1 9 9.28 769 9.28 833 9.28 896 9.28 960 9.29 024 9.29 087 9.29 i5o 9.29 214 9.29 277 9- 9- 9 i ; 9 9 9 9 29 601 29 668 29 734 29 800 29 866 29 932 29 998 3oo64 3o i3o 0.70 399 0.70 332 0.70 266 0.70 200 o. 70 i34 0.70 068 0.70 002 0.69 g36 0.69 870 9.99 9-99 9-99 9-99 9.99 9.99 9-99 9-99 9.99 167 165 162 1 60 i5 7 155 I 52 150 i4 7 49 48 47 46 45 44 43 42 4i 20 9 .29 34o 3 9 3o 195 0.69 805 9.99 145 40 21 22 23 24 26 27 28 2 9 9.29 4o3 9.29 466 9.29 529 9 . 29 5gi 9.29 654 9.29 716 9.29 779 9.29 84i 9.29 903 6 3 63 62 63 62 63 62 62 9 9 9 9 9- 9- 9- 9- 9 3o 261 3o 326 3o 391 3o45? 3o 522 3o58 7 3o 652 3o 717 3o 782 0.69 739 0.69 674 0.69 609 0.69 543 0.69 4?8 0.69 4i3 0.69 348 0.69 283 0.69 218 9.99 9-99 9-99 9-99 9.99 9.99 9-99 9.99 9.99 i4o i3 7 132 i3o 127 124 122 39 38 3? 36 35 34 33 32 3i 30 9. 29 966 6 3 9 3o846 0.69 1 54 9-99 "9 30 L. Cos. d. L Cotg. d. L. Tang 1 . L. Sin. d. 78 3O . PP .2 3 4 5 .6 .1 9 63 67 66 .i .2 3 4 '.6 '.B 9 65 6 4 63 .1 .2 3 4 ^6 '.S 9 62 3 6.8 ,3.6 20.4 27.2 34-o 40.8 47.6 54 4 6.7 6.6 '3-4 13-; 20. i 19.8 26.8 26.4 33-5 330 40.2 39.6 46.9 46.2 53.6 52.8 60. 3 ' 59.4 6.5 13.0 '9-5 26.0 32-5 39.0 45 5 52.0 6. 4 12.8 19 2 2 5 6 32.0 38-4 44.8 5' 2 57-6 6 3 12.6 18.9 25.2 3I - 37-8 44.1 50.4 5*7 6.2 .3 12.4 .6 18.6 .9 24.8 1.2 31.0 I.S 37-2 1.8 43-4 2.1 496 2.4 55 8 2-7 52 ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9. 29 966 9- 3o 846 0.69 1 54 9.99 119 30 3i 9. 3o 028 62 9 3o 91 1 ^ 0.69 089 9.99 117 29 32 9-3o 090 9- 3o 975 4 o.6 9 025 9-99 u4 28 33 9. 3o 1 5j 9- 3 1 o4o 65 0.68 960 9.99 I 12 27 62 6 4 3 34 9-3o 21 3 62 9- 3i io4 g. 0.68 896 9.99 109 26 35 9.30275 9- 3i 1 68 0.6 3 832 9-99 I 06 25 36 g.3o 336 9- 3i 233 65 0.68 767 9.99 104 24 62 6 4 3 3 7 g.3o 398 61 9- 3i 297 0.68 703 9.99 101 23 38 9-3o 459 9- 3i 36i 4 0.6 3 63 9 9-99 099 22 3 9 g.3o 521 3 1 4^5 6 4 0.68 676 9.99 096 3 2 I 40 9. 3o 5b2 61 9- 3r 489 64 0.68 5 1 1 9.99 093 3 20 4i g.3o 643 61 9- 3: 552 64 o.6i 3448 9-99 091 '9 42 9-3o 704 61 9- 3i 616 o.6i 3 384 9.99 088 18 43 9. 3o 765 61 9- 3i 679 6 4 0.68 32i 9.99 086 3 i? 44 9. 3o 826 61 9- 3i 743 6, 0.68 257 9.99 o83 16 45 g.3o 887 9- 3i 806 0.68 194 9.99 080 i5 46 9-3o 947 9- 3i 870 64 0.68 i3o 9.99078 i4 47 9. 3 1 008 60 9- 3i g3' 1 63 0.68 067 9.99 075 3 i3 48 9 . 3i 068 9- 3 1 996 3 0.68 oo4 9-99 072 12 49 9 .3i [29 9- 32 o5c ) 63 0.67 941 9.99 070 I I 50 9 .3, 189 61 9 32 I 22 63 0.67 878 9.99 067 10 5i 9 .3i 250 60 9 32 i85 63 0.67 815 9.99 o64 9 52 9 . 3i 3io 9 32 248 J 0.67 752 9.99 062 8 53 9 . 3 1 370 9 32 3i i 63 0.67 689 9-99 oSg 7 60 62 3 54 9-3i 43o 60 9 32 3?3 6 0.67 627 9-99 066 6 55 g.3i 49 9 3a436 3 0.67 564 9.99 o54 5 56 9 .3i 549 59 9 32498 0.67 5o2 9.99 o5i 3 4 60 63 3 5 7 9. 3i 609 60 9 32 56i 62 0.67 43g 9.99 o48 2 3 58 9. 3 1 669 9 32 623 0.67 377 9-99 o46- 2 5 9 9 .3i 728 59 9 32 685 0.67 315 9.99 o43 I 60 9 .3, 788 9 32 747 0.67 253 9.99 o4o L. Cos. d. L Cotg. d. L. Tang. L. Sin. d. ' 78. PP 65 64 63 62 61 60 59 3 .1 6.5 6.4 6.3 .1 6.2 6.1 6.0 .1 5-9 -3 2 12.8 12.6 .2 12.4 12.2 I2.O .2 11.8 0.6 3 19-5 19.2 18.9 3 18.6 .8.3 18.0 3 17.7 0.9 4 26.0 25.6 25.2 4 24.8 24.4 24.0 4 23.6 1.2 5 32.5 32- 3'-5 5 31.0 3-S 30.0 5 29-5 i-5 .6 39- 38.4 37-8 .6 37-2 36.6 36.0 .6 35-4 1.8 7 45-5 44.8 44.1 -7 43-4 42.7 42.0 7 41.3 2.1 .8 52.0 51.2 5-4 .8 49.6 48.8 48.0 .8 47-2 2-4 9 58.5 57-6 56.7 9 55-8 54.9 54.Q 53-' 2.7 53 12. , L. Sin. d. L Tang. d. L. Cotg. L. Cos. d. 9-3i 7 ss 9 .32 7 4 7 0.67 253 9.99 o4o 60 I 9-3i 84? 60 9 .32 8lO 62 0.67 190 9.99 o38 3 5 9 2 9-3i 907 9 32 872 0.67 128 9.99 o35 58 3 g.3i 966 59 9 32 933 62 0.67 067 9.99 o32 2 57 4 9. 32 025 59 9 32 995 62 0.67 005 9.99 o3o 56 5 g.32 o84 9 33o57 0.66 943 9.99027 55 6 g.32 i43 59 9 33 119 61 0.66881 9.99 024 54 7 g.32 202 59 9 33 180 62 0.66 820 9.99 O22 3 53 8 9. 32 261 rg 9 33 242 0.66 758 9.99019 52 9 9. 32 ii9 9 33 3o3 0.66 697 9.99 016 5i 10 g.32 378 9 33 365 0.66635 9.99 Ol3 50 1 1 9.32 437 58 9 33426 61 0.66 574 9.99011 49 12 9.32 495 9 33487 0.66 5i3 9-99 008 48 i3 9.32 553 5 9- 33 548 0.66 452 9.99 oo5 3 47 59 61 3 i4 9.32 612 58 9- 33 609 61 0.66 3gi 9.99 002 46 i5 9.32 670 9- 33 670 0.66 33o 9.99 ooo 45 16 9.32 728 5 9- 33 731 61 0.66 269 9.98 997 3 44 58 61 3 17 9.32 786 5 8 9 33 792 61 0.66 208 9.98 994 43 18 9.32 844 9 33853 0.66 147 9.98 991 42 '9 9.32 902 5 9 33 913 60 0.66 087 9.98 989 4i 20 9 . 32 960 9 33 974 0.66 026 9.98 986 3 40 21 9. 33 018 9- 34o34 61 o.65 966 9.98 9 83 J 3 9 22 9.33 075 9 34095 o.65 goS 9.98 980 38 23 9.33 i33 5 57 9 34 1 55 60 o.65 845 9.98 978 3 37 24 9-33 190 9 342i ) 61 0.65785 9.98 975 36 25 9-33 248 5 9- 34276 o.65 724 9.98 972 35 26 9.33 3o5 57 57 9- 34336 60 o.65 664 9.98 969 3 34 27 9-33 362 eg 9- 343 9 6 60 o.65 6o4 9.98 967 33 28 9.33 420 9- 34456 o.65 544 9.98 964 32 2 9 9-33477 57 9- 345i6 0.65484 9.98 961 3 3i 30 9-33 534 57 9- 34576 o.65 424 9.98 958 3 30 L. Cos. d. L Cotg. d. L. Tang-. L. Sin. d. ' 773O. PP 63 62 61 60 59 58 57 3 .1 6.3 6. 2 6.1 .1 6.0 5-9 5.8 .1 5-7 0-3 .2 12.6 12-4 12.2 2 I2.O n. 8 ii. 6 .2 11.4 0.6 3 18.9 18.6 18.3 3 18.0 '7-7 '7-4 3 17.1 0.9 4 25.2 24.8 244 4 24.0 23.6 23.2 4 22.8 1.2 5 3'-5 31.0 30.5 5 30.0 29.5 29.0 5 28.5 I.J .6 37-8 37.2 36.6 .6 36.0 35-4 34-8 .6 34-z 1-8 7 44-' 43 4 42-7 7 42.0 4'-3 40.6 7 39-9 2- .8 5-4 49.6 48.8 .8 48.0 47.2 46.4 .8 45-6 2.4 9 5 6 -7 55-8 . 54.9 9 S3- 1 S2.2 51.3 2.7 54 123O / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.33 534 57 56 57 57 57 56 57 56 56 57 56 56 56 56 56 56 55 56 55 56 55 56 55 55 55 55 55 55 55 55 9-34 5 7 6 59 60 60 59 60 59 59 59 60 59 59 59 59 58 59 59 58 59 58 59 58 58 58 58 58 58 58 58 58 57 o.65 424 9.98 958 3 2 3 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 30 3i 32 33 34 35 36 3? 38 39 9 . 33 5gi 9.3364? 9.33 704 9-33 761 9.33 818 9.33874 9 . 33 g3i 9.33 987 9.34 o43 9.34635 9.34695 9-34 755 9.348i4 9. 348 7 4 9 .34 9 33 9.34 992 9.35 o5i 9-35 in o.65 365 o.65 3o5 0.65245 o.65 186 o.65 126 o.65 067 o.65 008 o-64 949 o-64 889 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9 55 953 95 94? 944 94 1 9 38 936 933 29 28 2 7 26 25 24 23 22 21 40 9. 34 100 9 .36 170 o-64 83o 9. 98 g3o 20 4i 42 43 44 45 46 47 48 49 9.34 i56 9.34 212 9.34268 9.34324 9.34 38o 9.34436 9.34 491 9.34547 9.34 602 9. 35 229 9-35 288 9-35 347 9 .354o5 9-35464 9-35 523 9-35 58i 9-35 64o 9-35 698 o.64 771 o . 64 712 0.64653 o.64 595 0.64536 0.64477 o. 64 419 o.64 36o o.64 3o2 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 927 924 921 919 916 910 907 9 o4 '9 18 16 i5 M i3 12 I I 50 9-34 658 9.35 7 5 7 o. 64243 9.98 901 10 5i 52 53 54 55 56 5 7 . 58 5 9 9.34 71 3 9.34 769 9.34 824 9.34879 9.34989 9.35 o44 9-35 099 9 .35 1 54 9-35 815 9-35 873 9.35 g3i 9.35 989 9 . 36 o4? 9 .36 io5 9 .36 i63 9-36 221 9-36 279 o.64 i85 o.64 127 o . 64 069 o.64 on o . 63 gSS o.63 895 o.63 837 o.63 779 o.63 721 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 898 896 893 890 887 884 881 878 8 7 5 9 8 7 6 5 4 3 2 I 60 9/35 209 9-36 336 o. 63 664 9.98 8 7 2 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 1 77. PP .1 .2 3 4 5 .6 7 .8 60 59 58 57 56 .i .2 3 4 '.6 9 55 3 6.0 12. 18.0 24.0 30.0 36.0 42.0 48.0 54- 5-9 '7-7 23.6 29 5 35-4 4'-3 47-2 53- * 5.8 ii. 6 17.4 23.2 29.0 34-8 40.6 46.4 i 5-7 .2 11.4 3 i/- 1 .4 22.8 5 28.5 .6 34.2 7 39-9 .8 45.6 9 5'-3 5 .6 II. 2 16.8 22.4 28.0 33-6 39- a 44-8 50.4 5-5 -3 ii. o 0.6 16.5 0.9 22. 1.2 27-5 1-5 33-o 1.8 38-5 2.1 44.0 2.4 49-5 2.7 55 13. L. Sin d. L. Tang. d. L. Cotg. L. Cos. d. 9. 35 209 54 55 55 54 54 55 54 54 54 54 54 54 54 54 54 53 54 53 54 53 53 53 54 53 53 53 52 53 53 53 9.36 336 58 58 57 57 58 57 57 57 57 57 57 57 57 57 56 57 56 57 56 57 56 56 56 56 56 56 56 56 56 55 0.63664 9.98 872 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 60 I 2 3 4 5 6 7 8 9 9 .35 263 9.35 3i8 9. 35 373 9.35 427 9.3548i 9-35 536 g.35 5go 9.35 644 9-35 698 9-36 3g4 9.36452 9. 36 Sog 9.36566 9.36624 9-3668i 9-36 738 9 . 36 795 9-36 852 o. 0. o. 0. o. o. 0. o. o. 63 606 63 548 63 491 63434 63 376 63 3ig 63 262 63 205 63 1 48 9.98 869 9.98 867 9.98 864 9.98 861 9.98 858 9.98855 9.98 852 9.98849 9.98 846 5 9 58 5? 56 55 54 53 52 5i 10 9 . 35 7 52 9.36 909 0. 63 091 9.98 843 50 1 1 12 i3 i4 i5 16 17 18 '9 9-35 806 9.35 860 9-35 914 9-35 968 9. 36 022 9-36 076 g.36 129 g.36 182 9.36 236 9-36 966 9.37 O23 9.37 080 9 .3 7 i3 7 9.37 Ig3 9.37 250 9.37 3o6 9.37 363 9.37 419 0. o. o. o. o. o. o. o. o. 63o34 62977 62 920 62863 62 807 62 7 5o 62 6g4 62 637 62 58i 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 84o 83 7 834 83i 828 825 822 819 816 49 48 47 46 45 44 43 42 4i 20 9-36 289 9.37 4?6 0. 62 524 9.98 8i3 40 21 22 23 24 25 26 27 28 2 9 9-36 342 9 36 3g5 g.36 449 9-36 5o2 9.36555 g.36 608 9-35 660 9-36 713 9. 36 766 9.37 532 9 .3 7 588 9.37 700 9.37 7 56 9.37 812 9.37868 9.37 924 9.37 980 o. o. 0. o. 0. 0. 0. o. o. 62468 62 412 62 356 62 3oo 62 244 62 1 88 62 1 32 62 076 62 O2O 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 810 807 8o4 801 798 79 5 79 2 789 7 86 3 9 38 3? 36 35 34 33 32 3i 30 g.36 819 g.38o35 0. 61 965 9.98 7 83 30 L. Cos. | d. L. Cotg. d. L. Tang. L. Sin. d. ' 76 3O . PP 58 57 56 55 54 .1 .2 3 4 .6 53 3 i 5 8 .2 II. 6 3 '7-4 5-7 11.4 17.1 5-6 II. 2 16.8 i 5-5 .2 II. O 3 l6 -5 .4 22.0 5 27.5 6 33.0 7 38-5 .8 44.0 9 49- 5 10.8 16.2 21.6 27.0 32.4 37-8 43.2 10.6 '59 21.2 26.5 37- i 42.4 .6 9 1.2 1.5 2.1 24 2-7 .5 29.0 .6 34.8 .7 40.6 .8 46.4 28.5 34-2 39-9 45-6 5t-3 28.0 33-6 39.2 44.8 5 4 56 13 SO. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 . 36 8i 9 52 53 52 52 53 52 52 52 52 52 52 52 52 y 5' 52 5' 52 5' 52 5i 5i 5i 5i 5' 5' 5i 5' 5' 5' 9 38o35 56 56 55 55 56 55 55 56 55 55 55 55 55 54 55 55 54 55 55 54 54 55 54 54 54 54 54 54 54 54 0.61 965 9.98 783 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 4 30 3i 82 33 34 35 36 37 38 3 9 9 .36 871 9 . 36 9 a4 9 . 36 976 9. 37 028 9 .37 08 1 9.37 i33 9.37 i85 9 .3 7 2 3 7 9.37 289 9 9 9 9 9 9 9 9 9 38 091 38 147 38 202 38 25 7 38 3i3 38 368 38423 38479 38 534 0.61 909 0.61 853 0.61 798 0.61 743 0.61 687 0.61 632 0.61 577 0.61 52i 0.61 466 9.98 9.98 9 . 9 8 9.98 9.98 9 . 9 8 9 . 9 8 9.98 9.98 780 777 774 771 768 7 6 5 762 7 5 9 756 2 9 28 27 26 25 24 23 22 21 40 9. 37 34i 9 38 589 0.61 4i i 9.98 7 53 20 4i 42 43 44 45 46 47 48 4 9 9. 37 3g3 9.37445 9.37497 9.37 54 9 9.37 600 9.37 652 9 . 37 703 9.37 755 9. 37 806 9 9 9 9 9 9 9 9 9 38 644 38 6 99 38 7 54 38 808 38863 38 918 38 972 3g 027 3g 082 0.61 356 o. 61 3oi 0.61 246 0.61 192 0.61 137 0.61 082 0.61 028 0.60 973 0.60 918 9.98 750 9.98 746 9-98 ?43 9-98 740 9.98 737 9.98 734 9.98 731 9.98728 9.98 725 '9 18 17 16 i5 i4 i3 12 II 50 9.37 858 9 39 1 36 0.60 864 9.98 722 10 5i 52 53 54 55 56 5? 58 5 9 9.37909 9.37 960 9 . 38 01 i 9.38 062 9-38 n3 9.38 1 64 g.38 2i5 g.38 266 9.38 3 1 7 9 9 9 9 9 9 9 9 9 3g 190 3 9 245 3 9 299 3 9 353 39 407 3 9 46i 3 9 5i5 3g 56g 3g 623 0.60 810 0.60 755 0.60 701 0.60 64? 0.60 5g3 0.60 SSg 0.60 485 0.60 43i 0.60 377 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 719 7 i5 712 709 706 703 700 697 6 9 4 9 8 7 6 5 4 3 2 I 60 9.38 368 9 39677 0.60 323 9.98 690 L. Cos. d. L . Cotg. d. L, Tang. L. Sin. d. ' 76. PP .1 .2 3 4 .6 :i 56 55 54 .1 .2 3 4 .6 .8 53 52 51 4 3 5.6 II. 2 16.8 22.4 28.0 33-6 39-2 44-8 5-4 5-5 5-4 ii. o 10.8 16.5 16.2 22. 21.6 27.5 27.0 33.0 32.4 38.5 37.8 44.0 43.2 49.5 48.6 5-3 10.6 15.9 21.2 26.5 31-8 37-1 42.4 47-7 5-2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 5-1 10.2 '5-3 20.4 25.5 30.6 35-7 40.8 45. s) .1 .2 3 4 ii '.8 9 0.4 0.3 0.8 0.6 1.2 0.9 1.6 1.2 2.0 1.5 2.4 1.8 2.8 2.1 3-2 2.4 3-6 2.7 57 14 C , L. Sin.- d. L. Tang. d. L. Cotg. L. Cos. d. 9. 38 368 9 3 9 677 0.60 323 g.g8 6go 60 1 g.384i8 51 9 3g 73t 54 0.60 26g 9.98 687 3 5g 2 g.3846g 9 3g 785 0.60 2i5 9.98 684 58 3 g.38 5ig 9 3g 838 53 0.60 162 9.98 681 57 5' 54 3 4 9. 38 570 5 9 3g 8g2 0.60 1 08 9.98 678 3 56 5 g.38 620 9 3g g45 0.60 o5 9.98 675 55 6 9. 38 670 9 Sgggg 54 0.60 oo i 9.98 671 54 5' 53 3 7 9. 38 721 5 9 4o o52 o.5g g48 9.98668 53 8 g. 38 771 9 4o 1 06 o.5g 8g4 g.g8 665 52 9 g.38 821 9 4o i5g 53 o.5g 84i 9.98 662 5i 10 g.38 871 9 4o 212 53 o.5g 788 9.98 65g 50 1 1 g.38 g2i 9 4o 266 54 o.5g 734 9.98 656 4 9 12 9. 38 971 9 4o 3ig o.5g 681 9.98 652 48 i3 g.3g 02 1 9 4o 372 53 o.5g 628 g. 9 8 64g 3 47 i4 g.3g 071 5 9 4o 425 53 o.5g 575 9.98 646 3 46 i5 g.3g 121 9- 4o 4? 3 o.5g 522 9.98 643 45 16 9 .3 9 170 49 9 4o 53i 53 o.5g 46g g.g8 64o 3 44 , 7 g.3g 220 So 9 4o584 53 o.5g 4i6 9.98 636 4 43 18 g.3g 270 9 4o636 o.5g364 9.98 633 42 r 9 g. 3g 3ig 49 9 4o68g 53 o.5g 3n 9.98 63o 3 4i 20 g.3g 36g 5 9 4o 742 53 o.5g258 9.98 627 40 21 g.3g 4i8 49 9 4o 795 53 o.5g 2o5 9.98 6 2 3 4 3 9 22 g.3g 467 9- 4o84? o.5g i53 9.98 620 38 23 g.Sg 5i7 5 9- 4o 900 53 o.5g 100 9.98 617 i 3? 24 g.Sg 566 49 9 4o g52 5* o.5g o48 9.98 6i4 3 36 25 g.Sg 615 9- 4i 005 o.58 gg5 9.98 610 35 26 g.Sg 664 49 9 4i o5j 52 o.58 g43 9.98 607 J 34 27 9 .3 97 i3 49 9 4 1 log 52 o.58 891 9.98 6o4 3 33 28 g.3g 762 9- 4i 161 0.58 83g 9.98 60 1 32 29 g.Sg 8n 49 9 4l 21; 4 53 o.58 786 g.g8 5 97 4 3i 30 g. 3g 860 49 9 4i 266 52 o.58 734 9.98 5 9 4 3 30 L. Cos. d. L Cotg. d. L. Tang. L. Sin. d. ' 75 3O . PP 54 53 53 51 50 49 4 3 .1 5-4 53 52 .j 5-i 5.0 49 .1 4 -3 .2 10.8 10.6 10.4 2 10.2 IO.O 9.8 .2 .8 .6 3 16.2 15-9 '5-6 3 '5-3 15.0 14.7 3 1.2 .9 4 21.6 21.2 20.8 4 20.4 20. o 19.6 4 1.6 1.2 5 27.0 26. 5 26.0 5 255 25.0 24-5 5 2.O 1.5 .6 32-4 31.8 31.2 6 30.6 30.0 29.4 .6 2.4 1.8 7 37.8 37- * 36-4 7 35-7 35-o 34-3 .- 2.8 2.1 .8 43-2 42-4 4'-6 8 40.8 40.0 39-2 .8 3 .2 2.4 9 48.6 47-7 46.8 9 45 9 45.0 44.1 58 143O / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.39 860 9- 4i 266 o.58 734 9.98 594 30 3i 9.39909 49 9- 4i 3i8 52 o.58 682 9.98 591 3 29 32 9.399 58 Q 9- 4i 870 0.58 63o 9.98 588 28 33 9.40 006 9- 4i 422 o.58 578 9.98 584 27 t9 52 H 34 9 ,4o c 48 9- 4i 4?/ 52 o.58 526 9.98 58i 3 26 35 9.40 io3 9- 4i 5 2 6 o.584?4 9.98 578 25 36 9.40 i 52 9- 4i 5 7 8 o.58 422 9.98 574 24 a 51 3 37 9 .40 200 9- 4i 629 52 o.58 3?i 9.98 571 3 23 38 9.40 249 9- 4i 681 o. 58 3ig 9.98 568 22 3 9 9.40 297 9- 4i 733 o.58 267 9.98 565 21 40 g.4o 346 9- 4i 784 o.58 216 9.98 56i 20 4i 9.40 v 94 1 9- 4i 836 o.58 1 64 9.98 558 3 '9 42 9 .4o 44 2 9- 4i 887 o.58 1 13 9.98 55.5 18 43 9.40490 9- 4i 939 o.58 061 9.98 55i 17 B 51 3 44 g.4o 538 s 9- 4i 990 o.58 oio 9. 9 8 548 3 16 45 9.40 586 9- 42 o4i 0.57 959 9.98 54.5 i5 46 g.4o 634 (" 9- 42 093 0.57 907 9.98 54 1 i4 H 51 3 47 9-4o 682 ,8 9- 42 i44 o.5 7 856 9.98 538 3 i3 48 9-4o 7 3o 9- 42 ig5 0.57 805 9.98 53,5 12 49 9-4o 778 |fl 9- 42 246 51 0.57 754 9.98 53i I I 50 9.40 825 47 ,Q 9- 42 297 5 1 0.67 703 9.98 528 10 5i 9.40 873 18 9- 42 348 0.57 652 9.98 525 4 9 52 9.40 921 9- 42 399 0.57 601 9.98 521 8 53 g.4o 968 9- 42 45o 5' 0.57 550 9.98 5i8 7 V 51 54 9.41 016 9- 42 5oi 0.57 499 9.98 515 4 6 55 9.41 o63 9- 42 552 0.57448 9.98 5n 5 56 9-4i i ii |B 9- 42 6o3 Si 0.57 397 9.98 5o8 4 5 7 9.41 i58 47 9- 42 653 5 0.57 347 9.98 5o,5 4 3 58 9-4l 2 o5 9- 42 704 0.57 296 9.98 5oi 2 5 9 g.4l 252 47 9- 42 755 5i o. 57 245 9.98 498 I 60 9 . 4 1 3oo * u 9- 42 8o5 0.57 195 9.98 4 9 4 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. / 75. PF 52 51 50 49 48 47 4 3 .1 5 2 5 .i S-o .1 4.9 4 .8 4-7 .1 0.4 -3 .2 10.4 IO. 2 IO. O .2 9.8 96 9.4 .2 0.8 0.6 3 15.6 15-3 150 3 14.7 14.4 14.1 3 12 0.9 4 20.8 20.4 20. o 4 ' 19.6 19.2 18.8 4 1.6 1.2 5 26.0 25.5 25.0 5 24-5 24.0 23-5 5 2.O i 5 .6 31 2 30.6 30.0 .6 29.4 28.8 28.2 .6 24 1.8 7 36.4 35-7 35.0 7 34-3 33 6 32-9 7 2.8 2.1 .8 4 ..6 40.8 40.0 .8 39-2 38.4 37- 6 .8 3-2 2-4 9 46.8 45_9__ 45 9 44.1 43.2 42.3 ___ '-' 7 5 9 15 C ' L.Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.41 3oo 9.42 bo5 5' 5 5' 5 5 5' 5 So 5 5 5 5 5 5 50 49 5 5 49 5 49 5 49 5 49 49 49 So 49 49 0.57 195 9 .9*494 3 3 4 3 4 3 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 60 I 2 3 4 5 6 7 8 9 9.41 9-4i 9.41 9,41 9-4i 9-4i g.4i 9-4i 9-4i 34 7 3 9 4 44 1 483 535 682 628 6 7 5 722 47 47 47 47 47 46 47 47 46 47 46 47 46 47 46 46 47 46 9.42 856 9.42 906 9.42 957 9.43 007 9.43o57 9.43 108 9.43 1 58 9.43 208 9.43 258 0.57 i 44 o. 57 094 0.57 o43 o.56 993 o.56 9 43 0.66892 o.56 842 o.56 792 o.56 742 9 9 9 9 9 9 9 9 9 .98 491 .98488 .98484 .98481 .98477 .98474 .98471 .98467 .98464 5 9 58 5 7 56 55 54 53 52 5i 10 9-4i 768 9-43 3o8 o.56 692 9 .98460 50 1 1 12 i3 i4 i5 16 17 18 '9 9-4i 9.41 9.41 9.41 9.42 9.42 9.42 9.42 9.42 815 861 908 954 OOI 047 098 i4o 186 9-43358 9.434o8 9.43458 9 .435o8 9. 43 558 9.43 607 9.43 657 9-43 707 9.43756 o.56 642 o. 56 5 9 2 o.56 542 o.56 492 o.56 442 o.56 3g3 0.56343 o.56 293 o.56 244 9 9 9 y 9 '<; 9 9 9 .98457 .98453 .98450 .98447 . 9 8443 .98440 .98 436 .98433 .98 429 49 48 4? 46 45 44 43 42 4i 20 9.42 232 46 46 46 46 46 45 46 46 46 45 46 9.43 806 o.56 194 9 .98426 40 21 22 23 24 25 26 27 28 2 9 9.42 9.42 9.42 9.42 9.42 9.42 9 .4a 9.42 9.42 278 324. 870 4i6 46i 607 553 5 99 644 9-43855 9-43 905 9-43 954 9 . 44 oo4 9.44 o53 9-44 102 9-44 i5i 9-44 201 9-44 250 o.56 i4s o.56 og5 o.56 o46 o.55 996 o.55 947 0.55898 0.55849 o.55 799 o.55 750 9 9 9 9 9 9 9 9 9 .98 422 .98 419 .984x5 .98412 . 98 4og .98 4o5 .98 4O2 .98 398 .98 395 3 9 38 37 36 35 34 33 32 3i 30 9.42 690 9-44 299 o.55 701 9 .98 391 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 74 SO. PP .1 .2 3 4 :! 51 50 49 48 47 4 6 .1 .2 3 4 .6 .8 9 45 4 3 5-i IO. 2 '5-3 20.4 25-5 30.6 35-7 40.8 45 9 5.0 IO. O 15.0 20.0 250 30.0 35-0" 40.0 49 9.8 14.7 19.6 24- S 39.4 34-3 39- 44 I .1 . 3 4 '.6 .8 4 .8 9 .6 14.4 19.2 24.0 28.8 33-6 38.4 4S- 2 4-7 9-4 14.1 1 8.8 23-5 28.2 32.9 37-6 42 3 4-6 9.2 13- <* 18.4 .3.0 27.6 32.2 36.8 41 4 4-5 9.0 13-5 18.0 22.5 27.0 3' -5 30.0 Joy 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3-| 3-6 0.3 0.6 0.9 1.2 IS 1.8 2.1 2.4 2 -7 60 15 30 L. Sin. d. . Tang. d. L. Cotg. L. Cos. d. 30 9.42 690 9.44 299 o . 55 701 9 . 9 3 9 i 30 3 1 9.42 7 35 9 .44348 49 o.55 652 9.98 388 3 29 32 9.42 781 9 44 397 o. 55 6o3 9.98 384 28 33 9.42 826 9 .44446 o.55 554 9.98 38i 3 27 4 6 44 ^ 34 9.42 872 45 9 44 495 49 o.55 5o5 9.98 377 26 35 9.42 917 9 .44544 ,0 o.55 456 9.98 3 7 3 20 36 9.42 962 9 .44 592 o.554o8 9.98 370 3 24 46 44 37 9-43 008 9 .4464i 49 o.55 35 9 9.98 366 23 38 9 .43 o53 9 .44 690 o.55 3io 9.98 363 22 3 9 9.43098 45 9 .44738 4 o.55 262 9.98 35g 4 21 40 9-43 i43 45 9 .44 787 o.55 2i3 9.98 356 3 20 4i 9-43 1 88 45 9 .44836 o.55 1 64 9.98 352 4 '9 42 9.43233 9 .4488 i o.55 1 16 9.98 349 18 43 9-43 278 45 9 .44933 49 o.55 067 9 . 9 8 345 4 7 44 9-43 323 45 9 .4498 I 40 48 o.55 019 9.98 342 3 16 45 9-43 36? 9 .45 029 o.54 971 9.98 338 i5 46 9.43 412 45 9 .45078 49 o.54 922 9.98 334 4 i4 4 7 9.43457 45 9 .45 126 48 . 0.54874 9.98 33i 3 i3 48 9 .43 5o2 9 45 174 0.54826 9.98 327 12 49 9-43 546 44 9 .45 222 40 o.54 778 9.98 324 3 I I 50 9 .43 5gi 45 9-45 271 4y 0.54 729 9.98 32O 4 10 5i 9-43635 44 9 .45 319 ,a o.5468i 9.98 3i 7 3 9 52 9.4368o 9 .45 367 0.54633 9.98 3i3 8 53 9-43 724 44 9 .454i5 48 0.54585 9.98 3og 4 7 54 9-43 769 45 9 .45463 4*5 ,a 0.54537 9.98 3o6 3 6 55 9-43 8i3 9 -455n 0.54489 9.98 302 56 9.4385? 44 9 .45559 4 o.5444i 9.98 299 3 4 57 9 .43 901 44 9 .45 606 47 48 o.54 394 9.98 295 4 3 58 9-43 9 46 9 .45654 0.54346 9.98 291 2 5 9 9.43 990 44 9 .45 702 48 o.54 298 9.98 288 3 I 60 9-44 o34 44 9 .45 750 48 o. 54 25o 9.98 284 4 L. Cos. d. L . Cotg. d. L. Tang. L. Sin. d. ' 74 . PP 49 4 8 47 4 6 45 44 4 3 .! 4.9 - 4 .8 4-7 .1 4 .6 4-5 4-4 .1 0.4 3 .2 9.8 Q.b 9-4 .2 9.2 9.0 8.8 .2 0.8 0.6 3 14.7 14.4 14.1 3 13.8 13-5 13.2 3 1.2 0.9 4 19.6 ig.2 18.8 4 z8. 4 18.0 ,7.6 4 1.6 I 2 5 24 5 24.0 23-5 5 23.0 22.5 22. 5 2.O i 5 .6 29.4 28.8 28.2 .6 27.6 27.0 26.4 .6 2-4 i 8 7 34-3 33 6 32.9 7 32.2 3'-5 30.8 7 2.8 2 I .8 39 2 3-4 37-6 .8 36.8 36.0 35-2 .8 3-2 2 4 9 44-i 4?. 2 42. -^ 9 41.4 40. 5 9 3.6 2 7 16. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9-44 o34 9.45 750 o.54 z5o S .98 284 60 1 9.44 078 9-45 797 47 *.& O.54 203 9 .98 281 5 9 2 9.44 122 9-45845 0.54 i55 9 .98 277 58 3 9-44 l66 9.45 892 47 o.54 1 08 9 .98 273 57 44 48 9 4 9.44 2IO 43 9.45 940 o.54 060 9 .98 270 4 56 5 9-44253 9.45987 o.54oi3 9 .98 266 55 6 9-44 297 9.46035 4 0.53965 9 .98 262 54 44 47 3 7 9-44 34 1 9-46 082 o.53 918 9 .98 259 53 8 9-44385 9-46 i3o 4 o.53 870 9 .98255 52 9 9.44428 43 9-46 177 47 o.53 823 9 .98 25 1 5i 10 9-44 472 9.46 224 47 o.53 776 9 .98 248 50 1 1 9.44 5i6 9.46 271 47 o.53 729 9 ,98244 49 12 9 .44559 9.46 319 4 o.53 681 9 -98 240 48 i3 9-44 602 43 9.46 366 47 0.53634 9 .98 237 47 44 47 4 i4 9.44646 9.46 i ii3 o.53 58? g .98 233 46 i5 9.44689 9-46 46o 47 o.53 54o 9 .98 229 45 16 9.44733 44 9-46 607 47 o.53 493 9 .98 226 3 44 '7 9-44 776 43 9. 46 554 47 0.53446 9 .98 222 4 43 18 9-44 819 9.46 601 o.53 399 .98 218 42 '9 9-44 862 43 __ 9-46648 47 o.53 352 9 .98 215 4i 20 9-44 go5 43 9-46 694 46 o.53 3o6 9 .98 21 I 4 40 21 9.44948 43 9-46 74 1 47 o.53 2 5 9 9 .98 207 3 3o 22 9-44 992 9.46 788 O.53 212 .98 2O4 38 23 9-45 035 43 9.46 835 47 o.53 i65 9 .98 200 37 24 9.45077 42 9-46 88 1 46 o.53 119 9 .98 196 4 36 25 9.45 I2O 9 .46( >28 o.53 072 9 .98 192 35 26 9.45 i63 43 9.46975 47 o.53 O25 9 .98 189 34 27 9.45 206 43 9.47 021 46 o.52 979 9 . 9 8 185 4 33 28 9.45 249 9.4? 068 47 o.52 9 32 9 .98 181 32 29 9.45 292 43 9-47 i4 4 b o.52 886 9 .98 177 3i 30 9.45 334 42 9.47 160 46 o.52 84o 9 .98 174 30 L. Cos. d. L. Cotg. d. L. Tang. L.Sin. d. ' 73 30 . PP 48 47 46 45 44 43 ' 43 4 3 .1 4 .8 4-7 4-6 ., 4-5 4-4 4-3 .1 4-2 0.4 0.3 .2 9.6 9.4 9.2 .2 9.0 8.8 8.6 .2 8.4 0.8 0.6 3 M-4 . M i 13-8 3 '3'S 13-2 12.9 3 12.6 1.2 0.9 4 19.2 18.8 18.4 4 18.0 17.6 17.2 4 16.8 1.6 1.2 5 24.0 23-5 23.0 5 22.5 22. 21.5 5 21. 2.0 1.5 .6 28.8 28.2 27.6 .6 27.0 26. 4 25-8 .6 25.2 2.4 1.8 7 33-6 32-9 32.2 .7 31. s 30.8 30.1 7 29.4 2.8 2.1 .8 38.4 37-6 36.8 .8 36.0 352 V 4 .8 33-6 3.2 2.4 9 43-2 42.3 41.4 9 40. s 30.6 38.7 3.6 2.7 62 16 3O . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9-45 344 9.47 160 o.52 84o 9 98 174 30 3i 9-45 377 43 9.47 207 47 46 o. 52 793 9 .98 170 4 29 32 9-45 419 9.4? 253 o.52 747 9 .98 166 28 33 9.45 462 43 9.4? 299 46 o.52 701 9 .98 162 4 27 34 9-45 5o4 42 9-47 346 47 0.62 654 9 .98 i 59 3 26 35 9-45 54? 9.47 392 4 O.52 608 9 .98 165 25 36 9-45 58 9 9 . 47 438 46 o. 52 562 9 .98 i )i i 24 3? 9-45 632 43 9. 4? 484 4 b -6 o.52 5i6 9 .98 147 4 23 38 9-45 6 7 4 9.47 53o o.52 470 9 .98 1 44 22 39 9-45 716 42 9.47 576 46 O.52 424 9 .98 i4o 4 21 40 9-45 758 9.4? 622 46 o.52 378 9 .98 i36 4 20 4i 9-45 801 42 9.47 668 46 o.52 332 9 .98 i32 4 19 42 9-45 843 9-4? 1 i4 O.52 286 9 .98 129 18 43 9-45 885 9.4? 760 4 0.62 240 9 .98 125 4 17 42 46 4 44 9-45 927 42 9.47 806 46 o.52 194 9 .98 121 16 45 9.45 969 9.4? 852 o.52 i48 9 .98 117 i5 46 9-46 OI I 9.47897 45 o.52 io3 9 .98 n3 4 i4 42 46 3 47 9-46 o53 42 9.47 943 A.6 o.52 057 9 .98 no i3 48 9.46 095 9.47 9 8 9 O.52 i 9 .98 1 06 12 49 9-46 i 36 9.48 035 46 o. 5i 965 9 .98 102 4 I I 50 9-46 178 9.48 080 45 o . 5 1 920 9 .98 o ?8 4 10 5i 9-46 220 42 9.48 126 o.5i 874 9 .98 094 9 52 9-46 262 9.48 171 o.5i 829 9 .98 090 8 53 9-46 3o3 9.48 217 o.5i 783 9 .98 087 3 7 42 45 4 54 9-46 34.5 9.48 262 o.5i 738 9 .98 o83 6 55 9-46 386 9. 48 3 07 o.5i 693 9 .98 079 5 56 9-46 428 9.48 353 46 o.5i 64? 9 .98 075 4 4 4 1 45 4 5? g.46 469 9.48 398 o.5i 602 9 .98 071 3 58 9-46 5n 9-48443 o.5i 55 7 9 .98 067 2 5 9 9-46 552 4i 9.48489 46 o.5i 5i i 9 .98 o63 4 I 60 9-46 5 9 4 42 9.48 534 45 o.5i 466 9 .98 060 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 73. PP 47 46 45 44 43 43 41 4 3 .1 4.7 4.6 4-5 .1 4.4 4.3 4.2 .1 4.1 0.4 o 3 .2 9-4 9.2 9.0 .2 8.8 8.6 8.4 2 8.2 0.8 06 3 14.1 13.8 >3-5 3 13.2 12.9 12.6 3 12.3 1.2 9 4 18.8 18.4 18.0 4 17.6 17.2 16.8 4 16.4 1.6 I 2 5 23-5 23.0 22.5 5 22. 21.5 21. 5 20.5 2.0 1 $ .6 28.2 27.6 27.0 .6 26.4 25.8 25.2 .6 24.6 2.4 i 8 7 32.9 32.2 31.5 7 30.8 30.1 29.4 7 28.7 2.8 2 I .8 37-6 36.8 36.0 .8 35.2 34.4 33.6 .8 32.8 3.2 2 4 9 42-3 41.4 40.5 39.6 38.7 37.8 9 36.9 63 17 C > L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.46 5g4 41 41 41 41 42 41 41 4 41 41 40 41 41 41 41 40 4' 40 41 40 4' 40 4i 40 40 4i 40 40 40 40 9- 4-3 534 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 o.5i 466 9.98 060 60 I 2 3 4 5 6 7 8 9 9-46635 9.46 676 9-46 717 9-46 758 g.46 800 9.46 84 1 9.46 882 9:46 923 9.46 964 9- 9- 9- 9- 9- 9- 9- 9- 9- 48 579 48624 48669 48 714 48 7 5 9 488o4 48849 48894 48 939 o.5i 421 o.5i 376 o.5i 33i o.5i 286 o.5i 241 o.5i 196 o.5i i5i o.5 1 1 06 o.5i 061 9.98 o56 9.98 o52 9 . 98 o48 9.98 o44 9.98 o4o 9.98 o36 9.98 o32 9.98 029 9.98 025 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 5 9 58 57 56 55 54 53 52 5i 10 9.47 005 9- 48 9 84 o.5i 016 9.98 021 50 1 1 12 13 i4 i5 16 17 18 '9 9.47 o45 9.4? 086 9-47 127 9.47 1 63 9-4? 209 9.47 249 9.47 290 9.47 33o 9-47 371 9- 9- 9- 9- 9- 9- 9- 9- 9- 4g 029 49 073 49118 49 1 63 4g 207 49 252 4g 296 49 34 1 4g 385 o.5o 971 o.5o 927 o.5o882 o.5o83 7 o.5o 793 o.5o 748 o.5o 704 o.5o 65g o.5o 615 9.98 017 9.98 oi3 9.98 009 9.98 oo5 9.98 ooi 9.97997 9.97993 9.97989 9.97986 49 48 47 46 45 44 43 42 4i 20 9-4? 4i i 9- 49 43o o.5o 570 9-97 982 40 21 22 23 24 25 26 27 28 2 9 9.47 452 9.47 492 9.47533 9.47 5?3 9.47 6 1 3. 9.4? 654 9.47 6g4 9.47 7 3 4 9-47 774 9- 9- 9- 9- 9- 9- 9- 9- 9- 4g 474 4g 5ig 49 563 4g 607 49 65a 4g 696 49 740 4 9 7 84 49 828 o.5o 526 o.5o48i 0.60437 o.5o 3g3 o.5o348 o.5o 3o4 o.5o 260 o.5o 216 o.5o 172 9.97978 9.97 974 9.97970 9.97966 9.97 962 9.97958 9.97954 9.97950 9.97946 3 9 38 3? 36 35 34 33 32 3i 30 9-47 8i4 9- 49 872 o.5o 128 9.97942 30 L. Cos. d. L. Cotg. | d. L. Tang. L. Sin. d. 72 3O. PP .1 .2 3 4 .6 .8 45 44 43 .1 4* 41 40 .1 4 3 4-5 9.0 '3-5 18.0 22.5 27.0 3'-5 36.0 40. s 8.8 8.6 13.2 12.9 17.6 17.2 22.O 2I.S 26.4 25.8 30.8 30.1 35-2 34-4 r 4.1 4.0 o. 4 o. 3 o. 8 o. 6 1.2 0.9 1.6 1.2 2.0 1.5 2.4 1.8 2.8 2.1 3.2 2.4 3.6 2.7 3 4 '.6 :l 12.6 16.8 21.0 25.2 29.4 33-6 37-8 12.3 16.4 20.5 24.6 28.7 32.8 36.9 I2.O 16.0 20. o 24.0 28.0 32.0 3 4 5 .6 .8 64 17 30. > L. Sin. d. L . Tang. d. L. Cotg. L. Cos. d. 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 30 9-4? 8i4 40 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 9 .49 872 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 o.5o 128 9-97 942 30 3i 32 33 34 35 36 37 38 3 9 9.47 9.47 9.47 9-4? 9.48 9-48 9-48 9-48 9-48 854 8 9 4 934 974 oi4 o54 094 i33 I 7 3 9 9 9 9 9 9 9 9 9 .49 916 .49 960 . 5o oo4 .5o o48 . 5o 092 .5o i36 .5o 180 ,5o 223 . 5o 267 o.5o o84 o.5o o4o 0.49 996 0.49 952 o .49 908 0.49 864 0.49 820 0.49 777 0.49 733 9-97 9 38 9.97 934 9.97 9 3o 9.97926 9.97922 9.97 918 9-97 9'4 9.97910 9-97 9 6 29 28 27 26 25 24 23 22 21 40 9 .48 2l3 9 .5o 3n 0.49 689 9-97 902 20 4i 42 43 44 45 46 4 7 48 49 9-48 252 9-48 292 9-48 332 9-48 371 9-484U 9.48 45o 9 .48 490 9.48 52 9 9 .48 568 9 9 9 9 9 9 9 9 9 5o 355 5o 3 9 8 5o 442 5o485 5o 529 5o 572 5o 616 5o 65g 5o 703 0.49 645 0.49 602 0.49 558 0.49 5i5 o. 49 4?i 0.49 428 0.49 384 0.49 34i 0.49 297 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 898 8 9 4 890 886 882 878 8 7 4 870 866 '9 18 '7 16 i5 i4 i3 12 I I 50 9-48 607 9 5o 746 0.49 254 9-97 861 10 5i 52 53 54 55 56 57 58 5 9 9 .48 9-48 9.48 9-48 9-48 9-48 9-48 9-48 9-48 64? 686 725 8o3 842 881 920 9 5 9 9 9 9 9 9 9 9 9 9 5o 789 5o833 .50876 .5o 919 ,5o 962 . 5 1 oo5 .5i o48 . 5 1 092 .5i 135 O.49 211 0.49 167 0.49 124 0.49 08 1 0.49 o38 0.48 995 o.48 g52 o.48 908 0.48865 9.97 857 9.97 853 9.97849 9.97 845 9-97 84i 9.97837 9.97 833 9.97 829 9-97 825 9 8 7 6 5 4 3 2 I 60 9-48 998 9 .5i 178 o.48 822 9-97 821 L. Cos. d. L . Cotg. d. L. Tang. L. Sin. d. ' 72. PP .2 3 4 .6 .8 9 44 43 45 2 3 4 6 4 40 39 .1 .2 3 4 '.6 .8 9 5 4 4-4 8.8 13.2 ,7.6 22.0 26.4 30.8 35 2 4-3 8.6 12.9 17.2 21.5 25.8 30.1 34-4 4.2 8.4 12.6 16.8 21. 25.2 29.4 33-6 37.8 4.1 8.2 12.3 16.4 20.5 24.0 28.7 32.8 4.0 8.0 I2.O 16.0 20. o 24.0 28.0 32.0 36.0 3-9 7.8 11.7 15.6 19.5 23-4 27-3 31.2 35- ' 0.5 0.4 i.o 0.8 1-5 1.2 2.0 1.6 2-5 2.O 3.0 2.4 3-5 2.8 4-0 3.2 4-5 3-6 65 18. L.Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.4^ 998 9-5i 178 43 43 42 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 o.48 822 9 97 821 4 5 4 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 60 I 2 3 4 5 6 7 8 9 9.49 oSy 9.49 076 9.49 us 9.49 i53 9.49 192 9.49 z3i 9.49 269 9.49 3o8 9.49 347 39 39 38 39 39 38 39 39 9.61 221 9. 5 1 264 9. 5 1 3o6 9. 5 1 349 9.5i 392 9.61435 9-5i 4?8 g.5i 520 g.5i 563 o.48 779 o.48 736 o.48 694 o.4865i o.48 608 o.48 565 0.48522 o.4848o 0.48437 9 9 9 9 9 9 9 9 9 97817 97 812 97808 97 8o4 97 800 97796 97792 97788 97 7 84 59 58 57 56 55 54 53 52 5i 10 9.49 385 3 39 38 38 39 38 38 39 38 38 g.5i 606 o.48 3g4 9 97779 50 ii 12 i3 i4 i5 16 17 18 '9 9.49 424 9.49 462 9.49 5oo 9.49 53g 9.49 5 77 9.49 6 1 5 9.49654 9.49692 9.49 ?3o 9-5i 648 9-5i 691 g.5i 734 g.5i 776 9-5i 819 9-5i 861 9. 5 1 903 9. 5 1 946 9.5i 988 0.48352 0.48 3og 0.48266 0.48 224 o.48 181 0.48 i3g 0.48 097 o.48 o54 0.48 OI 2 9 9 9 9 9 9 9 9 9 97775 97771 97 767 977 63 97759 97754 97750 97 746 97 742 49 48 47 46 45 44 43 42 4i 20 9.49 768 38 9.02 o3i 0.47 969 9 97 7 3 ^ 40 21 22 23 24 25 26 27 28 2 9 9.49 806 9.49844 9.49882 9.49920 9.49958 9.49996 9.5oo34 g.So 072 g.So 1 10 38 38 38 38 38 38 38 38 38 38 9.52 073 9.52 n5 9.52 157 9. 52 200 9.52 242 9.52 284 9.62 326 9.52 368 9.52 4io 0.47 927 0.47885 0.47843 0.47 800 0.47753 0.47 716 0.47 674 0.47632 0.47 590 9 9 9 9 9 9 9 9 9 97734 97729 97725 97721 97717 97713 97708 97 704 97 700 5 4 4 4 4 5 4 4 4 3 9 38 3 7 36 35 34 33 32 3i 30 g.5o i48 9.52 452 0.47543 9 97 696 30 L. Cos. d. L. Cot K. d. L. Tang. L.Sin. d. 713O. PP i 2 3 4 I I 9 43 43 39 38 I .2 3 4 '.6 j| 5 4 4-3 8.6 12.9 17.2 21.5 5-8 30-1 34-4 38.7 4-2 8 -* 12. 6 16.8 21. 25.2 29.4 33-6 37-8 i 2 3 4 6 I 3 2 7.8 11.7 .5.6 19.5 23-4 27.3 3i-a *** 7 .6 11.4 IS-2 19.0 22.8 26.6 3-4 3^2 o-S I.O '5 2.0 2-5 3-o 3-5 4.0 4-5 o-4 0.8 1.2 1.6 2.O 2-4 2.8 3-2 3-6 66 18 3O L. Sin. d. L. Tang. d. L Cotg. L. Cos. d. 30 9. 5o i48 9.5-2 452 O 47548 9.97696 30 3i y .5o i85 38 9 .5 2 494 42 4? 5o6 9-97 69! 29 32 9. 5o 223 9.52 536 0. 47464 9-97 087 28 33 9-5o 261 3 9.52 578 O. 47 422 9-97 683 4 27 37 42 4 34 g.So 298 38 9-52 620 O. 4?38o 9.97679 26 35 9. 5o 336 38 9.52 661 O. 47339 9-97 6 7 4 25 36 9-5o 3 7 4 9.52 7o3 O. 47297 9-97 670 4 24 37 42 4 37 9. 5o 4i i 38 9 .52 745 42 O. 47255 9.97 666 23 38 9. 5o 449 9.52 787 O. 4? 2i3 9.97 662 22 39 9-5o486 37 9 .52 829 O. 47 171 9-97 65 7 5 21 40 9.5o 523 9.52 870 O. 47 i3o 9-97 653 20 4i g.5o 56i 3 9.62 912 O. 47088 9-97 64g 4 19 42 9. 5o 5g 8 9.52 953 O. 4? 047 9-97 645 18 43 9. 5o 635 37 9.52 995 42 O. 47005 9.97 64o 5 17 44 9-5o 673 3* 9.53 037 42 0. 46 9 63 9-97636 4 16 45 9-5o 710 9.53078 O 46 922 9.97 632 i5 46 9-5o 747 37 9.53 120 42 O. 4688o 9.97 628 i4 37 41 5 47 9.5o 784 9.53 161 O. 4683 9 9-97 623 i3 48 9-5o 821 9. 53 202 O. 46 79 8 9.97619 12 49 9 .5o858 37 9.53 244 4 2 O. 46 7 56 9-97 615 I I 50 g.5o 896 38 9 .53 285 4i 0. 46 715 9-97 610 10 5i 9. 5o 933 37 9.53 327 42 O. 46673 9-97 606 4 9 52 9-5o 970 9.53368 O. 46 632 9.97 602 8 53 9. 5 1 007 37 9.53 409 4' O. 465 9 i 9.97597 5 7 54 9-5i o43 30 9.5345o 4 1 O. 46 550 9-97 5 9 3 4 6 55 9.5i 080 37 9-53 492 O. 465o8 9.97 58g 5 56 9-5i 117 37 9.53 533 4 1 O. 46467 9.97 584 5 4 57 9. 5 1 1 54 37 9-53 5 7 4 4' O. 46426 9.97 58o 4 3 58 g.Si 191 37 9-53 6i5 O. 46385 9.97576 2 5 9 9.61 227 36 9-53 656 4' O. 46 344 9-97 5 7 i 5 I 60 9. 5 1 264 37 9.53 697 4' O. 46 3o3 9.97 567 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 71. PP 42 41 38 37 36 5 4 .1 4-2 4.1 3 .8 i 3-7 3 .6 .1 0.5 04 .2 8.4 8.3 7 .6 .2 7.4 7-2 .2 t.o 0.8 3 12.6 12.3 11.4 3 "-i 10.8 3 I.S 1.2 4 16.8 16.4 15-2 .4 14.8 14.4 4 2.O 1.6 5 21. 20.5 19.0 5 18.5 18.0 5 2.5 2.0 .6 25 2 24.6 22.8 .6 22.2 21.6 .6 3.0 2.4 7 29.4 28.7 26.6 7 25.9 25.2 7 3-5 2.8 .8 336 32.8 37-5 .6 21. 7 24.5 .8 28.0 9 3'-5 3-4 6.8 10.2 13-6 17.0 20.4 23 8 27.2 30.6 o-S I.O 15 2.0 2-5 3-5 4.0 45 70 2O 3O L. Sin. d. L. Tang. d. j L. Cotg. L. Cos. d. 30 9.54433 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 9 .5 7 2?4 38 39 38 39 38 38 39 38 38 39 38 38 38 38 39 38 38 38 38 38 38 38 38 38 38 38 37 38 38 38 O.42 726 9-97 l5 9 5 5 4 5 5 5 4 5 S S 4 5 5 5 5 4 S 5 5 5 4 5 5 5 5 4 5 5 5 5 30 3i 32 33 34 35 36 3? 38 39 9.54466 9. 54 5oo 9.54534 9.54567 9.54 601 9.54635 9.54668 9.54 702 9.54735 9.57 3i2 9.57 35i 9.57 389 9.57428 9.57 466 9.57 5o4 9.57 543 9.57 58i 9.57 619 0.42 688 0.42 649 0.42 6 1 1 0.42 572 0.42 534 0.42 496 0.42 457 0.42 4>9 0.42 38 1 9-97 9-97 8.97 9-97 9-97 9-97 9-97 9-97 9-97 1 54 149 i45 i35 i3o 126 121 116 29 28 27 26 25 24 23 22 21 40 9.54769 9.5 7 658 o. 42 342 9-97 in 20 42 43 44 45 46 4 7 48 49 9.54 802 9.54836 9.54869 9.54903 9.54936 9.54969 9. 55 oo3 9.55o36 9.55 069 9.57 696 9.57734 9.57772 9.57 810 9.57849 9.5 7 88 7 9.57925 9.57 963 9-58 ooi 0.42 3o4 0.42 266 0.42 228 0.42 190 0.42 i5i 0.42 1 1 3 0.42 075 0.42 037 0.4 1 999 9.97 107 9.97 1 02 9.97097 9.97092 9.97087 9.97 o83 9.97078 9.97073 9.97 068 '9 18 7 16 i5 i4 i3 12 I I 50 9-55 102 9.58 039 o. 4i 961 9.97 o63 10 5i 52 53 54 55 56 5? 58 59 9.55 i36 9-55 169 9-55 202 9.55 235 9.55268 9-55 3oi 9.55334 9.55 367 9-55 4oo 9. 58 077 9.58 115 9.58 i53 9-58 191 9-58 229 9-58 267 9.583o4 9.58 342 9.58 38o o. o. o. 0. o. o. 0. o. o. 4i 923 4i 885 41847 4i 809 4i 77' 4i ?33 4 1 696 4i 658 4i 620 9.97059 9.97054 9.97049 9.97044 9.97039 9.97035 9.97 o3o 9.97025 9.97 020 9 8 7 6 5 4 3 2 I 60 9.55433 9.584i8 o. 4i 582 9.97 01 5 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 69. PP 39 38 37 34 33 9 5 4 i 3-9 .2 7.8 3 "? 4 '5-6 S '9-5 .6 23.4 7 27.3 .8 31.2 9 35-< 11.4 '5-2 19.0 22.8 26.6 3-4 34-2 3-7 7-4 n. i 14.8 18.5 22.2 259 29.6 33-3 I 3-4 .2 6.8 3 10.2 4 '3-6 .6 20.4 7 23.8 .8- 27.2 .9 30. 6 u 9-9 16.5 19.8 23-' 26.4 29.7 0.5 I.O 2.0 2-5 3-5 4.0 4-5 0.4 0.8 1.2 1.6 2.0 2.4 2.8 32 3 .6 21. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.55 433 g.58 4i8 o.4i 582 9 g7 oi5 60 I g.55466 33 g.58 455 37 18 o.4i 545 9 g7 oio 5 5 9 2 g.55 4gg 9. 584 9 3 o.4i 507 9- g7 oo5 58 3 g.55 532 33 9. 58 53i 3 8 o.4i 46g 9- g7 ooi 57 32 38 S 4 g.55 564 33 9. 58 56g o.4i 43i 9- g6 gg6 5 56 5 g.55 5g7 g.58 606 o.4i 3 94 9- g6ggi bb 6 g.55 63o 33 g.58 644 3 8 o.4i 356 9- g6g86 54 33 37 S 7 g.55 663 g.5868i o.4i 3 '9 9- g6 g8i 5 53 8 g.55 6g5 g. 58 719 3 o.4i 281 9 g6 9 76 52 9 g.55 728 33 g.58 7 5 7 3 8 o.4i 243 9 9 6 97i 5i 10 g.55 761 g.58 79 4 37 o.4i 206 9 g6 g66 50 1 1 g . 55 7g3 3 2 g.58 832 3 8 o.4i 168 9 g6 9 62 5 49 12 g.55 826 g.5886g o.4i i 3i 9 9 6g57 48 i3 g.55 858 32 g.58 907 3 8 o.4i og3 9- g6 g52 47 i4 g.55 8g I 33 g.58 9 44 37 o.4i o56 9- 9 6g47 b 5 46 i5 g.55 g23 g.58 9 81 o.4i 019 9 g6 g42 45 16 g.55 g56 33 g. 5g oig 38 o.4o g 81 9 9 6 9 3 7 44 17 g.55 g8 8 32 g.Sg o56 37 o.4o g44 9 g6 g32 b 43 18 g. 56 021 g.5gog4 3 o.4o 906 9 g6 g27 42 '9 9. 56 o53 32 g.Sg i 3i 37 o.4o 869 9 g6 g22 4i 20 g.56o85 32 g. 5g 168 37 o.4o 832 g.g6 gi7 S 40 21 9. 56 118 33 g . 5g 2o5 37 o.4o 7g5 9 g6 gi2 5 3 9 22 g. 56 i5o g. 5g 243 3 o.4o. 757 9 g6 go7 38 23 g.56 182 32 g.5g 280 37 o.4o 720 9 g6 go3 37 24 g. 56 215 33 g.5g 317 37 o.4o683 9 96 898 b 5 36 25 g.56 247 32 g.Sg 354 37 o.4o 646 9 96 8g3 35 26 g.56 27g 32 g . 5g 3gi 37 o.4o 609 9 96888 34 2 7 g.56 3i t 32 g.Sg 42g 3 8 o.4o 571 9 g6883 b s 33 28 g.56 343 g.Sg 466 37 o.4o 534 9 96 878 32 2 9 g. 56 375 32 g . 5g 5o3 37 o.4o 4g? 9 96873 3i 30 9. 56 4o8 33 g. 5g 54o 37 o.4o 46o 9 96 868 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 68 3O . PP 38 37 33 33 5 4 .1 3 .8 3-7 ., 3-3 3-2 .1 0.5 0.4 .2 7 .6 7-4 .2 6.6 6.4 .2 I.O 0.8 3 11.4 n. i 3 9-9 9.6 3 '5 1.2 4 IS .3 14.8 4 '3-2 12.8 4 3.0 1.6 5 19 o 18.5 5 16.5 16.0 S 2-5 2.O .6 22.8 22.2 .6 19.8 19.2 .6 3- 2-4 7 26.6 25-9 7 23-' 22.4 7 3-5 2-8 .8 30.4 29.6 .8 26.4 25.6 .8 4.0 32 9 34- a 9 4-? 3-6 72 21 3O L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9-56 4oS 32 32 32 32 32 3" 32 S 2 32 9. 5g 54o 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 o.4o 460 9 .96 S68 5 5 5 5 5 S 5 S 5 5 5 5 5 5 5 5 5 5 6 5 5 S S 5 5 5 5 5 5 S 30 3i 32 33 34 35 36 3? 38 3 9 9.56 44o 9-56 472 9-56 5o4 9.56536 g.56 568 9-56 599 g.56 63i 9.56663 9-56 6g5 9 .5 9 5 77 9.59 6i4 9.59 65i 9.59688 9 .5 9 725 9.59 762 9.59799 9.59 835 9.59 872 o.4o 423 o.4o 386 o.4o 349 o.4o 3i2 o.4o 275 o.4o 238 O.4O 2OI o.4o 165 o.4o 128 9 9 9 9 9 9 9 9 9 . 9 6863 96858 .96853 .96848 .96843 96838 96833 96 828 96823 29 28 27 26 25 24 23 22 2 I 40 9-56 727 9- 5 9 99 o.4o 091 9 96 818 20 4i 42 43 44 45 46 4? 48 4 9 g.56 759 9. 56 790 9. 56 822 g.56854 9.56886 9-56 917 9-56 949 g.56 980 9.57 012 31 32 32 32 3i 32 31 3 2 9.59 946 9. 5g 983 9.60 019 9.60 o56 9.60 og3 9.60 i3o 9.60 166 9.60 2o3 9 . 60 24o o.4o o54 o.4o 017 0.39 981 0.39 944 0.39 907 o. 3g 870 0.39 834 0.39 797 0.39 760 9 9 9 9 9 9 9 9 9 96 8i3 96 808 96 8o3 96 798 96 793 96 788 96 783 96 778 96 772 '9 18 '7 16 i5 i4 i3 12 I I 50 9.57 044 9 .60 276 0.39 724 9 96 767 10 5i 52 53 54 55 56 5? 58 5 9 9.57 075 9.57 107 9.57 1 38 9.57 169 9.57 2OI 9.57 232 9,57 264 9.57295 9.57 826 32 31 3' 32 3' 32 3i 3 1 9.60 3i3 9.60 34g 9.60 386 9.60 422 9.60 459 9.60 495 9.60 532 9.60 568 9. 60 605 0.39 687 0.39651 0.39 6i4 o.3g 578 o.3g 54i o.3g 505 0.39468 o. 3g 432 o. 3g 3g5 9 9- 9- 9 9 9 9 9 9 96 762 96757 96 752 96 747 96 742 96 737 96 732 96 727 96 722 9 8 7 6 5 4 3 2 I 60 9.5 7 358 3 2 9.60 64 1 o.3g 35g 9 96 717 L. Cos. d. L. Cotg. d. L, Tang. L. Sin. d. ' 08. PP .1 .2 3 4 '.6 '.8 '? 37 36 .1 .2 3 4 '.6 .8 9 3* 31 .1 .2 3 4 '.6 9 6 5 3-7 7-4 n. I .4.8 18.5 22.2 25.9 29.6 33-3 3.6 7-2 10.8 14.4 18.0 21.6 25.2 28.8 3 2 -4 3-2 6-4 9.6 12.8 16.0 19.2 22.4 25.6 11 9-3 12 .4 '5-5 18.6 21.7 24.8 27.9 0.6 1.2 1.8 2.4 3- 3-6 4-2 4.8 5-4 o-5 I.O 1.5 2.0 2-5 3.0 3-5 4.0 4-5 73 22. L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9- 5 7 358 9- 60 64 1 ,6 o. 39 359 9.96 717 60 I 9.57 389 31 9- 60 677 37 0.39 323 9.96 711 5 5 9 2 9.5 7 420 9- 60 714 16 o. 39 286 9.96 706 58 3 9.57 45 1 9- 60 750 0.39 25o 9.96 701 57 31 3 S 4 9.57 482 32 9- 60 786 37 0.39 214 9.96 696 56 5 9.57 5i4 9- 60823 16 O.39 177 9.96 691 55 b 9< 5 7 545 9- 60859 0.39 i4i 9.96686 54 3' 30 5 7 9- 5 7 576 31 9- 60 895 36 0.39 105 9.96 681 5 53 8 9.57 607 9- 60 g3i 16 0.39 069 9.96 676 52 9 9.07 638 9- 60 967 o.3g o33 9.96 670 Dl 10 9.57 669 9- 6 1 oo4 36 o.38 996 9.96 665 50 1 1 9.57 700 31 9- 6 1 o4o o.38 960 9.96 660 49 12 9.57 ?3i 9- 61 076 36 0.38 924 9.96 655 48 i3 9.57 762 3' 9- 6l 1 12 36 0.38888 9.96 650 5 5 4 7 i4 9.57 79 3 31 9- 61 1 48 36 0.38852 9.96 645 46 t5 9' 5 7 824 9- 61 1 84 16 o.38 816 9.96 64o 45 16 9.57 855 3 9- 6l 220 36 o.38 780 9.96 634 5 44 17 9.57 885 31 9- 61 256 36 o.38 744 9.96 629 43 18 9.57 916 9- 61 292 76 o.38 708 9.96 624 42 ! 9 9.57 947 9- 61 328 ,6 o.38 672 9.96 619 4i 20 9.57 978 9- 61 364 76 0.38636 9.96 6i4 5 40 21 9.58 008 31 9- 6 1 4oo 36 o.38 600 9.96 608 3 9 22 9. 58 o3g 9- 61 436 o.38 564 9.96 6o3 38 23 9-58 070 3 1 9- 61 472 3 o.38 528 9.96 5 9 8 5 37 31 36 5 24 9.58 101 9- 6 1 5o8 36 0.38492 9.96 5g3 36 2!) 9-58 i3i 9.61 544 0.38456 9.96 588 35 26 9.58 162 3' 9.61 579 35 0.38421 9.96 582 34 3 36 5 27 9. 58 192 9.61 6i5 o.38 385 9.96 5 77 33 28 9. 58 223 9.61 65i o.38 349 9.96 572 32 29 9 .58 253 3 9.61 687 3" o.38 3i3 9.96 56 7 * 3i 30 g.58 284 3 1 9.61 722 o.38 278 9.96 562 5 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 67 3O . PP 37 36 35 39 31 30 6 5 i 3-7 3-6 35 I 3-2 3-' 3 ,1 0.6 0.5 .2 74 7.2 7.0 .2 64 6.3 6.0 a I 2 I.O 3 n. i 10 8 10.5 3 9.6 9-3 9.0 3 1.8 1.5 4 14.8 14.4 14 4 12.8 12.4 12.O 4 24 20 5 18.5 18.0 17.5 5 16.0 '5-5 15-0 5 3.0 2.5 .6 22.2 21.6 31. 6 19.2 18.6 18.0 .6 3.6 3.0 7 25.9 25-2 4 5 7 22 4 21 7 21. . 7 42 3-5 .8 29.6 28. 8 28 o .8 25.6 248 24.0 .8 48 4 9 31 3 3 2 4 3'5 9 27.0 9 54 4-5 22 3O . , L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 .58 284 3 3 3 3' 30 30 3 30 3 30 3 3 30 3 3 3 3 3 30 3 3 3" 29 30 3 3 9.61 722 3 6 36 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35 35 36 35 35 36 35 35 35 36 35 35 35 35 o.38 278 9.96 562 6 S. s 5 6 5 5 5 6 5 5 6 5 5 5 6 S 5 6 5 5 6 '5 5 6 's 5 6 5 5 30 3i 32 33 34 35 36 37 38 39 9 9 9 9 9 9 9 9 9 .58 3i4 .58 345 .58 3 7 5 .584o6 .58 436 .5846? .58497 .58 52 7 .58 55 7 9.61 758 9.61 794 9.61 83o g.6r 865 g ,61 go i g .61 g36 g ,61 972 9.62 008 g .62 o43 o.38 242 o.38 206 o.38 170 o.38 i35 o. 38 ogg o.38 o64 0.38 028 0.37 gg2 0.37 967 g.g6 556 g.g6 55i g.g6 546 g .g6 54i 9.96 535 g.g6 53o g.g6 525 g.g6 620 g.g6 5i4 29 28 27 26 25 24 23 22 2 I 40 9 .58 588 g.62 079 0.37 921 9-96 5og 20 4i 42 43 44 45 46 4 7 48 49 9 9 9 9 9 9 9 9 9 .586i8 .58 648 .586 7 8 .58 7og .58 739 .58 769 .58 799 .58 829 .58 85g 9.62 u 4 g.62 150 g.62 i85 g.62 221 g.62 256 9.62 292 9.62 327 g.62 362 g.62 3g8 0.37 886 0.37 85o 0.37 815 0.37 779 0.37 744 0.37 708 0.37673 0.37 638 0.37 602 9.96 5o4 g.g6 4g8 g.g6 4g3 9 .g6488 g. 96 483 9.96477 g.g6 4?2 g.g6 467 9.96 46i '9 18 16 i5 i4 i3 12 1 1 50 9 .58 88c ) g.62 433 0.37 56 7 g.g6 456 10 5i 52 53 54 55 56 57 58 5g 9 9 9 9 9 9 9 9 .58 gig .58 g4g .58 979 .5g oog .5g o6g -5g og8 .5g 128 .5 9 i58 g.62 468 g.62 5o4 g.62 53g 9.62 674 9.62 6og g.62 645 9.62 680 g.62 715 g.62 760 0.37 532 o.3 7 4g6 0.37 46i 0.37 426 0.37 3gi 0.37 355 0.37 32O 0.37 285 0.37 250 g.g6 45i g.g6 445 9.96440 9.96 435 9.96 429 9.96 424 g.g6 4ig 9-96 4i3 g.g6 4o8 9 8 7 6 5 4 3 2 1 60 9 .5g 1 88 9.62 785 o. 37 215 g.g6 4o3 L. Cos. d. L. Cotg-. d. L. Tang. L. Sin. d. ' 67. PP 36 35 31 30 .1 3.0 2 D.O 3 9- .4 12. 5 '5-0 6 18.0 .7 21.0 .8 24.0 29 .1 .2 3 4 '.6 .8 6 5 0.5 I.O 2.0 25 3-5 4 3 .6 3-5 2.9 5.8 8.7 ii 6 14.5 '7-4 20.3 23.2 26. i 0.6 1.2 1.8 2.4 3- 3-6 4.2 4.8 3 4 6 .8 10.8 '44 18.0 21.6 28.8 10.5 14.0 '7-5 21 O 24.5 28.0 Sj-3 9-3 12 .4 15-5 18.6 21.7 24.8 27.9 23. L. Sin. d. L. Tang 1 , d. L . Cotg. L. Cos. d. 9 .5 9 188 9.62 75 37215 9.90 4o3 60 I 9.59 218 9.62 820 35 o 37 180 9.96 397 5 9 2 9.59 247 9.62 855 o 37 145 9.96 392 58 3 9.59277 9.62 890 37 1 10 9.96 387 D 7 3 36 4 9.59 307 9.62 926 35 O. 3 7 o 7 4 9.96 38i 56 b 9.59 336 9.62 961 O. 3 7 o3g 9.96 376 bb 6 9-5 9 366 3 9.62 996 o. 37 oo4 9.96 370 54 3 35 s 7 9. 5g 3g6 9-63 o3i 35 o. 36 969 9.96365 53 8 g.Sg 425 9. 63 066 o. 36 934 9.96 36o 52 9 9. 5g 455 3 9-63 101 o 36 899 9.96 354 5i 10 9.5 9 484 29 9.63 i35 o. 36 865 9.96 349 50 1 1 9. 5g 5i4 3 9-63 170 o. 36 83o 9.96 343 49 12 9. 5g 543 9-63 2o5 Oi 36 795 9.96 338 48 i3 g.Sg 573 3 g.63 240 35 o. 36 760 9.96 333 47 29 35 6 i4 9. 5g 602 9.63 2 7 5 o. 36725 9.96 327 46 i5 9. 5g 632 9-63 3io .0. 36 690 9.96 322 45 16 9. 59 661 29 9.63 345 35 o. 36655 9.96 3i6 44 i7 9.59 690 29 9.63 379 34 o. 3662! 9.96 3i i 5 6 43 18 9. 59 720 9-634i4 o. 36586 9.96 3o5 42 '9 9.59749 29 9-63 44g 35 o. 36 55i 9.96 3oo 5 4i 20 9.59778 29 9.63484 35 o. 36 5i6 9.96 294 40 21 9.59 808 9.63 5. 9 34 o. 3648i 9.96 289 5 39 22 9. 59 837 9-63 553 o. 36447 9.96 284 38 23 9.59866 29 9. 63 588 o. 364i2 9.96 278 37 29 35 5 24 9.59 895 g.63 623 34 o. 363 77 9.96 2 7 3 6 36 25 9.59924 9-63 65 7 0. 36343 9.96 267 35 26 9.59954 3 9-63 692 o. 363o8 9.96 262 5 34 29 34 6 27 g.Sg 983 9 .63 726 35 X). 36 274 9.96 256 33 28 9.60 OI2 9.63 761 o. 36 23g 9.96 25l 32 29 9.60 o4i 29 9 .63 79 6 35 o. 36 204 9.96 245 3i 30 9 .60 070 29 9. 63 83o o. 36 170 9.96240 s 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 66 3O . PP 36 35 34 30 29 6 5 3-6 3-5 3-4 .1 3.0 2-9 .1 0.6 0.5 .2 7.2 7.0 6.8 .2 6.0 5.8 .2 1.2 I.O .3 10.8 10.5 10.2 3 9- 8-7 3 1.8 1.5 4 '4-4 14.0 ,3.6 .4 12. ii. 6 4 2.4 2.0 .5 18.0 J7-5 17.0 5 '5-o 14-5 5 3 2.5 .6 21.6 21. O 20.4 .6 18.0 6 3-6 3.0 7 2 5- 2 24-5 23.8 .7 21. 20.3 7 4-2 35 .8 28.8 28.0 27.2 ,8 24.0 23.2 .8 4.8 4.0 9 324 31.5 30.6 .9 27.0 26 i 9 5-4 4-5 76 23 3O ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 .60 070 29 29 29 29 29 29 29 29 29 9 .63 83o 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 "dT o.36 1 70 9.96 240 6 5 6 5 6 5 6 5 6 5 6 5 6 6 5 6 5 6 5 6 6 5 6 5 6 6 5 6 5 6 30 3i 32 33 34 35 36 3? 38 3 9 9 9 9 9 9 9 9 9 9 . 60 099 .60 128 .60 i5y .60 186 .60 215 .60 244 .60 273 .60 3o2 .60 33i 9 . 63 863 9-63 899 9-63 934 9-63 968 9.64 oo3 9 . 64 037 9.64 072 9.64 1 06 9 .64 i4o o.36 i35 o.36 101 o.36 066 o. 36 o32 o. 35 997 o.35 963 o.35 928 o.35 894 0.35 860 9 .96 234 9.96 229 9.96 223 9.96 218 9 .96 212 9.96 207 9.96 201 9.96 196 9.96 190 29 28 27 26 25 24 23 22 21 40 9 .60 35g 9 .64 175 0.35 825 9.96 185 20 4i 42 43 44 45 46 47 48 49 9 9 9 9 9 9 9 9 9 .60 388 .60 4i? .60 446 .60 4?4 .6o5o3 .60 532 .60 56i .6o58 9 .60618 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 9.64 209 9.64 243 9.64 278 9 .64 3i2 9-64346 9-6438i 9.64 415 9.64 449 9-64483 o.35 791 o.35 757 o. 35 722 0.35688 0.35654 o.35 619 0.35585 o.3555i o.35 5i 7 9.96 I 79 9.96 I74 9.96 l68 9.96 162 9.96 157 9 .96 i5i 9.96 i46 9 .96 i4o 9.96 185 '9 18 7 16 i5 i4 i3 12 I I 50 9 .60 646 9.64 5 1 7 0.35483 9.96 129 10 5i 52 53 54 55 56 5 7 58 5 9 9 9 9 9 9 9 9 9 9 .60 675 .60 704 .60 732 .60 761 .60 789 .60818 .60846 .60875 .60 903 9.64552 9-64586 9.64 620 9.64654 9.64688 9 .64 722 9-64 ?56 9-64 790 9.64824 0.35448 o.35 4i4 o.35 38o o.35 346 o.35 3i2 o.35 278 o.35 244 o.35 210 o.35 176 9.96 123 9.96 1 18 9.96 112 9.96 1O7 9.96 ioi 9 .96 095 9.96 090 9.96 084 9.96079 9 8 7 6 5 4 3 2 I 60 .60 9 3r 9.64858 o.35 142 9.96 073 L. Cos. d. L. Cotg. L. Tang. L. Sin. d. ' o. PP .1 .2 3 4 !e '.8 9 35 34 29 28 .1 .2 3 4 ^6 7 .8 9 6 5 3-5 7.0 J 5 14.0 '75 21. 24-5 Z8.0 3J-S 3-4 6.8 IO.2 13-6 I 7 .0 20.4 2 3 .8 27.2 30.6 .1 .2 3 4 .6 !s 9 2.9 5-8 8.7 ii.6 14.5 17.4 20 3 23.2 26.1 2.8 5.6 8.4 II. 2 14.0 16.8 19.6 22.4 25.2 0.6 1.2 1.8 2.4 3- 3-6 4.2 4.8 5-4 5 l.O i-5 2.O 25 3- 3-5 4.0 4-5 77 24 C ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 . 60 93 1 9-64858 o. 35 142 9.96 073 60 I 9.60 960 28 9-64 892 34 o. 35 108 9.96 067 5 5 9 2 9.60 9 8 8 28 9.64 926 0. 35o 7 4 9.96 062 g 68 3 9.61 016 9.64 960 34 0. 35 o4o 9.96 o56 5 7 29 34 4 9.61 o4 5 28 9-64 994 o. 35 006 9.96 o5o 5 bb 5 9.61 073 28 9 .65 028 o. 34 972 9.96045 g bb 6 9.61 101 9. 65 062 34 0. 34938 9.96 o3 9 b4 28 34 S 7 9.61 129 9-65 096 0. 34 904 9.96 o34 6 53 8 9.61 i58 9. 65 i3o 0. 34870 9.96 028 b2 9 9.61 186 28 9.65 1 64 34 o. 34836 9.96 022 5i 10 9.61 21 4 g.65 197 33 o. 348o3 9.96 017 g 50 1 1 9.61 242 28 9-65 23l 34 o. 34 769 9.96 01 I ft 49 12 9.61 270 9-65 265 o. 34735 9.96 oo5 48 i3 9.61 298 9 .65 299 34 o. 34 701 9.96 ooo 47 i4 9.61 326 28 28 9 .65 333 34 o. 34667 9 . 9 5 994 6 46 i5 9.61 354 9. 65 366 o. 34634 9 . 9 5 988 45 16 9.61 382 g.65 4oo 34 o. 34 600 9-9 5 982 44 17 9.61 4i i 29 9". 6 5 434 34 o. 34566 9.95 977 t f, 43 18 9.61 438 9. 65 467 o. 34533 9.95 971 g 42 '9 9.61 466 9.65 5oi 34 o. 34 499 9.95 9 65 4i 20 9.61 4g4 9.65 535 34 0. 34465 9 . 9 5 960 g 40 21 9.61 522 28 9-65 568 33 o. 34432 9 . 9 5 9 54 6 39 22 9.61 55o 9. 65 602 34 o. 343 9 8 9 . 9 5 948 g 38 23 9.61 578 9 .65 636 34 o. 34364 9 . 9 5 942 37 24 9.61 606 28 28 9 .65 669 33 o. 3433i 9 . 9 5 9 3 7 S 6 36 25 9.61 634 9-65 7o3 34 o. 34 297 9.95 g3i g 35 26 9.61 662 28 9. 65 736 33 o. 34264 9 . 9 5 925 34 27 9.61 68 9 27 28 9 .65 77 34 o. 34 23o 9 . 9 5 920 b 6 33 28 9.61 717 9. 65 8o3 33 o. 34 197 9 . 9 5 914 g 32 29 9.61 745 28 9 .6583 7 34 o. 34 1 63 9 . 9 5 908 g Si 30 9.61 773 9.65 870 33 o. 34 i3o 9 -9 s 902 30 L. Cos. i d. L. Cotg. d. L. Tancr. L. Sin. I d. ' 65 3O'. PP 34 33 29 38 97 6 5 i 3-4 3-3 2 9 .1 2.8 2.7 .1 0.6 0.5 .2 6.8 66 5.8 2 5-6 54 .2 1.2 I.O 3 10.2 9-9 8.7 3 8.4 8.1 3 1.8 1.5 4 '3-6 13.2 n.6 4 II. 2 10.8 4 2.4 2.0 5 17.0 16.5 M 5 5 14.0 '3-5 5 3.0 2.5 .6 20.4 19.8 J7-4 6 16.8 16.2 .6 3.6 3.0 7 238 23 i 20.3 7 9-6 18.9 7 4-2 3-5 .8 27.2 26.4 23.2 8 22.4 21.6 .8 4.8 4.0 .0 70.6 24.7 5-4 4-5 78 24 3O / L. Sin. d. L. Tang. d. L. Cotg. | L. Cos. d. 30 9 bi 773 27 28 28 27 28 28 27 28 27 28 9 .b5 870 34 33 34 33 34 33 33 34 33 33 34 33 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 o. 34 i 3o 9.96 902 5 6 6 6 6 5 6 6 6 6 5 6 6 6 6 S 6 6 6 6 6 5 6 6 6 6 6 6 6 5 30 3i 32 33 34 35 36 3? 38 39 9 9 9 9 9 9 9 9 9 bi 800 61 828 61 856 61 883 61911 61 g3g 61 966 61 994 62 02 1 9-65 904 9-65 937 9-65 971 9.66 oo4 9.66 o38 9.66 071 9.66 io4 9.66 i38 9.66 171 0.34 096 o.34o63 o.34 029 o.33 996 o.33 962 o.33 929 0.33 896 o.33 862 o.33 829 9.95897 9. gS 891 9 . 9 5 885 9.95 879' 9.95 873 9.95 868 9.96 862 9 <9 5 856 9.95 85o 29 28 27 26 25 24 23 22 2 I 40 9 62 o4g 27 28 27 28 27 28 27 27 28 9.66 204 o. 33 796 9. 9 5 844 20 4i 42 43 44 45 46 4? 48 49 9 9 9 9 9 9 9 9 9 62 076 62 io4 62 i3i 62 i5g 62 186 62 214 62 241 62 268 62 296 9.66 238 9.66 271 9.66 3o4 9.66 337 9.66 371 9.66 4o4 9 .66 437 9.66 4?o 9.66 5o3 o.33 762 0.33 729 o.33 696 0.33663 0.33 629 o.33 696 0.33563 o.33 53o o.33 497 9.95 83g 9.95 833 9.95 827 9.95 821 9.95 8i5 9.95 810 9 .95 8o4 9.95 798 9 . 9 5 792 ] 9 IS '7 16 i5 i4 i3 12 I I 50 9 62 323 9.66 537 0.33463 9.95 786 10 5i 52 53 54 55 56 5? 58 5 9 9 9 9 9 9 9 9 9 9 62 35o 62 877 62 405 62 432 62 45g 62486 62 5i3 62 54 1 62 568 27 28 27 27 27 27 28 27 27 9.66 570 9.66 6o3 9. 66 636 9.66 669 9.66 702 9.66 735 9.66 768 9.66 801 9.66834 o.3343o o.33 397 o.33 364 o.33 33i o.33 298 o.33 265 o.33 232 o.33 199 o.33 166 9.95 780 9-9 5 775 9 . 9 5 769 9.95 763 9 . 9 5 757 9.95 75i 9.96 745 9 >9 5 739 9.95 733 9 8 7 6 5 4 3 2 I 60 9 62 595 9.66867 o.33 i33 9.95 728 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. / 65. PP .1 .2 3 4 .'e ',8 9 34 33 .1 .2 3 4 .6 .8 9 28 37 .1 .2 3 4 6 6 5 3-4 6.8 10.2 13-6 17.0 20.4 23.8 27.2 30.6 6.6 9.9 13.2 16.5 19.8 23-1 26.4 29.7 2.8 5.6 8.4 11.2 14.0 1 6. 8 19.6 22.4 2.7 5-4 8.x 10.8 '3-5 1 6. 2 18.9 21.6 24-3 0.6 o. 5 1.2 I.O 1.8 1.5 2-4 2.0 3.0 2.5 3.6 3.0 4-2 3-5 4.8 4.0 5-4 4-5 79 25. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .62 595 27 27 27 27 27 27 2 7 27 27 9 .t6 867 33 33 33 33 33 33 33 33 32 33 33 33 33 32 33 33 33 32 33 33 32 33 33 32 33 32 33 33 32 33 o.33 i33 9.95 728 6 6 6 6 6 6 6 6 6 6 5 6 6 6 6 6 6 6 6 60 I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 .62 622 .62 64g .62 676 .62 703 .62 730 .62 757 .62 784 .62 811 .62838 9.66 900 9.66 gSS 9.66 966 9.66999 9 .67 o32 9.67065 9.67 098 9.67 i3i 9.67 i63 o.33 100 o.33 067 o.33 o34 o.33 ooi 0.32 968 0.32 935 o.32 902 O.32 869 o.32 837 9 .95 722 9.95 716 9.95 710 9.95 704 9.95 698 9 .gS 692 9.95 686 9.95 680 9.95 674 5 9 58 5? 56 55 54 53 52 5i 10 9 .62 865 9.67 196 o.32 8o4 9.95 668 50 1 1 I 2 [3 i4 i5 16 '7 18 '9 9 9 9 9 9 9 9 9 9 .62 892 .62 918 .62 945 .62 972 .62 999 .63026 ,63o52 .63 079 .63 1 06 26 27 27 27 27 26 27 27 2 7 26 27 27 26 7 26 27 26 27 9.67 229 9.67 262 9.67295 9.67 327 9.67 36o 9.67 393 9.67 426 9.67 458 9.67 491 o.32 771 o.32 738 o.32 705 o.32 673 O.32 640 o.32 607 0.32 5?4 0.32 542 o.32 5og 9.95 663 9.95 657 9.95 65i 9 . 9 5645 9.95 63g 9.95 633 9.95 627 9.95 621 9.95 615 49 48 47 46 45 44 43 42 4i 20 9 .63 i33 9.67 524 o.32 476 9.95 609 40 21 23 24 25 26 27 28 29 9 9 9 9 9 9 9 9 9 .63 i5 9 .63 186 .63213 .63239 .63266 .63 292 .63 3ig .63 345 .63 3 7 2 9.67 556 9.67 589 9.67 622 9.67 654 9.67 687 9.67 719 9.67 752 9.67 785 9.67 817 o.32 444 o.32 4i i o.32 378 o.32 346 o.32 3i3 O.32 28l 0.32 248 0.32 2l5 o.32 i83 9.95 6o3 9.95 597 9.95 5gi 9.95585 9 . 9 5 579 9 .g5 573 9.95 567 9.95 56i 9.95 555 6 6 6 6 6 6 6 6 6 3 9 38 3? 36 35 34 33 32 3i 30 9 .63 3 9 8 9.67 850 o. 32 i5o 9.95 54g 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 64 3O . PP .i .2 3 4 .6 !i 9 33 32 27 26 .1 .2 3 4 .6 .S 6 5 33 6.6 9.9 13.2 16.5 19.8 33.1 26.4 29.7 3-2 6-4 9.6 12.8 16.0 19.2 22. J 25.6 .1 .2 3 4 6 !s 2-7 5-4 8.1 10.8 13-5 16.2 18.9 21. D 24.3 2.6 52 7.8 10.4 13.0 15.6 18.2 20.8 2^-4 0.6 1.2 1.8 2-4 3-o 3-6 42 4.8 S I.O '5 2.O 2-5 3- 3-S 4.0 4' 5 80 25 3O . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 63 3 9 S 9.67 850 o.32 i5o 9.95 549 30 3i 9 63 425 27 26 9.67 882 32 O. 32 I l8 9.95 543 g 29 32 9 6345i 9.67 915 o.32 o85 9.95 537 28 33 9 63 4?8 27 9.67 947 32 o.32 o53 9.95 53i 27 26 33 6 34 9 63 5o4 9.67 980 O.32 O2O 9.95 525 26 35 9 63 53i 9.68 012 o.3i 98 8 9.95 5ig 25 36 9 63 55? 9.68044 32 o.3i 96 6 9.95 5i3 24 37 9 63 583 26 9.68 077 33 o.3i 923 9.95 507 b 23 38 9 63 610 9.68 109 o.3i 891 9.95 5oo 22 3 9 9 63 636 26 9.68 142 33 o.3i 858 9 . 95 494 21 40 9 63 662 9.68 174 o.3i 826 9.95 488 20 4i 9 63 689 26 9.68 206 33 o.3i 794 9.95 482 6 19 42 9 63 715 26 9.68 239 i o.3i 761 9.95 476 18 43 9 63 74 1 26 9.68 271 3 2 o.3i 729 9.95 470 6 i? 44 9 63 767 27 9.68 3o3 o.3i 697 9.95 464 6 16 45 9 63 794 26 9.68 336 o.3 1 664 9.95 458 i5 46 9 63 820 26 9.68 368 3 2 3 2 o.3i 632 9.95 452 6 i4 47 9 63846 26 9.68 4oo o.3i 600 9.95 446 fi i3 48 9 63872 9.68432 o.3i 568 9.95 44o 12 4 9 9 63 898 26 9.68 465 33 o.3i 535 9 . 9 5434 I t 50 9 63 924 26 9.68 497 S 2 o. 3i 5o3 9.95 427 10 5i 9 63 o5o 26 9.68 529 3 2 o.3i 471 9.95 421 6 9 52 9 .63 976 26 9.68 56i o.3i 439 9.95 4i5 8 53 9 .64 002 9.68 59; 1 32 o.3i 407 9.95 409 7 26 33 54 9 .64 028 26 9.68 626 o.3i 374 9.95 4o3 6 6 55 9 .64o54 9.68658 o.3i 342 9 . 9 5 397 5 56 9 .64 080 9.68 690 3 2 o.3i 3io 9.95 391 4 26 32 7 5? 9 .64 1 06 26 9.68 722 o.3i 278 9.95 384 6 3 58 9 .64 1 32 9.68 754 o.3 1 246 9-9 5 3 7 8 2 5 9 9 .64 1 58 9.68 786 32 o.3i 214 9.95 372 I 60 9 .64 1 84 9.68 818 32 o.3i 182 9.95 366 L. Cos. d. L. Cotg. d. L, Tang. L. Sin. d. ' 64. PP 33 32 27 26 7 6 . x 3-3 3-2 .1 2-7 2.6 ., 0.7 0.6 .2 6.6 6. 4 .2 5-4 5.2 .2 M 1.2 3 9-9 9 .6 3 8.1 7-8 3 2.1 1.8 4 13.2 12.8 4 10.8 104 4 2.8 2.4 5 16.5 16.0 5 13-5 13.0 5 3-5 3.0 .6 19.8 19.2 .6 16.2 15.6 .6 4-2 3-6 7 23- i 22.4 7 18.9 18.2 7 4-9 4-2 .8 26.4 25.6 .8 21.6 20.8 .8 5-6 4.8 9 24.3 23.4 ^3 5-4 81 26. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .64 184 26 9.68 818 o.3i 182 9.95 366 g CO I 9 .64 210 26 9.68 85o 32 o.3i 150 9.95 366 6 5 9 2 V .64236 26 9.68 882 o.3i 118 9 . 9 5 354 68 3 9 .64 262 26 9.68 914 32 32 o.3i 086 9.95 348 7 57 4 9 .64288 25 9.68 946 o.3i o54 9.95 34i 6 56 5 9 .643i3 26 9.68978 o.3i 022 9.95 335 55 6 9 .64339 26 9.69 oio 3 3 32 o. 3o 990 9.95 329 6 54 7 9 .64365 26 9.69 O42 o.3o 9 5S 9.95 323 6 53 8 9 .64 391 26 9.69 074 o.3o 926 9-9 5 3l 7 62 9 9 .64 4i7 9.69 I 06 3 2 o.3o 894 9.95 3io 5i 10 9 .64442 26 9.69 1 38 32 o.3o 862 9.95 3o4 g 50 ii 9 .64468 26 9.69 170 32 o.3o83o 9.95 298 fi 4 9 12 9 .64494 9.69 202 o.3o 798 9.95 292 48 i3 9 .64 5 1 9 25 9.69 234 32 o.3o 766 9.95 286 4? i4 9 .64 545 26 9.69 266 32 o.3o 734 9.95279 7 6 46 i5 9 .64571 9.69 29^ I | o.3o 702 9.95 273 45 16 9 .64596 25 26 9.69 329 3' o.3o 671 9.95 267 6 44 17 9 .64 622 9.69 36i o.3o 6 39 9.95 261 43 18 9 .64647 9.69 3g' J o.3o 607 9.95 254 42 '9 9 .64673 9.69425 32 o.3o 5 J^ 9.95 248 4i 20 9 .646 9 8 25 9.69 457 32 o.3o 543 . 9 . 9 5242> 40 21 9 .64 724 9.6948* J o.3o 5i2 9.95 236 3 9 22 9 .64 749 9.69 52( > o.3o48o 9.95 229 38 23 9 .64 775 9.69 552 32 o.3o448 9.95 223 37 25 32 h 24 9 .64 800 26 9.69 584 o.3o 4 iG 9.95 217 36 25 9 .64826 9.69 6i5 o.3o 385 9.95 211 35 26 9 .6485i 25 9.69 647 S 2 o.3o 353 9.95 204 7 34 26 3 2 h 27 9 .64877 9.69679 o.3o 32i g.gS 198 33 28 .64 902 9.69 710 o.3o 290 g.gS 192 32 2 9 9 .64 927 25 9.69 742 32 o.3o258 9.95 i85 7 3i 30 9 .64 g53 9.69 774 32 o.3o 226 9.95 179 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 63 3O . PP 38 31 26 9 7 6 .i 3.2 3-' ., 2.6 2-5 I 07 0.6 .2 6.4 6.2 2 5-2 5.0 2 1.2 3 9.6 9-3 3 7.8 7-5 3 2.1 1.8 4 12.8 12.4 4 10.4 IO.O 4 2.8 2-4 5 16.0 '5-5 5 13.0 12.5 5 3-5 .6 19.2 1 8.6 6 15.6 150 6 4-2 3-6 7 22.4 21.7 7 18.2 17-5 7 49 4-2 .8 25.6 24.8 8 20.8 20. o 8 5-6 4.8 9 21-4 9 82 26 30 . / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.64953 9.69 774 o. 3o 226 9. 9 5 179 6 30 3i 9.64 978 9.69 8o5 3 1 o. 3o 195 9- 9 5 I 7 3 6 29 32 9-65 oo3 9.69 837 o. 3o i63 9. 9 5 167 28 33 9.65 029 9.69 868 3i o. 3o i3z 9 >9 5 1 60 27 34 9-65 o54 25 9.69 900 32 o. 3o 100 9 . 9 5 1 54 fi 26 35 9-65 079 9.69 932 o. 3oo68 9 . 9 5 i48 25 36 9-65 io4 25 26 9.69 963 3 1 3 2 o. 3o 037 9 . 9 5 i4i 6 24 37 9 .65 i3o 9.69995 o. 3o oo5 9 . 9 5 i3g fi 23 38 g.65 i5g 9.70 026 o. 29974 9 . 9 5 129 22 3 9 g.65 180 9.70 o58 3 2 o. 29 942 9 . 9 5 122 6 21 40 g.65 2o5 9.70 089 0. 29 91 1 9 . 9 5 116 6 20 4i 9-65 23o 9.70 121 o. 29879- 9 . 9 5 no 7 19 42 9-65 255 9.70 ID2 0. 29 848 9. 9 5 io3 18 43 g.65 281 9.70 1 84 32 o. 29 816 9 . 9 5 097 17 44 9.65 3o6 25 9.70 2i5 o. 29785 9 . 9 5 090 7 6 16 45 9.65 33i 9.70 247 o. 29 7 53 9 . 9 5 o84 i5 46 9.65 356 25 9.70 278 3 1 o. 29 722 9 . 9 5 078 i4 25 31 7 4 7 9-65 38i 9.70 309 o. 29 691 9-9 5 071 fi i3 48 9 .65 4o6 9.70 34i o. 29 65g 9 . 9 5 065 6 12 49 9 .6543i 25 9.70 3 7 2 3 1 o. 29 628 9-9 5 oSg I I 50 9 .65456 9.70 4o4 o. 29 5g6 9.95 o52 6 10 5i 9 .6548i 25 9.70435 3 1 0. 29 565 9.95 o46 7 9 52 9 .65 5o6 9. 70 466 o. 29 534 9 . 9 5 o3g g 8 53 9 .65 53i 25 9.70498 3 2 o. 29 5o2 9.95 o33 7 25 3 1 54 9 .65 556 9.70 529 o. 29 4?i 9 . 9 5 027 7 6 55 9 .65 58o 9.70 56o o. 29 44o 9 . 9 5 020 5 56 9 .656o5 25 9.70 592 32 o. 29 408 9 . 9 5 oi4 4 57 9 ,6563o 25 9. 70 623 o. 29377 g.gS 007 y fi 3 58 9 .65 655 9. 70 654 o. 29 346 9 . 9 5 OOI 6 2 5 9 9. 65 680 23 9.70 685 3 1 o. 29 315 9-94995 I 60 9. 65 705 2 s 9.70 717 o. 29 283 9.94988 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. r 63. PP 32 31 26 *5 34 7 6 .1 3-2 3-i 2.6 .1 2.5 2.4 .1 o. 7 o. 6 .2 6.4 6.2 5-2 2 5.0 4 .8 .2 1.4 1.2 3 9.6 9-3 7.8 3 7-5 7-2 3 2.1 1.8 4 12.8 12.4 10.4 .4 10.0 9.6 4 2.8 2.4 S 16.0 13.0 5 12.5 I2.O 5 3-5 3- .6 19.2 18.6 15.6 .6 15.0 14.4 .6 4.2 3.6 7 22.4 21.7 18.2 7 17-5 16.8 7 49 4-2 .8 25.6 24.8 20.8 8 20.0 19.2 .8 5-6 4-8 9 23-4 .9 22.5 21.6 9 6-3 5-4 83 27. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9-65 705 24 25 25 25 24 25 25 24 25 25 24 25 24 25 25 24 25 24 25 24 2 4 25 24 25 24 24 25 24 2 4 9.70717 31 31 31 3> 32 3' 3i 3' 3' 3i 3i 3' 3' 32 3' 31 3i 3' 3> 3' 3' 3' 3 3' 3' 3i 3i 3i 3' 3' o .29283 9.949*8 6 7 6 7 6 7 6 7 6 7 6 6 7 6 7 6 7 7 6 60 I 2 3 4 5 6 7 8 9 9. 65 729 9.65 7 54 9. 65 779 9-65 8o4 9-65 828 9-65853 9.63 878 9-65 902 9.65 927 9.70 748 9.70779 9.70 810 9.70 84i 9.70873 9.70904 9.70935 9.70 966 9.70997 o o o o o o o o . 29 25a .29 221 .29 190 .29 1 59 .29 127 .29 096 .29 o65 .29034 .29 oo3 9.94 982 9.94975 9.94969 9.94962 9.94 956 9.94949 9.94943 9.94936 9.94 93o 5 9 58 57 56 55 54 53 52 5i 10 9-65 962 9.71 028 o .28 972 9 . 9 4 923 50 i 1 12 i3 i4 i5 16 17 18 '9 9.65 976 9.66 ooi 9.66 O25 9.66 o5o 9.66075 9.66 099 9.66 124 9.66 i48 9.66 178 9.71 059 9.71 090 9.71 121 9.71 i53 9.71 184 9.71 215 9.71 246 9.71 277 9.71 3o8 o o o o o o o o .28941 ,28 910 28879 28847 28 816 28 785 28 754 28 723 28 692 9.94917 9.94911 9.94904 9.94898 9.94 891 9.94885 9.94878 9.94871 9.94865 4 9 48 47 46 45 44 43 42 4i 20 9.66 197 9.71 339 o. 28661 9-94858 1 40 21 22 23 24 25 26 27 28 29 9.66 221 9.66246 9-66 270 9-66 295 9.66 319 9.66 343 9-66368 9.66 392 9-664i6 9.71 9.71 9.71 9.71 9.71 9.71 9.71 9.71 9.71 370 4i 43 1 462 493 524 555 586 617 o. o. o. o. o. o. o. o. o. 28 63o 28 599 28569 28 538 28 507 28476 28445 28414 28383 9.94 852 9-94845 9.94839 9.94832 9.94826 9.94819 9-948i3 9.94 806 9.94799 7 6 7 6 7 6 7 7 6 39 38 3? 36 35 34 33 32 3i 30 9.66 44i 9.71 648 o. 28 352 9 . 9 4 793 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 62 3O . PP .1 .2 3 4 'e .'s 9 3 31 30 .1 .2 3 4 '.6 .8 5 4 .1 .2 3 4 .6 '.8 9 7 6 3-2 6.4 9.6 12.8 16.0 19.2 22.4 256 11 93 12 .4 '5-5 18.6 21.7 24.8 27.9 3- 6.0 9.0 12.0 IS.O 18.0 21. 24.0 27.0 2-5 5-o 7-5 IO.O 12.5 15.0 '7-5 20. o 2.4 4 .8 7-2 9.6 I2.O 14.4 168 19.2 21.6 0.7 '4 2.1 2.8 3-5 42 4-9 5-6 1.2 1.8 2.4 3- 3.6 42 4.8 ^4 84 27 3O . ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 ,6644i 9.71 648 0.28 352 9.9479 3 30 3i 9.66465 9.71 679 3 0.28 321 9.94 786 6 29 32 9 .66489 9.71 709 0.28 291 9.94 780 28 33 9 .665i3 9.71 ?4 31 0.28 260 9.94773 I 27 24 31 b 34 9 .6653 7 25 9.71 771 31 0.28 229 9.94767 7 26 35 9 .66 562 9.71 802 o. 28 198 9.94 760 25 36 9.66 586 9.71 833 31 0.28 167 9.94753 24 24 3 3? 9 .66 610 9.71 863 0.28 137 9.94747 23 38 9.66634 9.71 894 0.28 106 9.94 740 22 3 9 9.66658 24 9.71 9 2 5 3i 0.28 076 9 . 9 4 ?34 21 40 9 .66682 9.71 9 55 0.28 045 9.94727 20 4i 9. 66 706 24 9.71 986 31 0.28 oi4 9.94 720 fi i9 42 9.66 731 9.72 017 0.27983 9 . 94 7 1 4 18 43 9.66 755 24 9.72 o48 3i 0.27 952 9.94707 1 '7 24 3 7 44 9.66 779 9.72 078 0.27 922 9.94 700 6 16 45 9.66 8o3 9.72 109 0.27 891 9.94694 is 46 9.66 827 24 9.72 i4o 31 0.27 860 9.94 687 i4 24 3 7 47 9.6685i 9.72 170 0.27 83o 9.94 680 f, i3 48 9.66875 9.72 201 0.27 799 9.94 674 12 49 9.66 899 24 9.72 23l 3 o. 27 769 9-94 667 I I 50 9.66 922 23 9.72 262 3 1 o. 27 788 9.94 660 6 10 5.1 9.66 946 24 9.72 293 3 1 o. 27 707 9-94654 9 52 9.66 970 9.72 323 0. 27 677 9.94,647 8 53 9.66994 24 9.72 354 3 1 o. 27 646 9.94 64o 1 7 54 55 9.67 018 9.67 O42 24 24 9.72 9.72 384 4i5 3 3 1 o. 0. 27616 27 585 9. 9 4634 9. 94 627 7 6 5 56 9.67 066 24 9.72 445 3 o. 27 555 9.94 620 4 5 7 9.67 090 2 4 9.72 4 7 6 3i o. 27 524 9.94 6i4 3 58 9.67 1 13 9.72 5o6 o. 27 494 9.94607 2 59 9.67 187 24 9.72 53 7 3i o. 27 463 9.94 600 J I 60 9.67 161 24 9.72 56 7 3 0. 2 7 433 9.94 5g3 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 62. PP 31 30 25 24 *3 7 6 .1 3- 1 3-o 2-5 .1 2.4 2-3 .1 0.7 0.6 .2 6.2 6.0 .2 4 .8 4.6 .2 1.4 1.2 3 9-3 9.0 7-5 3 7-2 6.9 3 2.1 1.8 4 12.4 12.0 IO.O .4 9.6 9.2 4 2.8 2-4 5 iS-5 I 5 .0 12.5 5 12. o 11.5 5 3-5 3- .6 18.6 18.0 15.0 .6 14.4 13-8 .6 4.2 3.6 .7 21.7 21.0 J 7-5 .7 16.8 16.1 7 4.9 4-2 .8 24.8 24.0 20. o .8 19.2 18.4 .8 5-6 4.8 9 27.9 27.0 22.1; .9 2r.6 20.7 9 6-3 5-4 85 28. L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .67 161 9.72 567 31 3 3i 3 3 30 3 3i 3 3i 3 3 3' 30 30 3i 30 3 30 3i 30 3 30 3 3' 30 3 3 30 3 0.27 433 9.94 SgS 6 7 7 6 7 7 7 6 7 7 7 6 7 7 7 7 6 7 7 7 7 6 7 7 7 7 7 6 7 7 60 I 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 .67 185 .67 208 .67 232 .67 256 .67 280 .67 3o3 .6 7 3a 7 .67 35o 6 7 3 7 4 23 24 24 24 23 24 23 24 9.72 598 9.72 628 9.72 65g 9.72 689 9.72 720 9.72 750 9.72 780 9. 72 81 1 9.72 84i 0.27 402 0.27 372 0.27 34i 0.27 3i i 0.27 280 0.27 250 0.27 220 0.27 189 0.27 169 9.94587 9.94 58o 9.94 573 9.94 567 9.94 56o 9.94 553 9.94 546 9.94 54o 9.94 533 5 9 58 5? 56 55 54 53 52 Si 10 9 .67 3 9 8 9.72 872 0.27 128 9.94 526 50 1 1 12 i3 i4 i5 16 i? 18 '9 9 9 9 9 9 9 9 9 9 .67 421, .67445 .67468 .67 492 .67 5i5 .67 53g .6 7 562 .67 586 .67 609 24 23 24 23 24 23 24 23 9.72 902 9.72 932 9.72 963 9.72993 9.73 023 9.73 o54 9.73 084 9.73 n4 9.73 144 0.27 098 '0.27 068 0.27 037 0.27 007 0.26 977 0.26 946 0.26 916 0.26 886 0.26 856 9.94 Sig 9.94 5i3 9.94 5o6 9.94499 9.94 492 9 . 9 4485 9.94479 9.94 472 9-94465 49 48 47 46 45 44 43 42, 4i 20 9 .67 633 9 . 7 3 175 0.26 8a5 9. 9 4458 40 21 22 23 24 25 26 27 28 29 9 9 9 9 9 9 9 9 9 .67 656 .67 680 .67 703 .67 726 .67 750 .6 777 3 .67 796 .67 820 .67 843 24 23 23 24 23 23 24 23 9.73 205 9.73 235 9.73 266 9.73 295 9.73 326 9.73 356 9.73 386 9.73416 9.73 446 0.26 795 0.26 765 0.26 735 0.26 705 0.26 674 0.26 644 0.26 6i4 0.26 584 0.26 554 9. 94 45 1 9.94445 9-94438 9.94 43i 9.94 424 9.94417 9-94410 9.94 4o4 9.94 3 9 7 3 9 38 3? 36 35 34 33 32 3i 30 9 .67 866 9.73 476 0.26 524 9.94 390 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 61 3O. PP .1 .2 3 4 .6 !s 9 3 30 *4 23 .1 .2 3 4 .6 i 9 7 6 11 93 12.4 J5-5 18.6 21.7 24.8 27.9 3- 6.0 9.0 I2.O 15.0 18.0 21. 24.0 27.0 .1 .2 3 4 '.6 :l 2.4 4-8 7-2 9.6 I2.O 14.4 16.8 I 9 .2 2.3 4.6 6. 9 9.2 "5 .3-8 16.1 18.4 0.7 1.4 2.1 2.8 3-5 4-2 4.9 5-6 6.3 0.6 1.2 1.8 2-4 3 3-6 4.2 4.8 54 86 28 3O . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.67866 9.73 476 3i 3 3 3 3 3 3 3 30 3 3 30 3 3 3 3 3<> 3 3 3 3 3 29 30 30 3 3 30 29 3 0.26 624 9.94 3go 7 7 7 7 7 6 7 7 7 7 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 7 7 7 7 7 30 3i 32 33 34 35 36 3? 38 3 9 9.67 890 9.67 gi3 9.67 936 9.67959 9 .67 982 9.68 006 9 .68 029 9.68062 9.68 075 23 23 23 23 24 23 23 23 9. 73 507 9.73537 9.73 567 9.73597 9.73 627 9.73 657 9.78 687 9.73 717 9 . 7 3 7 4? 0.26 493 0.26463 0.26433 0.26 4o3 0.26 3?3 0.26343 0.26 3i3 0.26 283 0.26253 9-94383 9.94 376 9.94 369 9.94 362 9.94355 9.94349 9 . 94 342 9.94335 9.94 328 29 28 27 26 25 24 23 22 21 40 9.68 098 9 . 7 3 777 0.26 223 9.94 32i 20 4i 42 43 44 45 46 4? 48 49 9.68 121 9.68 i44 9.68 167 9.68 190 9.68 2i3 9.68 237 9 .68 260 9.68 283 9.68 3o5 23 23 23 23 23 24 23 23 22 23 23 23 23 23 23 23 23 23 22 2 3 9.73 807 9.73837 9.73 867 9.73897 9.73927 9.73957 9.78987 9.74017 9.74047 0.26 193 0.26 i63 0.26 i33 0.26 io3 0.26 073 0.26 o43 0.26 oi3 o.25 983 O.25 953 9.94 3i4 9.94 3o7 9.94 3oo 9.94 293 9.94 286 9.94279 9.94 273 9.94 266 9.94 25g '9 1 8 17 16 i5 i4 i3 12 II 50 9.68 328 9.74077 0.26 923 9.94 252 10 5i 52 53 54 55 56 5? 58 5 9 9.68 35i 9.68 374 9.68 397 9.68 420 9-68443 9.68466 9.68489 9.68 5i2 9.68 534 9.74 107 9-?4 i3? 9.74 166 9.74 196 9.74 226 9.74 256 9.74 286 9.74 3i6 9.74345 o.25 893 0.25863 0.25834 o.258o4 0.26 774 O.25 744 o.25 714 0.26 684 o.25 65 9.94245 9.94 238 9.94 23i 9.94224 9.94 217 9.94 2IO 9.94 2o3 9.94 196 9-94 189 9 8 7 6 5 4 3 2 I 60 9.68 557 9- ?43 7 5 0.25 625 9.94 l82 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 61. PP .1 .2 3 4 :i 7 .8 9 3 30 29 .1 .2 3 4 '.6 .8 *4 23 22 7 6 06 1.2 1.8 2.4 3-o 3-6 4.2 4.8 5-4 3-i 6.2 9-3 124 in 21.7 2 4 .8 27. 3-0 2.9 6.0 5.8 9.0 8.7 12.0 II. 6 15-0 14-5 18.0 17.4 21.0 20.3 24.0 23.2 27.0 26.1 2.4 4 .8 7.2 9.6 I2.O 14.4 16.8 19.2 21.6 2 '3 4.6 6.9 9.2 "5 13-8 16.1 18.4 2.2 4.4 6.6 8.8 II. 13.2 '5-4 17.6 19.8 i 2 3 4 6 8 0.7 >-4 2.1 2.8 3-5 4.2 4-9 5-6 6.3 87 2 c ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .68 557 9.74 375 3 3 3 29 3 3 29 3 3 30 29 3 3 29 3 3 29 3 29 3 29 3 3 29 3 29 3 3 29 o.25 625 9.94 182 7 7 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 60 I 2 3 4 5 6 7 8 9 y y y y y y y 9 y .6858o .686o3 .68625 .68648 .68 671 .68694 .68 716 .68 739 .68 762 23 22 23 23 23 22 23 23 9.74405 9.74435 9.74465 9.74494 9.74 524 9-74554 9.74 583 9.74 6i3 9.74 643 o.25 5g5 o.25 565 o.25 535 o.25 5o6 o.25 476 o.25 446 o.25 417 0.25 387 0.25 357 9 . 9 4 175 9.94 168 9.94 161 9.94 i54 9.94 i4? 9.94 i4o 9.94 1 33 9.94 126 9-94 119 5 9 58 56 55 54 53 52 Si 10 y .68 784 9.74 673 O.25 327 9.94 H2 50 1 1 12 i3 i4 i5 16 17 18 '9 y y 9 9 9 9 9 y y .68 807 .68 829 .68 852 .68875 .68 897 .68 920 .68 942 .68 965 .68 987 22 23 23 22 23 22 23 22 9.74 702 9.74 732 9.74 762 9-74 79 1 9. 74 821 9 . 7 485i 9.74 880 9.74 910 9.74 939 o.25 298 0.25 268 o.25 238 o. 25 209 o.25 179 o.25 149 O.25 120 o.25 090 o.25 061 9.94 105 9.94 098 9.94 090 9.94 o83 9.94 076 9.94 069 9.94 062 9.94 o55 9.94 o48 49 48 47 46 45 44 43 42 4i 20 9 .69 oio 9.74969 o.25 o3i 9.94 o4i 40 21 22 23 24 25 26 27 28 2 9 y 9 y y y y y y 9 .69 o32 .69 o55 .69077 .69 too .69 122 .69 144 .69 167 .69 189 .69 212 23 22 23 22 22 23 22 9.74998 9. 75 028 9.75 o58 9.75 087 9 . 7 5 117 9.75 i46 9.75 176 9.75 2o5 9 . 7 5 235 O.25 OO2 0.24 972 0.24 942 0.24 gi3 0.24883 0.24854 0.24 824 0.24 795 0.24 765 9.94 o34 9.94 027 9.94 020 9.94 OI2 9.94 oo5 9.93 998 9.93991 9.93 984 9.93 977 7 7 7 8 7 7 7 7 7 7 39 38 37 36 35 34 33 32 3i 30 9 . 69 234 9.75 264 0.24 736 9.93 970 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. - 6O 3O . PP .1 .2 3 4 .6 9 30 39 33 33 .1 .2 3 4 '.6 9 8 7 3-o 6.0 9.0 12.0 15.0 1 8.0 21.0 24.0 27.0 2-9 5-8 8.7 n.6 14.5 '7-4 20.3 23.2 .1 .2 3 4 '.6 2.3 46 6.9 9-2 ii-S 13-8 16.1 .8.4 20.7 2.3 4-4 6.6 8.8 II. 13.2 15.4 ,7.6 10.8 0.8 1.6 2-4 3-2 4.0 4.8 I' 6 6.4 0.7 M 2.1 2.8 3-5 42 5^ 6.3 29 3O . ' L. Sin. d. L. Tang . d. L. Cotg. L. Cos. d. 30 9.69 a34 22 2 3 22 22 22 23 22 22 22 9. 75 264 3 29 3<> 29 29 3 29 3 29 29 3 29 30 29 29 3 29 29 29 3 29 29 29 3 29 29 29 30 29 29 0.24 ?36 9.93 970 7 8 7 7 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 8 7 7 7 8 7 30 3i 32 33 34 35 36 3? 38 3 9 9 .69 256 9.69279 9.69 3oi 9. 69 323 9.69 345 9.69 368 9.69 Sgo 9.69 412 9.69434 9.75 294 9 . 75 323 9.75 353 9.75 382 9.75 4i i 9.75 44i 9.75 470 9 . 7 5 500 9.75 529 0.24 706 0.24 677 0.24 64? 0.24 618 0.24 58g 0.24 55 9 0.24 53o 0.24 5oo 0.24 471 9.93 963 9.93 955 9.93 948 9.93 941 9.93 934 9.93927 9.93 920 9.93 912 9.93 905 29 28 27 26 25 24 23 22 21 40 9.69 456 23 22 22 22 22 22 22 22 22 9 . 7 5 558 0.24 44a 9.93 898 20 4i 42 43 44 45 46 47 48 49 9.69479 9.69 5oi 9 . 69 523 9.69 545 9.69 56y 9.69 58g 9.69 61 1 9.69 633 9.69 655 9 . 7 5 588 9.75 617 9.75 647 9.75 676 9 . 75 705 9 . 7 5 735 9.75 764 9.75 7 9 3 9. 75 822 0.24 4i2 0.24383 0.24 353 O.24 324 0.24 295 0.24 265 0.24 236 0.24 207 0.24 178 9.93 891 9.93 884 9.93 876 9.93 869 9.93 862 9 . 9 3855 9-93847 9.93 84o 9 . 9 3 833 '9 18 7 16 i5 i4 i3 12 II 50 9.69 677 9.75 852 0.24 1 48 9 .93 826 10 5i 52 53 54 55 56 5? 58 5 9 9.69699 9.69 721 9.69 743 9.69 765 9.69 787 9.69 809 9.69831 9.69 853 9.69 875 22 22 22 22 22 22 22 22 9.75 881 9.75 910 9 . 7 5 939 9.75969 9.75998 9. 76 027 9. 76 o56 9. 76 086 9.76 115 0.24 119 0.24 090 0.24 061 0.24 o3i o .24 002 o.23 973 o.23 944 o.23 914 o.23 885 9.93 819 9.98 81 1 9.93 8o4 9.93 797 9 . 9 3 789 9.93 782 9-9 3 775 9.93 768 9.93 760 9 8 7 6 5 4 3 2 I 60 9.69 897 9.76 144 o.23 856 9.93 753 L. Cos. ! d. L. Cotg. d. L. Tang. L. Sin. d. ' 6O. PP 30 29 23 22 .1 .2 3 4 '.6 :l 9 8 7 .1 3.0 2 6.0 3 9- .4 12.0 5 '5-o .6 iS.o 7 21.0 .8 24.0 .9 27.0 2.g 5-8 8.7 ii. 6 M-5 '7-4 20.3 23.2 26.1 .1 .2 3 4 .6 '.8 2 -3 4-6 6-9 9.2 11. 5 "3-8 16.1 18.4 2.2 4-4 6.6 88 II. 13.2 I5 1 17.6 19.8 0.8 1.6 2-4 3- 2 4.0 4.8 5.6 6-4 7.2 0.7 1.4 2.1 2.8 3-5 4.2 4-f 5-6 6.3 30. ' L. Sin. d. L. Tang. d. L .Cotg. L. Cos. d. 9.69 897 22 22 22 21 22 22 22 22 21 22 22 22 21 22 22 21 22 21 22 22 21 22 21 22 21 22 21 22 21 22 9.76 i44 29 29 29 3 29 29 29 29 29 29 29 29 29 29 29 29 3 29 29 28 29 29 29 29 29 29 29 29 29 29 o 2 3 856 9 . 9 3 753 7 8 7 7 7 8 7 7 8 7 7 8 7 8 7 7 8 7 7 60 I 2 3 4 5 6 7 8 9 9.69919 9.69 94 1 9.69 963 9.69 984 9 .70 006 9.70 028 9.70 050 9.70 072 9.70 093 9.76 173 9.76 202 9.76 23i 9.76 261 9.76 290 9.76 319 9.76 348 9.76377 9 .76 4o6 o o. o. o. o. o. o. o. e. 23 827 23 798 23 769 23 739 23 710 2368i 23652 23623 23 5g4 9.93 746 9.93 7 38 9.93 7 3i 9.93 724 9 . 9 3 717 9.93709 9.93 702 9.93 695 9.93 687 5 9 58 5? 56 55 54 53 D2 5i 10 9. 70 1 1 5 9.76435 o. 2356s 9 . 9 3 680 50 1 1 12 i3 i4 i5 16 i? 18 '9 9.70 i3 7 9.70 169 9.70 180 9.70 202 9.70 224 9.70 245 9.70 267 9.70 288 9.70 3 10 9.76 464 9.76493 9.76 522 9.76 55i 9.76 58o 9.76 609 9.76 639 9.76 668 9.76697 o. o. o. 'o, o. o. o. o. 23 536 23 507 23478 23 449 23 42O 23 391 23 36 1 23 332 23 3o3 9.93 673 9.93 665 9-93658 9.93 65o 9.93 643 9-93636 9.93 628 9.93 621 9.93 6i4 49 48 47 46 45 44 43 42 4i 20 9. 70 332 9.76 725 o. 23 275 9.93 606 40 21 22 23 24 25 26 27 28 29 9.70 353 9. 70 375 9. 70 396 9.70 4 1 8 9.70439 9.70 46i 9.70 482 9. 70 5o4 9.70 525 9.76 754 9.76 783 9.76 812 9v06_84i 9.76 870 9.76899 9.76 928 9.76957 9.76 986 o. o. o. o. o. o. o. o. o. 23246 23 217 23 188 23 i5g 23 i3o 23 IOI 23 072 23o43 23 Ol4 9.93599 9.93 591 9.93 584 9.935^7 9. 93 56g 9.93 562 9.93 554 9.93 547 9.93 53g 7 8 7 7 8 7 8 7 8 3 9 38 37 36 35 34 33 32 3i 30 9.70 547 9-77 015 o. 22 gSS 9 . 9 3 532 f 30 L. Cos. d. L. Cotg. d. ; L. Tang. L.Sin. d. 59 3O . PP 30 39 38 .1 .2 3 4 :! .s 9 33 31 i 2 3 4 6 i 9 8 7 .1 3.0 .2 6.O 3 9- 4 12.0 5 '5- .6 18.0 7 21.0 8 24.0 .9 27.0 2.0 K it.6 14.5 17.4 20.3 2J.2 2.8 5.6 8.4 II. 2 14.0 16.8 19.6 22.4 2.2 4-4 6.6 8.8 II. 13.2 '5-4 17.6 19.8 2.1 4-2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 o. 8 o. 7 1.6 1.4 2.4 2.1 3-2 2.8 4-o 35 4.8 4.2 S- 6 4-9 6.4 5.6 90 3O 30 . 1 L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9- 70 547 9-77 015 29 O.22 985 9.93 532 30 3i 9- 70 568 22 9.77 o44 29 O.22 gS 6 9.93 525 8 29 32 9- 70 5go 9.77073 28 O.22 927 9.93 517 28 33 9- 70 61 1 9.77 101 O.22 89 9 9.93 5io 27 29 34 9- 70 633 21 9. 77 i3o 29 0.22 870 9.93 5o2 26 35 9- 70 654 9-77 '$9 O.22 8^ i 9.93495 g 25 36 9- 70 675 9.77188 O.22 8l2 9.93487 24 29 7 37 9- 70 697 21 9.77217 29 0.22 783 9.93480 8 23 38 9- 70 718 9.77 246 O.22 754 9.93 472 22 3 9 9- 70 739 9. 77 27^ O.22 726 9.93 465 g 21 40 9 70 761 9.77 3o3 O.22 697 9.93 457 20 4i 9 70 782 21 9.77 332 O.22 668 9.93450 8 '9 42 9- 70 8o3 9.77 36i O.22 6^ 9 o . o3 442 18 43 9- 70 824 9.77 3 9 o 29 O.22 6lO 9.93435 i? 28 44 9- 70 846 21 9.77 4i8 O.22 582 9.93 427 7 16 45 9 70867 9.77447 0.22 553 9 .93 420 g i5 46 9 70888 9.77476 29 O.22 524 9.93412 i4 29 7 4? 9 70 909 22 9.77505 28 O.22 49 5 9.93 405 8 i3 48 9 70 931 9.77 533 O.22 467 9.93 397 12 49 9 70 g52 9.77 562 29 0.22438 9 .93 390 g I I 50 9 70973 9.77591 29 28 O.22 409 9.93 382 10 5i 9 70 994 9.77 619 29 0.22 38l 9.93 375 8 9 52 9 71 oi5 9-77 648 O.22 352 9.93 367 8 53 9 71 o36 9.77677 29 O.22 323 9.93 36o 7 22 29 54 9 71 o58 9.77706 28 O.22 294 9.93 352 8 6 55 9 71 079 9-77 7 3 4 O.22 266 9.93 344 5 56 9 71 100 9-77763 29 O.22 237 9.93 337 4 21 28 57 9 71 121 9.77791 O.22 209 9.93 329 7 3 58 9 71 142 9.77 820 O.22 l8o 9-9 3 322 g 2 5 9 9 71 i63 9-77 8 4g 29 O.22 l I 9.93 3i4 I 60 9 71 i84 9-77 8 77 O.22 123 9.93 307 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 59. PP 29 28 22 31 8 7 .1 2.9 2.8 I 2.2 2.1 ,i 0.8 0.7 .2 5-8 5.6 2 4.4 4.2 ,2 1.6 1.4 3 8.7 8-4 3 6.6 6-3 3 24 2.1 4 n.6 II. 2 4 8.8 8.4 4 3.2 2.8 5 14.5 14.0 5 II. c-5 5 4.0 3.5 .6 17.4 16.8 6 13.2 2.6 .6 4.8 4.2 7 20.3 19.6 7 iS-4 4-7 7 5.6 4.9 .8 23.2 22.4 8 17.6 6.8 .8 6.4 5-6 26.1 25.2 9 19.8 8.9 9 7.2 6.3 9 1 31. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.71 1 84 9-7777 O.22 123 9.93 307 60 I 9.71 205 21 9.77906 29 0.22 094 9.93 299 8 5 9 2 9.71 226 9.7793 r ) 28 0.22 o65 9.93 291 58 3 9.71 247 9.77963 0.22 o3 7 9.93 284 ; 57 29 H 4 9.71 268 21 9.77992 28 O.22 OO8 9.93 276 56 5 9.71 289 9.78 020 O.2I 980 9.93 269 55 6 9.71 3io 9.78 049 O.2I 95 I 9.93 261 54 8 7 9.71 33i 21 9.78077 29 0.21 923 9.93 253 53 8 9.71 352 9.78 1 06 O.2I 894 9.93 246 52 9 9.71 3 7 3 9.78135 29 0.21 865 9.93 238 5i 10 9.71 393 9.78 i63 0.21 837 9.93 23o 50 1 1 9.71 4i4 21 9.78 192 28 0.21 808 9.93 223 7 8 49 12 9.71 435 9. 78 220 O.2I 780 9 . 93 2 1 5 48 i3 9.71 456 21 9.78 24 9 28 O.2I 761 9.93 207 47 i4 9.71 4?7 21 9.78277 29 0.21 723 9.93 2OO 46 i5 9.71 498 9.78 3o6 28 O.2I 694 9.93 192 45 16 9.71 519 9.78 334 0.21 666 9.93 1 84 44 29 17 9.71 539 21 9-78363 28 0.21 637 9.93 177 g 43 18 9.71 56o 9.78 391 28 0.21 609 9.93 169 42 9 9 .71 5Si 9.78 419 0.21 58i 9.93 161 4i 20 9 .71 602 9.78448 28 0.21 552 9.93 i54 7 40 21 9 .71 622 21 9.78476 29 0.21 524 9.93 1 46 8 3 9 22 9 .71 643 9.78 505 28 O.2I 495 9.93 i38 38 23 9 . 71 664 9.78 533 O.2I 467 9.93 i3i 7 3? 21 29 8 24 9 .71685 9.78 562 28 0.21 438 9.93 123 8 36 25 9 .71 705 9.78 590 28 0.21 4 [O 9.93 1 1 5 35 26 9 .71 726 9.78618 0.21 382 9.93 108 1 34 21 29 8 27 9 7 1 747 9.78647 28 0.21 353 9.93 too g 33 28 9 .71 767 9. 7 86 7 5 0.21 325 9.93 092 32 29 9 .71 788 9.78 704 28 O.2I 296 9.93084 3i 30 9 .71 809 9.78 732 0.21 268 9.93 077 7 30 L. Cos. d. L. Cotg . d. L. Tang. L. Sin. d. ' 58 3O . PP 29 28 21 20 8 7 .i 2.9 2.8 .1 2.1 2.0 .1 0.8 0.7 .2 5-8 S.6 .2 4.2 4.0 .2 1.6 1.4 3 8.7 84 3 6.3 6.0 3 2.4 2.1 4 ii. 6 II. 2 4 8.4 8.0 4 3-2 2.8 5 14-5 14.0 5 10.5 IO.O 5 4.0 3-5 .6 17.4 16.8 .6 12.6 12.0 .6 4.8 42 7 20.3 19.6 7 14.7 14.0 7 56 49 .8 23.2 22.4 .8 16.8 16.0 .8 6.4 5.5 9 26. i 18.9 18.0 9 6.3 92 31 3O . / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 .71 809 9 .78 732 0.21 268 9.93 077 g 30 3i 9 .71 829 9. 78 760 O.2I 240 9 .93 069 8 29 32 9 .71 850 9.78 789 0.21 211 9.93 061 g 28 33 9 . 71 870 9.78 817 0.21 183 9-93o53 27 21 28 7 34 9 .71 891 9.78 845 O.2I I 55 9.98 o46 8 26 35 9 .71 911 9-78874 O.2I 126 9.93 o38 g 25 36 9 .71 982 9. 78 902 O.2I O 98 9.93 o3o 24 2O 37 9 .71 g52 9.78 g3o O.2I 070 9.93 022 8 23 38 9 .71 973 9.78 9 5 9 0.21 o4i 9.93 oi4 22 3 9 9 .71 994 9.78987 28 0.21 Ol3 9.93 007 8 21 40 9 .72 oi4 9.79 oi5 O.2O 985 9.92 999 g 20 4i 9 .72 o34 9.79 o43 0.2O 957 9.92991 8 '9 42 9 .72 055 9.79072 O.2O 928 9.92 983 18 43 9 .72 075 21 9.79 100 28 O.2O 900 9.92976 8 i? 44 9 .72 096 9.79 128 28 O.2O 872 9.92 968 8 16 45 9 .72 1 16 9.79 i56 O.2O 844 9.92 960 8 i5 46 9 .72 i3 7 9.79 185 29 O.2O 8l 5 9.92 952 i4 20 28 4? 9 .72 167 9.79 2i3 28 O.2O 787 9.9 2 944 R i3 48 9 .72 177 9-79 241 O.2O 759 9.92 986 12 49 9 .72 198 9.79269 0.20 781 9.92 929 8 I I 50 9 .72 218 9-79 2 97 0.20 703 9.92 921 g 10 5i 9 .72 238 9.79 326 29 28 O.2O 674 9.92 918 8 9 52 9 .72 25g 9.79 354 0.20 646 9 .92 go5 8 8 53 9 .72 279 9.79 382 0.20 618 9.92897 7 2O 54 9 .72 299 9.79410 28 O.2O 5( ) 9.92 889 8 6 55 9 .72 32O 9-79438 O.2O 562 9.92 881 5 56 9 .72 34o 9.79 466 0.2O 534 9.92 874 4 2O 29 57 9 .72 36o 9.79495 28 0.20 5o5 9.92 866 8 3 58 9 .72 38i 9. 79 523 O.2O 4 r 11 9.92 858 g 2 5 9 9 .72401 9.79 55i 28 O.2O 449 9.92 850 8 I 60 9 .72 4ai 9-79 5 79 O.2O 421 9.92 842 L. Cos. d. L. Cotg . d. L. Tang. L. Sin. d. ' 58. PP 29 28 21 20 8 7 .! 2.9 2-8 .1 2.1 2.O .1 0.8 0.7 .2 5-8 S.6 .2 4.2 4.0 .2 1.6 1.4 3 8.7 8.4 3 6-3 6.0 3 2.4 2.1 4 ii. 6 II. 2 4 8.4 8.0 4 3-2 2.8 .5 14.5 I4.O .5 10.5 0.0 .5 4.0 3-5 .6 17.4 16.8 .6 12.6 2.O .6 4.8 4.2 7 20.3 19.6 7 14.7 4.0 . 7 5-6 4.9 .8 23.2 22.4 .8 16.8 6.0 .8 6.4 5-6 9 26.1 9 18.9 8.0 9 7-2 6-3 93 32 C ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9. 72 421 9-79 5 79 O.2O 4^1 9.92 842 i 60 I 9.72 44i 20 9.79607 28 O.2O 393 9.92 834 f 5 9 2 9.72 46 1 9- 79 635 O.2O 365 9.92 826 c 58 3 9.72 482 20 9- 79663 28 O.2O 337 9.92 818 *7 4 9.72 5O2 20 9.79691 28 0.20 Sog 9.92 810 7 56 5 9.72 522 9- 79 7'9 0.20 28l 9.92 8o3 i 55 6 9.72 542 9- 79747 0.20 253 9.92 795 54 29 7 9.72 562 20 9- 79 77 6 28 O.2O 224 9.92 787 f 53 8 9.72 582 9- 79 8o4 O.2O 196 9.92 779 t 52 9 9.72 602 9- 79 832 O.2O 168 9.92 771 i 5i 10 9.72 622 9- 79 860 O.2O l4o 9.92 7 63 I 50 1 1 9.72 643 2O 9- 7988 8 28 O.2O 112 9.92 755 f 4 9 12 9.72 663 9- 79916 0.20 084 9.92 747 48 i3 9.72 683 9- 79 944 0.20 o56 9.92 7 3 9 47 28 i4 9.72 7o3 9- 79972 28 O.2O O28 9.92 7 3i f 46 i5 9.72 723 9.80 ooo O.2O OOO 9.92 723 45 16 9.72 743 9.80 028 o. 19 972 9.92 7 .5 44 20 28 i 17 9.72 763 9.80 o56 28 0.19 944 9.92 707 f 43 18 9.72 783 9.80084 o. 19 916 9.92 609 42 '9 9.72 8o3 9- 80 I 12 28 0.19 888 9.92 691 4i 20 9.72 823 9- 80 i4 o o. 19 860 9.92 683 40 21 9.72 843 9.80 168 0.19 832 9.92 6 7 5 f 3 9 22 9.72 863 9.80 igS o. 19 805 9.92 667 38 23 9.72 883 9.80 223 o. 19 777 9.92 65 9 37 24 9.72 902 '9 9.80 25l 28 0.19 749 9.92 65i J 36 25 9.72 922 9.80 279 o. 19 721 9.92 643 35 26 9.72 942 9 . 80 3o,7 o. 19 6 9 3 9.92 635 34 20 27 9.72 962 9. 80 335 28 0.19 665 9.92 627 f 33 28 9.72 982 9- 80 363 0.19 637 9.92 619 32 29 9.73 002 9- 3o 3 9 i- 0.19 609 9.92 611 3i 30 9.73 022 9 .8o4i Q o. 19 58i 9.92 6o3 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 57 3O . PP 29 28 27 21 20 19 8 7 i 2.9 2.8 3.7 i 2.1 2.0 1.9 .1 08 0.7 2 5-8 5-6 5-4 a 4.2 4.0 3-8 .2 1.6 1.4 3 8.7 8.4 8.1 3 6-3 6.0 5-7 3 2.4 2.1 4 ti.6 1 1. 2 IO 8 4 8.4 8.0 7-6 4 3-2 2.8 5 '4-5 14.0 13.5 5 0.5 o.o 95 5 4.0 3-5 6 '7-4 16.8 1 6. 2 6 2.6 3.O 11.4 .6 4.8 4-2 7 20.3 19.6 18.9 7 4-7 4-0 13-3 7 5-6 4-9 8 23.2 22.4 21.6 8 6.8 6.0 15-2 .8 6.4 5-6 9 26.1 2S-2 24.3 8.9 17.1 6.3 32 3O L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9- 73 022 9.80 419 28 27 28 28 28 28 28 28 27 28 28 28 28 27 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 28 o. 19 58i 9.92 6o3 8 8 8 8 8 8 9 8 8 g 30 3i 32 33 34 35 36 3? 38 39 9- 9- 9- 9- 9- 9- 9- 9- 9- 73 061 73o8i 73 101 73 121 73 i4o 73 160 73 180 73 200 20 20 2O 2O '9 20 20 20 '9 20 20 19 2O 20 19 2O 2O 9 20 '9 20 '9 20 '9 2O 2O '9 20 9.80447 9.8o4?4 9.80 5o2 9.8o53o 9. 80 558 9. 80 586 9.80 6i4 9.80 642 9.80 669 0.19 553 0.19 526 0.19 498 o. 19 470 0.19 442 o. 19 4i4 0.19386 0.19358 0.19331 9.92 595 9.92 587 9.92579 9.92 571 9.92 563 9.92 555 9.92 546 9-92538 9.92 53o 29 28 27 26 25 23 22 21 40 9- 73 219 9.80 697 o.i 9 3o3 9.92 522 20 4i 42 43 44 45 46 47 48 4 9 9 9 9 9 9 9 9 9 9 "732 3o 7 3 2 5O 73 278 73 298 73 3i8 7 333 7 73 357 73 377 73 396 9.80 725 9.80 753 9.80781 9.80808 9. 80 836 9.80864 9.80 892 9.80 919 9.80947 o. 19 275 0.19 247 o. 19 219 0.19 192 0.19 i64 0.19 i36 0.19 108 o. 19 08 1 o. 19 o53 9.92 5i4 9.92 5o6 9.92498 9.92 490 9.92 482 9.92473 9.92 465 9.92457 9.92 449 8 8 8 8 9 8 8 8 8 8 8 9 8 8 8 8 8 9 8 '9 18 17 16 i5 i4 i3 12 II 50 9 . 7 34i6 9.80975 o. 19025 9.92 44i 10 5i 52 53 54 55 56 5 7 58 5 9 9 9 9 9 9 9 9 9 9 73435 73455 73494 735i3 73533 73552 73 572 73 591 9.81 oo3 9.81 o3o 9.81 o58 9.81 086 9.81 n3 9.81 i4i 9.81 169 9.81 i 96 9.81 224 0.18 997 0.18 970 0.18 942 0.18914 0.18887 0.18859 o.i883i o.i88o4 o. 18 776 9.92433 9.92425 9.92 4i6 9.92 4o8 9.92 4oo 9.92 392 9.92 384 9.92 376 9.92 367 9 8 7 6 5 4 3 2 I 60 9 73 61 1 9.81 252 0.18 748 9.92 359 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 57. PP .1 .2 3 4 .6 .3 9 38 27 20 19 .1 .2 3 4 7 9 9 8 2.8 5-6 8.4 II 2 14.0 16.8 ig.6 22.4 2.7 .1 5-4 -2 8.1 .3 10.8 .4 '3-5 -5 16.2 .6 18.9 .7 21.6 .8 24-3 -9 2.O 4-0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 1.9 3-8 5-7 7.6 9-5 11.4 '3-3 '5-2 17.1 0.9 0.8 1.8 1.6 2-7 2.4 3.6 3.2 4-5 4-o 5-4 4-8 6.3 5-6 7-2 6.4 8.1 7.2 33. ' L. Sin. d. L. Tang. d. L. Cotg. L Cos. d. 9.73 61 1 9.81 252 0.18 748 9.92 35g 60 I 9.73 63o 20 9.81 279 28 o. 1 8 721 9.92 35i 8 5 9 2 9.73 650 9.81 307 28 0.18 693 9.92 343 58 3 9.73 669 9.81 335 o.i8665 9.92335 57 27 9 4 9.73 689 9.81 362 28 0.18 638 9.92 326 8 56 5 9.73 708 9.81 3go 28 0.18 610 9.92 3i8 55 6 9.73 727 9.81 4i8 0.18 582 9.92 3io 54 20 27 8 7 9 . 7 3 747 9.81 445 28 0.18 555 9.92 3o2 53 8 9.73 766 9 81 473 o. 18 527 9.92 293 52 9 9 . 7 3 7 85 '9 9.81 5oo 28 o. 18 500 9.92 285 g 5i 10 9.73 805 9.81 528 28 0.18 472 9.92 277 50 1 1 9.73 824 19 9.81 556 27 0.18 444 9.92 269 49 12 9 . 7 3 843 9.81 583 28 0.18417 9.92 260 48 i3 9-73863 9.81 61 1 0.18 389 9.92 252 47 '9 2 7 i4 9.73 882 19 9.81 638 28 0.18 362 9.92 244 46 i5 9.73 901 9.81 666 0.18 334 9.92 235 45 16 9.73 921 9.81 693 0.18 307 9.92 227 44 '9 8 17 9.73 940 19 9.81 721 o. 18 279 9.92 219 fi 43 18 9 . 7 3 9 5 9 9.81 748 O.l8 252 9.92 211 42 '9 9.73978 9.81 776 o. 18 224 9.92 2O2 y 4i 20 9 .73 997 9.81 8o3 o. 18 197 9.92 194 g 40 21 9 .74017 9.81 83i 0.18 169 9.92 186 9 3 9 22 9 .74o36 9.81 858 o. 18 142 9.92 177 g 38 23 9 .74 o55 '9 9.81 886 0.18 1 14 9.92 169 37 19 27 24 9 .74074 9.81 gi3 28 0.18 087 9.92 161 9 36 25 9 .74 093 9.81 94 I 0.180 5 9 9.92 i52 35 26 9 . 7 4n3 9.81 968 27 o. 18 o32 9.92 1 44 34 28 27 9 74 i32 9.81 996 o. 1 8 oo4 9.92 i36 9 33 28 9 . 7 4 i5i 9.82 023 0.17977 9.92 127 g 32 2 9 9 ?4 170 9.82 o5i 0.17 949 9.92 119 g 3i 30 9 74 189 9.82 078 2 7 0.17 922 9.92 in 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 56 3O . PP 38 *7 ao 19 9 8 .1 2.8 2.7 i 2.O I.O i 0.9 0.8 .2 5.6 5-4 2 4.0 3.8 2 1.8 1.6 3 8-4 3 6.0 5-7 3 2.7 2-4 4 II. 2 10.8 4 8.0 7.6 4 3-6 3-2 5 I4.O 13-5 5 o.o 9-5 5 4-5 4.0 .6 1 6. 8 16.3 6 2.0 "4 6 5-4 4.8 7 19.6 18.9 7 4-0 13-3 7 6.3 5-6 .8 22.4 21.6 8 6.0 15.2 8 7.2 6.4 9 25.2 24.3 8.0 17.1 8.1 7.2 96 33 30 . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.74 189 9.82 078 28 0.17 922 9.92 i ii 30 3i 9.74 208 19 9.82 106 27 0.178 94 9.92 IO2 9 8 29 32 9.74 227 9.82 i33 28 o. 17 867 9.92094 28 33 9 .74 246 9.82 161 0.17 8 *9 9. 9 2 086 27 19 27 9 34 9 .74 265 19 9.82 188 27 o. 17 812 9.92077 g 26 35 9 .74284 9.82 2i5 28 0.17785 9.92 069 25 36 9 .74 3o3 9.82 243 o.i 77 5 7 9.92 060 9 24 '9 27 K 3? 9 .74 322 19 9.82 270 28 o. 17 730 9.92 o52 8 23 38 9 74341 9.82298 0.17 702 9.92 o44 22 3 9 9 .74 36o 9.82 325 0.17675 9.92 o35 9 21 40 9 .74379 9.82 352 28 0.17 648 9 . 9 2 027 20 4i 9 . 7 43 9 8 9.82 38o 0.17 620 9 . 9 2 018 9 8 i 9 42 9 .74417 9.82 407 0.17 5( ;3 9.92 oio 18 43 9 . 7 4436 '9 9.82435 0.17 565 9.92 002 17 19 27 9 44 9 .74455 9.82 462 0.17 538 9.91 99 3 8 16 45 9 .74 4?4 9.82489 0.17 5u 9.91 985 i5 46 9 .74493 19 9.82 617 0.17483 9.91 976 9 i4 19 27 K 47 9 .74 5 12 9.82 544 o. 17 456 9.91 968 i3 48 9 .74 53i 9.82 571 0.17 429 9.91 959 12 49 9 74 54g 9.82 599 o. 17 4oi 9 . 9 i 9 5i I I 50 9 74568 J 9 9.82 626 0.17 374 9.91 942 9 10 5i 9 7458 7 '9 9.82 653 27 28 0.17 347 9.91 934 9 52 9 .74 606 9.82681 0.17 3j 9 9.91 925 8 53 9 .74625 '9 9.82 708 27 0.17 292 9.91 917 7 54 9 74644 '9 9.82 735 27 0.17 265 9.91 908 9 8 6 55 9 .74 662 9.82 762 0.17238 9.91 900 5 56 9 . 7 468i '9 9.82 790 0.17 210 9.91 891 9 4 57 9 .74 700 '9 9.82817 27 0.17 i83 9.91 883 3 58 9 .74719 19 9.82 844 0.17 i56 9.91 874 2 5 9 9 18 9.82 871 27 0.17 129 9 . 9 i 866 I 60 9 .747^6 9.82 899 0.17 loi 9 . 9 i 857 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 56. PP 28 27 19 18 9 8 .1 2.8 2.7 .1 i. 9 1.8 , 0.9 0.8 .2 5-6 5-4 .2 3-8 3-6 2 1.8 1.6 3 8.4 8.1 3 5-7 5-4 3 2.7 2.4 4 II. 2 10.8 4 7-6 7-2 4 3-6 3-2 5 14.0 '3-5 5 9-5 9.0 5 45 4.0 .6 16.8 1 6. 2 .6 11.4 10.8 6 5-4 4.8 7 19.6 18.9 . 7 '3-3 12.6 7 6-3 5-6 .8 22.4 21.6 .8 14.4 8 7-2 6.4 9 25.2 24-3 17.1 16.2 9 S.j 7.2 97 34. . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 74 ?56 9.82 899 27 27 27 28 27 27 27 28 27 27 27 27 27 28 27 27 27 27 27 27 28 27 27 27 7 2 7 27 27 27 27 0.17 101 9.91 857 8 9 8 9 8 9 8 9 8 9 9 8 9 8 9 9 8 9 8 9 9 8 9 9 8 9 9 8 9 9 60 I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 74 775 ?4 794 .74812 . 7 483i .74850 .74868 .74887 .74 906 .74 924 '9 18 '9 '9 18 '9 '9 18 9.82 926 9.82 953 9.82 980 9-83 008 9-83 oSg 9. 83 062 9.83089 9-83 117 9 .83 1 44 o. 17 074 0.17 047 0.17 O2O o. 16 992 o. 16 965 o. 16 938 o. 16 911 0.16 883 0.16 856 9.91 84g 9.91 84o 9.91 832 9.91 823 9.91 815 9.91 806 9.91 798 9.91 789 9.91 781 5 9 58 57 56 55 54 53 52 Si 10 9 .74943 18 9-83 171 o. 16 829 9.91 772 50 1 1 12 i3 i4 i5 16 '7 18 '9 9 9 9 9 9 9 9 9 9 ?4 961 .74 980 74999 .75 017 .75 o36 .75 o54 .75 073 .75 091 .75 no 19 '9 18 19 18 '9 18 '9 18 9-83 198 9-83 225 9.83252 9.83 280 9. 83 307 9.83334 9-83 36i 9.83 388 9-834(5 o. 1 6 802 o. 16 775 0.16 748 o. 16 720 0.16 693 0.16 666 o. 16 63g o. 16 612 0.16 585 9.91 763 9.91 7 55 9.91 746 9.91 738 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 49 48 47 46 45 44 43 42 4i 20 9 .75 128 9.83442 0.16 558 9.91 686 40 21 22 23 24 25 26 27 28 2 9 9 9 9 9 9 9 9 9 9 .75 147 .75 i65 . 7 5 1 84 .75 202 .75 221 .75 239 .75258 .75 276 .75 294 18 '9 18 '9 18 '9 18 18 9. 83 4?o 9.83497 9.83 524 9.83 55i 9 .835 7 8 9.836o5 9-83 632 9.83 65g 9.83686 o. 16 53o o. 16 5o3 o. 16 476 o. 16 449 o. 16 422 o. 16 395 0.16 368 o.i634i 0.16 3i4 9.91 677 9.91 669 9.91 660 9.91 65i 9.91 643 9.91 634 9.91 626 9.91 617 9.91 608 3 9 38 37 36 35 34 33 32 3i 30 9 . 7 5 3i3 9-83 713 o. 16 287 9.91 599 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. t 55 3O . PP .1 .2 3 4 .6 .8 9 23 37 9 18 .1 .2 3 4 .6 .S 9 9 8 2.8 5.6 8.4 II. 2 14.0 1 6.8 19.6 22.4 2.7 5-4 8.1 10.8 '3-5 16.2 18.9 21.6 24-"! i a 3 4 6 I- 2 3-8 5-7 7.6 9-5 11.4 '3-3 15.2 17. i 1.8 36 5-4 7-2 9.0 10.8 12.6 J4-4 16. 2 0.0 i.S 2.7 3-6 4-5 5-4 6-3 7.2 8.1 0.8 1.6 2-4 32 4 2 4.8 5.6 6.4 7.2 34 3O' f L.Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 .75 3i3 18 9 .83 7 i3 o. 16 287 9.91 5 99 8 30 3i 9 .76 33i 19 9-83 ?4o 28 0. 16 260 9.91 5 9 i 9 29 32 9 .75 350 18 9.83 768 o. l6 232 9.91 582 28 33 9 .75 368 18 9.83 795 27 o. 16 2o5 9.91 5 7 3 8 27 34 9 .75 386 19 9-83 822 27 o. 16 178 9.91 56^ 9 26 35 V 7 5 405 9-83 84 9 o. 16 i5i 9.91 556 25 36 9 . 75 4s 3 9-83 876 o. 1 6 124 9.91 547 24 27 9 ^7 9 75 44 i it 9.83 903 27 o. 1 6 097 9.91 538 8 23 38 9 .75459 9-83 93o o. 16 070 9.91 53o 22 3 9 9 .75478 r 9 18 9-83 957 27 o. 16 o43 9.91 521 21 40 9 .75 4y6 it 9.83 984 o. 16 o 1 6 9.91 5l2 8 20 4i 9 .75 5i4 9 .84 01 I o. i5 989 9.91 5o4 9 '9 42 9 .76 533 18 9 .84o38 o. i5 962 9.91 495 18 43 9 . 7 5 55i 9-84 065 o. i5 935 9.91 486 J? 27 9 44 9 .75 56 9 18 9-84 092 o. i5 908 9.91 477 8 16 45 9 .75 58? 18 9-84 119 o. i588i 9.91 46 9 i5 46 9 .75 6o5 9-84 1 46 27 o. i5854 9.91 46o i4 4 7 9 .75 624 18 9-84 I 7 3 27 o. i5 827 9.91 45i 9 i3 48 9 . 75 642 9-84 200 o. 1 5 800 9.91 442 12 49 9 .75 660 18 9-84 227 27 o. i5 77 3 9.91 433 8 I I 50 9 .75 678 18 9-84 254 27 26 o. i5 746 9.91 425 10 5i 9 .75 696 18 9-84 280 27 o. 1 5 720 9.91 4i6 Q 9 52 9 .75 714 9-84 307 o. 1 5 693 9.91 407 8 53 9.75 733 '9 9-84 334 o. i5 666 9.91 3 9 8 7 18 27 9 54 9 . 7 5 7 5i 18 9-84 36i o. i5 63g 9.91 38 9 8 6 55 9.75 769 9-84 388 o. i5 612 9.91 38i 5 56 9.75 787 9.844i5 27 o. i5 585 9.91 372 4 ll 27 9 57 9.75 805 18 9-84 442 o. 1 5 558 9.91 363 9 3 58 9.75 823 9.84469 o. i5 53i 9.91 354 2 5 9 9 . 7 584i 9.84496 27 o. 1 5 5o4 9.91 345 I 60 9 . 7 585 9 9-84 523 27 o. i5 477 9.91 336 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 1 55. PP a 97 26 19 18 9 8 .1 2.8 2-7 2.6 .1 I.O 1.8 .1 0.9 0.8 .2 5.6 5-4 5.2 .2 3.8 3-6 .2 1.8 1.6 3 8.4 8.1 7.8 3 5-7 5-4 3 2.7 2.4 4 II. 2 10.8 10.4 4 7-6 7.2 4 3.6 3.2 5 14.0 13-5 13.0 5 9-5 9.0 5 4-5 4- .6 16.8 16. 2 156 .6 11.4 10.8 .6 5-4 4-8 7 19.6 18.9 18.2 7 13.3 12.6 . 7 6-3 5.6 .8 22.4 21.6 20.8 .8 15.2 14.4 .8 7.2 6.4 9 25.2 24.3 23.4 9 '7-i 16.2 9 8.1 7.2 99 35. t L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 . 7 55 9 it 18 18 18 18 18 18 18 18 9.84 523 27 26 27 27 27 27 27 27 26 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 27 26 27 27 o.i5 477 9.91 336 8 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 60 1 2 3 4 5 6 7 8 9 9 . 7 58 77 9 . 7 58 9 5 9.75913 9.75931 9.75949 9.75967 9 . 7 5 9 S5 9.76 oo3 9. 76 O2I 9.84550 9.84576 9-846o3 9-8463o 9.84 657 9-84684 9.84 711 9.84?38 9-84 ?64 o. i5 45o o. i5 424 o. 1 5 397 o. i5 370 o.i 5 343 o.i5 3i6 o. i5 289 o. i5 262 o.i5 236 9.91 328 9.91 319 9.91 3io 9.91 3oi 9.91 292 9.91 2*3 9.91 274 9.91 266 9.91 25 7 5 9 58 57 56 55 54 53 52 5i 10 9 .76 o3g 18 18 18 18 18 '7 18 18 18 18 18 9.84 791 o. 1 5 209 9.91 248 50 1 1 12 i3 i4 i5 16 17 18 '9 9 9 9 9 9 9 9 9 9 . 76 057 .76075 .76 093 .76 in .76 129 .76 i46 .76 i64 .76 182 .76 200 9.84818 9-84845 9.84872 9.84899 9.84 925 9.84 952 9.84979 9.85 006 9.85o33 o. i5 182 o. i5 1 55 o.i5 128 o. i5 101 o.i5o75 o. 1 5 o48 0. l5 021 0.14994 o. i4 967 9.91 239 9.91 23o 9.91 221 9.91 212 9.91 2o3 9.91 194 9.91 i85 9.91 176 9.91 167 49 48 4? 46 45 44 43 42 4i 20 9 .76 218 9.85 059 o. 1 4 941 9.91 i58 40 21 22 23 24 25 26 2 7 28 2 9 9 9 9 9 9 9 9 9 9 .76236 .76253 .76 271 .76 289 .76 307 .76 324 .76 342 .76 36o .76378 '7 18 18 18 '7 18 18 18 9. 85 086 9.85 ii3 9-85 i4o g.85 166 9-85 193 9-85 220 9-85 247 9.85 273 9.85 3oo o. i4 9'4 0.14887 o. 14 860 o.i4834 o. i4 807 o.i4 780 0.14753 o.i4 727 o. 1 4 700 9.91 i4g 9.91 i4i 9.91 i32 9.91 123 9.91 n4 9.91 105 9.91 096 9.91 087 9.91 078 3 9 38 3? 36 35 34 33 32 3i 30 9 .76395 9.85 327 0.14673 9.91 069 30 L. Cos. d. L. Cot?. d. L. Tang. L. Sin. d. ' 54 3O . P? . i .1 3 4 5 .6 .8 '? 37 26 18 17 .1 .2 3 4 .6 .8 9 9 8 27 5-4 8 i 10.8 '3-5 1 6. 2 iS.q 21.6 MJ 2.6 5-2 7.8 10.4 13.0 15.6 18.2 20.8 21 ( .1 .2 3 4 '.6 :l 1.8 3-6 5-4 7- a 9.0 10.8 12.6 '44 16.2 1 7 3-4 5-i 6.8 8-S 10.2 "9 13.6 'S-3 0.9 1.8 2-7 3-6 45 5-4 6.3 7.2 0.8 1.6 2.4 3-2 4.0 4.8 5-6 6.4 7-2 35 30 '. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 . 76 3y5 9.85 327 27 26 2 ? 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 26 o. 1 4 673 9.91 069 9 9 9 9 10 9 9 9 9 9 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 IO 9 9 9 9 30 3i 32 33 34 35 36 3 7 38 3 9 9 9 9 9 9 9 9 9 9 764i3 .76431 . 7 6448 .76466 .76484 .76 5oi .76 5ig . 7 653 7 .76 554 18 '7 18 18 '7 18 18 '7 18 9 .85 354 9.85 38o 9-85 407 9.85434 9-85 46o 9.85487 9.855i4 9-85 54o 9-85 567 o. i4 646 o. :4 620 o. i4 5 9 3 o.i4566 o. 1 4 54o o. 14 5i3 o.i4486 o. i4 46o o.i4433 9.91 060 9.91 o5i 9.91 O42 9.91 o33 9.91 O23 9.91 oi4 9.91 oo5 9.90996 9.90987 29 28 27 26 25 24 22 2 I 40 9 .76 572 9.85 5 9 4 o. i4 4o6 9.90978 20 4i 42 43 44 45 46 4 7 48 49 9 9 9 9 9 9 9 9 9 . 76 5go . 76 607 .76 625 .76 642 .76 660 .76677 .76695 .76 712 .76 73o 17 18 '7 18 7 18 i? 18 9.85 620 9.85 647 9-85 674 g.85 700 9-85 727 9-85 7 54 9.85 780 9.85 807 9.85 834 o.i4 38o o.i4353 o. i4 326 o. i4 3oo o. i4 278 0.14 246 o. i4 220 o. i4 193 o. 1 4 166 9.90969 9.90 960 9.90 9 5i 9.90 942 9.90 933 9.90 924 9.90915 9.90 906 9.90 896 '9 18 '7 16 i5 i4 i3 12 I I 50 9 .76 747 9.85 860 o. 1 4 i4o 9.90 887 10 5i 52 53 54 55 56 57 58 5 9 9 9 9 9 9 9 9 9 9 .76 765 .76 782 .76 800 .76 817 .76 835 .76 852 .76 870 .76 887 .76 904 '7 18 17 18 '7 18 17 '7 iR 9.85 887 9.85 918 9-85 9 4o g.85 967 9-85 993 9.86 020 9.86 o46 9.86 073 9.86 100 o.i4 u3 o. i4 087 o. 1 4 060 o.i4 o33 o. 1 4 007 o. i3 980 o.i3 954 o. i3 927 o. 1 3 900 9.90 878 9.90 869 9.90 860 9.90 85i 9.90 842 9.90 832 9.90 823 9.90 8i4 9.90 805 9 8 7 6 5 4 3 2 I 60 . 76 Q22 9.86 126 o.i3 874 9.90796 L. Cos. d. L. Cotg . d. L. Tang. L. Sin. d. f 54. PP .1 .2 3 4 5 .6 .8 q 7 26 18 17 .1 .2 3 4 '.6 9 IO 9 2.7 5-4 8.1 10.8 '3-5 16.2 18.9 21.6 24-3 2.6 5-2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 .1 .2 3 4 :i 9 1.8 3-6 5-4 7.2 9.0 10.8 13.6 H.4 IO.2 '7 3-4 5-' 6.8 8-5 IO.2 ,1.0 13.6 1^ .3 1.0 2.0 3- 4.0 5-o 6.0 7.0 8.0 0.0 0.9 1.8 2-7 3-6 4-5 5-4 6-3 7.2 8.1 101 36. ' L.Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .76 922 9.86 126 27 26 27 26 27 26 27 26 27 27 26 27 26 27 26 27 26 26 37 26 27 26 27 26 27 26 27 26 26 27 o.i3 874 9.90 796 9 10 9 9 9 9 10 9 9 9 10 9 9 9 10 9 9 9 10 60 I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 . 7 6 9 3 9 .76957 .76974 .76991 .77009 .77 026 77 43 .77 061 .77078 18 '7 i? 18 >7 '7 18 '7 9.86 i53 9.86 179 9.86 206 9.86282 9.86259 9 .86285 9.86 3i2 9.86338 9.86365 o.i3 847 o.i382i o.i3 794 0.18 768 0.18 741 0.18 715 o.h3688 o.i 3 662 0.13635 9.90 787 9.90777 9.90 768 9.90759 9.90750 9.90 741 9.90 781 9.90 722 9.90 71 3 5 9 58 5? 56 55 54 53 52 5i 10 9 .77095 >7 '7 18 '7 17 '7 18 '7 '7 i? 9.86 392 'o.i3 608 9.90 704 50 1 1 12 i3 i4 i5 16 '7 18 '9 9 9 9 9 9 9 9 9 9 77 112 .77 i3o 77 '47 77 i64 .77 181 77 '99 .77 216 .77233 .77 25o 9.86418 9.86445 9.86471 9.86498 9.86 524 9 .8655i 9.86577 9.866o3 9.8663o o.i3 682 o.i3555 0.18 529 o. 1 3 5o2 0.18476 o. 1 3 449 0.18428 0.18 397 o. i3 870 9.90694 9.90 685 9.90 676 9.90 667 9.90 657 9.90 648 9.90 689 9.90 63o 9.90 620 49 48 47 46 45 44 43 42 4i 20 9 .77268 '7 '7 i? '7 '7 '7 17 18 >7 '7 9.86656 o.i 3 344 9.90 61 1 y 40 21 22 23 24 25 26 27 28 29 9 9 9 9 9 9 9 9 9 77 285 .77 3o2 .77819 .77 336 .77353 77 3 7 .77887 774o5 .77422 9.86688 9.86 709 9.86 736 9.86 762 9.86 789 9.86 8i5 9.86 842 9.86868 9.86894 o.i3 317 o.i 3 291 0.18 264 0.18288 0. l3 21 I 0.18 185 0.18 i58 0.18 182 o . 1 3 1 06 9.90 602 9.90 5g2 9-90583 9.90 574 9.90 665 9.90 555 9.90 546 9.90 537 9.90 527 y 10 9 9 9 10 9 9 10 9 3 9 38 37 36 35 34 33 32 3i 30 9 . 77 43 9 9.86 921 o. i3 079 9.90 5 1 8 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 53 3O . PP .1 .2 3 4 .6 :l 9 27 26 18 17 I 2 3 4 1 I 10 9 2.7 i; 10.8 '3-5 16.2 18.9 21.6 ..i- ; 2.6 S 'o 7-8 10.4 13.0 15-6 18.2 20.8 2T..A i 2 3 4 5 6 8 9 1.8 3-6 5-4 7.2 90 10.8 12.6 14-4 '7 3-4 5-t 6.8 8.5 IO.3 II.9 13-6 I.O 2.0 3 4.0 5- 6.0 7- 8.0 0.9 2.7 3.6 45 5-4 6-3 7.2 8 i 102 36 3O ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 77439 9 . S6 92 1 26 o. 1 3 079 9.90 5i8 30 3i 9 77456 17 9.86 947 27 o. i3 o53 9 . 90 5og 2 9 32 9 7?4?3 9.86974 26 o. i3 026 9.90499 28 33 9 77490 9.87 ooo o. 1 3 ooo 9.90 490 9 27 17 27 IO 34 9 77607 17 9.87 027 26 o. 12 973 9.90 480 26 35 9 77 524 9.87053 26 o. 12 947 9.90471 2D 36 9 77 54i 9.87079 O. 12 921 9.90 462 y 24 17 27 10 37 9 77558 9.87 106 26 0.12 894 9.90 452 23 38 9 77 5 ? 5 9.87 i32 26 0,12 868 9.90443 22 3 9 9 77 592 9.87 1 58 0.12 842 9.90434 9 21 40 9 77 609 9.87 185 26 o. 12 8i5 9.90424 20 4i 9 77626 9.87 21 I O. 12 789 9.90415 9 '9 42 9 77 643 9.87238 O. 12 762 9.90 4o5 18 43 9 77 660 '7 9.87 264 O. 12 736 9.90 396 y 17 17 26 10 44 9 77677 9.87 290 O. 12 710 9.90386 16 45 9 77 6g4 9 .8 7 3i 7 0.12 683 9.90377 i5 46 9 77711 '7 9.87 343 O. 12 667 9.90368 9 i4 17 26 IO 47 9 77728 16 9.87 369 o. 12 63i 9. 90 358 i3 48 9 77 744 9.87 396 o. 12 6o4 9.90 349 12 49 9 77 ?6t '7 9 .87 422 O. 12 578 9.90 339 I I 50 9 77 77 8 '7 9. 87 448 O. 12 552 9.90 33o 9 10 5i 9 77 79 5 17 9.87475 27 26 O. 12 525 9.90 32o 9 52 9 77812 9.87 5oi O. 12 499 9.90 3 1 1 8 53 9 77 829 17 9.87 527 O. 12 4?3 9.90 3oi 7 54 9 77846 '7 16 9.8 7 554 27 26 o. 12 446 9.90 292 9 6 55 9 77 862 9.87 58o O. 12 42O 9.90 282 5 56 9 .77879 '7 9.87 606 o. 12 3g4 9.90 273 9 4 5? 9 . 77 896 '7 9.87633 27 26 O. 12 367 9.90 263 IO 3 58 9 .77913 '7 9.87 65$ 0.12 34 I 9.90 254 2 5 9 9 . 7 7 9 3o 17 9.87 685 o. 12 315 9.90 244 I 60 9 .77 946 9.87711 O. 12 289 9.90235 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 53. PP 37 26 17 16 10 g .1 2.7 2.6 .1 '7 1.6 .1 i.o 0.9 2 5-4 5-2 .2 3-4 3.2 .2 2.O 1.8 3 8.1 M 3 4.8 3 3.0 2.7 4 10.8 10.4 .4 6.8 6.4 4 4-o 3-6 5 '3-5 13.0 5 8.5 8.0 5 5.0 4.5 .6 16.2 15.6 .6 IO. 2 96 .6 6.0 5.4 7 18.9 18.2 7 ii 9 II. 2 -7 7.0 6.3 .8 21.6 20.8 .8 136 12.8 .8 8.0 7.2 9 24-3 23.4 9 15-3 144 90 8.1 io3 37. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 77 946 9.87 711 0.12 289 9.90 235 60 I 9 77963 17 9.87 738 26 O. 12 262 9.90 225 9 5 9 2 9 77980 9.87 764 O. 12 236 9.90 21 6 58 3 9 77997 16 9.87790 27 0. 12 2IO 9.90 2O6 p 5? 4 9 .78013 17 9.87817 26 0.12 i83 9.90197 IO 56 5 9 .78 o3o 9.87843 26 0. 12 l5 7 9.90 187 55 6 9 .78 047 16 9.87 869 26 0. 12 l2 i 9.90 178 10 54 7 9 .78 o63 17 9.87895 27 0.12 105 9.90 1 68 9 53 8 9 .78080 9.87 922 0.12 078 9.90 i5g 52 9 9 .78097 16 9.87948 O. 12 o52 9.90 149 5i 10 9 .78 n3 9.87974 O. 12 O26 9.90 i3g 50 ii 9 .78 i3o '7 9.88 ooo O. 12 OOO 9.90 i3o 49 12 9 .78 147 16 9.88 027 o.ii 973 9.90 1 20 48 i3 9 .78 i63 9.88o53 o.ii 947 9.90 in y 47 '7 26 IO i4 9 .78 180 17 9.88 079 26 o. II 921 9.90 101 46 i5 9 .78 197 16 9.88 io5 o. ii 895 9.90 091 45 16 9 .78 2i3 9.88 i3i o.ii 869 9.90 082 y 44 '7 27 10 17 9 .78 23o 16 9.88 i58 26 O.II 842 9.90 072 43 18 9 .78 246 9.88 i84 0.11 8l6 9.90 o63 42 '9 9 .78 263 9.88 2IO o.ii 790 9.90 o53 4i 20 9 .78 280 16 9.88 236 o.ii 764 9.90 o43 40 21 9 .78 296 9.88 262 o.ii 7! 53 9.90 o34 y 3 9 22 9 783i3 16 9.88 289 o.ii 711 9.90 024 38 23 9 .78 329 9.88 3i i > o.ii 685 9.90 oi4 37 '7 26 9 24 9 .78346 16 9.8834i 26 o.ii 659 9.90005 36 25 9 . 7 836 2 9.88 367 o.ii 633 9.89995 35 26 9 .78 379 9.88 3 9 ; i o.ii 607 9.8 99 85 34 27 9 . 7 83 9 5 17 9.88420 27 26 o.ii 58o 9.89976 9 33 28 9 .78 412 16 9.88446 o.ii 554 9.8 9 966 32 2 9 9 .78428 9.88472 O.II 528 9.8 9 9 56 3i 30 9 .78445 9.88 498 O.II 502 9.8 994? y 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 52 30 . PP 27 26 17 16 IO 9 .1 2.7 2.6 i '7 1.6 .1 I.O 0.9 .2 54 5-2 2 3-4 3-2 .2 2.0 j.8 3 7.8 3 4.8 3 3-o 27 4 10.8 10.4 4 6.8 6-4 4 4.0 3.6 5 '3-5 13.0 8.5 8.0 5 5 4 5 .6 16.2 15.6 6 10.2 9.6 .6 6.0 5-4 .7 18.9 18.2 7 II-9 II. 2 7 7.0 63 .8 21.6 20.8 8 I 3 .6 12.8 .8 8.0 7.2 9 24-3 2 3 4 '4-4 9.0 8.1 io4 37 30 ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 78445 9.88498 O. I I 5O2 9.89947 30 3i 9 78461 9.88 524 26 0. I I 4?6 9-8c ) 93? 29 32 9 78478 9.88 55o o. 1 1 450 9-8( 3927 28 33 9- 78494 16 9.88 577 27 26 O. I I 423 9.89 918 y IO 2 7 34 9- 78 5io 9.88 6o3 26 o. 1 1 397 9.8< 3908 26 35 9 78 527 16 9.88 629 o. 1 1 371 9.89898 25 36 9 78 543 9.88 655 O.I I 345 9.89888 24 17 26 9 37 9- 78 56o 16 9.88 681 26 o. 1 1 319 9.8 ? 8 79 23 38 9- 78 576 9.88 707 o. 1 1 293 9.89869 22 3 9 9 78 592 9.88 733 26 o. 1 1 207 9.8 ? 85 9 21 40 9 78 609 16 9.88 759 O. I I 24l 9.8 ? 84 9 20 4i 9 78 625 9.88 786 O. I I 2l4 9.8 9 84o y '9 42 9 78 642 16 9.88 812 26 o.n 188 9.8 983o 18 43 9 78658 16 9.88838 26 o. 1 1 162 9.8 9 820 IO 17 44 9 78674 9.88864 26 o.n i36 9.8 9 810 16 45 9 78 691 16 9.88890 i o.n no 9.8 9 801 i5 46 9 78 707 9.88 916 0. II 084 9.8 9791 i4 26 IO 47 9 78723 16 9.88 942 26 o.n o58 9.8 9 781 i3 48 9 7 8 7 3 9 9.88 966 1 O. I I O32 9.8 9 77 1 12 4 9 9 78 756 '7 9.88994 26 o. 1 1 006 9.89 761 I I 50 9 78 772 16 9.89 020 26 o.io 980 9.89 752 y 10 5i 9 78 788 17 9.89 o46 27 o. 10 954 9.8 9 742 9 52 9 .78 805 9.89 073 26 o. 10 927 9.8 9 732 8 53 9 78 821 16 9.89099 26 o.io 901 9.89 722 10 7 54 9 78837 16 9.89 125 26 o.io 875 9.8 9712 6 55 9 78853 16 9.89 161 26 o.io 849 9.8 9 702 5 56 9 78 869 9.89 177 O. IO 823 9.8 9 693 4 17 IO 57 9 78886 16 9.89 2o3 26 o. 10 797 9.8 9683 3 58 9 .78 902 9.89 229 26 o.io 771 9.8 9 6 7 3 2 5 9 9 .78 918 9.89 255 26 O.IO 745 9.8 9 663 I 60 9 .78934 9.89 281 o. 10 719 9.8 9 653 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 52. PP 27 26 17 16 10 9 .1 2.7 2.6 .1 1.7 1.6 .1 I.O 0.9 .2 5-4 5-2 .2 3-4 3-2 .2 2.O 1.8 3 8.1 7.8 3 5-' 4-8 3 3-0 2.7 4 10.8 10.4 4 6.8 6.4 4 4.0 3-6 '3-5 1 6.2 13.0 iS.6 J 8-5 10.2 8.0 9.6 .6 5-0 6.0 4-5 5-4 7 18.9 18.2 7 II.9 II. 2 7 7.0 6.3 .8 21.6 20.8 .8 I 3 .6 12.8 .8 8.0 7-2 243 23-4 9 '5-3 '4-4 9.0 8.1 io5 38 C ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.78 934 16 9.89 28l 26 o. 10 719 9.89 653 60 I 9.78 g5o 17 9.89 307 26 o. 10693 9.89 643 10 5 9 2 9.78 967 16 9.89 333 26 o. 10 667 9.89 633 58 3 9.78 983 16 9.89 35 9 26 o. 10 64 1 9.89 624 IO *7 4 9.78999 16 9.89 385 26 o. 10615 9.89 6i4 IO 56 5 9.79015 16 9.89 4u 26 o. 10 589 9.89 6o4 55 6 9.79 o3i 16 9.89437 26 0. 10 563 9.89 5 9 4 10 54 7 9.79047 16 9.89 463 26 o. 10 S3? 9.89 584 10 53 8 9.79 o63 9.89 48 9 26 o. 10 5i i 9.89 5 7 4 52 9 9-79 79 9.89 5i5 26 o. 10 485 9.89 564 Si 10 9-79 9 5 16 9.89 54 1 26 o. 10 459 9.89 554 50 1 1 9.79 in 17 9.89 56y 26 o. 10433 9.89 544 IO 49 12 9.79 128 16 9.89 5 9 3 26 o. 10 407 9.89 534 48 i3 9.79 i44 16 9.89 619 26 o. 10 38 1 9.89 524 IO 47 i4 9.79 160 16 9 .8 9 645 26 o. 10 355 9.89 5i4 46 i5 9.79 176 16 9.89 671 26 o. 10 329 9.89 5o4 45 16 9.79 192 16 9 .8 9 697 26 o. 10 3o3 9.89 495 IO 44 '7 9.79 208 16 9.89 7 23 26 o. 10 277 9.89 485 IO 43 18 9.79 224 16 9.89 749 o. IO 25l 9.89 475 42 [9 9.79 24 o 16 9.89 775 26 o. 10 225 9.89 465 4i 20 9.79 256 16 9.89 801 o. 10 199 9.89 455 40 21 9.79272 16 9.89 827 26 o. 10 173 9.89445 10 3 9 22 9.79 288 16 9.89 853 o. 10 i47 9.89435 38 23 9.79 3o4 9.89879 o. IO 121 9.89425 37 15 26 24 9.79 3ig 16 9 .8 9 95 26 o. 10 og5 9.89 415 IO 36 25 9.79335 9.89 o. 10 069 9.89 405 35 26 9.79 35i 16 9 .8 9 9 5 7 26 0. 10 o43 9.89395 IO 34 27 9.79367 16 9.89 9 83 26 0. 10 017 9.89 385 IO 33 28 9.79 383 16 9.90 009 0. 09991 9.89 375 32 29 9-79 3 99 16 9.90035 o. 09 965 9.89 364 3i 30 9.79 415 9.90 061 o. o 99 3 9 9.89 354 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 513O. pp 36 17 16 15 n 10 9 .1 26 '7 1.6 .1 i-S n .1 I.O 0.9 .2 5.2 3-4 3.2 .2 3.0 2.2 .2 2.O 1.8 3 7-8 5-' 4-8 3 4-5 3-3 3 3.0 2-7 4 '0-4 6.8 6.4 .4 6.0 4-4 4 4.0 3-6 5 '3-0 8.5 8.0 5 75 5-5 5 5- 4-5 .6 15.6 IO. 2 9.6 .6 9.0 6.6 .6 6.0 5-4 .7 18.2 II-9 II. 2 7 '0-5 7-7 . 7 7.0 6-3 .8 20.8 13-6 12.8 .8 12.0 8.8 .8 8.0 7-2 9 23-4 53 M4 9 !; 8.1 106 38 3O'. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.79415 16 9.90 061 o . 09 939 9.89 354 30 3i 9 7943i 16 9.90 086 26 0.09 914 9.8 9344 2 9 32 9 .79447 16 9.90 112 26 0.09 86 8 9.8 9 334 28 33 9 .79 463 IS 9.90 i38 26 0.09 862 9.8 9 324 IO 27 34 9 .79478 16 9.90 1 64 26 0.09836 9.8 9 3i4 26 35 9 .79 4g4 9.90 190 26 0.09 8 10 9.8 9 3o4 25 36 9 .79 5io 16 9.90 216 26 0.09 784 9.8 9 294 IO 24 3 7 9 .79 526 16 9.90 242 26 0.09 758 9.8 9 284 23 38 9 .79 542 9.90 268 26 0.09 782 9.8 9 274 22 3 9 9 .79 558 9.90 294 26 0.09 706 9.8 9 264 81 40 9 79 5 7 3 16 9.90 320 26 0.09 680 9. 89 254 20 4i 9 .79 58 9 16 9.90 346 25 0.09 654 9.8 9 244 '9 42 9 .79605 16 9.90 371 0.09 629 9.8 9233 18 43 9 .79 621 15 9.90397 26 0.09 6o3 9.8 9 223 IO 7 44 9 .79 636 16 9.90 423 26 0.09 577 9.8 9 2i3 16 45 9 . 79 652 16 9.90449 0.09 55i 9.8 9 2o3 i5 46 9 .79 668 16 9.90475 26 0.09 525 9.8 9 193 10 i4 47 9 .79 684 15 9.90 5or 26 0.09 499 9.8 9 i83 i3 48 9 .79699 9.90 527 0.09 4?3 9.8 9 !7 3 12 49 9 79 7 l5 16 9.90 553 0.09447 9.8 9 162 I I 50 9 .79 731 9.90 578 25 26 0.09 422 9.89 i52 10 5i 9 79 746 16 9.90 6o4 26 o . 09 396 9.8 9 i42 9 52 9 .79 762 9.90 63o 26 0.09 370 9.8 9 i3 2 8 53 9 79 778 15 9.90 656 26 0.09 344 9.8 9 122 IO 7 54 9 79 79 3 16 9.90 682 26 o . 09 3 1 8 9.8 9112 6 55 9 .79809 9.90 708 26 0.09 292 9.8 9 ioi 5 56 9 .79825 9.90 734 0.09 266 9.8 9091 4 15 25 IO 5? 9 .79 84o 16 9.90759 26 0.09 24 1 9.8 9 08 1 3 58 9 .79 856 9.90 785 o . 09 2 1 5 9.89 071 2 5 9 9 .79 872 9.90 811 0.09 i ^9 9.8 9 060 I 60 9 .79887 9.90 887 0.09 1 63 9.8 9 o5o L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 51. PP 26 25 16 15 ii IO .1 2.6 2-5 .1 1.6 i. 5 .1 i.i I.O .2 5-2 5.0 .2 3-2 3-o .2 2.2 2.0 . 3 7.8 7-5 3 4.8 4-5 3 3-3 3- 4 10.4 10.0 4 6.4 6.0 4 4-4 4.0 5 13.0 12.5 5 8.0 7-5 5 5-5 5-0 .6 15.6 15.0 .6 9.6 9.0 .6 6.6 6.0 . 7 18.2 17-5 7 II. 2 10.5 7 7-7 7.0 .8 20.8 20. o .8 12.8 I2.O .8 8.8 8.0 9 2^.4 '.!2. S 9 14.4 13.5 9 9-9 Q.O 107 ! L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .79887 16 9.90 837 26 0.09 i63 9.89 o5o 60 i 9 .79903 15 9.90 863 26 0.09 137 9.8 9 o4o IO 5 9 2 9 .79918 16 9 . 90 88 ; 0.09 in 9.89 o3o 58 3 9 .79934 16 9.90914 26 0.09 086 9.8 9 020 ii 3 7 4 9 .79950 15 9.90 940 26 0.09 060 9.89 009 TO 56 5 9 .79965 16 9.90 966 26 0.09 o34 9.88999 55 6 9 .79981 15 9.90992 26 o . 09 008 9.8 8 989 II 54 7 9 . 79 996 16 9.91 018 25 0.08 9 3 a 9.8 8978 10 53 8 9 .80 012 9.91 o43 26 0.08 957 9.8 8968 02 9 9 .80 O27 16 9.91 069 26 0.08 9 >i 9.8 8958 Si 10 9 .80043 15 9.91 og5 26 0.08 905 9.8 8948 50 1 1 9 .8oo58 16 9.91 121 26 0.08 879 9.88 937 49 12 9 .80 074 9.91 i4' i 0.08 853 9.8 8927 48 i3 9 .80 089 16 9.91 172 26 0.08 828 9.8 8917 II 4? t4 9 .80 105 15 9.91 19* ] 26 0.08 802 9.8 8906 46 i5 9 .80 120 16 9.91 224 26 0.08 776 9.8 8896 45 16 9 .80 i36 '5 9.91 250 26 0.08 750 9.8 8886 II 44 17 9 .80 i5i 15 9.91 276 2 5 0.08 724 9.8 88 7 5 IO 43 18 9 .80 166 16 9.91 3oi 26 0.08 699 9.8 8865 42 '9 9 .80 182 9.91 3a- i 26 0.08 673 9.88 855 4i 20 9 .80 197 16 9.91 353 26 0.08 647 9.8 8844 40 21 9 .80 2i3 15 9.91 379 25 0.08 621 9-88834 3 9 22 9 .80 228 16 9.91 4o4 0.08 5( fi 9.88*824 38 23 9 .80244 9.91 43o 0.08 570 9-888i3 3? 15 26 10 24 9 .80 25g 15 9.91 456 26 0.08 544 9.8 88o3 36 25 9 .80 274 mtf 9.91 482 0.08 5i8 9.8 8 79 3 35 26 9 .80 290 9.91 507 2 S 0.08 4 f> 9.8 8 782 34 IS 26 IO 2 7 9 .80 3o5 15 9.91 533 26 0.08 467 9.88772 33 28 9 .80 32o mtt 9.91 55c 1 0.08 44i 9.8 8 761 32 2 9 9 .80 336 9.91 585 0.08 4i5 9.8 8 7 5i 3i 30 .80 35i 9.91 610 '5 o . 08 390 9.8 8 741 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 50 30 . PP 26 as it 15 n IO .1 2.6 3-5 i 1.6 J-5 .1 i.i I O 2 5-2 S-o 2 32 3-o .2 2.2 20 3 7.8 7-5 3 4.8 4-5 3 3-3 3- 4 10.4 1 0.0 4 6.4 6.0 4 4-4 4.0 5 13.0 12 .5 5 8.0 7-5 -5 5 5 5.0 .6 15.6 IS.O 6 9,6 9.0 .6 6.6. 6.0 . 7 1 8.9 '7-5 7 II. 2 10.5 7 77 7.0 .8 20.8 20. 8 12.8 I2.O .8 8.8 8.0 9 234 22.5 '4-4 '3-5 1 08 39 3O . L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 .8o35i 9.91 610 26 26 26 25 26 26 26 25 26 26 25 26 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 26 26 25 0.08 390 9.88 741 30 3i 32 33 34 35 36 3? 38 3 9 9 9 9 9 9 9 9 9 9 .80 366 80 382 .80397 . 80 412 80428 80443 80 458 8o4?3 80489 16 15 IS 16 i5 15 15 16 9.91 636 9.91 662 9.91 688 9.91 718 9.91 739 9.91 765 9.91 791 9.91 816 9.91 842 0.08 364 0.08 338 0.08 3i2 0.08 287 0.08 261 0.08 235 0.08 209 0.08 1 84 0.^8 1 58 9.88 730 9.88 720 9.88 709 9.88 699 9.88 688 9.88 678 9.88668 9.8865? 9.8864? 10 II IO II 10 IO II IO 29 28 27 26 25 24 23 22 2 I 40 9 80 5o4 9.91 868 0.08 i32 9.8 3636 20 4i 42 43 44 45 46 4? 48 49 9 9 9 9 9 9 9 9 9 80 5ig 80 534 80 Sgo .80 565 .8o58o .SoSgS .80 610 .80625 .8o64i IS 15 16 15 IS iS IS 15 16 9.91 893 9.91 919 9.91 945 9.91 971 9.91 996 9.92 022 9.92 o48 9.92 073 9.92099 0.08 107 0.08 081 0.08 o55 0.08 029 0.08 oo4 0.07 978 0.07 952 0.07 927 0.07 901 9.88 626 9-886i5 9^88605 9.88 594 9.88584 9.88573 9.88563 9.88552 9.88 542 II IO II 10 II IO 11 IO II 10 II II IO II 10 II 10 II II 19 18 i? 16 i5 i4 i3 12 I I 50 9 .80 656 15 '5 15 15 15 15 IS 16 IS 'S 9.92 125 0.07 875 9.8 8 53: 10 5i 52 53 54 55 56 5? 58 5 9 9 9 9 9 9 9 9 9 9 .80671 .80686 .80 701 .80 716 .80731 .80746 .80 762 .80 777 .80 792 9.92 i5o 9.92 176 9.92 202 9.92 227 9.92 253 9.92 279 9.92 3o4 9.92 33o 9.92 356 0.07 850 0.07 824 0.07 798 0.07 773 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 9.88 52i 9.88 5io 9.88499 9.88 489 9.88478 9.88468 9.88457 9-88447 9.88436 9 8 7 6 5 4 3 2 I 60 9 .80807 IS 9.92 38 1 0.07 619 9.88425 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. / 5O. PP .1 .2 3 4 .6 .8 9 26 5 16 15 .1 .2 3 4 :! .B 9 ii IO 2.6 5-2 7.8 10.4 13.0 156 18.2 20.8 23.4 2.5 5-o 7-5 IO.O 12.5 15-0 '7-S 20.0 22.5 .i .2 3 4 .6 '.S 9 1.6 3- 2 4.8 6.4 8.0 9.6 1 1.2 12.8 14.4 '5 3- 45 6.0 7-5 9.0 10.5 12. 13-5 i.i 2.2 3-3 4-4 1.1 i!fi 9-9 1.0 2.0 3-0 4.0 5-o 6.0 7.0 8.0 9- 109 40. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.80 807 15 15 15 15 15 15 15 15 '5 'S 15 >5 % 15 15 '5 15 14 15 '5 'S IS . iS IS IS 14 15 15 IS IS 14 9.92 38i 26 26 23 26 26 25 26 26 25 26 25 26 26 25 26 26 25 26 25 26 26 23 26 25 26 26 25 26 25 26 0.07 619 9.88425 IO II 10 II II IO II II 10 II II 10 II II IO II II 10 II GO I 2 3 4 5 6 7 8 9 9.80822 9.80837 9.80 852 9.80867 9.80882 9.80 897 9.80 912 9.80 927 9.80 942 9.92 407 9.92433 9.92 458 9.92484 9.92 5io 9.92 535 9.92 56i 9.92 587 9.92 612 0.07 5g3 0.07 567 0.07 542 0.07 5i6 0.07 490 0.07 465 0.07 43g 0.07 4i3 0.07 388 9.88415 9-884o4 9.88 3 9 4 9.88383 9.88 372 9.88 362 9-8835i 9.88 34o 9.88 33o 5 9 58 57 56 55 54 53 52 5i 10 9 .80 957 9.92 638 0.07 362 9.8 8 3i 9 50 1 1 12 i3 i4 i5 16 17 18 1 9 9 9 9 9 9 9 9 9 9 .80 972 .80987 .81 002 .81 017 .81 o32 .81 047 .81 061 .81 076 .81 091 9.92 663 9.92 689 9.92 715 9.92 740 9.92 766 9.92 792 9.92 817 9.92 843 9.92 868 0.07 337 0.07 3n 0.07 285 0.07 260 0.07 234 0.07 208 0.07 i83 0.07 167 0.07 i32 9.88 3o8 9.88 298 9.88 287 9.88276 9.88 266 9.88 255 9.88244 9.88 234 9.88 223 49 48 4? 46 45 44 43 42 4i 20 9 .81 106 9.92 894 0.07 1 06 9.8 8 212 40 21 22 23 24 25 26 27 28 2 9 9 9 9 9 9 9 9 9 9 .8l 121 .81 i36 .81 i5i .81 166 .81 180 .81 196 .8l 2IO .8( 225 . 81 240 9.92 920 9.92 945 9.92 971 9.92996 9 .93 022 9-93o48 9.93 073 9.93099 9.93 124 0.07 080 0.07 oSg 0.07 029 0.07 oo4 0.06 978 0.06 952 0.06 927 0.06 901 0.06 876 9.88 2OI 9-88 igi 9.88 I 80 9.88 169 9.88 i58 9.88 i48 9.88 137 9.88 126 9.88 u5 10 II II II 10 II II II 3 9 38 3 7 36 35 34 33 32 3i 30 9 .81 254 9.93 150 0.06 85o 9.8 8 105 30 L. Cos. d. L. Cotg. d. L. Tang-. L. Sin. d. ' 49 3O . PP .1 .2 3 4 !e :2 9 36 25 15 14 .1 .2 3 4 .6 .8 9 ii 10 2.6 5-2 7-8 10.4 13.0 '5-6 18.2 20.8 23.4 2.5 5-0 7-5 IO.O 12-5 15-0 '7-5 2O.O 22.5 .1 .2 3 4 :I .'s i-S 3-o 4-5 6.0 7-5 9- io-5 12. '3-5 i-4 2.8 4-2 56 7.0 8.4 9.8 II. 2 12.6 i.i 2.2 3-3 4-4 5-5 6.6 H 0.9 1.0 2.0 3- 4-0 5-o 6.0 7.0 8.0 4O 3O . / L. Sin. d. L. Tang. d. i L. Cotg. Cos. d. ii ii ii ii IO II II II II II II IO II II II II II II II 30 9 .81 254 15 '5 '5 '5 14 15 15 M 15 15 9.93 150 25 26 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 0.06 85o 9.88 105 30 3i 33 34 35 36 3? 38 3 9 9 9 9 9 9 9 9 9 9 . 8 1 269 .81 284 .81 299 .81 3i4 .81 328 .81 343 .81 358 .81 3y2 .81 38 7 9.93 175 9.93 2OI 9.93 227 9.93 252 9.93 278 9.93 3o3 9.93 3ag 9.93 354 9.93 38o 0.06 825 0.06 799 0.06 773 0.06 748 0.06 722 0.06 697 0.06 671 0.06 646 0.06 620 9.88 9 4 9.88 o83 9.88 072 9.88 06 1 9.88 o5i 9.88 o4o 9.88 029 9.88 018 9.88 007 29 28 27 26 25 24 23 22 21 40 9 .8l 402 9.93 4o6 0.06 5g4 9.87 996 20 4i 42 43 44 45 46 4? 48 49 9 9 9 9 9 9 9 9 9 .81 417 .81 43i .81 446 .81 46i .81475 .81 490 .81 505 .81 5i 9 .81 534 M 15 15 M 15 15 M 15 9.93 43i 9.93 457 9.93482 9.93 5o8 9.93 533 9.93 55g 9 . 9 3 584 9.93 610 9.93 636 0.06 569 0.06 543 0.06 5i8 0.06 492 0.06 467 0.06 44i 0.06 4i6 0.06 390 0.06 364 9.87985 9- 8 7975 9.87 964 9.87953 9.87 942 9.87 9 3i 9.87 920 9.87 909 9.87 898 19 1 8 17 16 i5 i4 i3 12 I I 50 9 .81 549 9 .93 661 0.06 339 9.87 887 10 5i 52 53 54 55 56 5? 58 5 9 9 9 9 9 9 9 9 9 9 .81 563 .81 5 7 8 .81 592 .81 607 .81 622 .81 636 .81 65i .81 665 .81 680 5 14 15 15 '4 15 '4 15 9.93 687 9.93 712 9.93 738 9.93 763 9 . 9 3 789 9.93 8i4 9.98 84o 9.93 865 9.93 891 0.06 3i3 0.06288 0.06 262 0.06 237 O.O6 21 I O.O6 l86 O.O6 I 60 0.06 i35 0.06 109 9.87877 9.87 866 9.87 855 9-87844 9.87 833 9.87 822 9.87 811 9.87 800 9.87 789 II II II II II II II II 9 8 7 6 5 4 3 2 I 60 9 .81 6g4 9.93 916 0.06 084 9.87 778 L. Cos. d. L. Cotg. | d. L. Tang. L. Sin. d. f 49. PP .1 .2 3 4 .6 .8 26 25 5 M .1 .2 3 4 .6 9 ii 10 2.6 52 7.8 10.4 13.0 15-6 l8.2 20.8 23.4 2-5 5-0 75 IO.O 12 5 15.0 '7-5 20.0 .1 .2 3 4 .6 .8 -0 i-S 3.0 4-5 6.0 7-5 9.0 10.5 I2.O '3-5 1.4 2.8 4.2 5-6 7.0 8.4 9.8 II. 2 i.i 2.2 3-3 44 5-5 6.6 11 9-9 1.0 2.0 3- 4.0 5- 6.0 7.0 8.0 0.0 41. ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9.81 694 9.93 916 26 0.06 o84 9.87 77 8 60 I 9.81 709 14 9.93 942 25 0.06 o58 9.87 767 5 Q 2 9.81 723 9.93967 26 0.06 o33 9.87 756 58 3 9.81 738 9.93993 0.06 007 9.87745 57 14 25 ii 4 9.81 762 is 9.94 018 26 o.o5 982 9-87734 56 5 9.81 767 9.94044 0.05966 9.87 723 55 6 9.81 781 15 9.94069 26 o.o5 931 9.87 712 ii 54 7 9.81 796 14 9.94095 25 o.o5 905 9.87 701 53 8 9.81 810 9.94 120 26 o.o5 880 9.87 690 52 9 9.8! 825 9.94 1 46 o.o5854 9.87679 5i 10 9.81 83g 15 9.94 17' 26 o.o5 829 9.87 668 50 ii 9.81 854 14 9.94 197 25 o.o5 8o3 9 .8 7 65 7 ii 49 12 9.81 868 9.94 222 26 o.o5 778 9.87 646 48 i3 9.81 882 15 9.94248 25 o.o5 752 9.87635 ii 4? i4 9.81 897 *4 9.94 273 26 o.o5 727 9.87624 ii 46 i5 9.81 911 9.94299 o.o5 701 9.87613 45 16 9.81 926 M 9.94 324 26 o.o5 676 9.87 601 ii 44 17 9.81 940 15 9.94 350 25 o.o5 65o 9.87 590 ii 43 18 9.81955 9 . 9 43 7 5 26 o.o5 625 9.87579 42 J 9 9.81 969 9.94401 o.o5 599 9. 87 568 4i 20 9.81 983 9.94426 26 o.o5 574 ^.87.557 40 21 9 .81 998 14 9.944^2 25 o.o5 548 9.87 546 ii 39 22 9 .82 OI2 9.94477 26 o.o5 523 9.87535 38 23 9 .82 O26 9.94 5o3 o.o5 497 9.87 524 37 15 25 24 9 .82o4l 14 9.94 528 26 o.o5 472 9.87513 12 36 25 9 .82 055 9.94 554 o.o5 446 9.87 5oi 35 26 9 .82 069 9.94579 o.o5 421 9.87490 34 15 2 5 27 9 .82084 14 9.94604 26 o.o5 396 9.87479 II 33 28 9 .82 098 9.94 63o o.o5 370 9.87468 32 29 9 .82 112 9.94 655 26 o.o5 345 9.87457 3i 30 9 .82 126 9.94 681 o.o5 319 9.87446 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. 48 3O . PP 36 *5 5 M 12 II .x 2.6 25 .1 i-5 fc l .1 1.2 I.I .2 5-2 5-o .3 3-o 2.8 .2 2.4 2.2 3 7.8 7-5 3 45 4-2 3 3-6 3-3 4 10.4 10. 4 6.0 5.6 4 4.8 4-4 5 13.0 12-5 5 7-5 7.0 5 6.0 5-5 .6 15-6 15-0 .6 9.0 8-4 .6 7-2 6.6 7 ill '75 . 7 10. s 9.8 7 8.4 7-7 .8 20.8 20. o / -8 I2.O 1 1. 2 .8 9.6 8.8 '? 2^.4 22.5 9 13.5 12.6 9 10.8 9.9 41 3O . ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.82 126 9.94 68 1 o.o5 319 9.87446 30 3i 9.82 i4i 9.94 706 26 o.o5 294 9.87434 29 32 9.82 155 9.94 73 2 o.o5 268 9.87 423 28 33 9 .82 169 9.94757 o.o5 243 9.87 412 27 15 34 9 .82 1 84 14 9.94783 25 o.o5 217 9.87 4oi 26 35 9 .82 198 9 . 9 4 80 I o.o5 192 9.87 390 25 36 9 .82 212 9. 9 4834 o.o5 166 9.87 378 24 14 25 II 37 9 .82 226 14 9.94 85g 25 o.o5 i 4i 9.87 367 23 38 9 .82 240 9. 9 4884 o.o5 116 9.87 356 22 3 9 9 .82 255 9.94 910 o.o5 090 9.87345 II 21 40 9 .82 26 9 9.94 935 26 o.o5 065 9.87 334 20 4i 9 .82283 '4 9.94 961 2 5 o.o5 o3g 9.87 322 i 9 42 9 .82 297 9.94 986 26 o.o5 oi4 9.87 3n 1 8 43 9 .82 3n 9.95 012 o.o4 9 38 9.87 3oo 7 '5 25 12 44 9 .82 326 14 9- 9 5 o3 7 o.o4 963 9.87 288 16 45 9 .82 34o 9 .95 062 o.o4 938 9.87 277 i5 46 9 .82 354 9.95 088 o.o4 912 9.87 266 i4 25 II 4? 9 .82 368 9.95 1 13 26 o.o4 887 9.87 255 i3 48 9 .82 382 9 . 9 5 i3< ) o.o486i 9.87 243 12 49 9 .82 396 14 9.95 i64 25 26 o.o4 836 9.87 232 I I 50 9.82 4io J 4 9.95 190 o . o4 8 1 o 9.87 221 10 5i 9 .82424 15 9.95 2i5 2 5 o.o4 785 9.87 209 9 52 9 .82439 9.95 240 26 o.o4 760 9.87 I 9 8 8 53 9 .82453 9 .95 266 o.o4 734 9.87187 7 14 25 12 54 9 .82467 9.95 291 26 o.o4 709 9.87 175 6 55 q .82481 9- 9 5 317 o.o4 683 9.87 i64 5 56 9 .82 495 '4 14 9.95 34s i 26 o.o4658 9.87 i53 12 4 5 7 9 .82 .609 9.95 368 2 5 o.o4 632 9.87 i4i 3 58 9 .82 523 9.95 3g! \ o.o4 607 9.87 i3o 2 5 9 9 .82 537 '4 9 . 9 5 4i \ 26 o.o4 582 9.87 119 I 60 9 .82 55i 14 9.95 444 o.o4 556 9.87 107 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 48. PP 26 *5 15 14 12 ii .1 2.6 2-5 x i-5 1.4 .1 1.2 i.i .2 5-2 2 3.0 2.8 .2 2.4 2.2 3 7.8 7-5 3 4-5 4-2 3 3.6 3-3 4 10.4 IO.O 4 6.0 5.6 4 4-8 4-4 5 13.0 12.5 5 7-5 7.0 -5 6.0 5-5 .6 15.6 15-0 .6 9.0 8.4 .6 7.2 6.6 7 18.2 '7-5 7 10.5 9.8 7 8.4 7-7 .8 20.8 20. o .8 I2.O II. 2 .8 9.6 8.8 9 23-4 9 13.5 12.6 9 10 8 9-9 L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .82 55i 14 14 14 M '4 14 H '4 M 9.95 444 25 26 25 25 26 25 26 25 *5 26 25 25 26 25 26 25 25 26 25 26 25 25 26 25 25 26 25 26 25 25 o.o4556 9.87 107 ii ii 12 II 12 II II 12 II 60 I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 .82 565 .82 579 .82 5g3 .82 607 .82 621 .82635 .82 64g .82 663 .82 677 9.95 46y 9.95495 g.gS 52O 9.95 545 9.95 571 9.95 596 9.95 622 9.95 64? 9.95 672 o.o4 53i o.o4 5o5 o.o4 48o o.o4 455 o.o4 429 o.o4 4o4 0.04378 o.o4353 0.04 328 9.87 096 9.87 085 9.87073 9.87 062 9.87 o5o 9.87 o3g 9.87028 9.87 016 9.87005 5 9 58 5 7 56 55 54 53 52 Si 10 9 .82 691 J 4 M '4 14 M H 14 13 14 14 9.95 698 o.o4 3o2 9.86993 50 1 1 12 i3 i4 i5 16 i? 18 1 9 9 9 9 9 9 9 9 9 9 .82 705 .82 719 .82 733 .82 7 4? .82 761 .82775 .82 788 .82 802 .82 816 9.95 723 9.95748 9-9 5 774 9.95799 9.95825 9.95 85o 9.96 8y5 9.95 901 9.95 926 0.04 277 O.o4 252 o.o4 226 o.o4 20 1 o.o4 i?5 o.o4 150 o.o4 125 o.o4 099 o.o4 074 9.86 982 9.86 970 9.86 959 9.86 947 9.86936 9.86 924 9.86 gi3 9.86 902 9.86890 12 II 13 II 12 II II 12 49 48 4? 46 45 44 43 42 4i 20 9 .82 83o 14 9.95 952 o.o4 o48 9.86 879 40 21 22 23 24 25 26 2 7 28 2 9 9 9 9 9 9 9 9 9 9 .82 844 .82858 .82 872 .82886 .82899 .82 913 .82 927 .82 g4i .82955 14 '4 *3 '4 14 14 '4 '4 9.95977 9.96 002 9.96 028 9.96 o53 9.96 078 9.96 io4 9.96 129 9.96 165 9.96 1 80 o.o4 O23 o.o3 998 o.o3 972 o.o3 947 o.o3 922 0.08896 o.o3 871 o.o3845 o.o3 820 9.86 867 9.86855 9-86844 9.86 832 9.86 821 9.86809 9.86798 9.86 786 9.86 775 12 II 12 II 13 II 12 II 3 9 38 37 36 35 34 33 32 3i 30 9 .82 968 J 3 9 .96 2o5 o.o3 795 9.86 763 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 47 30'. PP .1 .2 3 4 .6 : 8 7 9 36 as M 13 .1 .2 3 4 .6 !s 9 13 ii 2.6 5 "o 7.8 10.4 13.0 15.6 18.2 20.8 -: 4 2-5 5 75 10. 12.5 15.0 '7-5 20. o i 2 3 4 .6 '.B 9 ;i 4-2 5.6 7.0 8.4 9.8 II. 2 '3 2.6 3-9 5-2 6.5 7.8 9.1 10.4 ii. 7 1.2 2-4 3-6 4-8 6.0 72 8-4 9.6 10.8 i.i 2.2 33 4-4 5-5 6.6 7-7 8.8 42 30 L. Sin. d. L. Tang. d. L. Cotg. Cos. d. 30 9 .82 968 9 .96 2o5 26 o.o3 795 9.86 763 30 3i 9 .82 982 J 4 9.96 23l 2 5 o.o3 769 9.86 762 29 32 9 .82 996 9.96 256 o.o3 744 9.86 740 28 33 9 .83 oio 13 9.96 281 26 o.o3 719 9.86.728 II 27 34 9 83023 14 9.96 307 25 o.o3 693 9.8 6 717 26 35 9 83 o3y 9.96 332 o.o3 668 9.8 6 705 2D 36 9 83 o5i 14 9.96 357 26 o.o3 643 9.86 694 12 24 3? 9 83o65 13 9.96 383 25 o.o3 617 9.86 682 23 38 9 83078 9 .96 4o8 o.o3 592 9.86 670 22 3 9 9 83 092 9.96433 26 o.o3 567 9.86 659 21 40 9 83 106 9.96 459 o.o3 54i 9.86647 20 *i 9 83 120 13 9. 96 484 26 o.o3 5i6 9.8 6635 J 9 42 9 83 i33 9.96 5io o.o3 490 9.86624 18 43 9 83 i4y 9.96 535 o.o3 465 9.8 6612 17 14 25 12 44 9 83 161 13 9.96 56o 26 o.o3 44o 9.8 6 600 16 45 9 83 i 7 4 9.96 586 o.o34i4 9.86 58g i5 46 9 83 188 9.96 6 1 1 25 o.o3 389 9.86577 i4 14 2 5 12 47 9 83 202 13 9.96 636 26 o.o3 364 g.86565 i3 48 9 83 2i5 9.96 662 o.o3 338 g.86554 12 49 9 83 229 9.96 687 25 o.o3 3i3 9.86 542 I I 50 9 83 242 9.96 712 o.o3 288 9.8 6 53o 10 5i 9 83 2 56 9.96 738 o.o3 262 9.8 65i8 9 52 9 832yo 9.96 763 o.o3 237 9.8 6 507 8 53 9 83283 13 9.96 788 25 o.o3 212 9.86495 7 14 26 12 54 9 83297 9.96 8i4 o.o3 186 9.86483 6 55 9 .833io 9.96 83g o.o3 161 9.86 472 5 56 9 .83324 M 9.96864 25 o.o3 i36 9.86460 4 14 26 12 5? 9 .83338 9.96 8gc ft o.o3 1 10 9.86448 3 58 9 .83 35i 9.96915 o.o3o85 9-86436 2 5 9 9 .83 365 14 9.96 940 25 o.o3 060 9.86 425 I 60 9 .833-8 13 9.96 966 o.o3 o34 9.864i3 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 47. PP 26 25 M 13 12 II .1 2.6 2-5 .1 1.4 '3 .1 1.2 I.I .2 5-2 5-0 .2 2.8 2.6 .2 2.4 2.2 3 7.8 7-5 3 4.2 3-9 3 3- 6 3-3 -4 10.4 10. 4 5.6 5-2 4 4.8 4.4 5 13.0 12.5 5 7.0 6-5 5 6.0 5.5 .6 15.6 15.0 .6 8.4 7.8 .6 7.2 6.6 7 18.2 '7-5 7 9.8 9.1 7 8.4 7.7 .8 20.8 20. o 8 II. 2 10.4 .8 9.6 8.8 9 23.4 22.5 9 12.6 11.7 9 jo. 8 9.9 n5 43. L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .83 3 7 8 9.96 966 25 S 26 25 25 26 25 25 25 26 25 25 26 2 S 25 26 25 25 26 25 25 26 25 25 25 26 25 25 26 25 o.o3 o34 9.864i3 12 12 12 II 12 12 12 12 12 II 12 12 12 12 12 12 12 II 12 12 12 12 12 12 12 12 12 12 12 60 I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 .83 392 .834o5 .83419 .83432 .83446' .83459 .834?3 .83486 .83 500 >4 '3 M 13 M 13 '4 '3 '4 9.96991 9.97 016 9.97 o4a 9.97067 9.97092 9.97 118 9.97 i43 9.97 168 9-97 i9 3 o.o3 009 0.02 984 0.02 958 0.02 933 O.O2 908 0.02 882 0.02 857 O.O2 832 O.O2 807 9.86 4oi 9.86 389 9.86377 9.86366 9-86354 9.86 342 9.86 33o 9-863i8 9.86 3o6 5 9 58 57 56 55 54 53 52 5i 10 9 .83 5i3 '3 9.97 219 O.O2 781 9.86 295 50 1 1 12 i3 i4 i5 16 17 18 '9 9 9 9 9 9 9 9 9 9 .83 52 7 .8354o .83554 .83 56 7 .83 58i .83 5g4 .83 608 .83 621 .83634 13 '4 '3 M '3 '4 '3 '3 9.97 244 9.97 269 9.97295 9.97 32o 9.97345 9-97371 9.97 3 9 6 9.97 421 9.97447 0.02 756 0.02 73i 0.02 705 O.O2 680 O.O2 65g O.O2 629 0.02 6o4 O.O2 579 0.02 553 9.86283 9.86 271 9.86 259 9.86 247 9.86 235 9.86 223 9.86 211 9.86 2OO 9.86 188 4 9 48 47 46 45 44 43 42 4i 20 9 .83 648 '4 9.97472 0.02 528 9.86 176 40 21 22 23 24 25 26 27 28 2 9 9 9 9 9 9 9 9 9 9 .83 661 .83674 .83688 .83 701 .83715 .83 728 .83 7 4i .83 755 .83 768 '3 '4 '3 14 '3 '3 14 3 9.97497 9.97 523 9.97 548 9.97573 9.97598 9.97 624 9.97649 9.97 674 9.97 700 0.02 5o3 O.O2 477 0.02 452 O.O2 427 O.O2 4O2 O.O2 376 0.02 35i 0.02 326 0.02 3oo 9.86 1 64 9.86 i52 9.86 i4o 9.86 128 9.86 116 9.86 104 9.86 092 9.86 080 9.86068 3 9 38 37 36 35 34 33 32 3r 30 9 .83 781 9-97 7 2 5 O.O2 275 9.86 o56 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' 46 3O . PP .1 .2 3 4 !e i g 26 5 14 3 .1 .2 3 4 '.6 .8 13 II 2.6 5-2 7.8 10.4 13.0 ,5.6 18.2 20.8 23.4 a- 5 S-o 7-5 IO.O "5 15-0 '7-5 20. o .1 .2 3 4 :I .8 1.4 2.8 4.2 5.6 1 8.4 9.8 II. 2 .2.6 i-3 2.6 3-9 5-2 M 7.8 9.1 10.4 it. 7 1.2 2.4 3-6 4.8 6.0 7.2 8.4 * I.I 2.2 3-3 4-4 55 6.6 7-7 8.8 116 4.3 3O / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9 83 781 9-97 7 2 5 O.O2 275 9- 86 o56 30 3 1 9 83 795 '4 9.97 750 25 26 O.O2 250 9- 86o44 29 32 9 83 808 9.97 776 O.O2 224 9- 86 o32 28 33 9 83 821 '3 9.97 801 25 O.O2 199 9- 86 020 27 13 25 12 34 9 83834 9.97 826 O.O2 I 74 9- 86008 26 35 9 83848 9.97 85i 0.02 149 9- 85 996 25 36 9 83 861 13 9.97877 O.O2 123 9- 85 984 24 J 3 2 5 12 ^7 9 838 7 4 9.97902 O.O2 098 9 85 972 23 38 9 83 887 9.97 927 O.O2 O73 9 85 960 22 39 9 83 901 9.97 9 53 0.02 047 9 85 9 48 21 40 9 83 yi4 9-97 97 8 O.O2 O22 9 85 9 36 20 4i 9 S3 927 9.98 oo3 2 5 26 o.oi 997 9 85 924 '9 42 9 83 940 9.98 029 o.oi 971 9 85 912 18 43 9 83 954 14 9.98 o54 25 o.oi 946 9 85 900 17 13 25 12 44 9 83967 9.9 8079 o.oi 921 9- 85888 16 45 9 83 980 9.9 8 io4 o.oi 896 9- 85 876 i5 46 9 83 993 '3 9 .98 i3o o.oi 870 9 85864 i4 47 9 84 006 '3 9.9 8 i55 25 o.oi 845 9- 8585i i3 48 9 84 020 9.98 180 o.oi 820 9 85 839 12 49 9 84o33 '3 9.9 8 206 o.oi 794 9 85 827 1 I 50 9 84 o46 9.9 8 2 3i o.oi 769 9 85 8i5 10 Si 9 84 o5g 13 9.98 256 o.oi 744 9 858o3 9 52 9 84 072 9.98 281 o.oi 719 9 85 791 8 53 9 84o85 13 9-9 8 307 o.oi 6g3 9 85 779 7 54 9 84 098 13 9.9 8 332 25 o.oi 668 9 85 766 13 6 55 9 .84 112 9 .98 357 o.oi 643 9 85 7 54 5 56 9 .84 125 '3 9.9 8 383 o.oi 617 9 85 742 4 13 25 12 57 9 .84 i38 9-9 84o8 o.oi 592 9 85 730 3 58 9 .84 i5i 9-9 8433 o.oi 567 9 85 718 2 5 9 9 .84 164 13 9.9 8458 2 S 26 O.OI 542 9 85 706 I 60 9 .84 177 9.98 484 o.oi 5i6 9 85 6 9 3 L. Cos. ; d. L. Cotg. d. L. Tang. L. Sin. d. " 46. PP 26 55 M 13 12 .1 2.6 2-5 .1 1.4 '3 .1 1.2 .2 5-2 .2 2.8 2.6 .2 2-4 3 7.8 7-5 3 4-2 3-9 3 3-6 4 10.4 IO.O 4 5.6 5-2 4 4 .8 5 12.5 5 7.0 6.5 5 6.0 .6 15.6 15.0 .6 8.4 7.8 .6 7.2 7 18.2 17-5 7 9.8 9.1 7 8.4 .8 20.8 20. o .8 II. 2 10.4 .8 9.6 23.4 22.5 .9 12.6 TI-7 9 10.8 117 44. / L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 9 .84 177 '3 '3 '3 '3 '3 '3 M '3 '3 13 '3 13 '3 '3 '3 12 '3 '3 '3 '3 '3 '3 '3 13 13 '3 '3 12 '3 '3 9. 98 484 25 25 26 25 25 25 26 25 25 26 25 25 23 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 o.oi 5i6 9 85 6y3 12 12 12 12 '3 12 12 12 13 12 12 12 '3 la 12 13 12 12 '3 12 12 '3 12 12 13 12 3 12 12 '3 GO I 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 .84 190 .84 2o3 .84216 .84 229 .84242 .84255 .84 269 .84282 .84295 9.98 5og 9.98 534 9.98 56o 9.98 585 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 o.oi 491 o.oi 466 o.oi 44" o.oi 4t5 o.oi 390 o.oi 365 o.oi 33g o.oi 3i4 o.oi 289 9 9 9 9 9 9 9 9 9 8568i 85 669 8565 7 85 645 85 632 85 620 85 608 85 5 9 6 85 583 5y 58 5? 56 55 54 53 52 5i 10 9 .84 3o8 9.98 737 o.oi 263 9 85 5 7 i 50 1 1 12 i3 i4 i5 16 '7 18 9 9 9 9 9 9 9 9 9 9 .84 32i .84334 .84347 .8436o .843 7 3 .84 385 .84398 .844n .84424 9.98 762 9.98 787 9.98 812 9.98 838 9.98 863 9.98888 9.98 gi3 9.98939 9.98 964 o.oi 238 O.OI 2l3 o.oi 188 o.oi 162 o.oi 1 37 O.OI 112 o.oi 087 o.oi 06 1 o.oi o36 9 9 9 9 9 9 9 9 9 85 SSg 85 547 85 534 .85 522 .85 5io .85497 .85485 .854?3 .85 46o 4 9 48 4? 46 45 44 43 42 4i 20 9 .8443 7 9.9 8 989 O.OI OI I 9 .85448 40 21 22 23 24 25 26 27 28 2 9 9 9 9 9 9 9 9 9 9 .8445o .84463 .84476 .8448 9 .84 5o2 .845i5 .84 528 .8454o .84 553 9.99015 9.99 o4o 9.99 o65 9.99090 9.99 i i 6 9.99 i4i 9.99 166 9.99 191 9.99 217 o.oo 985 o.oo 960 o.oo 935 o . oo 9 1 o o.oo 884 o.oo 85g o.oo834 o.oo 809 o.oo 783 9 9 9 9 9 9 9 9 9 .85 436 .85423 .85 4n .85 399 .85 386 .85 3 7 4 .85 36i .85 349 .85 33 7 3 9 38 37 36 35 34 33 32 Q, 30 9 .84 566 9.99 242 o.oo 758 9 .85 324 30 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. < 45 3O . PP .1 .2 3 4 !e : 7 8 9 26 5 .1 .2 3 4 '.6 .8 9 M 13 .1 .2 3 4 5 .6 .8 9 13 2.6 5-2 7-8 10.4 "3-? 15.6 i8.z 20.8 23.4 2.5 S-o 7-5 IO.O '2-5 15.0 17-5 20. o 22.5 '4 2.8 4-2 5.6 7.0 8.4 9.8 II. 2 .2.6 1:1 3-9 5-2 6.5 7.8 9-i 10.4 11.7 1.2 2-4 3-6 4.8 6.0 7-2 8 -* 9-6 10.8 118 44 3O I L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 9.84566 9.99 242 o.oo 758 9 .85 324 30 3i 9.84579 13 9.99267 26 o.oo 733 9 .853i2 29 32 9.84 592 9 99293 o.oo 707 9 .85 299 28 33 9.84605 9. 99 3i8 o.oo 682 9 .85 287 27 '3 13 34 9.84618 12 9.99843 25 o.oo 667 9 .85 274 26 35 9.8463o 9.99 368 26 o.oo 632 9 .85 262 25 36 g.84643 9-99 3 94 o.oo 606 9 .85 250 24 13 25 13 3? g.84656 9.99419 25 o.oo 58 1 9 .85 237 23 38 9.84669 9.99 444 o.oo 556 9 .85 225 22 3 9 9-84 682 J 3 9.99469 26 o.oo 53i 9 .85 212 3 21 40 9 .846 9 4 9.99495 o.oo 5o5 9 .85 200 20 4i 9.84 707 '3 9.99 620 25 o.oo 48o 9 .85 187 $ 42 9.84 720 9.99 545 o.oo 455 9 .85 175 18 43 9 .84 ?33 12 9.99 5 7 o 26 o.oo 43o 9 .85 162 'j 12 17 44 9.34745 '3 9.99696 o.oo 4o4 9 .85 150 16 45 9.84768 9.99 621 o.oo 379 9 .85 i3 7 i5 46 9.84 771 9.99 646 o.oo 354 9 .85i25 i4 '3 26 13 4? 9.84784 12 9.99672 O.OO 328 9 .85 112 i3 48 9 .84796 9.99697 o.oo 3o3 9 .85 100 12 49 9 .84809 9.99 722 25 o.oo 278 9 .85 087 '3 I I 50 9 .84 822 9.99 7 4? 25 26 o.oo 253 9 .85 074 10 5i 9 .84835 12 9.99773 25 o.oo 227 9 .85 062 13 9 52 9 .8484? 9-99 79 8 O.OO 2O2 9 .85 049 8 53 9 .8486o 9.99828 o.oo 177 9 .85 037 7 13 25 *3 54 9 .84873 9.99848 26 o.oo i5a 9 .85024 6 55 9 .84885 9.99874 o.oo 126 9 .85oi2 5 56 9 .84898 '3 9.99899 O.OO IOI 9 84 999 '3 4 J 3 25 J 3 5 7 9 .84911 9-999 2 4 o.oo 076 9 .84986 3 58 9 .84 923 9.99949 o.oo o5i 9 .84974 2 5 9 9 .84 9 36 '3 9-99975 o.oo 026 9 .84961 13 I 60 9 .84949 o.oo ooo 25 o . oo ooo 9 .84949 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. / 45. PP 26 25 14 '3 12 .1 2.6 2-5 .1 1.4 '3 .1 1.2 .2 5.2 5- .2 2.8 2.6 .2 2.4 3 7.8 7-5 3 4.2 3-9 3 3-6 4 10.4 IO.O 4 5.6 5-2 . -4 4.8 S 13.0 12-5 5 7.0 6.5 5 6.0 .6 15.6 15-0 .6 8.4 7.8 .6 7.2 7 18.2 17-5 7 9.8 9.1 7 8.4 .8 208 20. o .8 II. 2 10.4 .8 9.6 9 23.4 22.5 9 12.6 11.7 .9 10.8 119 TABLE III FIVE-PLACE LOGARITHMS OF THE SINE AND TANGENT OF SMALL ANGLES THE SINE AND TANGENT TO EVERY SECOND FROM O TO 8' J TO EVERY TEN SECONDS FROM O TO 2. THE COSINE AND COTANGENT TO EVERY SECOND FROM 833 3088 0671 1960 3212 0802 2087 3335 0932 2213 3458 1062 2339 358o 1191 2465 3702 1320 2590 3824 '449 2715 3946 '577 2840 4067 1705 2964 4188 20 10 o 54 6 o 10 20 4l88 5378 6536 4308 5495 6650 44*8 5612 6764 4548 5728 6877 4668 584! 6991 4787 59 6 ' 7104 496 6076 7216 5024 6192 7329 5 '42 6307 744' 5260 6421 7552 5378 6536 7664 50 40 3 30 40 50 7664 8763 9836 7775 8872 9942 7886 8980 *47 7997 9088 8107 9196 4,0257 8217 933 4,0362 8327 9410 4,0467 8437 95'7 *57' 8546 9623 8655 9730 4,0779 8763 9836 4,0882 20 IO o 53 7 o IO 20 7.30882 1904 2903 0986 2005 3001 1089 2106 3100 1191 2206 3198 1294 2306 3296 1396 2406 3393 1498 2506 349' 1600 2606 3S88 1702 2705 3685 1803 2804 3782 1904 2903 3879 5 40 3 3 40 5" 3879 483? 576? 3975 4928 5860 4071 5022 5952 4167 56 6044 4263 5209 6i35 4359 5303 6227 4454 63^8 4549 5489 6409 4644 5582 6500 4739 5675 6591 4833 5767 6682 20 10 o 52 10" 9" 8" 7" 6" 5" 4" 3" 2" 1" 0" " LOGARITHMIC COSINE AND COTANGENT. 89 C FUNCTIONS OP SMALL ANGLES. L. Sin. L. Tang. , L. Sin. L. Tang. o IO 20 3o 4o 5o 5.68 55 7 5.98 660 6.16 270 6.28 7 63 6.38454 5.6855 7 5.98 660 6. 16 270 6.28763 6.38454 o 60 5o 4o 3o 20 IO 73o 4o 5o 7.33879 7-34833 7.35 767 7-33 879 7-34833 7.35767 3o 20 IO 8 o IO 20 3o 4o 5o 7.36 682 7.37577 7.38454 7.39 3i4 7-4o i58 7.40985 7.36 b2 7 .3 7 5 77 7.38455 7-393I5 7-4o i5S 7.40985 o 52 5o 4o 3o 20 10 1 o 10 20 3o 4o 5o 6.46 373 6.53 067 6.58866 6.63 982 6.68 55? 6. 72 697 6.46 3 7 3 6.53 067 6.58866 6. 63 982 6.68557 6.72 697 o 59 5o 4o 3o 20 IO 9 o IO 20 3o 4o 5o 7-4i 797 7.42 594 7.433 7 6 7.44 i4s 7.44900 7-45643 7.41 797 7.42 594 7.433 7 6 7-44 i45 7-44 900 7-45643 o 51 5o 4o 3o 20 IO 2 o 10 20 3o 4o 5o 6.76 4 7 6 6 79 g52 6.83 170 6.86 167 6.88 969 6.91 602 6.76476 6.79 952 6.83 170 6.86 167 6.88969 6.91 602 o 58 5o 4o 3o 20 IO 10 o 10 20 3o 4o 5o 7.46373 7.47 090 7.47797 7-48491 7-49 176 7.49 84g 7 .463 7 3 7 -4 7 091 7-47797 7.48492 7 .49 i 7 6 7.49 849 o 50 5o 4o 3o 20 IO 3 o IO 20 3o 4o 5o 6.94 085 6.96433 6.98 660 7.00779 7.02 800 7.04 780 6.94 085 6.96433 6.98 660 7.00779 7.02 800 7.04 73o o 57 5o 4o 3o 20 IO 11 o IO 20 3o 4o 5o 7.5o 5i2 7-5i 165 7.5i 808 7.52 44a 7.53 067 7.53683 7-5o 5i2 7-5i i65 7. 5 1 809 7-52443 7-53 067 7-53683 o 49 5o 4o 3o 20 10 4 o IO 20 3o 4o 5o 7.06 379 7.08 35i 7.10 o5<5 7.11 6g4 7-i3 2 7 3 7.14 797 7.06 579 7.08 352 7. 10 oSg 7.11 694 7. i3 2 7 3 7.14 797 o 56 5o 4o 3o 20 IO 12 o IO 20 3o 4o 5o 7.54 291 7. 54 890 7.5548i 7-56 o64 7.5663 9 7.57 206 7-54 291 7 .54 890 7.5548i 7-56 o64 7 .5663 9 7.57 207 o 48 5o 4o 3o 20 IO 5 o IO 20 3o 4o 5o 7.16 270 7.17 6g4 7.19 072 7.20 409 7.21 705 7.22 964 7. 16 270 7.17 6g4 7.19073 7.20 409 7.21 70 r > 7.22 964 o 55 5o 4o 3o 20 10 13 o IO 20 3o 4o 5o 7.57 767 7-58 320 7-58866 7.59 4o6 7.59939 7.60 465 7.57 767 7.58 320 7 .5886 7 7.59 4o6 7.59939 7.60 466 o 47 5o 4o 3o 20 IO 6 o IO 20 3o 4o 5o 7.24 188 7.25 378 7.26 536 7.27 664 7.28 763 7.29 836 7.24 188 7.25 378 7.26536 7.87 664 7.28 764 7.29 836 o 54 5o 4o 3o 20 IO 14 o IO 20 3o 4o 5o 7.60 985 7.61 499 7.62 007 7.62 5og 7.63 006 7-(3 4g6 7 .6o 986 7 .6i 500 7.62 008 7 .62 5io 7 .63 006 7 .63 497 o 46 5o 4o 3o 20 10 7 o 10 20 3o 7.3o 882 7. 3 1 904 7.32 go3 7.33 879 7.30882 7. 3 1 904 7.32 go3 7.33 879 o 53 5o 4o 3o52 15 o 7.63 982 7.63 982 o 45 L. Cos. L. Cotg. L. Cos. L. Cotg. " ' 12 3 89. FUNCTIONS OF SMALL ANGLES. 0. , n L. Sin. L. Tang. , L. Sin. L. Tang. 15 o 10 20 7. (53 982 7-6446i 7.64936 7.63 982 7.64 462 7.64 937 o 45 5o 4o 22 3<> 4o 5o 7.81 5gi 7.81 911 7.82 229 7.81 5yi 7.81 912 7.82 23o 3o 20 10 3o 4o 5o 7.654o6 7.65 870 7.66 33o 7-65 4o6 7.65 871 7.66 33o 3o 20 10 23 o IO 20 7.82545 7.82 35g 7-83 170 7.82 546 7.82 860 7.83 171 o 37 5o 4o 16 o 10 20 7.66 784 7.67235 7.67680 7.66785 7.67 235 7.67 680 o 44 5o 4o 3o 4o 5o 7-83479 7.83786 7.84 091. 7.83480 7. 83 787 7 .84 092 3o 20 IO 3o 4o 5o 7.68 121 7.68557 7.68989 7.68 121 7.68558 7.68 990 3o 20 to 24 o IO 20 7.84 3 9 3 7-846 9 4 7-84 992 7.843 9 4 7.84695 7.84 993 o 36 5o 4o 17 o 10 20 7.69 47 7.69841 7.70 26l 7.69 4i8 7.69 842 7.70 261 o 43 5o 4o 3o 4o 5o 7.85 289 7.85583 7.85876 7-85 290 7.85 584 7. 85 877 3o 20 IO 3o 4o 5o 7.70 676 7.71 088 7.71 4g6 7.70677 7.71 088 7.71 496 3o 20 10 25 o 10 20 7.86 166 7.86455 7 86 741 7.86 167 7.86456 7.86743 o 35 5o 4o 18 o 10 20 7.71 gOO 7.72 3oo 7.72697 7.71 900 7.72 3oi 7.72697 o 42 5o 4o 3o 4o 5o 7.87 026 7.87 309 7 .87 Sgo 7.87 027 7.87 3io 7.87591 3o 20 IO 3o 4o 5o 7. 73 090 7. 7 3479 7-73865 7. 73 090 7.73 48o 7.73866 3o 20 IO 26 o 10 20 7.87 870 7.88 i4? 7.88423 7.87871 7.88 1 48 7.88424 o 34 5o 4o 19 o 10 20 7.74 248 7.74 627 7.75 oo3 7.74 248 7.74 628 7.75 oo4 o 41 5o 4o 3o 4o 5o 7.88697 7.88969 7.89 240 7.88698 7.88 970 7.89 24 1 3o 20 10 3o 4o 5o 7 . 7 53 7 6 7.75745 7.76 112 7.75377 7.75 746 7.76 u3 3o 29 IO 27 o IO 20 7.89 509 7.89776 7.90 o4 1 7.89 5io 7.89777 7.90 o43 o 33 5o 4o 20 o 10 20 7.76475 7-76836 7-77 i9 3 7.76 476 7.76837 7.77 i 9 4 o 40 5o 4o 3o 4o 5o 7.90 3o5 7.90 568 7.90 829 7.90 307 7 .90 569 7.90 83o 3o 20 IO 3o 4o 5o 7-77548 7.77899 7.78248 7-77 5 49 7.77900 7.78 249 3o 20 IO 28 o 10 20 7.91 088 7.91 346 7.91 602 7.91 089 7.91 34? 7.91 6o3 o 32 5o 4o 21 o 10 20 7.78 5 9 4 7.789^ 7.79278 7.78 5g5 7.78938 7.79279 o 39 5o 4o 3o 4o 5o 7.91 857 7.92 1 10 7.92 362 7.91 858 7.92 iii 7.92 363 3o 20 IO 3o 4o 5o 7.79 6 1 6 7.79952 7.80284 7.79617 7.79952 7.80 285 3o 20 10 29 o 10 20 7.92 612 7.92 861 7.93 108 7.92 61 3 7.92 862 7.93 1 10 o 31 5o 4o 22 o 10 20 7.80615 7.80 g4a 7.81 268 7.8o6i5 7.80943 7.81 269 o 38 5o 4o 3o 4o 5o 7.93354 7.93 5gg 7.93 842 7.93 356 7.93 601 7-93844 3o 20 10 3o 7.81 5 9 i 7.81 5gt 3o37 30 o 7-<;4 o84 7.94 086 o 30 L; COS. L. Cotg. " ' L. Cos. L. Cotg. " ' 124 FUNCTIONS OF SMALL ANGLES. 0. , ' L. Sin. L. Tang. , L. Sin. L. Tang. 30 o 10 20 3o 4o 5o 7.94 o84 7.94325 7.94 564 7.94 802 7.95 o3g 7.96 274 7.94 086 7.94 326 7.94 566 7.94 8o4 7 .gS o4o 7.95 276 o 30 5o 4o 3o 20 10 37 3o 4o 5o 8.o3 775 8.o3 967 8.o4 i Sg 8 .o3 777 8 .o3 970 8.o4 162 3o 20 10 38 o 10 20 3o 4o 5o 8.o435o 8.o454o 8.o4 729 8.04918 8.o5 io5 8.o5 292 8.o4 353 8.o4543 8.o4 732 8.o4 921 8.o5 1 08 8.o5 295 o 22 5o 4o 3o 20 10 31 o 10 20 3o 4o 5o 7 .96 5o8 7-9 5 ?4i 7.95^73 7.96 2o3 7.96 432 7.96 660 7.95 5io 7.95 743 7.95974 7.96 205 7.96434 7 .96 662 o 29 5o 4o 3o 20 10 39 o 10 20 3o 4o 5o 8.o5 478 8.o5 663 8.o5848 8.o6o3i 8.06 214 8.06 396 8.o5 48 1 8.o5666 8.o585i 8.o6o34 8.06 217 3.o6 399 o 21 5o 4o 3o 20 IO 32 o 10 20 3o 4o 5o 7.96 887 7-97 n3 7.97 33 7 7.97 56o 7 .97 782 7.98 oo3 7.96 889 7.97 ii'4 7.97339 7.97 562 7.97784 7.98 oo5 o 28 5o 4o 3o 20 10 40 o 10 20 3o 4o 5o 8.06578 8.06 758 8.06938 8.07 117 8.07 295 8.07473 8.06 58i 8.06 761 8.06 94 1 8.07 120 8.07 298 8.07 476 o 20 5o 4o 3o 20 10 33 o 10 20 3o 4o 5o 7.98 ^23 7.98 442 7.98 660 7.98 876 7.99092 7 .99 3o6 7 .98 225 7.98444 7.98 662 7.98878 7.99094 7.99 3o8 o 27 5o 4o 3o 20 10 41 o 10 20 3o 4o 5o 8.07 650 8.07826 8.08 002 8.08 176 8.08 35o 8.08 524 8. 07 653 8.07 829 8.08 005 8.08 180 8.08 354 8.08 527 o 19 5o 4o 3o 20 10 34 o 10 20 3o 4o 5o 7 .99 620 7.99 732 7.99943 8.00 i54 8.00 363 8.00 671 7.99 522 7.99 7 34 7.99946 8.00 1 56 8.oo365 8.00 574 o 26 5o 4o 3o 20 10 42 o 10 20 3o 4o 5o 8.08696 8.08868 8 . 09 o4o 8.09 210 8.09 38o 8.09 550 8.08 700 8.08 872 8.09043 8.09 214 8.09 384 8.09553 o 18 5o 4o 3o 20 IO 35 o 10 20 3o 4o 5o 8.00 779 8.00 985 8.01 190 8.01 395 8.01 698 8.01 801 8.00 781 8.00 987 8.01 193 8.01 397 8.01 600 8.01 8o3 o 25 5o 4o 3o 20 JO 43 o 10 20 3o 4o 5o 8.09 718 8.09886 8.ioo54 8. 10 220 8.io386 8.10 552 8.09 722 8.09 890 8. 10 067 8. 10 224 8. 10 3go 8.io555 o 17 5o 4o 3o 20 10 36 o 10 20 3o 4o 5o 8 .02 002 8.02 203 8. 02 402 8. 02 601 8. 02 799 8. 02 996 8.02 oo4 8. 02 2o5 8. 02 405 8. 02 6o4 8.02801 8.02 998 o 24 5o 4o 3o 20 10 44 o IO 20 3o 4o 5o 8.10717 8.10881 8.11 o44 8. ii 207 8. ii 370 8. ii 53i 8. 10 720 8.io884 8 . 1 1 o48 8. II 211 8. ii 373 8. ii 535 o 16 5o 4o 3o 20 IO 37 o 10 20 3o 8.o3 192 8.o3 387 8.o3 58i 8.o3 775 8.o3 194 8.o3 3go 8.o3584 8 .o3 777 o 23 5o 4o 3o22 45 o 8. 1 1 6g3 8. 1 1 696 3 15 L. Cos. L. Cotg. " ' L. Cos. L. Cotg. " ' 125 89. FUNCTIONS OF SMALL ANGLES. 0. , L. Sin. L.Tang. , L. Sin. L. Tang. 45 o 10 20 8. ii 693 8. ii 853 8. 12 oi3 8. 1 1 696 8. ii 85? 8.12 017 o 15 5o 4o 523o 4o 5o 8.18 ^87 8.i8524 8.18662 8.18 392 8.18 53o 8.18 667 3o 20 IO 3o 4o 5o 8.12 172 8.12 33i 8.12 489 8.12 176 8.12335 8.12493 3o 20 10 53 o IO 20 8.18 798 8.18935 8.19 071 8.18 8o4 8.18940 8.19 076 o 7 5o 4o 46 o IO 20 8.12647 8.12 8o4 8.12 961 8. 12 65i 8.12 808 8.12 965 o 14 5o 4o 3o 4o 5o 8.19 206 8.19 34i 8.19 4?6 8.19 212 8.19 347 8.I948J 3o 20 10 3o 4o 5o 8.i3 117 8.i3 272 8.13427 8.i3 121 8.i3 276 8.i343i 3o 20 IO 54 o IO 20 8.19610 8.19 744 8.19877 8.19 bib 8.19 749 8.19 883 o 6 5o 4o 47 o 10 20 8.i3 58i 8.i3 7 35 8.i3888 8.i3 56 8.i3 739 8.13892 o 13 5o 4o 3o 4o 5o 8. 20 oio 8.20 i43 8. 20 275 8.20 016 8 . 20 i4c; 8.20 281 3o 20 10 3o 4o 5o 8.i4o4i 8.i4 193 8.i4344 8.i4o45 8.i4 197 8.i4348 3o 20 IO 55 o IO 20 8. 20 407 8.20538 8.20 669 8.2o4i3 8.20 544 8.20675 o 5 5o 4o 48 o 10 20 8.14495 8.i4646 8. i4 796 8. i4 500 8.i465o 8.i48oo o 12 5o 4o 3o 4o 5o 8. 20 800 8. 20 g3o 8.21 060 8.20806 8.20936 8.21 066 3o 20 IO 3o 4o 5o 8.14945 8. 1 5 094 8.i5 243 8.14950 8. i5 099 8.i5 247 3o 20 IO 56 o 10 20 8.21 189 8.21 319 8.21 447 8.21 195 8.21 324 8.21 453 o 4 5o 4o 49 o IO 20 8.i5 391 8.i5 538 8.i5685 8.i5 3 9 5 8.i5 543 8. 1 5 690 o 11 5o 4o 3o 4o 5o 8.21 576 8.21 703 8.21 83i 8.21 58i 8.21 709 8.21 83 7 3o 20 IO 3o 4o 5o 8.15832 8.i5 978 8.l6 123 8.15836 8.15982 8.16 128 3o 20 IO 57 o IO 20 8.21 958 8.22085 8.22 211 8.21 964 8.22 091 8.22 217 o 3 5o 4o 50 o 10 20 8.16 268 8.i64t3 8.t655 7 8.16273 8.16417 8.i656i o 10 5o 4o 3o 4o 5o 8.22 33 7 8.22463 8.22 588 8.22 343 8.22 469 8.22 595 3o 20 IO 3o 4o 5o 8.16 700 8.i6843 8.16986 8.16 705 8.16848 8.16 991 3o 20 IO 58 o 10 20 8.22 713 8.22838 8.22 962 8.22 720 8.22 844 8.22968 o 2 5o 4o 51 o IO 20 8.17 128 8.17 270 8.17411 8.17 i33 8.17275 8.17416 o 9 5o 4o 3o 4o 5o 8. 23 086 8.23 2IO 8.23333 8.23 092 8.23 216 8.23 33 9 3o 20 IO 3o 4o 5o 8.i 7 55 2 8.17 692 8.i 7 83 2 8.17 557 8.17 697 8.i 7 83 7 3o 20 10 59 o IO 20 8. 23456 8.23578 8.23 700 8.23462 8.23 585 8.23 707 o 1 5o 4o 52 o 10 20 8.17971 8. 18 no 8.18 249 8.17 976 8.18 n5 8.18 254 o 8 5o 4o 3o 4o 5o 8.23 822 8.23 944 8.24065 8.23829 8.23950 8.24 071 3o 20 10 3o 8.18 887 8.18 392 3o 7 60 o 8.24 186 8.24 192 o L. Cos. L. Cot jf. " ' L. Cos. L. Cotg. " ' 126 89. FUNCTIONS OP SMALL ANGLES. 1. / // L. Sin. L. Tang. , n L. Sin. L.Tang. o IO 20 8.24 186 8.24 3o6 8.24426 8.24 192 8.243i3 8.24433 o 60 5o 4o 1 60 4o 5o 8 . 29 3oo 8.29 407 8.29 5i4 8.29 3og 8.29 4'6 8.29 523 So 20 IO 3o 4o 5o 8.24546 8.24665 8.24 785 8.24553 8.24 672 8.24 791 3o 20 IO 8 o IO 20 8.29 621 8.29 727 8.29 833 8.29 629 8.29 736 8.29842 o 52 5o 4o 1 o IO 20 8.24 go3 8.25 022 8.25 i4o 8.24 910 8.25 029 8.25 147 o 59 5o 4o 3o 4o 5o 8.29 939 8.3oo44 8 .3o 150 8.29 947 8.3oo53 8.3o i58 3o 20 IO 3o 4o 5o 8. 25 2 58 8.25 3y5 8.25 4g3 8.25 265 8.25 38 2 8. 25 500 3o 20 IO 9 o IO 20 8.3o 255 8.3o359 8.3o464 8.30263 8.3o368 8.3o473 o 51 5o 4o 2 o IO 20 8.25 609 8.25 726 8.25 842 8.25 616 8.25 7 33 8.25849 o 58 5o 4o 3o 4o 5o 8.3o568 8.30672 8.3o 776 8.3o577 8.3o68i 8.30785 3o 20 10 3o 4o 5o 8.25958 8.26 074 8.26 189 8.25965 8.26081 8.26 196 3o 20 IO 10 o IO 20 8.30879 8.3o 983 8.3i 086 8.3o888 8.3o 992 8.3i 095 o 50 5o 4o 3 o IO 20 8.263o4 8.26 419 8.26533 8.26 3l2 8.26426 8.2654i o 57 So 4o 3o 4o 5o 8.3i 188 8.3i 291 8.3i 3g3 8.3i 198 8 . 3 1 3oo 8.3i 4o3 3o 20 IO 3o 4o 5o 8.26648 8.26761 8.26875 8.26655 8.26 769 8.26882 3o 20 10 11 o 10 20 8.3i 4g5 8.3i 597 8.3i 699 8.3i 505 8.3i 606 8.3i 708 o 49 5o 4o 4 o 10 20 8.26988 8.27 101 8.27 214 8.26 996 8.27 109 8.27 221 o 56 5o 4o 3o 4o 5o 8.3i 800 8.3i 901 8.32 002 8.3i 809 8.3i 911 8. 32 OI2 3o 20 IO 3o 4o 5o 8.27 326 8.27438 8.27 Sgo 8.27334 8.27446 8. 27 558 3o 20 IO 12 o IO 20 8.3a io3 8.32 2o3 8.32 3o3 8.32 112 8.32 2i3 8.32 3i3 o 48 5o 4o 5 o 10 20 8.27 Obi 8.27773 8. 27 883 8.27 669 8.27 780 8.27891 o 55 5o 4o 3o 4o 5o 8.324o3 8.32 5o3 8.32602 8.324i3 8.325i3 8.32 612 3o 20 IO 3o 4o 5o 8.27 994 8.28 io4 8.28 215 8.28 002 8.28 112 8.28223 3o 20 IO 13 o 10 20 8.32 702 8.32801 8.32899 8.32 711 8.32 811 8.32 909 o 47 5o 4o 6 o IO 20 8.28324 8.28434 8.28 543 8.28 332 8.28442 8.2855i o 54 5o 4o 3o 4o 5o 8.32998 8.33 096 8.33 195 8. 33 008 8.33 106 8.33 205 3o 20 IO 3o 4o 5o 8.28652 8.28 761 8.28869 8.28660 8.28 769 8.28 877 3o 20 10 14 o IO 20 8.33 292 8.33 3go 8.33488 8.33 3o2 8.334oo 8.33498 o 46 5o 4o 7 o IO 20 8.28 977 8.29085 8.29 198 8.28986 8.29 094 8.29 2OI o 53 5o 4o 3o 4o 5o 8.33585 8.33682 8.33779 8.335 9 5 8.33692 8.33789 3o 20 IO 3o 8.29 3oo 8.29 3og 3o52 15 o L338- 8.33886 o 45 L. Cos. L. Cotg. " ' L. Cos. L. Cotg. " ' 127 88. FUNCTIONS OP SMALL ANGLES. 1. , L. Sin. L. Tang. , L.Sin. L.Tang. 15 i> 10 20 3o 4o 5o 8.33 876 8.33 972 8.34o68 8.34 i64 8.34260 8.34355 8.33 88b 8.33982 8.34078 8.34 174 8.34270 8.34366 o 45 5o 4o 3o 20 10 223 4o 5o 8.38 oi4 8.38 101 8.38 189 8.38 026 8.38 n4 8.38 202 3o 20 10 23 o 10 20 3o 4o 5o 8.38 276 8.38363 8.3845o 8.3853 7 8.38624 8.38710 8.38 289 8.383 7 6 8.38463 8.38650 8.38 636 8.38 723 o 37 5o 4o 3o 20 10 16 o 10 20 3o 4o 5o 8.3445o 8.34546 8.3464o 8.34735 8.3483o 8.34924 8.3446i 8.34556 8.3465i 8. 34 ?46 8.3484o 8.34935 o 44 5o 4o 3o 20 IO 24 o 10 20 3o 4o 5o 8.38 796 8.38882 8. 38 968 8.39054 8.39 1 39 8.3 9 225 8.38 809 8. 38 8 9 5 8.38 981 8.39 067 8.39 1 53 8.3 9 238 o 36 5o 4o 3o 20 10 17 o IO 20 3o 4o 5o 8.35oi8 8.35 112 8.35 206 8. 35 299 8.35 392 8.3545 8.35 029 8.35 123 8.352i 7 8.35 3io 8.354o3 8.35497 o 43 5o 4o 3o 20 10 25 o 10 20 3o 4o 5o 8.3 9 3io 8.3g 395 8.39480 8.39 565 8. 3 9 649 8.3 97 34 8.39 323 8.39408 8.39493 8.39578 8.3 9 663 8.3 97 47 o 35 5o 4o 3o 20 IO 18 o 10 20 3o 4o 5o 8.35 578 8.35671 8.35764 8.35856 8.35 9 48 8.36o4o 8.35 590 8.35682 8.35 77 5 8.35867 8.35959 8.36o5i o 42 5o 4o 3o 20 10 26 o 10 20 3o 4o 5o 8.39 818 8.3g 902 8.39986 8 .4o 070 8.4o i53 8.40237 8.39832 8.39916 8.4o ooo 8.4oo83 8.4o 167 8.4o25i o 34 5o 4o 3o 20 IO 19 o 10 20 3o 4o 5o 8.36 i3i 8.36223 8.363i4 8.364o5 8.36496 8.36587 8.36 i43 8.36235 8.36326 8.36417 8.36 5o8 8.36 599 o 41 5o 4o 3o 20 IO 27 o 10 20 3o 4o 5o 8.4o 3ao 8.4o4o3 8.4o486 8.4o569 8.4o65i 8.4o 734 8.4o334 8.4o 417 8.4o 500 8.4o583 8.4o665 8.40748 o 33 5o 4o 3o 20 IO 20 o 10 20 3o 4o 5o 8. 36 678 8. 36 768 8.36858 8.36948 8.37038 8.37 128 8.36 689 8.36780 8.36870 8.36960 8.37050 8.37 i4o o 40 5o 4o 3o 20 10 28 o IO 20 3o 4o 5o 8.40816 8.40898 8.40980 8.4i 062 8.4i i44 8.4i 225 8.4o83o 8.40913 8.40995 8.4i 077 8.4i i58 8.4i 240 o 32 5o 4o 3o 20 10 21 o 10 20 3o 4o 5o 8.37 217 8.37306 8.37395 8.37484 8.37573 8.37662 8.37229 8.37 3i8 8.37408 8. 3 7 497 8.3 7 585 8.37674 o 39 5o 4o 3o 20 10 29 o IO 20 3o 4o 5o 8.4i 307 8.4i 388 8.4i 469 8.4i 55o 8.4i 63i 8.4t 711 8.4i 32i 8.4i 4o3 8.4i484 8.4i 565 8.4i 646 8.4i 726 o 31 5o 4o 3o 20 IO 22 o 10 20 3o 8.37 750 8.37838 8.37 926 8.38oi4 8.37762 8.3 7 85o 8.37938 8.38026 o 38 5o 4o 3o37 30 o 8.4i 792 8.4i 807 o 30 . L. Cos. L. Cotg. " ' L. Cos. L. Cotg. ' " 128 88. FUNCTIONS OF SMALL ANGLES. 1. , n L. Sin. L.Tang. i n L. Sin. L.Tang. 30 o 10 20 8.4i 792 8.4i 872 8.4i 962 8.4i 807 8.4i 887 8.4i 967 o 30 5o 4o 373o 4o 5o 8.45 267 8.45 34i 8.454i5 8.45 2a5 8.45 35 9 8.45433 3o 20 10 3o 4o 5o 8.42032 8.42 112 8.42 192 8.42 o48 8.42 J2" 8.42 207 3o 20 10 38 o JO 20 8.45489 8.45563 8.45637 8.45 So? 8.4558i 8.45655 o 22 5o 4o 31 o 10 20 8.42 272 8.4235i 8.4243o a. 42 207 8.42 366 8.42446 o 29 5o 4o 3o 4o 5o 8.45 710 8.45 7 84 8.4585 7 8.45 728 8.45 802 8.45875 3o 20 10 3o 4o 5o 8.42 5io 8.42689 8.42667 8.42 525 8.42 6o4 8.42683 3o 20 10 39 o JO 20 8.45930 8.46oo3 8.46 076 a. 45 948 8.46 O2I 8.46094 o 21 5o 4o 32 o JO 20 8.42 746 8.42 825 8.42 903 8.42 762 8.42 84o 8 .42 919 o 28 5o 4o 3o 4o 5o 8.46 149 8.46 222 8.46294 8.46 167 8.4624o 8.46 3i2 3o 20 10 3o 4o 5o 8.42 982 8. 43 060 8.43 i38 8.42 997 8.43075 8.43 1 54 3o 20 10 40 o JO 20 a. 46 366 8.4643 9 8.465ii 8.46385 8.4645 7 8.46629 o 20 5o 4o 33 o 10 20 a. 43 2iti 8.43 2 9 3 8.43 871 8.43232 8.43 309 8.43387 o 27 5o 4o 3o 4o 5o 8.46583 8.46655 8.46 727 8.46602 8.46674 8.46745 3o 20 JO 3o 4o 5o 8.43448 8.43 626 8.436o3 8.43464 8.43 542 8.436i 9 3o 20 JO 41 o JO 20 a. 46 799 8.46870 8.46942 8.46817 8.46889 8.46960 o 19 5o 4o 34 o JO 20 a. 4-1 6ao 8.43 7 5 7 8.43834 a. 43 6y6 8.43 773 8.4385o o 26 5o 4o 3o 4o 5o 8.47 oi3 8.47084 8.47 1 55 8.4? o32 8.47 io3 8.4? i ?4 3o 20 JO 3o 4o 5o 8 .43 910 8.43987 8.44o63 8.43 927 8.44oo3 8. 44 080 3o 20 10 42 o 10 20 a.4 1 ; 226 8.47 297 8.47368 8.47245 8.473i6 8. 4? 387 o 18 5o 4o 35 o 10 20 8.44 i3g 8.442i6 8.44 292 8.44 i56 8.44232 8.443o8 o 25 5o 4o 3o 4o 5o 8.47439 8.47 5og 8.47 58o 8. 4? 458 8. 4? 528 8.47 599 3o 20 JO 3o 4o 5o 8.44367 8.44443 8.445iQ 8.44384 8.4446o 8.44536 3o 20 JO 43 o 10 20 8.4? 650 8.4? 720 8.4? 790 8.47669 8.4? ?4o 8.47810 o 17 5o 4o 36 o 10 20 a. 445 9 4 8.44669 8.44745 8.446ii 8.44686 8.44 762 o 24 5o 4o 3o 4o 5o 8.47860 8.4? 93o 8.48ooo 8.47880 8.4795 8.48020 3o 20 JO 3o 4o 5o 8.44820 8.44895 8.44969 8.44837 8.44 912 8.44987 3o 20 JO 44 o JO 20 8.48069 8.48 1 39 8.48208 8.48090 8.48 i5 9 8.48 228 o 16 5o 4o 37 o 10 20 8.45o44 8.45 119 8.45 193 8.45o6i 8.45 i36 8.45 210 o 23 5o 4o 3o 4o 5o 8.48 278 8.4834? 8.484i6 8.48298 8.48 367 8.48436 3o 20 10 3o 8.45267 8.45 285 3o22 45 o 8.48485 8.48 5o5 o 15 L. Cos. L. Cotg. " ' L. Cos. L. Cotg. " ' 88. 129 FUNCTIONS OF SMALL ANGLES. 1. , L. Sin. L. Tang. , L. Sin. L.Tang. 45 o IO 20 3o 4o 5o S. 48 485 8.48554 8.48622 8.48691 8.48 760 8.48828 8.48 5o5 8.48 574 8.48643 8.48 711 8.48 780 8.48849 o 15 5o 4o 3o 20 IO ~o~14~ 5o 4o 3o 20 10 52 : J ><> 4o 5o 8.5i 48o 8.5i 544 8. 5 1 609 8.5i 5o3 8.5i 568 8.5i 632 3o 20 IO 53 o 10 20 3o 4o 5o 8.5i 673 8.5i 737 8.5i 801 8.5i 864 8.5i 928 8.5i 992 8.5i 696 8.5i 760 8.5i 824 8.5i 888 8.5i 9 52 8.52 oi5 7 5o 4o 3o 20 IO ^~ir 5o 4o 3o 20 IO 46 o 10 20 3o 4o 5o 8.48896 8.48965 8.49033 8.4g 101 8.49 169 8.49236 8.48917 8.48985 8.49053 8.49 121 8.49 189 8.49 257 54 o IO 20 3o 4o 5o 8.52 o55 8.52 119 8.52 182 8.52 245 8.52 3o8 8.52 371 8 . 52 079 8.52 i43 8.52 206 8.52 269 8.5 2 332 8.52 3 9 6 47 o JO 20 3o 4o 5o 8.4g 3o4 8.49372 8.49439 8.4g 5o6 8.49574 8.49641 8.49 325 8.49 3g3 8.49460 8.49528 8.49595 8.49662 o 13 5o 4o 3o 20 IO ~^~i2 5o 4o 3o 20 IO 55 o IO 20 3o 4o 5o 8.52434 8.52497 8.52 56o 8.52 623 8.52685 8.52 748 8.52459 8.52 522 8.52 584 8.52647 8.52 710 8.52 772 o 5 5o 4o 3o 20 10 48 o 10 20 3o 4o 5o 8.49 708 8 -49775 8.4g 84a 8.49908 8.4g 9?5 8.5o 042 8.49 729 8.4g 796 8.49863 8.49 g3o 8.49997 8.5oo63 56 o 10 20 3o 4o 5o 8.52810 8.52 872 8.52 985 8.52 997 8.53 oSg 8.53 121 8.5a 835 8.52897 8.52 960 8.53 022 8.53o84 8.53 i46 o 4 5o 4o 3o 20 10 49 o 10 20 3o 4o 5o 8.5o 108 8.5o 174 8.5o24i 8.5o3o7 8. 5o373 8. 5o439 8.5o i3o 8.5o 196 8.50263 8.5o 329 8.5o 395 8.5o46i o 11 5o 4o 3o 20 IO 57 o 10 20 3o 4o 5o 8.53 i83 8.53245 8.533o6 8.53368 8.53429 8.53491 8.53 208 8.53 270 8.53 332 8.53 3g3 8.53455 8.535i6 o 3 5o 4o 3o 20 IO 50 o IO 20 3o 4o 5o 8.5o5o4 8.5o57o 8.5o636 8.5o 701 8.5o 767 8.5o832 8.5o 527 8.5o593 8.5o658 8.5o 724 8.50789 8.5o855 o 10 5o 4o 3o 20 10 58 o IO 20 3o 4o 5o 8.53552 8.536i4 8.53675 8.53736 8.53 797 8.53858 8.53 578 8.5363 9 8.53 700 8.53762 8.53823 8.53884 2 5o 4o 3o 20 IO 51 o 10 20 3o 4o 5o 8.5o 897 8.50963 8.5i 028 8. 5 1 092 8.5i 167 8.5l 222 8.5o 920 8.50985 8.5i o5o 8.5i n5 8. 5 1 1 80 8. 5 1 245 o 9 5o 4o 3o 20 10 59 o IO 20 3o 4o' 5o 8.53919 8.53 979 8.54o4o 8.54 ioi 8.54i6i 8.54 222 8.53 945 8.54oo5 8. 54 066 8.54 127 8.54 187 8.54248 o 1 5o 4o 3o 20 10 52 o 10 20 3o 8.5i 287 8.5i 35i 8.5i4i6 8.5i 48o 8.5i 3io 8.5i 374 8.5i 439 8.5i 5o3 o 8 5o 4o 3o 7 60 o 8.54282 8.54 3o8 o L. Cos. L. Cotg. ' L. Cos. L. Cotg. 130 88. TABLE IV FOUR-PLACE NAPERIAN LOGARITHMS NAPERIAN LOGARITHMS. LOGARITHMS OF POWERS OF 10. Num. Log. Num. Log. 10 2 . 3O26 . i 3.6974 IOO 4.6062 .01 5.3 9 48 IOOO 6.9078 .001 7.0922 IOOOO 9.2108 .0001 10.7897 IOOOOO 1 1 . 6129 .00001 12.4871 IOOOOOO i3.8i55 .00000 I i4.i845 IOOOOOOO 16.1181 .000000 I 77.8819 IOOOOOOOO 18.4207 .00000001 19.5793 IOOOOOOOOO 20.7233 .00000000 I 21.9767 Num. Log. Num. Log. LOGARITHMS OF NUMBERS FROM i TO 10. N 1 2 3 4 5 6 7 8 9 1.0 o.oooo OIOO 0198 0296 o3g2 o488 o583 0677 0770 0862 i . i .2 .3 0.0953 0.1823 0.2624 io44 1906 2700 n33 1989 2776 1222 2070 2852 i3io 2l5l 2927 i3g8 223l 3ooi i484 23l I 3075 1670 2390 3i48 i655 2469 3221 i?4o 2546 3293 .4 .5 .6 0.3365 o.4o55 0.4700 3436 4l2I 4762 35o7 4i8 7 4824 35 77 4253 4886 3646 43i8 494? 3716 4383 5oo8 3 7 84 4447 5o68 3853 45n 5i28 3920 45 7 4 5i88 3 9 88 463 7 5247 7 .8 9 o.53o6 0.5878 0.6419 5365 5 9 33 64? i 5423 5 9 88 6523 548i 6o43 65 7 5 553 9 6098 6627 55 9 6 6i52 6678 5653 6206 6729 5710 6269 6780 5 7 66 63i3 683i 5822 6366 6881 2.0 o.6g3i 6981 7o3i 7080 7129 7178 7227 7 2 7 5 7324 7 3 7 N 1 2 3 4 5 6 7 8 9 132 NAPERIAN LOGARITHMS. N 1 o 3 4 5 6 7 8 9 2.0 2.1 2.2 2.3 0.6931 6981 7o3i 7080 7129 7178 7227 7275 7324 7372 0.7419 o. 7 885 0.8329 7467 7 9 3o 8372 7 5i4 7975 84i6 756i 8020 845 9 7608 8o65 85o2 7655 8109 8544 7701 8 1 54 858 7 774? 8198 8629 7793 8242 8671 73 9 8286 8 7 i3 2.4 2.5 2.6 0.8755 o.gi63 o. 9 555 8796 9203 g5 9 4 8838 9243 9632 8879 9282 9670 8920 9022 9708 8961 g36i 9746 9002 9400 973 9042 943 9 9821 9083 94 7 8 9858 9123 9 5 '7 9 8 9 5 2.7 2.8 2.9 3.0 3.i 3.2 3.3 o.ggSS i .0296 1.0647 9969 o332 0682 6006 0367 0716 6o43 o4o3 0750 6080 o438 0784 61 16 o4?3 0818 6i52 o5o8 o852 0188 o543 0886 0225 0578 0919 6260 06 1 3 ogSS i .0986 1019 io53 1086 1119 1 1 5i ii84 1217 1249 1282 i.i3i4 i.i632 i .1939 1 346 i663 1969 i3 7 8 1694 2OOO i4io 1725 2o3o i442 1756 2060 i4?4 1787 2090 i5o6 1817 21 19 i53 7 1 848 2149 i56g 1878 2179 1600 1909 2208 3.4 3.5 3.6 1.2238 r.2528 i .2809 2267 2556 2837 2296 2585 2865 2326 26i3 2892 2 355 2641 2920 2384 2669 2947 24l3 2698 2975 2442 2726 3002 2470 2754 3029 2499 2782 3o56 3-7 3.8 3. 9 4.0 4.i 4.2 4.3 i.3o83 i.335o i.36io 3no 33 7 6 3635 3i3 7 34o3 366i 3i64 3429 3686 3igi 3455 3712 32i8 348 1 3 7 3 7 3244 35o7 3762 3271 3533 3 7 88 32 97 3558 38i3 33 2 4 3584 3838 1.3863 3888 3gi3 3 9 38 3962 3 9 8 7 4012 4o36 4o6i 4o85 i.4no i.435i 1.4586 4i34 43 7 5 4609 4i5g 43 9 8 4633 4i83 4422 4656 4207 4446 4679 423i 446 9 4702 4255 4493 4725 4279 45i6 4748 43o3 454o 4 77 o 432 7 4563 4793 4-4 4.5 4.6 i.48i6 i .5o4i 1.5261 483g 5o63 5282 486 1 5o85 53o4 4884 5 1 07 5326 4907 5129 534? 4929 5i5i 536 9 4g5i 5i?3 5390 4974 5i 9 5 54 12 4996 5217 5433 Soig 523g 5454 4-7 4.8 4-9 5.0 5.i 5.2 5.3 1.5476 1.5686 1.5892 54 9 7 5 7 o 7 5gi3 55i8 5728 5g33 553 9 5?48 5 9 53 556o 5769 5 974 558i 5790 5994 56o2 58io 60 1 4 5623 583i 6o34 5644 585i 6o54 5665 5872 6074 i .6og4 6n4 6i34 6i54 6174 6ig4 6214 6233 6 2 53 6273 i .6292 i.648 7 1.6677 63i2 65o6 6696 6332 6525 6 7 i5 635i 6544 6 7 34 63?! 6563 6752 63go 6582 6771 6409 6601 6790 6429 6620 6808 6448 663g 6827 646 7 6658 6845 5.4 5.5 5.6 1.6864 1.7047 i .7228 6882 7066 7246 6901 7 o84 7263 6919 7102 7281 6 9 38 7120 7299 6 9 56 7 i38 7 3i 7 6974 7 i56 ?334 6993 7i74 7 352 701 1 7192 7 3 7 o 7029 7210 7 38 7 5-7 5.8 5-9 6.0 i.74o5 1.7579 r.775o 7422 7 5 9 6 7766 ?44o 7 6i3 7783 7457 763o 7800 7 4 7 5 7647 7817 7492 7664 ?834 7 5o 9 7681 7 85i 7 52 7 7699 7867 7 544 7716 7 884 7 56i 7733 79' 1.7918 79 34 79 5i 79 6 7 794 8001 8017 8o34 8o5o 8066 1 2 3 4 5 6 7 8 9 i33 NAPERIAN LOGARITHMS. N 1 2 3 4 5 6 7 8 9 6.0 i .7918 7934 7951 7967 79^4 8001 8017 8o34 8o5o 8066 6.1 6.2 6.3 .8o83 .8245 .84o5 8099 8262 8421 8116 8278 843 7 8i32 8294 8453 8i48 83io 846 9 8i65 8326 8485 8181 8342 85oo 8197 8358 85i6 82i3 83 7 4 853 2 8229 83 9 o 854 7 6.4 6.5 6.6 .8563 .8718 .8871 85 79 8733 8886 85g4 8 7 4 9 8901 8610 8764 8916 8625 8779 8g3i 864i 8 79 5 8 9 46 8656 8810 8961 8672 8825 8976 8687 884o 8991 8 7 o3 8856 9006 6.7 6.8 6.9 .9021 .9169 . 9 3i5 9o36 9184 933o 905 1 9199 9344 9066 92 1 3 gSSg 9081 9228 9 3 7 3 9 o 9 5 9242 9 38 7 91 10 9257 g4o2 9125 9272 94 1 6 9140 9286 g43o gi55 gSoi 9445 7.0 .9459 9 4?3 g488 9502 g5i6 953o 9544 9 55 9 9 5 7 3 9 58 7 7- 1 7-2 7 .3 .9601 9?4i .9879 9615 9755 9892 9629 9769 9906 9643 9782 9920 9 65 7 9796 9933 9671 9810 9947 9 685 9824 9961 9699 9 838 9974 97 i3 985i 9988 97 2 7 9 865 OOOl 7-4 7 .5 7.6 2.OOJ5 2.0149 2.0281 0028 0162 0295 0042 0176 o3o8 oo55 0189 0321 0069 O2O2 o334 0082 02 1 5 o347 0096 0229 o36o 0109 0242 0373 OI22 O255 o386 oi36 0268 0399 7-7 7.8 7-9 2 . O4 I 2 2.o54l 2.0669 o425 o554 0681 o438 0567 0694 o45i o58o 0707 o464 0592 0719 0477 o6o5 0732 0490 0618 0744 o5o3 o63i 7 5 7 o5i6 o643 0769 o528 o656 O 7 82 8.0 2 .0794 0807 0819 o832 o844 0857 0869 0882 0894 0906 8.1 8.2 8.3 2.0919 2 . I O4 I 2.II63 0931 io54 1 175 0943 1066 1187 og56 1078 1199 0968 1090 121 I 0980 I IO2 1223 0992 ii i4 1235 ioo5 1 126 1247 1017 1 1 38 1258 1029 n5o I2 7 8.4 8.5 8.6 2. 1282 2. l4oi 2.i5i8 1294 l4l2 1529 i3o6 1424 i54i i3i8 i436 i552 i33o 1 448 i564 I 342 i45 9 1576 i353 i4?i i58 7 1 365 i483 1 599 i3 77 i4g4 1610 i38 9 i5o6 1622 8.7 8.8 8.9 2.i633 2.1748 2.1861 1 645 i 7 5 9 1872 i656 1770 i883 1668 1782 1894 1679 1793 igoS 1691 1804 1917 1702 i8i5 1928 I 7 i3 1827 i 9 3 9 I 7 25 1 838 igSo i 7 36 i84 9 1961 9.0 2. 1972 1983 i 99 4 2006 2017 2028 2039 2060 2061 2O 7 2 9.1 9.2 9 .3 2.2083 2.2192 2.23OO 2094 22O3 23ll 2105 22l4 2^22 2116 2225 2332 2127 2235 2343 2i38 2246 2354 2148 225 7 2364 2i5g 2268 2375 2I 7 O 22 7 9 2386 2l8l 2289 2396 9 .4 9 .5 9.6 2.24O7 2.25l3 2.2618 2418 2523 2628 2428 2534 2638 2439 2 544 2649 245o 2 555 265g 2460 2565 2670 2471 2576 2680 2481 2586 2690 2492 25 97 2 7 OI 25O2 26o 7 2 7 II 9-7 9.8 9.9 2.2721 2.2824 2.2925 2732 2834 2935 2742 2844 2946 2752 2854 2g56 2762 2865 2966 2 77 3 2875 2976 2783 2885 2986 2 79 3 2895 2996 28o3 2 9 o5 3oo6 28l4 29l5 3oi6 10.0 2. 3O26 3i26 3224 3322 34i8 35i4 3609 3703 3 79 6 3888 N 1 2 3 4 5 6 7 8 9 1 34 TABLE V FOUR-PLACE LOGARITHMS OF NUMBERS FOUR-PLACE LOGARITHMS. N 1 2 3 4 5 6 7 8 9 10 oooo o43 086 128 170 212 253 94 334 3 7 4 1 1 4i4 453 492 53i 56 9 607 645 682 7i9 755 12 792 828 864 899 934 969 ioo4 io38 I0 7 2 1106 i3 1 1 39 i 7 3 206 271 3o3 335 367 3 99 43o i4 46 1 492 523 553 584 6i4 644 6 7 3 7 o3 732 i5 I 7 6i 79 818 847 8 7 5 90 3 9 3i 9 5 9 987 20l4 16 20 il 068 095 122 i48 i 7 5 2OI 227 253 2 79 17 3o4 33o 355 38o 4o5 43o 455 48o 5o4 529 18 553 577 60 1 625 648 6 7 2 6 9 5 718 7 42 7 65 '9 788 810 833 856 8 7 8 900 923 945 967 989 20 3oio 032 o54 075 096 lib 1 39 1 60 181 201 21 222 243 263 284 3o4 324 345 365 385 4o4 22 424 444 464 483 5O2 522 54 1 56o 5 79 5 9 8 23 617 636 655 674 692 7 1I 729 747 766 7 84 24 802 820 838 856 874 892 909 927 945 962 25 3 979 997 4oi4 4o3i 4o48 4o65 4082 4099 4n6 4 1 33 26 4i5o 166 i83 200 216 232 249 265 281 298 27 3i4 33o 346 862 378 3 9 3 4< >y 4 25 44o 456 28 472 48 7 502 5i8 533 548 564 5 79 5 9 4 609 29 624 639 654 669 683 698 7 i3 728 742 7 5 7 30 4771 786 800 8i4 829 843 85 7 87, 886 900 3i 9 4 928 942 9 55 969 9 83 997 5oi i 5o24 5o38 32 5o5i o65 079 092 io5 "9 I 32 i45 i5g I 7 2 33 i85 198 21 I 224 237 25o 263 276 289 302 34 3i5 328 34o 353 366 3 7 8 3 9 i 4o3 4i6 428 35 544 1 453 465 4 7 8 490 5O2 5i4 527 53 9 55i 36 563 5 7 5 58 7 5 99 611 623 635 647 658 6 7 o 37 682 6 9 4 7o5 717 729 74< > 7 52 7 63 77 5 786 38 798 809 821 832 843 855 866 877 888 899 39 911 922 933 944 9 55 966 977 9 88 999 6010 40 6021 o3i 042 M^^M^M o53 o64 075 o85 096 ^^^^^i 107 117 N 1 2 3 4 5 6 7 8 9 PP 38 32 28 as 99 21 19 18 17 :6 .1 3-8 3.2 2.8 a-5 ., 2.2 2.1 1.9 .1 1.8 '7 1.6 .2 6.4 5.6 5.0 .2 4-4 4-2 3-8 .2 3.6 3-4 3-2 3 n-4 9.6 8.4 7-5 3 6.6 6. 3 5-7 3 5-4 5-' 4.8 4 15.2 12 8 II. 2 1 0.0 4 8.8 8.4 7.6 4 7-2 6.8 6.4 5 19.0 1 6.0 14.0 12.5 5 II. O 10.5 9-5 5 9.0 8-5 8.0 .6 22.8 19.2 16.8 15-0 .6 13-8 13.6 11.4 .6 10.8 IO. 2 9.6 7 26.6 22.4 19.6 I7-S 7 '5-4 14.7 '3-3 . 7 12.6 II-9 II. 2 .8 3-4 25.6 22.4 20. o .8 17.6 16.8 15.2 .8 14.4 I 3 .6 12.8 9 34-2 28.8 25.2 22.5 .9 19.8 18.9 17.1 9 16.2 15.3 14.4 i36 FOUR-PLACE LOGARITHMS. N 1 2 3 4 5 6 7 8 9 40 6021 o3i 042 o53 o64 075 o5 096 107 117 4i 128 [3 i4 9 1 60 170 180 191 201 212 222 42 232 243 253 263 274 284 294 3o4 3i4 325 43 335 345 355 365 3 7 5 385 3 9 5 4o5 4i5 425 44 435 444 454 464 4 7 4 484 493 5o3 5i3 522 45 653 2 542 55i 56i 5 7 r 58o 590 5 99 609 618 46 628 63 7 646 656 665 6 7 5 684 6 9 3 702 712 47 721 780 7 3 9 ?4g 7 58 767 776 7 85 794 8o3 48 812 821 83o 83 9 848 85 7 866 8 7 5 884 8 9 3 49 902 911 9 2O 928 9 3 7 9 46 9 55 9 64 97 2 9 8i 50 6990 998 7007 7016 7 O24 7o33 7042 7o5o 7 o5 9 7067 5i 7076 084 o 9 3 101 I IO 118 126 i35 i43 l52 52 160 168 177 i85 I 9 3 202 210 218 226 235 53 243 s5i 267 2 7 5 284 292 3oo 3o8 3i6 54 324 332 34o 348 356 364 3 7 2 38o 388 3 9 6 55 ?4< 54 4l2 4i 9 427 435 443 45i 45 9 466 4 7 4 56 482 490 4 9 7 5o5 5i3 52O 528 5 36 543 55i 57 55 9 566 5 7 4 58 2 58 9 5 97 6o4 612 6i 9 627 58 634 642 64g 65 7 664 6 7 2 679 686 6 9 4 701 5 9 709 716 723 7 3i 7 38 7 45 752 760 767 774 60 7782 789 796 8o3 810 818 825 832 83 9 846 61 853 860 868 8 7 5 882 88c ) 8^ 6 . 93 9 IO 917 62 924 9 3i g38 945 9 52 9 5 9 966 97 3 9 8o 987 63 99 3 8000 8007 8oi4 8021 8028 8o35 8o4i 8o48 8o55 64 8062 069 075 082 089 096 102 109 116 122 65 8129 i36 1 42 i4 9 i56 162 169 176 182 i8 9 66 i 9 5 202 209 2l5 222 228 235 24 1 248 254 67 261 26 7 274 280 287 2 9 : \ 299 3o6 3l2 3i 9 68 325 33i 338 344 35i 35 7 363 870 3 7 6 382 69 388 3 9 5 4oi 4o 7 4i4 420 426 432 43 9 445 70 45i 45 7 463 470 4?6 482 488 4 94 5oo mmmmmmm 5o6 N 1 2 3 4 5 6 7 8 9 PP 15 14 '3 12 IX 10 9 8 7 6 0.8 0.6 .2 3- 2.8 2.6 2.4 .2 2.2 2.0 !s .2 0.7 1.4 1.2 3 45 4.2 3-9 3.6 3 3-3 3- 2.7 3 2.4 2,1 1.8 4 6.0 5-6 5-2 4.8 4 4-4 4.0 3-6 4 3- 2 2.8 2.4 5 7-5 7.0 6-5 6.0 5 5-5 4-5 5 4.0 3-5 3.0 .6 9.0 8.4 7.8 7.2 .6 6.6 6.0 5-4 .6 4.8 4.2 3-6 7 10.5 9.8 9.1 8.4 7 7-7 7.0 6.3 7 5-6 4.9 4.2 .8 12. II. 2 10.4 9.6 .8 8m .a 8.0 7.2 8 6-4 5-6 48 9 '3-5 12 6 ii 7 10.8 8.1 9 7.2 6.3 5-4 i3 7 FOUR-PLACE LOGARITHMS. N 1 2 3 4 5 6 7 8 9 70 7 1 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9 1 92 9 3 9 4 9 5 96 97 98 99 100 <^^MM N 845i 45 7 463 470 476 482 488 494 5oo 5o6 5i3 573 633 692 8 7 5i 808 865 921 976 5i 9 579 63 9 698 7 56 8i4 871 927 982 525 585 645 704 762 820 876 9 32 987 53i Soi 65i 710 768 8 2 5 882 9 38 99 3 53 7 5 9 7 657 716 774 83i 887 9 43 998 543 6o3 663 722 779 83 7 8 9 3 9 4 9 9 oo4 ~o58~ 549 6o 9 66 9 727 785 842 899 9 54 9009 555 6i5 6 7 5 7 33 791 848 904 960 9015 56i 621 681 739 797 854 910 9 65 9020 56 7 62 7 686 7 45 802 85 9 gi5 97i 9025 9 o3i o36 o42 o47 o53 o63 069 o 7 4 079 o85 1 38 191 243 9 2 9 4 345 3 9 5 445 4 9 4 090 i43 196 248 299 35o 4oo 45o 499 096 i4 9 2OI 253 3o4 355 4o5 455 5o4 IOI 1 54 206 258 3o 9 36o 4 i <> 46o 5o 9 106 i5 9 212 263 3i5 365 4i5 465 5i3 I 12 i65 217 26 9 320 370 420 46 9 5i8 117 170 222 274 325 3 7 5 425 4 7 4 523 122 i 7 5 22 7 279 33o 38o 43o 479 528 128 1 80 232 284 335 385 435 484 533 i33 186 238 28 9 34o 3 9 o 44o 48 9 538 9 542 547 552 55 7 562 566 671 5 7 6 58 1 586 5 9 o 638 685 7 3i 9777 823 868 912 9 56 595 643 689 736 782 827 872 9'7 9 6i 600 647 6 9 4 74 1 786 832 877 921 9 65 6o5 652 699 745 7 9 i 836 881 9 26 9 6 9 609 65? 7 o3 75o 79 5 84 1 886 gSo 974 6i4 661 708 754 800 845 8 9 o 9 34 978 619 666 7-3 7 5 9 8o5 85o 8 9 4 939 9 83 624 671 7'7 7 63 809 854 899 9 43 98? 628 6 7 5 722 768 8i4 85 9 9 o3 9 48 99 i 633 680 7 2 7 773 818 863 9 o8 9 52 99 6 oooo oo4 1 009 2 oi3 3 017 4 022 5 026 ^ 6 o3o -.. 7 o35 8 o4o 9 PP .2 3 4 .6 .8 9 7 6 .1 .2 3 4 '.6 .8 5 4 0.7 '4 2.1 2.8 3-5 4.2 4.9 5-6 6.3 0.6 1.2 1.8 2.4 3' 3.6 4-2 4.8 5-4 0.5 I.O '5 20 2-5 3.0 3-5 4.0 4-5 0.4 0.8 1.2 1.6 2.O 2.4 2.8 H 36 i38 TABLE VI FOUR-PLACE LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS TO EVERY TEN MINUTES FOUR-PLACE LOGARITHMIC FUNCTIONS. O ' L. Sin. d. L.Tang. d. L. Cotg. L. Cos. d. o 10 20 3o 4o 5o 7 7 7 E 3011 1760 1250 969 792 669 5 8o 5" 458 4'3 378 348 300 280 263 248 235 322 212 203 193 l8 S 177 I 7 I 5 8 152 '47 7 .463 7 7.7648 7 . 9409 8.o658 8.1627 3011 1761 1249 969 792 670 512 457 4'5 378 348 323 300 281 263 249 235 223 213 202 194 '85 ,78 171 165 '58 '54 148 2.5363 2.2352 2.O59I I .9342 1.8373 o.oooo o . oooo o.oooo o.oooo o.oooo o.oooo o o o o 90 5o 4o 3o 20 1 O .463 7 .7648 .9408 .o658 . 1627 1 o 10 20 3o 4o 5o 8 8 a 8 8 8 .2419 .3o88 .3668 .4i79 .463 7 .5o5o 8.2419 8.3089 8.3669 8.4i8i 8.4638 8.5o53 i. 7 58i i .691 i i. 633 i 1.5819 1.5362 9-9999 0.9999 9.9999 9.9999 9.9998 9.9998 I o o I o I o I o I o I I I I I o 89 5o 4o 3o 20 IO 2 o 10 20 3o 4o 5o 8.5428 8.5 77 6 8.6097 8.6397 8.6677 8.6940 8.543i 8.5 779 8.6101 8.64oi 8.6682 8.6945 .4569 .4221 .3899 .35 99 .33i8 .3o55 9.9997 9-9997 9.9996 9.9996 9.9995 9.9995 o 88 5o 4o 3o 20 10 3 o 10 20 3o 4o 5o 8 8 8 8 8 8 .7188 . 7 423 . 7 645 . 7 85 7 .825i 8 . 7 i 94 8. 7 652 8.7865 8.8067 8.8261 i .2806 I .25 7 I 1.2348 .2i35 .1933 .i 7 3 9 9.9994 9 . 999 3 9.9993 9.9992 9.9991 9.9990 o 87 5o 4o 3o 20 IO 4 o 10 20 3o 4o 5o 8 8 8 8 8 8 .8436 .86i3 .8 7 83 .8946 .9104 .9256 8.8446 8.8624 8.8795 8.8960 8.9118 8.9272 .i554 .1376 .1205 . io4o .0882 .0728 9.9989 9.9989 9.9988 9.9987 9.9986 9.9985 o I I I I 2 o 86 5o 4o 3o 20 IO 5 8 . 94o3 8.9420 i.o58o 9 . 998 3 o 85 L .Cos d. L. Cotg. d. L. Tang. L.Sin. d. ' PP .2 3 4 .6 .8 9 348 300 36 3 26.3 52.6 78.9 IOS-2 I3I-5 157.8 l8 4 ., 210.4 .1 .2 3 4 .1 : 8 9 *35 313 185 .1 .2 3 4 .6 9 171 158 147 34-8 69.6 104.4 139.2 174.0 208.8 243-6 278.4 30 60 90 120 ISO 1 80 210 240 47-C 94-c 117.5 141.0 164.5 i88.c 211.5 21., 42. c 63.5 85.2 106.5 I27-J 149.1 170.4 191.7 .8.5 37 o 55-5 74.0 92.5 III.O 129.5 148.0 166.5 17.1 342 5'-3 68.4 85.5 IO2.D II9.7 136.8 '53-9 15. 3i.< 47-' 63. 79- < 94- I 10. 126. 142. i 14.7 ) 29.4 \ 44 ' i 588 > 73-5 ! 88.2 i 102.9 I "7- 6 132.3 i4o FOUR-PLACE LOGARITHMIC FUNCTIONS. o i L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 5 o 10 20 3o 4o 5o 8 8 8 8 8 9 ,g4o3 . 9 545 .9682 .9816 .9945 .0070 142 137 '34 129 125 122 II 9 "5 "3 109 107 104 IO2 99 97 95 93 9' 89 87 85 84 82 80 79 78 76 75 73 73 8.9420 8. 9 563 8.9701 8. 9 836 8.9966 9.0093 143 138 '35 13 127 123 1 20 117 "4 in 108 105 104 101 98 97 94 93 9i 89 87 86 84 82 81 80 78 77 76 74 i. 0680 i .0437 i .0299 i .0164 i.oo34 o.ggo? g.gg83 9.9982 g.ggSi 9.9980 9-9979 9.9977 i i i i 2 I I 2 I I 2 I 2 2 I 2 2 I 2 2 2 2 2 2 2 2 2 2 2 2 o 85 5o 4o 3o 20 JO 6 o 10 20 3o 4o 5o 9.0192 9.o3i i 9.0426 g.o53g 9.0648 9.0755 9.0216 g.o336 g.o453 9.0667 9.0678 9.0786 o.g784 o.g664 o.g547 o.g433 o.g322 o.g2i4 9.9976 9-99?5 9.9973 9.9972 9-9971 9.9969 o 84 5o 4o 3o 20 IO 7 o 10 20 3o 4o 5o g.o85g 9.0961 9. 1060 9.1157 g. 1252 9.1345 9.0891 9.0996 9. 1096 9.1194 9. 1291 g.i385 o.giog 0.9006 o.8go4 0.8806 o.87og 0.8616 9.9968 g.gg66 9-9964 g.gg63 9.9961 g.ggSg o 83 5o 4o 3o 20 10 8 o 10 20 3o 4o 5o 9. i436 9. i525 9. 1612 9.1697 9.1781 9 .i863 g.i47^ g. 1669 g.i658 9.1745 g.i83i g. 1916 0.8622 o.843i 0.8342 0.8266 o.8i6g 0.8086 g.gg58 g .9966 g.gg54 9.gg52 9.9960 g.gg48 o 82 5o 4o 3o 20 10 9 o 10 20 3o 4o 5o 9.1943 9.2022 9.2100 9.2176 9.2261 9.2324 9.1997 9.2078 9.2168 9. 2236 g.23i3 9.2389 o.8oo3 o.7g22 0.7842 0.7764 o. 7687 0.761 1 9.9946 9-gg44 g.gg42 g.gg4o g.gg38 g.gg36 o 81 5o 4o 3o 20 10 10 O 9.2397 g.2463 0.7637 g.gg34 o 80 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' o PP .1 .2 3 4 5 .6 :! 9 138 "5 117 .1 .2 3 4 '.6 9 104 97 8 9 .1 .2 3 4 .6 .S 9 84 78 73 13.8 27.6 41.4 55-2 69.0 82.8 96.6 110.4 124.2 '2-5 25.0 37-5 50.0 62.5 75-0 87.5 1OO.O 112.5 11.7 23-4 35-1 46.8 58.5 70.2 Sr.g 93.6 -'5-3 10.4 20.8 31-2 4 ..6 52.0 62.4 72.8 83.2 93.6 9-7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.3 8. 9 .i 26.7 35-6 44-5 53-4 62.3 71.2 80. i. 8.4 1 6. 8 25.2 33-6 42.0 50.4 58.8 67.2 75-6 7.8 7.3 15.6 14.6 23.4 21.9 31.2 29.2 39- o 36-5 46.8 43.8 54.6 51.1 62.4 58.4 FOUR-PLACE LOGARITHMIC FUNCTIONS. ' L. Sin. d. L.Tang. d. L. Cotg. L. Cos. j d. 10 o 9 .23 97 9.2463 0.7637 9.9934 o 80 10 9 .2468 9.2636 0.7464 9.9931 3 5o 2 9 .2538 70 9 .26o 9 73 0.7391 9.9929 2 4o 68 7 1 3o 9 .2606 68 9.2680 0.7320 9.9927 3o 4o 9 .2674 9.2760 0.7260 9-9924 3 20 5o 9 .2740 9 .28i 9 09 0.7181 9.9922 2 IO 11 9 .2806 64 9 .2887 66 0.7113 9.9919 3 o 79 IO 9 .2870 9 .2 9 53 0.7047 9.9917 5o 20 9 .2 9 34 64 9.3020 t>7 0.6980 9.9914 3 4o 3o 9 .2997 63 61 9 .3o85 65 6* 0.6916 9.9912 2 3o 4o 9 .3o58 9. 3i 4 9 0.6861 9.9909 3 20 5o 9 .3u 9 9.3212 03 0.6788 9.9907 2 IO 12 9 .3i 79 9.3276 03 61 0.6726 9.9904 3 o 78 10 9 .3238 9 .3336 0.6664 9.9901 3 5o 20 9 .3296 50 9 .33 97 o.66o3 9.9899 2 4o 3o 9 .3353 57 9 .3458 61 0.6642 9.9896 3 3o 4o 9 .34io 9 .35i7 0.6483 9 .9893 3 20 5o 9 .3466 56 9 .3576 59 eft 0.6424 9.9890 3 IO 13 9 .3521 9. 3634 0.6366 9.9887 3 o 77 10 9 .3676 9.3691 o.63og 9.9 884 3 5o 20 9 .3629 54 9 .3 7 48 57 0.6262 9-9 881 3 4o 53 56 3o 9 .3682 9 .38o4 0.6196 9.9878 3o 4o. .3 7 34 9 .385 9 o.6i4i 9.9876 3 20 5o y .3 7 86 52 9 .3 9 i4 55 0.6086 9.9872 3 10 14 9 .3837 5 1 9 .3 9 68 54 o.6o32 9.9869 3 o 76 10 9 .3887 9 .4o2i 0.6979 9.9866 3 5o 20 9 .3 9 3 7 5 9.4074 53 0.6926 9.9863 3 4o 3o 9 .3 9 86 49 9.4127 53 o.58 7 3 9.9869 4 3o 4o 9 .4o35 49 9.4178 0.6822 9.9866 3 20 5o 9 .4o83 48 9.42 3o 52 0.6770 9.9863 3 IO 15 O 9 .4i3o 47 9.4281 5' 0.6719 9.9849 4 o 75 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' PP 71 68 66 64 61 58 55 53 Si .1 7-i 6.8 6.6 .1 6. 4 6.1 5-8 .1 5-5 5-3 5-i .2 14.2 13.6 13.2 .2 12.8 12.2 ii. 6 .2 II. IO.6 IO.2 3 21.3 20.4 19.8 3 19.2 l8. 3 17.4 3 16.5 '5-9 >5-3 4 28.4 27.2 26.4 4 25.6 24-4 23.2 4 22. 21.2 2O.4 5 35-5 34-o 33-o 5 ^2.0 3-5 29.0 5 27.5 26.5 2S.5 .6 42.6 40.8 39- 6 .6 sM 36.6 34-8 .6 33-o 31.8 30.6 7 49-7 47.6 46.2 . 7 44.8 42.7 40.6 7 38.5 37- 1 357 .8 56.8 54-4 52.8 .8 51.2 48.8 46.4 .8 44.0 42.4 40.8 61.2 59-4 9 57-6 54-9 52-2 142 FOUR-PLACE LOGARITHMIC FUNCTIONS. O ' L.Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 15 O g.4i3o 9.4281 0.6719 9.9849 o 75 IO 9.417 7 9 .433i 0.6669 9- ? 846 5o 20 9-4223 46 9.4 5-81 0.6619 9.9843 3 4o 3o 9.4269 4 6 9-443o 49 o. 6670 9.9839 4 3o 4o g.43i4 9.4479 0.6621 9.9836 20 5o 9.4369 45 9.4627 4 0.6473 9 .9832 4 IO 16 g.44o3 44 9.4676 0.5426 9- 3828 4 o 74 10 9.4447 9.4622 0.5378 9.9826 5o 20 9.4491 44 9.4669 47 o.533i 9.9821 4 4o 3o 9.4533 42 9.4716 47 xfi 0.6284 9.9817 4 3o 4o 9.4676 9.4762 0.5238 9.9814 20 ; )0 9.4618 4 2 9.4808 46 0.6192 9.9810 4 IO 17 9.4669 9. 4853 45 0.6147 9 .9806 4 o 73 10 9.4700 9.4898 0.6102 9- 3802 5o 20 9.474 9.4943 45 0.6067 9.9798 4 4o 3o 9.4781 40 9.4987 44 o.5oi3 9 979 4 4 3o 4o 9.4821 9-5o3i 0.4969 9.9790 20 5o 9.4861 40 9.6076 44 0.4926 9.9786 4 IO 18 o 9.4900 39 9.6118 43 0.4882 9.9782 4 o 72 10 9.4939 9.6161 o.483g 9.9778 5o 20 9.4977 38 9.6203 42 0.4797 9.9774 4 4o 3o 9.6016 3 9.6245 42 0.4766 9.9770 4 3o 4o 9. 6062 9.6287 o.47i3 9.9766 20 5o 9. 6090 38 9.5329 42 0.4671 9.9761 4 IO 19 o 9.6126 36 9.5370 4 1 o.463o 9.9767 71 IO 9.5i63 9.641 i 0.4689 9.9762 5o 20 9.6199 36 9.545i 40 0.4549 9.9748 4 4o 3o 9.6235 30 9.5491 40 0.4609 9.9743 5 3o 4o 9.6270 9 . 553i 0.4469 9.9739 20 5o 9.53o6 36 9.6671 40 0.4429 9-< J7^4 5 IO 20 9.534i 35 9.6611 40 0.4389 9.9730 4 o 70 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' O PP 49 47 45 44 43 41 40 38 36 .1 4-9 4-7 4-5 .1 4-4 4-3 4.1 .1 40 3.8 3-6 .2 9.8 9.4 9.0 .2 8.8 8.6 8.2 .2 80 7 .6 7-2 3 14.7 14.1 13-5 3 13.2 12.9 12.3 3 12 O 11.4 10.8 4 19.6 18.8 18.0 4 17.6 17.2 16.4 4 16 o 1^.2 14.4 5 24.5 23.5 22.5 5 22. 21.5 20.5 5 20 o ig.O 18.0 .6 29.4 28.2 27.0 .6 26.4 25.8 24.6 .6 340 22.8 21.6 7 34-3 32.9 3'-5 7 30.8 30.1 28.7 . 7 280 26.6 25.2 .8 39.2 37.6 36.0 .8 35-2 34-4 32.8 .8 32 o 30.4 28.8 9 44-1 42.3 40.5 39- 6 38.7 36.9 9 360 34.2 32-4 i43 FOUR-PLACE LOGARITHMIC FUNCTIONS. > L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 20 o 10 20 3o 4o 5o 9.5341 9.5376 9.5409 9.5443 9.5477 9.55io 34 34 34 34 33 33 33 33 32 32 3' 3 2 3i 3i 30 3' 30 3 29 3 29 29 2 9 28 28 28 28 28 27 27 9. 56i i 9.565o 9.5689 9.5727 9.5766 9.58o4 39 39 38 39 38 38 37 38 37 37 37 36 36 36 36 36 35 36 35 34 35 34 35 34 34 33 34 33 34 33 0.4389 o.435o o.43i i 0.4273 o.4a34 0.4196 9.9730 9.9725 9.9721 9.9716 9.9711 9.9706 5 4 5 5 o 70 5o 4o 3o 20 10 21 o 10 20 3o 4o 5o 9.5543 9 .55 7 6 9.5609 9.5fi4i 9 .56 7 3 9.5704 9.5842 9.0879 9.5917 9.5954 9.5991 9.6028 o-4i58 o.4i 21 o.4o83 o.4o46 0.4009 0.3972 9.9702 9.9697 9.9692 9.9687 9.9682 9.9677 5 5 5 5 5 5 5 6 5 5 5 o 69 5o 4o 3o 20 IO 22 o 10 20 3o 4o 5o 9.5736 9.5767 9.5798 9.5828 9.585g 9.5889 9.6064 9.6100 9.6i36 9.6172 9.6208 9.6243 0.3936 0.3900 0.3864 0.3828 0.3792 o.3 7 5 7 9.9672 9.9667 9.9661 9.9656 9.9651 9 .9646 o 68 5o 4o 3o 20 I O 23 o 10 20 3o 4o 5o 9.5919 9 .5 9 48 9.5978 9.6007 9-6o36 9-6o65 9.6279 9-63i4 9-6348 g.6383 9.6417 9-6452 0.3721 0.3686 0.3652 0.3617 0.3583 0.3548 9.9640 9.9635 9. 9629 9.9624 9.9618 9.9613 5 6 5 6 5 o 67 5o 4o 3o 20 10 24 o 10 20 3o 4o 5o 9.6093 9.6121 9.6149 9.6177 9.6205 9.6232 9.6486 9-652O 9-6553 9.6587 9.6620 g.6654 o.35i4 o.348o 0.3447 o. 34 i 3 o.338o 0.3346 9.9607 9.9602 9.9596 9.9590 9 . 9 584 9-9 5 79 5 6 6 6 5 o 66 5o 4o 3o 20 IO 25 9.6259 9.6687 o.33i3 9. 9 5 7 3 o 65 L. Cos. d. L.Cotg. d. L. Tang. L. Sin. d. ' O PP .i .2 3 4 '.6 7 .8 9 39 37 35 .i .2 3 4 '.6 .8 34 33 33 .1 .2 3 4 .6 .8 9 31 30 29 3-9 3-7 7-8 7-4 11.7 u. i 15.6 14.8 19-5 '8.5 23.4 22.2 27.3 25.9 3 1. 1 29.6 35- i 33- 3 3-5 7.0 10.5 14.0 '7-5 21. 24.5 28.0 3'-5 3-4 6.8 IO.2 13.6 I 7 .0 20.4 2 3 .8 27.2 30.6 II 9-9 13.2 16.5 19.8 23-' 26.4 32 6. 4 9 .6 12.8 16 o 19.2 22.4 5- 6 28.8 3-i 6.2 9-3 12.4 iS-5 18.6 21.7 24.8 27.0 3- 6.0 9.0 12.0 15.0 18.0 21. 24.0 27.0 2 'i 58 8.7 ii. 6 '45 17-4 20.3 23.2 26. i 1 44 FOUR-PLACE LOGARITHMIC FUNCTIONS. O ' L . Sin. d. L.Tang. d. L. Cotg. L. Cos. d. 25 o 10 20 3o 4o 5o 9 9 9 9 9 9 .6269 .6286 .63i3 ,634o .6366 .6392 27 27 27 26 26 26 26 26 25 26 25 24 25' 25 24 24 24 9.6687 9.6720 9.6762 9.6786 9.6817 9.6860 33 S 2 33 32 33 32 32 32 32 32 3' 3' 3 3 3 3 3 3 29 30 29 3 29 29 o.33i3 0.3280 0.3248 o.32i5 o.3i83 o.3i5o 9.9673 9.9667 9.9661 9.9655 9.9649 9.9643 6 6 6 6 6 o 65 5o 4o 3o 20 10 26 o 10 20 3o 4o 5o 9 9 9 9 9 9 .64i8 .6444 .6470 .6496 .6621 .6546 9.6882 9.6914 9.6946 9.6977 9.7009 9.7040 o.3ii8 o.3o86 o.3o54 o.3o23 0.2991 0.2960 9.9637 9.9630 9.9624 9.9618 9.9612 9.9606 7 6 6 6 7 6 7 6 7 6 7 o 64 5o 4o 3o 20 IO 27 o 10 20 3o 4o 5o 9 9 9 9 9 9 .6670 .65 9 5 .6620 .6644 .6668 . 6692 9.7072 9.7103 9.7134 9. 7166 9.7196 9.7226 0.2928 0.2897 0.2866 0.2835 0.2804 0.2774 9.9499 9.9492 9.9486 9.9479 9-94?3 9.9466 o 63 5o 4o 3o 20 10 28 o 10 20 3o 4o 5o 9 9 9 9 9 9 .6716 .6740 .6763 .6787 .6810 .6833 24 24 23 24 23 23 23 22 23 22 23 22 22 9.7267 9.7287 9 . 7 3i 7 9 . 7 348 9.7408 0.2743 0.2713 0.2683 0.2662 0.2622 0.2692 9.9469 9.9453 9.9446 9.9439 9.9432 9.9426 7 6 7 7 7 7 o 62 5o 4o 3o 20 10 29 o 10 20 3o 4o 5o 9 9 9 9 9 9 .6856 .6878 .6901 .6923 .6 9 46 .6968 9. 7 438 9.7467 9.7497 9.7626 9.7666 9 . 7 585 0.2662 0.2533 o.25o3 0.2474' O.2444 0.24 i 5 9.9418 9.9411 9.9404 9-9 3 97 9.9390 9-9383 7 7 7 7 7 o 61 5o 4o 3o 20 10 30 9 .6990 9.7614 0.2386 9.9376 o 60 L . Cos. d. L. Cotg. d. L.Tang. L. Sin. d. 1 O PP 28 27 26 25 24 23 22 7 6 o 6 .2 3 4 '.I .9 5-6 8.4 II. 2 14.0 16.8 19.6 22.4 25.2 5-4 8.1 10.8 '3-5 16.2 18.9 21.6 24.3 5-2 7-8 10.4 13.0 15-6 18.2 20.8 23.4 .2 3 4 9 5.0 4-8 7-5 7-2 10.0 9.6 I2.S I2.O 15.0 14.4 17.5 16.8 20.0 19.2 22.5 21.6 ? 9-2 "5 13-8 16.1 18.4 20.7 .2 3 4 '.6 '.& 4-4 6.6 8.8 II. 13.2 19.8 1.4 2.1 2.8 3-5 4.2 4.9 5-6 6-3 1.2 1.8 2.4 3' 3-6 4-2 4.8 5-4 i.45 FOUR-PLACE LOGARITHMIC FUNCTIONS O ' L. Sin. d. L. Tang. d. L. Cotg. L. Cos. d. 30 o IO 20 3o 4o 5o 9.6990 9.7012 9.7033 9.7065 9.7076 9.7097 22 21 22 21 21 21 21 21 21 2O 21 3O 2O 2O 20 20 2O 19 19 2O J 9 19 19 '9 18 19 ll '9 18 18 9.7614 9.7644 9.7673 9.7701 9 . 77 3o 9.7769 30 29 28 29 29 29 28 29 28 29 28 28 28 28 28 28 27 28 28 27 28 27 28 27 27 27 27 27 27 27 0.2386 0.2356 0.2327 0.2299 0.2270 0.224l 9.9375 9.9868 9.9361 9.9353 9.9346 9.9338 7 7 8 7 8 o 60 5o 4o 3o 20 IO 31 o IO 20 3o 4o 5o 9.7118 9.7189 9.7160 9.7181 9.7201 9. 7222 9.7788 9.7816 9.7845 9.7873 9.7902 9.7930 'O.22I2 0.2184 o.2i55 0.2127 0.2098 0.2070 9.9331 9.9323 9 . 9 3i5 9.9308 9.9300 9.9292 7 8 8 7 8 8 o 59 5o 4o 3o 20 10 32 o 10 20 3o 4o 5o 9.7242 9.7262 9.7282 9.7302 9.7322 9.7342 9.7968 9.7986 9.8014 9.8042 9.8070 9.8097 O.2O42 O.20l4 o. 1986 o. 1968 o. igSo o. 1903 9.9284 9.9276 9.9268 9.9260 9.9262 9.9244 8 8 8 8 8 o 58 5o 4o 3o 20 10 33 o IO 20 3o 4o 5o 9.7361 9.7380 9.7400 9.7419 9. 7 438 9.7457 9.8125 g.8i53 9.8180 9.8208 9.8235 9.8263 0.1875 0.1847 o. 1820 o. 1792 o. 1765 0.1737 9.9236 9.9228 9.9219 9.9211 9.9203 9.9194 8 9 8 8 9 o 57 5o 4o 3o 20 IO '34 o IO 20 3o 4o 5o 9.7476 9.7494 9.7613 9.7531 9.7550 9.7668 9.8290 9.8317 9 .8344 9.8371 9.8398 9.8426 0.1710 o.i683 o. i656 o. 1629 o. 1602 o. 1676 9.9186 9.9177 9.9169 9.9160 9.9161 9.9142 9 8 9 9 9 8 o 56 5o 4o 3o 20 IO 35 9.7586 9.845a o.i548 9.9134 o 55 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' PP .i 29 38 37 .1 33 21 30 .1 19 8 7 2. Q 2.8 t8 z 6 2.7 '2.2 2.1 2.0 1.9 3 8 0.8 1.6 0.7 3 4 '.6 '.8 9 .7 8.4 II. 6 II. 2 14.5 14.0 17.4 16.8 20. 3 19.6 23.2 22.4 26.1 25.2 ft 10.8 '3-5 16.2 18.9 21.6 24.3 3 4 '.6 :l 9 6.6 8.8 II. O 13.2 15-4 17.6 19.8 6-3 8.4 10.5 12. M-7 16.8 18.9 6.0 8.0 o.o 2.O 4.0 6.0 3 4 '.6 :l 9 5-7 7.6 9-5 11.4 J3-3 15.2 17.1 2-4 3-2 *% 4.8 5.6 6.4 7-2 2.1 2.8 3-5 4-2 4.9 56 6-3 1 46 FOUR PLACE LOGARITHMIC FUNCTIONS. O ' L. Sin. d. L.Tang. d. L. Cotg L. Cos. 9 9 9 9 9 9 10 9 9 IO 9 35 o 10 20 3o 4o 5o 9.7686 9. 7604 9.7622 9.7640 9.7667 9.7676 18 18 18 17 18 '7 18 '7 16 17 16 17 16 16 16 15 16 16 9.8462 9.8479 9.8606 9. 8533 9.8669 9.8686 27 27 27 26 27 27 26 27 26 26 27 26 26 27 26 26 26 26 26 26 26 26 26 26 26 25 26 26 25 26 o.i548 o. 1621 0.1494 o. 1467 o. i 44 i O. I-i I 4 ON ON ON ON ON ON 9134 9126 91 16 9107 9098 90^9 o 55 5o 4o 3o 20 IO 36 o 10 20 3o 4o 5o 9. 7692 9.7710 9.7727 9.7744 9.7761 9.7778 9-86i3 9.863g 9.8666 9.8692 9.8718 9.8746 0.1387 o. i36i o. i 334 o.i3o8 o. 1282 o. 1266 9.9080 9.9070 9.9061 9.9062 9.9042 9.9033 o 54 5o 4o 3o 20 10 37 o 10 20 3o 4o 5o 9.7796 9.781 i 9.7828 9-7844 9.7861 9.7877 9.8771 9.8797 9.8824 9.8860 9.8876 9.8902 o. 1229 O. I2O3 o. i 176 o. 1160 o. i 124 o. 1098 9.9023 9.9014 9 . 9004 9.8996 9.8986 9.8976 9 10 9 IO 10 o 53 5o 4o 3o 20 10 38 o 10 20 3o 4o 5o 9 9 9 9 9 9 .7893 .7910 . 7926 .7941 .7967 .7973 9.8928 9.8964 9.8980 9.9006 9.9032 9.9068 o. 1072 o. io46 o. 1020 0.0994 0.0968 0.0942 9.8966 9.8966 9.8945 9.8935 9.8926 9.8916 IO IO 10 IO 10 o 52 5o 4o 3o 20 10 39 o 10 20 3o 4o 5o 9 9 9 9 9 9 .7989 .8004 .8020 .8o35 .8060 .8066 15 16 15 16 9.9084 9.9110 9.9135 9.9161 9.9187 9.9212 0.0916 0.0890 o.o865 0.0839 o.o8i3 0.0788 9.8906 9.8896 9.884 9.8874 9.8864 9.8853 IO II 10 IO II o 51 5o 4o 3o 20 10 40 9 .8081 15 9.9238 0.0762 9.8843 o 50 L .Cos. d. L. Cotg. d. L.Tang. L. Sin. d. ' O PP .1 .2 3 4 9 26 25 18 .1 .2 3 4 '.6 '.S 9 '7 16 15 .2 3 4 .6 9 ii IO 9 2.6 5-2 7-8 10.4 13.0 15-6 18.2 20.8 23.4 2-5 5.0 7-5 IO.O 12.5 15.0 17-5 20. o 22.5 1.8 3-6 5-4 7.2 9.0 10.8 12.6 14.4 16 2 '7 3-4 6.8 8-5 10.2 II.O 13.6 1.6 32 4.8 6.4 8.0 9.6 II. 2 12.8 14.4 i-5 4-5 6.0 7-5 9.0 10.5 12.0 1 3- 5 i.i 2.2 3-3 4-4 u u 9.9 I.O 2.O 4.0 6!o 7.0 8.0 0.9 1.8 2.7 3-6 4-5 5-4 6.3 7.2 8.1 FOUR-PLACE LOGARITHMIC FUNCTIONS. O ' L. Sin. d. L.Tang. d. L. Cotg. L .Cos. d. 40 o 10 20 3o 4o 5o 9.8081 9.8096 9.8111 9.8125 9.8140 9.8r55 15 '5 M 15 '5 M '5 M '5 '4 '4 9.9238 9.9264 9.9289 9-93i5 9 . 9 34i 9.9366 26 25 26 26 25 26 25 26 25 26 25 25 26 25 26 25 25 26 25 25 25 25 25 25 26 25 25 25 26 S 0.0762 0.0736 0.071 1 o.o685 0.0659 o.o634 9 9 9 9 9 9 .8843 .8832 .8821 .8810 .8800 .8789 ii ii ii IO II II II II II 12 II o 50 5o 4o 3o 20 10 41 o 10 20 3o 4o 5o 9.8169 9.8184 9.8198 9.8213 9.8227 9.8241 9.9392 9.9417 9.9443 9.9468 9.9494 9 . 9 5i 9 0.0608 o.o583 o.o557 o.o532 o.o5o6 o.o48i 9 9 9 9 9 9 .8778 .8767 .8756 .8 7 45 .8733 .8722 o 49 5o 4o 3o 20 10 42 o IO 20 3o 4o 5o 9.8255 9.8269 9.8283 9.8297 9-83u 9-8324 M '4 '4 M '3 M '3 '4 '3 13 14 3 '3 '3 '3 12 13 13 9.9544 9.9370 9.9595 9.9621 9.9646 9.9671 o.o456 o.o43o o.o4o5 0.0379 o.o354 0.0329 9.8711 9.8699 9.8688 9.8676 9-8665 9.8653 12 II 12 II 12 o 48 5o 4o 3o 20 IO 43 o 10 20 3o 4o 5o 9.8338 9.835i 9.8365 9.8378 9.8391 9.84o5 9.9697 9.9722 9-974? 9.9772 9.9798 9.9823 o.o3o3 0.0278 0.0253 0.0228 O.O2O2 0.0177 9 9 9 9 9 9 .864i .8629 .8618 .8606 .85 9 4 .8582 12 II 12 12 12 '3 12 12 '3 12 '3 12 o 47 5o 4o 3o 20 10 44 o 10 20 3o 4o 5o 9.84(8 9.843i 9-8444 9.8457 9.8469 9.8482 9.9848 9.9874 9.9899 9.9924 9.9949 9.9975 O.Ol52 0.0126 O.OIOI 0.0076 o.ooSi O.OO25 9 9 9 9 9 9 .8569 -855 7 .8545 .8532 .8520 -85o7 o 46 5o 4o 3o 20 10 45 o 9.8495 o.oooo o . oooo 9 .8495 o 45 L. Cos. d. L. Cotg. d. L. Tang. L. Sin. d. ' O PP 36 *5 15 14 13 13 .1 ii 10 .1 2.6 2-5 1.5 .1 1.4 "3 1.2 i.i I.O 3 4 '.6 '.8 '? 78 10.4 '3 156 iS 2 20.8 23 4 7-5 10.0 12 5 150 '7 5 20. o 4-5 -3 6.0 .4 75 -5 9.0 .6 10.5 .7 12.0 .8 4-2 5-6 7.0 8.4 9.8 II. 2 12.6 3-9 5.2 6 -l 7.8 9.1 104 3.6 4.8 6.0 7-2 8.4 9 '2 10.8 3 4 .6 .S 3-3 4-4 5-5 6.6 11 9-9 3- 4.0 5-o 6.0 7 8.0 9.0 i48 TABLE VII FOUR-PLACE NATURAL TRIGONOMETRIC FUNCTIONS TO EVERY TEN MINUTES FOUR-PLACE NATURAL FUNCTIONS. ' Sin. d. Tang. d. Cotg. d. Cos. d. o 10 2O 3o 4o 5o , o.oooo 0.0029 o.oo58 0.0087 o.oi 16 o.oi45 29 29 29 29 29 o.oooo 0.0029 0.0068 0.0087 o.oi 16 o.oi45 29 29 29 29 29 30 29 29 29 29 29 29 29 29 3 29 29 29 29 29 3 29 29 29 3 29 29 29 3 29 infinit. 171.8864 114.6887 86.9398 68.7601 1st 6'. 477 J* 313 fc 22C. 188 ,6; 143 at 112 IOL 9<: Si 74 6h 62 5/ 5= 45 4: 4- 3< I I I o o .0000 . oo'oo .0000 .0000 9999 9999 o o I o I o I I I I I 2 I I 2 I 2 2 I 2 2 2 3 2 2 3 2 o 90 5o 4o 3o 20 10 1 o 10 20 3o 4o 5o 0.0176 O.O2O4 O.O233 0.0262 0.0291 O.O32O 29 29 29 29 29 0.0176 O.O2O4 0.0233 0.0262 0.0291 0. O320 67.2900 49. 1039 42.9641 38.1885 34.3678 3 i .2416 61 tf 5'' 01 '. 2 o o o .9998 .9998 9997 9997 .9996 .9996 o 89 5o 4o 3o 20 IO 2 o 10 20 3o 4o 5o 0.0378 0.0407 o.o436 o.o465 0.0494 29 29 29 29 29 29 29 29 29 29 3 29 29 29 29 29 29 29 29 0.0378 0.0407 o.o466 0.0496 28.6363 26.43i6 22.9038 21 .4?o4 2O.2O56 33 47 98 So 34 4 o o o o o o .9994 . 999 3 .9992 .9990 .9989 .9988 o 88 5o 4o 3o 20 10 3 o 10 20 3o 5o 0.0623 _ 0.0662 0.0681 0.0610 o.o64o 0.0669 0.0624 o.o553 0.0682 0.0612 o.o64 i 0.0670 19.081 I 18.0760 17. ID93 16.3499 i5.6o48 14.9244 6 1 57 '.'4 5i 04 37 4 u y8 "7 57 43 6, o o o .9986 .9986 .9933 .9981 .9980 .9978 o 87 5o 4o 3o 20 IO 4 o 10 20 3o 4o 5o 0.0698 0.0727 0.0766 0.0786 0.08 i 4 o.o843 0.0699 0.0729 0.0768 0.0787 0.0816 o.o846 14.3007 13.7267 i 3 . i 969 12.7062 12.2606 i i .8262 o o o I) .9976 9974 .9971 .9969 .9967 .9964 o 86 5o 4o 3o 20 10 5 o o. 0872 0.0876 i i .43oi (1 .9962 o 85 Cos. d. Cotg. d. Tang. d. Sin. d. ' O PP .1 .2 3 4 .8 9 26053 16380 "245 8194 623* 4907 .1 .2 3 4 5 .6 : 7 8 9 396: 30 29 2605 5211 7816 10421 13027 15632 18237 20842 23448 1638 3276 4914 6552 8190 9828 11466 13104 14742 1125 .1 2249 .2 3374 -3 4498 .4 5623 .5 6747 .6 7872 .7 8996 .8 IOI2I .9 819.4 1638.8 3277-6 4097.0 4916.4 5735-8 6555-2 '374 6 623.7 1247.4 1871.1 2494.8 3742-2 4365.9 4989.6 490.7 981.4 1472.1 1962.8 2453-5 2944.2 3434-9 3925.6 4416. 396.1 792.2 1188.3 1584-4 1980.5 2376.6 2772.7 3168.8 35*49 3.0 2.0 6.0 5.8 9.0 8.7 12. II. 6 iS-o M-5 18.0 17.4 21. 2O.3 24.0 23.2 270 26.1 160 FOUR-PLACE NATURAL FUNCTIONS. o / Sin. d. Tang. d. Cotg. d. Cos. d. 3 2 3 3 3 5 o IO 20 3o 4o 5o 0.0872 0.0901 0.0929 o.ogSS 0.0987 o. 1016 29 28 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 29 28 29 29 29 28 29 29 28 29 29 28 d. o o .0875 .0904 .0934 .0963 .0992 . 1 022 29 30 29 29 30 29 29 30 29 3 29 3 29 30 3 29 3 29 3 30 3 29 3 3 30 3 29 30 3 30 d. n.43oi II .0594 10.7119 10.3854 10.0780 9.78*2 37 34 32 30 28 *7 as a* 33 22 at 20 '9 it *7 16 16 *5 14 *3 13 12 12 II II IO to so 9 ^H d 37 75 53 74 ^8 38 ," 55 29 '4 ->7 3 26 +6 7i 00 33 -2 '3 57 06 58 IO 68 26 B6 5 81 mmm . O o o c .9962 99 5 9 99 5 7 . 99 54 -99 51 9948 o 85 5o 4o 3o 20 IO 6 o 10 20 3o 4o 5o o. o. 0. o. o. o. io45 1074 I io3 Il32 1 161 1 190 o . io5i . 1080 . i no . 1 1 3g . 1 169 .1198 9.5i44 9.2553 9.0098 8.7769 8.5555 8.345o o o 9945 9942 99 3 9 .9936 .9932 .9929 3 3 3 3 4 3 o 84 5o 4o 3o 20 IO 7 o 10 20 3o 4o 5o o. o. 0. o. o. o. 1219 1248 1276 i3o5 1 334 i363 .1228 . 1257 .1287 .1346 .i3 7 6 8.1443 7.9530 7.7704 7-5 9 58 7.4287 7.2687 0.9925 0.9922 0.9918 0.9914 0.991 1 0.9907 4 3 4 4 3 4 o 83 '5o 4o 3o 20 IO 8 o 10 20 3o 4o 5o o. o. 0. 0. 0. o. 1392 i449 i4?8 i5o7 i536 o o o o o .i4o5 .i435 .1465 . i495 . 1 524 .i554 .i584' .1614 .1644 .1673 . 1703 .i 7 33 7. i i 54 6.9682 6.8269 6.6912 6.56o6 6.4348 0.9903 0.9899 0.9894 0.9890 0.9886 0.9881 4 4 5 4 4 5 4 5 4 5 5 5 o 82 5o 4o 3o 20 10 9 o IO 20 3o 4o 5o o. o. o. o. o. 0. 1 564 i5g3 1622 i65o 1679 1708 6. 3 i 38 6. 1970 6.o844 5.9758 5.8708 5.7694 0.9877 0.9872 0.9868 0.9863 0.9858 0.9853 o 81 5o 4o 3o 20 10 10 o i o. 1736 Cos. o. 1763 Cotg. 5.6 7 i3 Tang. Ml .9848 ^cr Sin. HIM d. o 80 ommm^m* f PP 2738 1533 981 .1 .2 3 4 '.6 3 *9 28 .1 .2 3 5 4 3 .1 273.8 2 547- 6 3 821.4 -4 I09S-2 -5 1369-0 .6 1642.8 .7 1916.6 .8 2190.4 .9 2464.2 '53-3 306.6 459-9 613.2 766.5 919.8 1073.1 1226.4 1370.7 98.1 196.2 294-3 392-4 490.5 588.6 686.7 784.8 882.9 3-0 6.0 9.0 2.9 i: 7 2.8 5-6 8.4 0.5 I.O 0.4 0.8 1.2 0.6 -9 10.0 21.0 24.0 27.0 "7-4 20.3 23.2 26.1 14.0 16.8 19.6 22.4 25.2 .6 9 2.5 3- 3-5 4.0 4-5 2.O 2-4 2.8 3-2 3-6 '5 1.8 2.1 2-4 2-7 FOUR-PLACE NATURAL FUNCTIONS. o ' < Jin. (1. Tang d. Cot ?. C L Cos. d. 10 to 10 3o $0 5o O O o. o. o. o. 1736 1765 1794 1822 i85i 1880 29 29 28 29 29 o. 176 0.179 0.182 o.i85 0.188 0.191 3 3 3 3 3 4 3 3 3 30 3' 5.67 5.5? 5.48 5.3 9 5.3o 5.22 i3 64 45 55 9 3 57 9 9 a & 8 49 9 < > Sa ;'. 0.9848 0.9843 0.9838 0.9833 0.9827 0.9822 5 5 5 6 5 o 80 5o 4o 3o 20 IO 11 , o to 10 Jo io io o. o. o. o. o. 0. 1908 1937 i 9 65 1994- 2O22 2o5i 29 28 29 28 29 0.194 0.197 O.2OO O.2o3 O.2O6 0.209 4 4 4 5 5 5 3 3 3 1 3 3 5.i4 5.o6 4.98 4.91 4.84 4-77 46 58 94 52 3o 2 9 t t 7 y 7 f. 3S '4 22 11 3, 0.9816 0.9811 0.9805 0.9799 o. 979 3 0.9787 S 6 6 6 6 o 79 5o 4o 3o 20 IO 12 t 10 JO Jo fo )0 0. o. o. o. o. o. 2079 2108 2i36 2164 2193 2221 29 28 28 29 28 O.2I2 O.2l5 0.218 O.22I O.224 O.227 6 6 6 7 7 s 3 3 3i 3 3i 4.70 4.63 4.5 7 4.5i 4.44 4.38 46 82 36 07 94 97 6 6 6 A 5 4 t* 20 3 '7 b 0.9781 0.9775 0.9769 0.9763 0.9757 0.9750 6 6 6 6 7 o 78 5o 4o t 3o 20 10 13 i JO io fo o. o. o. o. o. o. 2260 2278 23o6 2334 2363 23 9 i 29 28 28 28 29 28 O.23o 0.233 0.237 O.24o 0.243 O.246 9 9 I 2 2 30 3' 3' 3' 30 4.33 4.27 4.21 4.16 4. 1 1 4.06 i5 47 9 3 53 26 1 1 5 5 5 5 S '.8 M to IJ 5 0.9744 0.9737 0.9730 0.9724 0.9717 0.9710 7 7 6 7 7 o 77 5o 4o 3o 20 10 14 i \ 10 Jo |o io o. o. o. o. 0. o. 2419 2447 2476 25o4 2.532- 256o 28 29 28 28 28 O.249 O.252 o.255 o.s58 0.261 0.264 i 4 5 6 7 S 31 3" 3' 3' 3> 4.01 3.96 3.91 3.86 3.82 3. 77 38 17 36 5? 38 5o 4 4 4< 4! 44 )' b Sj 9 s 0.9703 0.9696 0.9689 0.9681 0.9674 0.9667 7 7 7 8 7 7 o 76 5o 4o 3o 20 IO 15 o. 2588 0.267 9 3.73 21 0.9659 o 75 c OS. d. Cotg d. Tan S- d . Sin. d. ' PP 7 43 448 31 30 29 a8 7 6 5 .i .2 3 4 '.6 IB g 3 14 K ~< 35 44 Ji 4.2 8.4 2.6 ,6.8 I.O 5-2 9-4 3.6 7.8 44.8 89.6 134-4 179.2 224.0 268.8 313-6 3584 6. 9- 12. 15- 18. 21. 24- i .1 2 .2 3 -3 4 -4 5 -5 5 .6 7 -7 8 .8 i i i 2 2 3-o 6.0 9.0 2.O 5-o 8.0 I.O 4-o 2.9 * 5-8 8-7 n.4 M-5 17-4 20.3 *3- 2 2. s'. II. 14- 16. 19. 22. s 6 4 2 ) 3 6 4 1 i 0.7 2 1.4 3 2.1 4 2.8 5 3-5 6 4.2 7 4-9 8 5.6 0.6 1.2 1.8 2-4 3- 3.6 4-2 4-8 -5 I.O '5 2.O 2-5 3-o 3-5 4.0 152 FOUR-PLACE NATURAL FUNCTIONS. O ' Sin. d. Tang. d. Cotg, d. Cos. d. 15 o 10 20 3o 4o 5o o. 0. o. 0. o. o. 2588 2616 2644 2672 2700 2728 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 27 28 28 27 28 =7 28 27 27 28 27 o. 2679 0.271 1 0.2742 0.2773 o.28o5 0.2836 32 31 32 31 3' 3 2 32 3' 32 3 1 32 3 2 32 32 32 3 2 S 2 33 32 S 2 33 32 33 S 2 33 33 33 33 3.7321 3.6891 3.64?o 3.6o59 3.5656 3.526i 4; 3* 3' 3 35 * 3! 3! 31 3: 3: y 31 9 3< 3 r 2( a ai 2S 5 5 at at 2 2_ a; I >3 B 7 9 i 5 7 o 3 3 B 5 9 3 >7 12 7 i 7 i 7 2 3 3 9 5 o 0.9659 0.9652 0.9644 0.9635 0.9628 0.9621 7 8 8 S 7 o 75 5o 4o 3o 20 IO 16 o 10 20 3o 4o 5o o. 0. o. o. o. 0. 2 7 56 2784 2812 2840 2868 2896 0.2867 o. 2899 0.2931 0.2962 0.2994 o. 3o26 3. 48 7 4 3.4495 3.3769 3. 34o2 3.3o52 0.9613 0.9605 0.9596 0.9688 0.9580 0.9572 8 9 8 8 8 o 74 5o 4o 3o 20 IO 17 o IO 20 3o 4o 5o o. o. o. o. o. o. 2924 2952 2979 3007 3o35 3o62 o. 3o8g 0.3I2I o.3i53 o.3i85 0.3217 3.2709 3.2371 3.2o4i 3.1716 3. 1 397 3.io84 0.9553 o. 9 555 0.9546 0.9537 0.9528 0.9520 9 8 9 9 9 8 9 9 IO 9 9 9 o 73 5o 4o 3o 20 10 18 o 10 20 3o 5o o. o. o. o. 0. o. 3ogo 3n8 3i45 3i 7 3 32OI 3228 0.3249 o.328i o.33i4 0.3346 0.3378 o.34n 3.0777 3.0178 2.9887 2.9600 2.9319 o.g5i i 0.9602 0.9492 0-9483 0.9474 o 72 5o 4o 3o 20 IO 19 o 10 20 3o 4o 5o o. o. o. o. 0. o. 3256 3 2 83 33n 3338 3365 33 9 3 0.3443 0.3476 o.35o8 o.354i 0.3574 0.3607 2.9042 2.8770 2.85O2 2.8239 2.7980 2.7725 0.9455 0.9446 0.9436 0.9426 0.9417 0.9407 9 IO IO 9 IO o 71 5o 4o 3o 20 10 20 o. 3420 o.364o 2.7475 0.9397 o 70 Cos. d. Cotg. d. Tang. d Sin. d. ' PP .1 ,2 3 4 '.6 :l 9 255 33 3* 31 28 37 ,i .2 3 4 .6 !i 9 IO 9 8 25-5 5.o 76.5 102.0 127-5 153-0 178.5 204.0 6.6 9-9 %l 19.8 23-1 26.4 3.2 .1 6.4 .2 9.6 .3 12.8 .4 16.0 .5 19.2 .6 22.4 .7 25.6 .8 28.8 .9 6.' 2 9-3 12.4 '5-5 1 8.6 21.7 24.8 2.8 5-6 8-4 II. 2 14.0 1 6. 8 19.6 22.4 2-7 5-4 8.1 10.8 13 5 16.2 18.9 21.6 24-3 I.O 2.0 3.0 4.0 6.0 7.0 8.0 0.9 1.8 2-7 3-6 4-5 5-4 6-3 7.2 8.1 0.8 1.6 2-4 3-2 4.0 48 5-6 6.4 i53 FOUR-PLACE NATURAL FUNCTIONS. O f Sin. d. as 27 27 27 28 27 27 27 27 27 27 27 27 27 27 27 27 26 27 27 26 27 27 26 27 26 27 26 27 26 Tang . d. Cotg. d. Cos. d. 20 o 10 20 3o 4o 5o o. o. o. o. 'o. 0. 3420 3448 34 7 5 35o2 3539 355 7 o. 364o o.36 7 3 o.3 7 o6 o.3 7 39 o.3 77 2 o.38o5 33 33 33 33 33 34 33 34 33 34 33 34 34 34 34 34 34 2. 7 4 7 5 2. 7 228 2.6985 2.6 7 46 2. 65 i i 2.6279 -'. -'- 2_ 2; 2; 2: th 2! 21 21 21 2C 2C 2C 2C '9 19 iC '9 if 18 if 18 '7 '7 '7 '7 16 16 16 7 3 9 5 2 S t I 9 4 2 9 3 o 7 5 i 6 5 i 11 7 4 3 n S i o. 9 3 97 o.9~3 7 -9 3 77 o.936 7 0.9356 0.9346 10 n IO ' 70 5o 4o 3o 20 10 21 o 10 20 3o 4o 5o o. o. 0. o. o. o. 3584 36n 3638 3665 3692 3719 i i I ( < .383 9 .38 7 2 .3go6 .3 9 3 9 .3 97 3 .4oo6 2.6o5i 2.5826 2 .56o5 2.5386 2.5l 7 2 2 .4960 0.9336 o. 9 325 o.gSiS 0.9304 0.9293 0.9283 II 10 II II IO o 69 5o 4o 3o 20 IO 22 o 10 20 3o 4o 5o o. o. b. o. o. o. 3 7 46 3 77 3 38oo 382 7 3854 388i ( < 4o4o 4o 7 4 .4108 .4142 4i 7 6 .4210 2.4 7 5i . 2.4545 2.4342 2.4i4z 2.3945 2.3 7 5o 0^9272 0.9261 0.9250 0.9239 0.9228 0.9216 II II II II 12 o 68 5o 4o 3o 20 10 23 o 10 20 3o 4o 5o o. o. o. o. o. o. 3 97 3 9 34 3961 3 9 8 7 4oi4 4o4i b o .4245 42 7 9 .43i4 .4348 .4383 44i 7 34 35 34 35 34 35 35 35 35 35 36 35 2.3559 2.3369 2.3i83 2.2998 2.28l 7 2.263 7 0.9205 0.9194 0.9182 o.9i 7 i 0.91 59 o.9i4 7 II 12 II 12 12 o 67 5o 4o 3o 20 IO 24 o 10 20 3o 4o 5o o. o. o. o. o. o. 4o6 7 4094 4l2O 4i4 7 4'i?3 4200 o .4452 448 7 .4522 .455 7 .4592 .4628 2.246O 2.2286 2.21 l3 2.1 9 43 2.1 77 5 2. 1609 o o o o o .9135 .9124 .9112 .9100 .9088 . 9 o 7 5 II 12 12 12 '3 12 o 66 5o 4o 3o 20 IO 25 0. 4226 (1 .4663 2.1445 o .9063 o 65 Cos. d. cotg. d. Tang. d. Sin. d. ' O PP .1 .2 3 4 '.6 7 .8 9 177 35 34 3-4 6.8 IO.2 .3-6 17.0 20.4 2 3 .8 27.2 30.6 ii .2 3 4 .(, .8 33 97 26 ,i .2 3 4 !e .8 12 ii 10 17.7 35-4 53-i 70.8 88.5 106.2 123.9 141.6 '59-3 3-5 7.0 10.5 14.0 '7-5 21.0 24-5 28.0 3-3 6.6 9-9 '3-2 .6.5 19.8 23.1 26.4 2.7 5-4 8.1 10.8 '3-5 16.2 18.9 21.6 24.3 2.6 5 "o 7 .8 10.4 I 3 .0 I 5 .6 18.2 208 23-4 1.2 2.4 3-6 4.8 6.0 7.2 8.4 9.6 10.8 i.i 2.2 3-3 4-4 u 7-7 8.8 9-9 I.O 2.0 3 4.0 5 6.0 7.0 8.0 9.0 1 54 FOUR-PLACE NATURAL FUNCTIONS. O ' Sin. d. Tang. 1 d. Cotg. d. Cos. d. 25 o 10 20 3o 4o 5o 0. o. o. o. 0. o. 4226 4253 4279 43o5 433i 4358 27 26 26 26 27 26 26 26 26 26 26 26 26 26 25 26 26 26 25 . 26 26 25 26 25 26 25 25 26 25 25 0.4663 0.4699 o .4?34 0.4770 o.48o6 o.484i 36 35 36 36 35 36 36 37 36 36 37 36 37 37 37 37 37 37 37 38 38 37 38 38 38 38 39 38 39 39 2. I 445 2.1283 2. 1123 2.0965 2.0809 2 .o655 162 160 158 '56 i54 152 150 '49 '47 MS 144 142 140 139 137 136 '34 133 . IS' 130 128 127 126 "5 123 121 121 119 116 o o t .9063 .9038 .9026 .9013 .9001 12 '3 12 '3 12 o 65 5o 4o 3o 20 IO 26 o 10 20 3o 4o 5o o. o. o. o. ' O. 0. 4384 44 10 4436 4462 4488 45i4 0.4877 o.4gi3 o . 495o 0.4986 0.5o22 o.5o5g 2.o5o3 2.O2O4 2.0057 1.9912 1.9768 0.8988 0.8975 0.8962 0.8949 o.8 9 36 0.8923 13 '3 '3 13 13 '3 '3 '3 14 '3 M 14 '3 '4 M M '4 '4 '4 '5 '4 o 64 5o 4o 3o 20 IO 27 o 10 20 3o 4o 5o o. o. o. 0. o. o. 454o 4566 4592 4617 4643 4669 o . SogS o.5 i 32 o.5i6g o.52o6 0.5243 0.5280 i .9626 i .9486 1.9347 i .9210 i .9074 0.8910 0.8897 0.8884 0.8870 0.8857 0.8843 o 63 5o 4o 3o 20 IO 28 o 10 20 3o 4o 5o 0. o. o. o. 0. o. 4720 4?46 4772 4797 4823 o.53i7 0.5354 0.5392 o. 543o 7-2 5 21.5 .6 25.8 7 3-i 8 34-4 9 38.7 4.2 8. 4 12.6 16.8 21.0 25.2 29.4 33-6 I 1 8.2 12.3 ,6.4 20.5 24.6 28.7 32.8 36.9 4.0 8.0 12.0 16.0 20. o 24.0 28.0 32.0 2.5 S.o 7-5 IO.O 12.5 15.0 '7-5 2O.O 2. 4 .1 4.8 7-2 -3 9.6 .4 12. .5 14.4 .6 16.8 .7 19.2 .8 21.6 .9 '7 3-4 5-1 6.8 8-5 IO.2 II.9 I 3 .6 1.6 11 & 9.6 II. 2 12.8 14-4 '5 3- 45 6.0 75 9.0 10.5 12.0 '3-5 1 56 FOUR PLACE NATURAL FUNCTIONS. o / Sin. d. Tang. d. Cotg. d. Cos. d. 35 o 10 20 3o 4o 5o o. o. 0. o. 0. o. 5 7 36 6760 5 7 83 6807 583i 5854 24 23 24 24 23 24 23 24 23 24 23 23 23 24 23 23 23 o. 7002 0.7046 0.7089 0.7133 0.7177 0.7221 44 43 44 44 44 44 45 45 45 45 45 46 45 46 46 47 46 1.4281 1.4193 i .4io6 i .4019 i .3g34 1.3848 88 87 87 85 86 84 84 83 83 82 81 81 80 79 79 78 78 77 76 76 75 75 74 0.8192 0.8175 o.8i58 o.8i4i 0.8124 0.8107 17 J 7 i? !7 '7 o 55 5o 4o 3o 20 IO 36 o 10 20 3o . 4o 5o o. o. o. 0. o. o. 58 7 8 5goi 5920 5 9 48 5 97 2 5995 0.7265 0.7310 'o. 7 355 0.7400 0.7445 0.7490 1.3764 i.368o i .3597 i.35i4 1.3432 i.335i 0.8090 0.8073 o.8o56 0.8039 0.8021 o.8oo4 J 7 *7 '7 18 i? o 54 5o 4o 3o 20 IO 37 o 10 20 3o 4o 5o o. o. o. 0. o. o. 6018 6o4i 6o65 6088 61 1 1 6 1 34 0.7536 0.7581 0.7627 0.7673 0.7720 0.7766 i .3270 i .3190 i.3m i.3o32 i .2954 1.2876 0.7986 o. 7969 0.7951 0.7934 0.7916 0.7898 17 18 '7 18 18 o 53 5o 4o 3o 20 IO 38 o 10 20 3o 4o 5o o. o. o. o. o. 0. 6i5 7 6180 6202 ' 6225 6248 6271 23 23 22 23 23 2 3 22 23 22 23 22 23 0.7813 0.7860 0.7907 0.7954 0.8002 o.8o5o 47 47 47 47 48 48 1.2799 i .2723 i .2647 i .2572 1.2497 1.2423 (. i o .7880 7?62 .7844 .7826 .7808 779 18 18 18 18 18 o 52 5o 4o 3o 20 10 39 o 10 20 3o 4o 5o o. o. o. o. o. o. 6293 63i6 6338 636i 6383 64o6 0.8098 o.8i46 0.8195 0.8243 0.8292 0.8342 40 48 49 48 49 5 I .2349 I .2276 I .22O3 I.2l3l i . 2o5g 1.1988 74 73 73 72 72 7i o .7771 . 77 53 . 77 35 .7716 .7698 .7679 *9 18 18 J 9 18 19 o 51 5o ' 4o 3o 20 IO 40 O o. 6428 0.8391 49 i . 1918 70 c .7660 '9 o 50 Cos. d. Cotg. d. Tang. d. Sin. d. ' O PP .1 .2 3 4 '.6 .8 9 48 47 4 6 45 44 *3 .1 .2 3 4 3 1 23 19 18 4-8 9.6 14.4 19.2 24.0 28.8 33-6 38.4 43.2 4-7 9-4 14.1 "18.8 23-5 28.2 32-9 37-6 4 .6 92 .2 I 3 .8 . 3 l8. 4 .4 23.0 .5 27.6 .6 32.2 .7 36.8 .8 41.4 .9 4-5 9.0 '3-5 18.0 22.5 27.0 3'-5 36.0 4 4 8.8 13.2 17.6 22. 26.4 30.8 35-2 2-3 4.6 6.9 9.2 "5 13.8 16.1 .8.4 2.2 4-4 6.6 8.8 II. O '3-2 15.4 17.6 19.8 i. 9 3-8 5-7 7-6 9-5 11.4 13.3 '5.2 17.1 1.8 3-6 5-4 7-2 9.0 10.8 12.6 14.4 16.2 i5 7 FOUR PL ACE NATURAL FUNCTIONS. O ' Sin. d. Tang. d. Cotg. d. Cos. d. 40 o 10 20 3o 4o 5o o. o. o. o . o. 0. 6428 645o 64?2 64g4 65i 7 653 9 22 22 22 23 22 22 22 21 22 22 22 22 21 22 21 22 21 21 22 21 21 21 2O 21 21 21 2O 21 0.8391 o.844i 0.8491 o.854i o.SSgi 0.8642 5 5 5 SO Si 5' 5' 52 Si 52 53 52 53 53 53 54 54 54 55 55 55 55 56 56 56 57 57 57 58 58 i . 1918 I.I84? 1.1778 i . 1708 i . i64o i . 1571 7i 69 70 68 69 67 68 67 66 66 66 65 65 64 64 63 64 62 63 62 61 61 61 61 60 60 59 59 59 58 0.7660 0.7642 0.7623 0.7604 0.7585 o. 7566 18 '9 '9 '9 19 o 50 5o 4o 3o 20 10 41 o IO 20 3o 4o 5o o. o. 0. 0. 0. o. 656i 6583 66o4 6626 6648 6670 0.8693 0.8744 0.8796 0.8847 0.8899 0.8952 i . i5o4 i.i436 1.1369 i.i3o3 i . 1237 i . 1171 0.754? 0.7528 0.7509 0.7490 0.7470 o.745i '9 '9 '9 '9 20 '9 o 49 5o 4o 3o 20 IO 42 o 10 20 3o 4o 5o 0. o. o. o. o. 0. 6691 6713 6 7 34 6 7 56 6777 6799 0.9004 0.9057 0.9110 0.9163 0.9217 0.9271 i . i i 06 i . io4i 1.0977 i . 09 i 3 i .o85o 1.0786 o . 74,3 1 0.7412 0.7392 0.7373 0.7353 0.7333 '9 20 '9 20 20 o 48 5o 4o 3o 20 IO 43 o 10 20 3o 4o 5o 0. o. o. o. o. o. 6820 684 1 6862 6884 6906 6926 0.9325 o.g38o 0.9435 0.9490 0.9545 0.9601 i .0724 i .0661 i .0699 i.o538 1.0477 i . o4 i 6 0.7314 0.7294 0.7274 0.7254 0.7234 0.7214 '9 20 20 20 20 20 o 47 5o 4o 3o 20 10 44 o IO 20 3o 4o 5o o. o. o. o. o. 0. 6g4 7 6967 6988 7009 7o3o 7060 0.9657 0.9713 0.9770 0.9827 0.9884 0.9942 i -o355 i .0295 i .O235 i .0176 i .01 17 i .oo58 0.7193 0.7173 o.7i53 0.7133 0.7112 0.7092 20 20 2O 21 20 o 46 5o 4o 3o 20 10 45 o. 7071 i .0000 i .0000 0.7071 o 45 Cos. d. Cotg. d. Tang. d. Sin. d. ' o PP .1 .2 3 4 '.6 '.8 9 57 55 54 53 51 23 31 30 '9 5-7 11.4 17.1 22.8 28.5 34-2 39-9 45-6 gj 5-5 II. 16.5 22. 27-5 33-o 38.5 44-0 5-4 -i 10.8 .2 16.2 .3 21.6 .4 27.0 .5 32.4 .6 37-8 .7 43-2 -8 48.6 .9 5-3 10.6 15.9 21.2 26.5 3 1.8 37-i 42-4 5-' IO. 2 '5-3 20.4 2 5-5 30.0 35-7 40.8 2.2 .1 2.1 4.4 .2 4-2 6.6 .3 6.3 8.8 .4 8.4 ii. o .5 10.5 13.2 .6 12.6 15.4 .7 14.7 17.6 .8 16.8 19.8 .9 18.9 2.O 4.0 6.0 8.0 IO.O 12.0 14.0 16.0 18.0 1.9 3-8 5-7 7.6 9-5 ii. 4 '3-3 '5-2 .7.1 i58 TABLE VIII. SQUARES AND SQUARE ROOTS OP NUMBERS. SQUARES OF INTEGERS FROM 10 TO 100. N 1 2 3 4 5 6 7 8 9 10 IOO 121 1 44 169 196 226 256 289 324 36i 20 4oo 44 1 484 629 576 626 676 729 74 84i 3o 900 961 1024 1089 n56 1225 1296 i36g 1 444 l52I 4o 1600 1681 1764 1849 ig36 2O25 2116 2209 23o4 24OI 5o 2600 2601 2704 2809 2916 3o25 3i36 324g 3364 348 1 60 36oo 3 7 2I 3844 3969 4096 4226 4356 448 9 4624 4?6i 70 4900 5o4i 5i84 5329 54?6 5625 5 77 6 5 9 2 9 6o84 6241 80 64oo 656i 6724 6889 7066 7225 7 3 9 6 7 56 9 7744 7921 9 8100 8281 8464 8649 8836 9025 9216 9409 9604 9801 SQUARE ROOTS OF NUMBERS FROM TO 10; AT INTERVALS OF .1. N .0 .1 .2 .44? .3 .4 .5 .6 .7 .837 .8 .9 o .3i6 .548 .632 .707 775 .894 .949 i 2 3 1. 000 i.4i4 1.732 1/049 1.449 1.761 i. og5 1.483 1.789 i.i4o 1.517 1.817 i.i83 1.549 1.844 1.225 i. 58 1 1.871 1.265 i. 612 1.897 i.3o4 1.643 1.924 1.342 i.6 7 3 1.949 1.878 1.703 '975 4 5 6 2.OOO 2.236 2.449 2.025 2.258 2.470 2.o4g 2.280 2.490 2.074 2.3O2 2.5lO 2.098 2.324 2.53o 2. 121 2.345 2.55o 2.145 2.366 2.569 2.168 2.387 2.588 2.191 2.4o8 2.608 2.2l4 2.429 2.627 7 8 9 2.646 2.828 3.000 2.665 2.846 3.017 2.683 2.864 3.o33 2.702 2.881 3.o5o 2.720 2.898 3.o66 2. 7 3 9 2.915 3.082 2.757 2.933 3.098 2.775 2.950 3.n4 2.793 2.966 3.i3o 2.811 2.983 3.i46 SQUARE ROOTS OF INTEGERS FROM 10 TO 100. N 1 2 3 4 5 6 7 8 9 10 3.162 3.3 1 7 3.464 3.6o6 3.742 3.8 7 3 4.ooo 4.123 4.243 4.35 9 20 4.472 4.583 4.690 4.796 4.899 S.ooo 5.099 5.196 5.292 5.385 3o 5-477 5.568 5.65 7 5. 7 45 5.83i 5.916 6.000 6.o83 6.i64 6.245 4o 6.325 6.4o3 6.48i 6.55 7 6.633 6.708 6.782 6.856 6.928 7.000 5o 7.071 7-i4i 7.21 1 7.280 7.348 7.4i6 7-483 7.55o 7.616 7.681 60 7.746 7.810 7.874 7.937 8.000 8.062 8.124 8.i85 8.246 8.307 70 8.367 8.426 8.485 8.544 8.602 8.660 8.718 8-775 8.832 8.888 80 8. 9 44 9.000 g.o55 9.1 10 9.i65 9.220 9.274 9.327 9.38i 9-434 9 9-487 9.539 9.592 9-644 9.695 9-747 9.798 9-849 9.899 g.gSo TABLE IX. THE HYPERBOLIC AND EXPONENTIAL FUNCTIONS OF NUMBERS FROM TO 2.5, AT INTERVALS OF .1. C cosh ./ sinh ./ tanh oc e* e~' . I .2 .3 i .000 o i .000 i .000 i .oo5 I .020 1. 045 . IOO .201 .3o5 . IOO .197 .291 i . io5 1 .221 i.35o .905 .819 74i .4 .5 .6 1.081 1.128 i.i85 .4u .521 .63 7 .38o .462 .53 7 i .492 i .649 i .822 .670 .607 .549 7 .8 9 1.0 i . i I .2 1.3 1.255 i.33 7 1.433 7 5 9 .888 i .027 .6o4 .664 .716 2.0l4 2.226 2.460 497 .449 .407 1.543 i .175 .762 2.718 .368 i .669 i'.8n 1.971 j.336 i .5og 1.698 .801 .834 .862 3 ,oo4 3. 32o 3.669 .333 .3oi .2 7 3 i.4 i.5 1.6 2. l5l 2.352 2.577 1.904 2. 129 2.376 .885 .goS .922 4.o55 4.482 4. 9 53 .247 .223 .202 i-7 1.8 1.9 2.0 2. I 2.2 2.3 2.828 3. 107 3.4i8 2.646 2.942 3.268 .935 947 956 5.4?4 6.o5o 6.686 .i83 .i65 .i5o 3.762 3.627 .964 7 .38 9 -.i35 4.i44 4.568 5.o37 4.022 4.45 7 4.937 .970 .976 .980 8.]66 9.025 9-974 . 122 .III . IOO 2.4 2.5 5.55 7 6.i32 5.466 6.o5o .984 .987 I I .023 12. l82 .091 .082 1 60 TABLE X CONSTANTS MEASURES AND WEIGHTS AND OTHER CONSTANTS MEASURES AND WEIGHTS English Measures Metric M 'ensures LENGTH LENGTH 12 inches (in.) = i foot (ft.). 10 millimeters (mm.) = i centimeter (cm.). 3 feet = i yard (yd.). 10 centimeters = i decimeter (dcm.). 16^ feet = i rod (rd.). 10 decimeters = i meter (m.). 5280 feet = i mife (m.). 10 meters = i dekameter (dkm.). 6080.3 feet = i nautical mile. 10 dekameters = i hektometer (hkm.). 5.}$ yards = i rod. 10 hektometers = i kilometer (km.). 4 rods -i chain (ch.). i foot =30.48 centimeters. ( = 39-37 inches, i meter \^ ( = 3.2808 feet. i yard = .9144 meter. i kilometer = 0.6214 mile. i mile = 1.6093 kilometers. SURFACE SURFACE 144 sq. inches = - 1 sq. foot. 100 sq. millimeters = i sq. centimeter. 9 sq. feet = i sq. yard. too sq. centimeters = i sq. decimeter. 3oJ sq. yards = i sq. rod. 160 sq. rods = i acre. ( = i sq. meter. 100 sq. decimeters < (=i centare (ca.). 43560 sq. feet = i acre. 100 sq. meters = i are (a.). 640 acres = i sq. mile. loo ares = t hektare (hka.). i sq. inch =6.4516 sq. centimeters. i sq. centimeter = 0.1550 sq. inch. i sq. foot = 0.0929 sq. meter. 1= 1.196 sq. yards. i sq. yard =0.8361 sq. meter. , , = 10.764 sq. feet. i acre =0.4047 hectare. i are = 1076.48 sq. feet. i hektare = 2.471 acres. VOLUME VOLUME 1728 cu. inches = i cu. foot. looo cu. millimeters = i cu. centimeter. 27 cu. feet = i cu. yard. 1000 cu. centimeters = i cu. decimeter. 128 cu. feet = i cord (cd). ( = i cu. meter, .ooocu. decimeters { = igtere(st) i cu. inch = 16.387 cu. centimeters. i cu. foot = 0.028 cu. meter. i cu. cantimeter = 0.06 1 cu. inch. i cu. yard = 0.7646 cu. meter. (= 35-3M cu. feet. i cord = 3.625 steres. {= 1.308 cu. yards. i stere = 0.2759 cord. CAPACITY CAPACITY i liq. gal. = 3. 785 liters = 231 cu. in. 100 centiliters (cl.) = i liter (1.). i dry gal. = 4-404 liters = 268.8 cu. in. loo liters = i hektoliter (hkl). i bushel =0.3524 hkl. =2150.42 cu. in. i liter = 1.0567 liq. qts. = i cu. dcm. AVOIRDUPOIS WEIGHT METRIC WEIGHT 16 ounces (oz.) = i pound (lb.). looo grams (gm.) = i kilogram (kilo.). loo Ibs. = i hundredweight (cwt.). looo kilograms = i tonneau (t.). 20 hundredweight = i ton (T.). i gram = 15-432 grains. i pound = .4536 kilo. = 7000 grains. i kilogram = 2.2046 pounds. i ton =.9071 tonneau (t). i tonneau = 1.1023 tons. TROY WEIGHT i pound = 5760 grains = 12 ounces. 162 MEASURES AND WEIGHTS Continued 60 seconds (")= i minute ('). 60 minutes = i degree (). 90 degrees = i right angle. radians = i right angle. It 3.141 27T = 6.2831853 41T =12. 5663706 = 1.0471976 - 4 "" = 4.1887902 -^ = 0.7853982 -r = 0.5235988 = 0.3183099 *= 9.8696044 I = 0.1013212 Vr = 1.7724539 = 0.5641896 VTT CONSTANTS log it =0.4971499 log 2?r =0.7981799 log 47r =1.0992099 log =0.1961199 log = 0.0200286 log =0.6220886 3 log =9.8950899 10 log = 9.7189986 10 log =9. 5028501 10 log 7r = 0.9942997 log t = 9.0057003 10 logVw =0.2485749 log ' =9.7514251 10 Vit = i. 1447299 e =2,718281828459 M =0.4342945 log* =0.4342945 log M = 9.6377843 10 ^=2.3025851 log = 0.3622157 ( = 57- 295779 5 Radian j =. 3437-747' ' = 206264.8" log 57.2957795 = 1.7581226 log 3437-747 = 3-5362739 log 206264.8 = 5.3144251 i degree =0.0174533 radians i minute = 0.0002909 radians i second = 0.0000048 radians log 0.0174533 = 8.2418774 10 log 0.0002909 = 6.4637261 10 log 0.0000048 = 4.6857749 10 i63 QA 531 Phillips-Elements of trigonometry; plane and spherical UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. Form L9-116m-8,'62(D1237s8)444 Form L-9 23m-2,'43(5205) ^UNIVERSITY of CALTFORMTA K JOL72