'€ V h II Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofplanesOOcrocrich o O^qIu AX ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY BY C. W. CROCKETT PROFESSOR OF MATHEMATICS AND ASTRONOMY RENSSELAER POLYTECHNIC INSTITUTE >>Hc NEW YORK :• CINCINNATI : • CHICAGO AMERICAN BOOK COMPANY y\ /<;- Copyright, 1896, by AMERICAN BOOK COMPANY. OBOCKXTT. PLANE AND SPHES. TBIGONOM. W. P. 2 PREFACE. This work has been prepared for the use of beginners in the study of trigonometry. Assuming that a high degree of proficiency cannot be expected from such students, the author has limited himself to the selection of simple proofs of the formulas, not striving after original demonstrations. Geo- metrical proofs have been added in many cases, experience having shown that the student is assisted by them to a clearer understanding of the subject. The student is expected, in technical institutions, to acquire facility in the use of the tables. All of the numerical exam- ples have been computed by the author, with special attention to correctness in the last decimal place, and the arrangement of the computations has been carefully considered. Five-place tables have been adopted, and the angles in the examples are given to the nearest tenth of a minute, because the instruments ordinarily used by engineers are read by the vernier only to the nearest minute of arc, while the angle corresponding to a computed function may be found usually to the nearest tenth of a minute by the use of five-place tables. Credit is due particularly to the works of Chauvenet, Snow- ball, Beasley, Woodhouse, Newcomb, and Todhunter, although many others have been consulted. A number of the illustra- tive examples in Art. Ill were taken from Gillespie's "Land Surveying," the numerical values being assigned by the author of this work. The author cannot hope that among so many examples there are no errors ; he therefore requests those finding such to kindly notify him. Rensselaer Polytechnic Institutb, Troy, N. Y. 8 1 ^:>i)45 GREEK ALPHABET. A, a, B, A r, y, A, S E, 6 z, ? H, V @, « I, '. K, K A, \ M, M a, a Alpha /3, p Beta 7, 7 Gamma Delta . - Epsilon Zeta , Eta Theta Iota Kappa ........ Lambda Mu N, 1/ . . . . • Nu H, ? Xi O, Omicron n, TT ........ Pi V, p Rho 2, cr, 9 Sigma T, T Tau T, V Upsilon ^, (^ PJd X, X ^A*' ^, >/r Psi 12, o) Omega CONTENTS. PART ONE. PLANE AND ANALYTICAL TRIGONOMETRY. CHAP. I. MEA.SURBMKNT OF ANGLES: TRIGONOMETRIC FUNCTIONS OF Angles less than 90°. PAOB Trigonometry defined 7 Directed lines ; angles 7 Measurement of angles 8 Trigonometric ratios in right triangles 10 Tables of the ratios 12 Ratios for 30°, 45°, 60°' . . . . v 12 Computation of the ratios when one is given 13 Measurement of angles in the field 14 Illustrations of the applications of the ratios 16 CHAP. 11. Right Plane Triangles. Facts derived from geometry 17 Solution defined 17 Formulas employed in the solution of right triangles 17 Relations between the sides and the angles ...*... 18 Methods of solution 19 Isosceles triangles 23 Special methods 23 CHAP. III. Trigonometric Functions of Ant Angle. Generation of angles 30 General measure of an angle 31 Coordinates ............ 53 General definitions of the trigonometric functions 35 Geometrical representation of the functions 37 Changes in the values of the functions 40 Limiting values of the functions ......... 42 Graphical representation of the functions 42 Two angles correspond to any given function 43 CHAP, IV. Relations between the Functions of One Angle. Relations between the functions of one angle, and their applications . . 46 Solution of trigonometric equations containing one an^ile .... 52 Functions of angles greater than 360° 54 Functions of 90° ± x, 270° ± x, 180° ± ?/, 300° - y, and - // . . .54 The trigonometric tables 60 Transformations 61 CHAP. V. Relations between Functions of Several Angles. Functions oi x + y and of x — y 64 Functions of 2 X in terms of functions of a; 71 Functions of x in terms of functions of 2 ic 72 Multiple angles 74 To change the product of functions into the sum of functions ... 74 To change the sum of functions into the product of functions ... 76 Circular functions 77 To prove that tan aj > x > sin oj, when a; < ^ tt 79 To prove that sin x, tan x, and x approach equality as x approaches zero . 79 6 I CONTENTS. PAGE Development of sin x, cos x, and tan x 80 Computation of the trigonometric functions 82 Approximate assumptions 84 Transformations 85 CHAP. VI. Trigonometric Equations. Equations containing multiple angles 88 Special cases 89 CHAP. VII. Oblique Plane Triangles. Facts derived from geometry 97 The sine proportion 97 c^ = a^ -}- b"^ - 2 ab cosy 98 Methods of solution . 99 Metliods of solution, using right triangles 107 Areas 110 Illustrative examples 112 PART TWO. SPHERICAL TRIGONOMETRY. CHAP. VIII. Definitions and Constructions. Spherical trigonometry defined " . . 123 Representation of trihedral angles . . 123 Limitation of values 124 Definitions, relations, and constructions 124 The polar triangle 126 Facts derived from geometry 127 Construction of triangles 127 CHAP. IX. General Formulas. cos a = cos b cos c + sin 6 sin c cos a 130 cosa =— COS/3C0S7 + sin^sin7Cosa 132 The sine proportion ........... 133 Additional formulas ........... 134 CHAP. X. Right Spherical Triangles. Formulas Napier's rules Species . 135 136 .138 Methods of solution 139 Isosceles triangles ........... 143 Quadrantal triangles 144 CHAP. XI. Oblique Spherical Triangles. To find an angle, given the three sides 146 To find a side, given the three angles 148 Napier's analogies 150 Gauss's equations 152 Species 153 Methods of solution . 155 Methods of solution, using right triangles ', ' . 168 CHAP. XII. Applications of Spherical Trigonometry. Distance between two points on the earth's surface Miscellaneous applications Spherical excess . Legendre's theorem Astronomical definitions Astronomical applications 174 175 181 184 187 189 PART ONE. PLAJ{E AKB AJ^ALYTICAL TRIGOKOMETRY. CHAPTER I. MEASUREMENT OF ANGLES; TRIGONOMETRIC FUNCTIONS OF ANGLES LESS THAN NINETY DEGREES. 1. Analytical Trigonometry treats of the relations of lines and angles by algebraic methods. In Plane and Spherical Trigonometry, these relations are applied to the solution of plane and spherical triangles. 2. Directed Lines; Angles. — A directed line is one whose beginning, direction, and length are known. The direction of the line is indicated by the order of the letters in its symbol \ for instance, the line AB is drawn from A to B. If one direc- tion along the line is considered positive, the opposite direction will be negative ; thus, if the line AB is positive, the line BA will be negative, their numerical measures being equal, or line AB = — line BA. An angle is the figure formed by two intersecting lines, the point of intersection being the vertex. The angle between any two given lines, whether intersect- ing or not intersecting,* is defined to be the same as the angle formed by two lines drawn through any point parallel to and in the same direction as the given lines. Hence an angle may be defined as the difference in direction of two directed lines. * That is, parallel or in space. 7 8 PLANE AND ANALYTICAL TRIGONOMETRY. 3. Measurement of Angles. — Two methods of measuring angles are in common use, — the sexagesimal and the circular or natural methods. 4. Sexagesimal Measure.* — The circumference of a circle described about the vertex of the angle as a center is divided into 360 equal parts, and the angle at the center subtended by one of these parts is taken as the unit. The length of one of these divisions of the circle will depend upon its radius ; but the corresponding angle at the center will be independent of the radius, since it is -jj^ of four right angles. This unit angle, called a degree^ is divided into 60 parts called minutes^ each of which is subdivided into 60 parts called seconds. These are marked °, ', " ; thus 43° 14' 35". 2 is read, "43 de- grees, 14 minutes, and 35.2 seconds." How many degrees are there in 1. Two tbirds of four riglit angles ? Ans. 240°. 2. Two fifths of three right angles ? Ans. 108°. 3. Five sixths of two right angles ? Ans. 150°. 5. The Circular or Natural Measure. — From geometry we know that in any two concentric circles the arcs intercepted by any angle at the center are to each other as- the radii of the circles. Therefore, if ACB be any central angle, we have arc ^5 'dvcA'B' CA CA' (1) Hence the length of the intercepted arc divided by the radius is a number that is always the same for the same angle, no matter what the radius may be. We also know that in any circle any two central angles are to each other as their intercepted arcs, and therefore as the quotients of their intercepted arcs divided by the radius. We can, then, use these quotients to measure the angles. V * From sezagesimus, sixtieth. MEASUREMENT OF ANGLES. 9 The circular measure of an angle is the quotient obtained by dividing the length of its intercepted arc, in a circle whose center is at the vertex of the angle, by the radius of the circle. Thus, if c is the circular measure of the angle, I its intercepted arc, and r the radius, we have «=- (2) If the radius of the circle is unity, c = «. (3) Hence the circular measure is represented by the length of the intercepted arc in the circle whose radius is unity. The angle whose circular measure is one, that is, whose intercepted arc is equal to the radius, is called the radian. 1. The length of the intercepted arc of a central angle is 4 feet in a circle whose radius is 2 feet; the length of the intercepted arc of another central angle is 20 meters in a circle whose radius is 5 meters. Show that the second angle is twice as large as the first. 2. In a circle with a radius of 10 inches, the intercepted arc of a central angle is 5 inches, and that of an angle whose vertex is on the circumference is 10 inches. Find their circular measures. Ans. \. 6. Relation between the Two Measures. — Two right angles are measured by 180°, and also by 7rr* ^ r = tt, since irr is the semicircumference of a circle whose radius is r. Hence, using the equality sign to represent " corresponds to," we have 180° = 1x1^11 circular measure 5 (1) . •. 1° = -^ in circular measure. (2) 180 Again, tt in circular measure = 180° ; (3) . •• 1 in circular measure = i^^. C4) .-. 1 in circular measure = 57°. 29577 95 + (5) .-. 1 in circular measure = 206 264''.806. (6) 1. What is the circular measure of 120° ? Ans. 120 x -^ = I 2. What is the circular measure of 10° 10' 10" ? 180 ' The circular measure of 1° is -— , and that of 1" is But 180 180 X 60 X 60 10° 10' 10" = 36610". . •. Circular measure of 10° 10' 10" = ^^^^^ "^ 180 X QO X 60 ♦ IT denotes the ratio of the circumference of a circle to its diameter, and is the number 3.14159 265+. 10 PLANE AND ANALYTICAL TRIGONOMETRY. 3. What is the sexagesimal measure of the angle whose circular measure ^^^'^^ w = 180°; .-. ^7r=60°. 4. What is the sexagesimal measure of the angle whose circular measure ^^^ TT .. • • 1 180° , - , 120° Unity in circular measure = ; . •. f corresponds to TT IT 6. What are the sexagesimal and circular measures corresponding to f of three right angles ? Ans. 60° ; \ ir. 6. The sexagesimal measures of two angles are 22° 30' and 43° 14' 3". 1350 IT 155643 tt Show that their circular measures are and 180 X 60 180 X 60 X 60 7. The circular measures of three angles are y^j tt, | tt, and J^ v. Show that their sexagesimal measures are 15°, 40°, and 3° 36'. 8. The circular measures of three angles are i, f , and |. Show that their . , 45° 300° , 40° sexagesimal measures are — , , and _ . TT IT TT 9. Find the sexagesimal and circular measures corresponding to (a) Seven tenths of four right angles. Ans. 252° ; |ir. (6) Five fourths of two right angles. Ans. 225° ; | t. (c) Two thirds of one right angle. Ans. 60° ; | tt. 7. Centesimal Measure. — In this system, proposed by the French, the right angle is divided into 100 parts called grades, each of which is subdivided into 100 parts called minutes, each minute being divided into 100 parts called seconds; marked ^ \ '\ 8. Trigonometric Ratios. — Let the sides of a right-angled triangle be denoted as shown in Fig. 2. The trigonometric ratios may be defined as follows: The sine of an angle = -^ P- ; written sin -4 = - hypotenuse h n^^ . o I side adiacent .^^ .a ihe cosine oi an angle = -.; ~ ; written cos A = ~ nypotenuse h The tangent of an angle = -r-^ ^ -; written tan A = - side adjacent « The cotangent of an angle = ^^ — t— : written cot ^ = - ^ side opposite o The secant of an angle = . ^^ ,. -; written sec^ = - side adjacent « The cosecant of an angle = -^^ r— ; written cosec A = - side opposite o These fundamental equations should be thoroughly memorized. O) MEASUREMENT OF ANGLES. 11 Fig. 2. Fig. 8. 9. The Ratios are Constant for Any One Angle. — In Fig. 3 let BAO and BAF be two angles differing by a quantity as sin^ll as Ave please. At any two points B and I) on AB, draw BF and BG- perpendicular to AB ; with ^ as a center, and radius A (7, describe the arc OIT, and draw Lff perpendicular to AB. The triangles BAO and BAF are similar. BC^BF AC AF BO BF a constant sin a;. = ---- = a constant AB AB side opposite hypotenuse side opposite , — — = tan Xm side adjacent AC AF , , hypotenuse — -— = — — = a constant = ,•: -, . = sec x, AB AB side adjacent 10. The Values of the Ratios differ for Different Angles. — From Fig. 3 we have, since AH = AO^ BO . • LH LH sin. = — andsin^ = -^= — ; BO . ^ BF tan X = — — ; and tan y = ; AB ^ AB AO . AF sec :c = — and sec ^ = — . 11. The Angle may he constructed when One of the Ratios is known. — Let sin x = ^. With any convenient radius AO, describe a circle about A as a center. Draw AB perpendic- ular to AB, and on it lay off AB= ^AO; draw i> (7 parallel to AB until FiQ. 4. 12 PLANE AND ANALYTICAL TRIGONOMETRY. it intersects the circle at ; join A and (7, and BA C will be the required angle, since . ^,ri BC AD 1 sin BA C = -— - = -TT^ = - . AC AC 2 Let tan a; = J. Lay off any convenient distance AB ; at B draw BC perpendicular to AB, and lay off J9(7= ^ AB ; join A and (7, and ^J. C will be the required angle, since tani?^(7=4^=f. Let sec x = 2. Lay off any convenient distance AB ; erect the perpendicular line BC; with a radius J. (7= 2 J.^ describe an arc cutting BC at C; join ^ and C, and jB^Cwill be the required angle, since secj&A(7=4^=2. Let the student construct the angle whose cosine is |, the angle whose cotangent is 5, and the angle whose cosecant is 4. 12. We therefore conclude that to any one angle there will correspond a special value of each of these ratios, that the value of each ratio will differ for different angles, and that, if any one of these ratios is given, the angle may be constructed. 13. Tables of Sines, Cosin*es, etc. — The values of these ratios for angles between 0° and 90° have been computed, and are given in tables so arranged that the values corresponding to any angle may be readily found. The tables of natural sines., etc., contain the actual values of these ratios ; while the tables of logarithmic sines, etc., contain their logarithms. 14. Ratios for 30% 45°, 60°. (rt) Ratios for 45°. — In Fig. 5 let the angle A = 45° ; then J5=90°-^ = 45°. .-. AC — CB, since they are opposite equal angles. Let AC= a; then CB = a^ and AB = Va^ + a^ = a V2. MEASUREMENT OF ANGLES. 13 ^°^'''°=3l=^' cot45'=^|=l; cosec46o=^=V-2. (5) Ratios for 30° and 60°. — In the equilateral triangle ABC (Fig. 6), let .A.S = a ; draw D^ perpendicular to AO ; AC will be bisected at i>, making J.i> = Ja, and the angle ^^i)=angle i>^(7=30°. Also DB = Va2 - 1 ^2 = 1 a Vs. sin ABi> = sin 30^^ ^^ 2 tan 30° = cos 30° = ^=—; cot 30° AB 2 DB V3 DB V3 ^ = V3 siriZU^=sin60° = :^=^; tan60° = — = >/3 ^^2 ^Z> 2)^ AD cos60° = — = i; *cot60° = ^^2 Z)^ V3 Note that the sines of 30°, 45°, and 60°, are spectively. sec 30° = ^ =— DB V3 cosec 30° = ^ = 2. AD sec 60° = :^^ = 2 : cosec 60° = :^= A. DB V3 ^(v/l,/^V2, and^VS re- 15. The Ratios are not Independent of Each Other ; for we have from Fig;. 2, so that if two of the three quantities A, o, and a, are given, the third can be found. Hence if we know one of the ratios, that is, the relative values of two of the three elements, we can de- termine the relative value of the third element, and from it the other ratios. 14 PLANE AND ANALYTICAL TRIGONOMETRY. Thus if tan a: = |, and the other ratios are required, we have tana: ; let = 3, a = 4 ; then A = 5. a 4' ' ••• sin 2: = 3 5' COS X a 4 ^5' cot 2: 4 ^3' sec 2: _h_ a 5 cosec X - 5 3* Having given the ratic on the left, find the ratios on the right : sin a;. cos a?. tan a. cot 85. seca;. cosec 85. o.-^ ^ 8 15 8 15 17 17 1. sin X = — — 17 17 15 8 15 8 5 12 12 5 13 13 2. cos x = — — 13 13 5 12 5 12 8. tan a; = — 7 24 24 25 25 24 25 25 7 24 7 4. cot a = 2 1 V5 2 V5 1 2 , — IVB V5 29 21 20 21 20 29 b. seca; = — 20 29 29 20 21 21 6. cosec x = 3 1 3 1--. Iv^ 2V2 \^^ — 16. Measurement of Angles in the Field. — In Fig. 7, FG-HK represents a fixed graduated circle, and ABDE a circle resting on the plate FGrHK^ and capable of moving about a pivot at O'; J and are two small rods fixed to ABBE^ and perpendicular to~the planes of the circles ; and il[f is a mark on the circle ABBE in the same line with /, (7, and 0. If we wish to measure the horizontal angle between two distant objects, two church towers, for ex- ample, we proceed as follows : first place the circles in a horizontal posi- tion ; revolve the circle ABBE, look- ing along the line 10, until the line of sight passes through one of the objects, and note the reading MEASUREMENT OF ANGLES. 16 of the circle opposite the mark M; then revolve the circle ABBE, being careful not to move FGHK, until the line of sight passes through the second object, and note the new read- ing of the circle opposite the mark M. The difference between the two readings will be the angular distance required. 17. The Engineers' Transit, shown in Fig. 8, is used in measuring horizontal and vertical angles. The lower circle is provided with two levels, by which its horizontality is tested. Fig. 8. The rods I and are replaced by the telescope with a system of intersecting wires in the common focus of the object glass and eyepiece, the telescope being capable of rotation about an axis parallel to the horizontal circle. The circle fixed to the axis of the telescope is vertical when the plate bearing the upright supports is horizontal. 16 PLANE AND ANALYTICAL TRIGONOMETRY. 18. Illustrations of the Application of the Ratios.* 1. A rope fastened to the top of a vertical pole 60 feet high, and to a stake driven in the ground, is inclined at an angle of 30°. How far is the stake from the bottom of the pole ? How long is the rope ? -^ = tan 30° = —. AC V3 ••• ^^ = V3a^ = 60V3feet. Fig. 9. OB . ono 1 — — = sm 30° = -. AB 2 .-. ^j5 = 2 (7^ = 120 feet. 2. The angle at the vertex of a right circular cone is 60°, and the slant height is 10 inches. What is the altitude and the radius of the base of the cone ? Fig. 11. OB _„ ono V3 _ = cos30 =— . OB = ^AB 5 V3 inches. ^=sin 30^ = 1 . •. AC= ^ AB = 5 inches. 3. The top of a ladder 30 feet long rests on the upper edge of a wall 15 feet high. What is the inclination of the ladder ? sin CAB = 05^15^1. AB 30 2' but sin30° = V. .'. (7^^ = 30°. In these cases the ratios corresponding to the angles were known from Art. 14. Usually it will be necessary to refer to the tables in solving problems involving the ratios. It is assumed that the ground is horizontal. CHAPTER II. RIGHT TLANE TRIANGLES. 19. It has been shown in Geometry that a right-angled tri- angle can be constructed when two elements * besides the right angle are known, one of the known elements being a side. We also know that (1) The hypotenuse is greater than either of the other ' two sides. (2) The hypotenuse is less than the sum of the other two -^ sides. (3) The sum of the two acute angles must be 90°. '^ (4) The greater side is opposite the greater angle. ' (5) The square on the hypotenuse is equal to the sum of "^ the squares on the other two sides. 20. A triangle is said to be solved when, having some of the elements given, the others have been found by some process. 21. The Solution of a Right Triangle is effected by means of the trigonometric ratios. Each equa- tion, as sin ^ = -, contains three c quantities ; and two of them must be known in order that the third may be found. Hence in any particular case we use the equations that con- tain the two given elements ; thus, if ^^ a and h are given, we use tan J. =y via-^ii to find A, and then e may be found from either sin A =- or COS A = -, c * The elements of a triangle are the three sides and the three angles. CROCK. TRIG. — 2 17 18 PLANE AND ANALYTICAL TRIGONOMETRY. The equations used in the solution of right triangles are Fia. 18 sin A = - = cos B, c cos A= - = sin B, c tan J. = - = cot-B. cot A = - — tan B. a A + B = 90°. ^ = ^2 + ^2. (1) 22. From the Trigonometric Ratios we have tanJ.= ; .*. a = 5tanJ., cot B = -; .'. a = b cot B^ (1) or, any side of a right triangle is equal to the other side multiplied by the tangent of the angle opposite^ or by the cotangent of the angle adjacent^ to the side itself. sin^ = — ; .*. a = (? sin ^, c eosB = -; .*. a = c cos B, c (2) or, any side is equal to the hypotenuse multiplied by the sine of the opposite angle^ or by the cosine of the adjacent angle. sec J. = -; '=h sec^, cosec^=-; .-. c=5cosecJ5, b (3) or, the hypotenuse is equal to a side multiplied by the secant of the adjacent angle^ or by the cosecant of the opposite angle. Note. — The secant of an angle is the reciprocal of its cosine, and the cose- cant is the reciprocal of its sine ; hence the logarithm of the secant is the arith- metical complement of that of the cosine, and the logarithm of the cosecant is the A. C. of that of the sine, or log sec X = colog cos x, and log cosec x = colog sin x. RIGHT PLANE TRIANGLES. 23. Case I. Given c and A, Formulas: a = (? sin A, h = c cos A, ^=90° -A 19 1. Solve the triangle when c = 1.0034, and A = 42^ lO'.S. .-. B = 90° - ^ = 47°49'.7. (a) By natural functions. a = c sin ^ = 1.0034 x 0.67136 = 0.67364. 6 = ccos^ = 1.0034 X 0.74114 = 0.74366. (b) By the use of logarithms. a = c sin ^ ; . •. log a = log c + log sin A. 6 = c cos ^ ; . •. log 6 = log c + log cos A. Always write first all the formulas that will be used in the problem; then write them in a form adapted to logarithmic computation ; then refer to the tables and write the logarithms in their proper places. Thus in this case we arrange the work as follows: logc= logc = + log sin ^ = + log cos A = .-. loga= .-. log 6 = .-. a= .'. 6 = The positive signs preceding log sin A and log cos A indicate that they are to be added to log c. We now find the angle A in the table of logarithmic func- tions and take from the table both log sin A and log cos A, writing them in their proper places. Then we refer to the table of logarithms of numbers and find log c, writing it oppo- site log c. Then we add the proper quantities to find log a and log 5, finally looking in the table of the logarithms of numbers for the numbers corresponding to the computed values of log a and log b. The arrangement on the right is preferable, since it saves log sin A = (1) logc = (3) log cos A = (2) .-. loga = (l) + (3) (4) a = (6) .'. log 6 =(2)4- (3) (5) b = (7) 20 PLANE AND ANALYTICAL TRIGONOMETRY. the writing of one line. The numbers in the parentheses indi- cate the order in which the quantities should be found. 0.00147 9.82695 logc: 10 + log cos ^ log6 = 9.87137 6 = 0.74365 0.00147 or log sin A = 9.82G95 - 10 9.86990-10 logc = 0.00147 log cos ^ = 9.86990 - 10 10 Check : c + 6 = 1.74705 c-b = 0.25975 b) loga = 9.82842 - 10 a = 0.67363 log6 = 9.87137 -10 6 = 0.74365 log (c+ 6) = 0.24230 log (c - &) = 9.41456 ~ 10 .-. loga2 = 9.65686 loga = 9.82843 Exact agreement is not expected, since the tables give the values of the functions only to the nearest unit in the fifth decimal place. The — lO is usually omitted, and sin A is written for log sin J., when there is no danger of confusion. 2. Solve the triangle when c = 34.687, and B = 49° 8'.4. Ans. A = 40° 51'.6 ; b = 26.234 ; a = 22.6925. 3. Solve the triangle when c = 305, and A = 63° 31'.14, using the natural functions. ^^^^ ^ ^ 273.00 ; h = 136.00. 4. Solve the triangle when c = 205, and B = 49°33'.01, using the natural Ans. a = 133.00 ; b = 156.00. functions. 24. Case II. Given c and a. Formulas sin A — -. c b = a cot A = c cos A. 1. Solve the triangle when c = 8.7982, and a = 3.1292. .*. logsin^ = loga — logc; logft = loga + logcot^ = logc + logcos A log a = 0.49544 log a = 0.49544 log c = 0.94439 -logc = 0.94439 +logcot^ = 0.41958 +logcos^ = 9.97063 - 10 log sin ^ = 9.55105 - A = 20° 50M ^ = 69° 9'.9 10 log 6 = 0.91502 b = 8.2228 log6 = 0.91502 b = 8.2228 RIGHT PLANE TRIANGLES. 21 log cot ^ = 0.41958 (5) loga = 0.49544 (1) -logc = 0.94439 (2) log cos ^ = 9.97063 (6) logsin^ = 9.55105 (l)-(2) A = 20° 50'.1 (4) i? = 69° 9'.9 (l) + (5) (2) + (6) b = 8.2228 Check : b^=(c- a){c-\- a) c-a= 5.6690, log (c - a) c-ha = 11.9274 0.75361 log (c + a) =1.07656 log 62 1.83006 log 6 =0.91603 log6 = 0.91602 { 2. Solve the triangle when c = 369.27, and b = 235.64. Ans. A = 50° 20'.9 ; 5 = 39° 39M ; a = 284.31. 3. Solve the triangle when c = 281, and a = 160, using the natural functions. Ans. A = 34° 42'.5 ; b = 231.00 or 231.01. 4. Solve the triangle when c = 365, and b = 76, using the natural functions. Ans. A = 77° 58'.93 ; a = 357.00. 25. Case III. Given a and b. Formulas: tan^ = c = sin A cos A 5 = 90° - A. 1. Solve the triangle when a = 169.03, and b = 203.44. . •. log tan ^ = log a - log 6 ; log c = log a - logsin A = \ogb — log cos^. log a = 2.22796 -log 6 = 2.30843 log tan A log a = 2.22796 log sin ^ = 9. 80555 10 log 6 = 2.30843 log cos A = 9.88602 - 10 9.91953-10 ^ = 39°43'.3 J5 = 50°16'.7 logc = 2.42241 c = 264.49 logc = 2.42241 c = 264.49 or *loga = 2.22796 (1) logsin^ = 9.80555 (5) log cos ^ = 9. 88602 (6) log6 = 2.30843 (2) log tan ^ = 9.91953 (3) al = 39°43'.3 (4) J5 = 50°16'.7 logc = 2.42241 c = 264.49 1(2)- -(5) -(6) Check : a2 = c2 - 62 c + 6 = 467.93 c-6= 61.06 log(c + 6)= 2.67018 log(c- 6)= 1.78569 .-. loga^ =4.45587 log a =2.22794 * This form is preferable. 22 PLANE AND ANALYTICAL TRIGONOMETRY. 2. Solve the triangle when a = 4.8199, and b = 2.6492. Ans. A = 61° 12'.8 , B = 28" il'.l ; c = 5.4999. 3. Solve the triangle when a = 60, and 6 = 91, using the natural functions. Ans. A = 33° 23'.9 ; c = 109.00. 4. Solve the triangle when a = 72, and b = 65, using the natural functions. Ans. A = 47° 55'.5 ; c = 97.000. B 26. Case IV. Given a and A, h = a cot A. Formulas: c = B sin A cos A 90° - A, 1. Solve the triangle when a = 613.35, and A = 40° 12'.6. .-. 5 = 90° -^=49° 47 '.4. log & = log a + log cot A. log c = log a — log sin A = log b — log cos A. loga = 2.78770 cot^ = 0.07295 log& = 2.86065 b = 725.52 loga = 2.78770 logsin^ = 9.80996 -10 logc = 2.97774 c = 950.04 log6 = 2.86065 log cos^ = 9.88291 - 10 logc = 2.97774 c = 950.04 or log sin ^ = 9.80996 (1) log a = 2.78770 (3) log cot ^ = 0.07295 (2) logc = 2.97774 (3)-(l) c = 950.04 log6 = 2.86065 (3) + (2) b = 725.52 Check: a2=(c + &)(c-fe) c + b = 1675.56, log (c + b)= 3.22416 c-b= 224.52, log (c - 6) = 2.35126 log«''^ = 5.57542 loga = 2.78771 2. Solve the triangle when a = 3.6378, and B = 69° 23'. 5. Ans. A = 20° 36'.5 ; b = 9.6738 ; c = 10.335. 3. Solve the triangle when b = 160, and A = 55° 17'.48, using the natural functions. Ans. c = 281.00 ; a = 231.00. 4. Solve the triangle when a = 340, and A = 60° 55'. 85, using the natural functions. Ans. c = 389.00 ; b = 189.00. RIGHT PLANE TRIANGLES. 23 27. Isosceles Triangles. — If a perpendicular to the base is drawn from the vertex, it will bisect the base and the angle at the vertex, forming two equal right tri- angles. ZABI)=ZI)BC=l^; AB = BO; 1. Solve the triangle when h = 2.1452, and j8 = 121° 14'.6. .-. ^i?= 1.0726; ^i52) = 60°37'.3; o = 90°-^)8=29°22'.7. ^"^sinT^' •■• loga = logi6-logsinJ)8. log ^6 ==0.03044 log sin i)a = 9.94022 -10 loga = 0.09022 a = 1.2309 \ogp = log J 6 + log cot J 0. log^& = 0.03044 + log cot ^ /8 = 9. 75049 - 10 logp = 9.78093- 10 p = 0.60385 2. Solve the triangle when a = 52° 10'. 2, and a = 600.2. Ans. /8 = 75°39'.6; ^6=368.12; ;> = 474.07. 28. Given c and b (Special Method). — When b nearly equals c, the angle found from the formula cos J. = - is uncertain, the c tabular difference for the cosine being so small that a small error in cos^ would produce a large error in A. In the figure, AB bisects the angle A^ and BU is perpen- dicular to AB; .'. BE= CJ). Let OB = x=BB; .-. tan4a = - 2 b Also, CB=a= CB + BB= OB-^ BUsec a a (1) a = X -{- X sec a ; a ab X = 1 H- sec a ' X 1 + + 4' (2) 24 PLANE AND ANALYTICAL TRIGONOMETRY. From (1) and (2), tan -|- u c -^ b c h)(c-b}. . •. tan J + 6 ^l ic + bf ■ ^ C -\- Suppose that we wish to find the greatest distance at sea at which a moun- tain 4.3 miles high can be seen, the earth being considered as a sphere witli a radius of 3963.3 miles, and the distance being measured as a -A chord. Let 5^=4.3, and CB=CB = 'dmS.S ; BD being the distance required. Then cos DCA = — , giving log cos DCA = 9.9m52 ; CA and DCA as found from the tables might have any value between 2° 40'.5 and 2° 42'.5. Using (3), we have CA-CD= 4.3; log = 0.63347 CA + CD = 7930.9 ; log = 3.89932 2 )6.73415 - 10 log tan IDCA = 8.36708 - 10 bpl. r= 3.53620 log (I Z>C^)'= 1.90328 .-. I DCA = 80'm5; .-. DCA =2° iO'.Ol. Then BD = 2CDsmhDCA will give the chord BD. The arc BD is found from the proportion : 360° : DCA = 2Tr X 3963.3 : arc BD. Note. — Eq. (3) follows directly from (4), Art. 69 : tan i a = \ 52ifi; where cos a ^ ^1+COSa' 29. Given a and b (Special Method). — ^ When a and b are nearly equal, the angle a may be determined more accurately, as follows : Draw AI), making CAD = 45°, and DU perpendicular to AB. Then tan BAU = tan(a - 4o°) = AH But DU = DB cos a = COB- CD') cos a (a — 5) cos a (a - b)b RIGHT PLANE TRIANGLES. 26 and AE = AB — EB = AB — BB sin a = c — ^ ^- = — — c c ^ b^ + ab c BE ^ (a-b)b ^ a-b AE ab + b'^ a + h .-. tan(«-45°) = ^. (1) a -{• If b were greater than a, the formula would be tan(45°-«)=^^^. (2) f Note. — Eq. (1) may be found from the relation proved in Art. 100 : a - 6 _ tan Ha - ^)^ ^j^^-^ i ^^ ^ ^)= 45°, and K^ - /3)= a - 46°. a + 6 tanHa + /3) EXAMPLES. Note. — The angle between the line of sight and a horizontal plane is called an angle of elevation when the point sighted on is above the horizontal plane, and an angle of depression when it is below the horizontal plane. 1. The shadow of a vertical pole 30 feet high is 40 feet long. Find the elevation of the sun above the horizon. Ans. 36° 52 '.2. 2. The vertical central pole of a circular tent is 20 feet high, and its top is fastened by ropes 40 feet long to stakes set in the ground, the ground being horizontal. How far are the stakes from the foot of the pole, and what is the inclination of the ropes to the ground ? Ans. 34.641 feet ; 30°. 8. The top of a lighthouse is 200 feet above the sea level, and the angle of depression to a buoy is 9° 52 '.8. Find the horizontal distance of the buoy from the lighthouse. Ans. 1148.3 feet. 4. The horizontal distance from a point to the vertical wall of a tower is 1000 feet, and the angle of elevation of the top is 4° 15'.2. Find the height of the top of the wall above the point. Ans. 74.370 feet. 5. Two points A and B are on the opposite banks of a stream*. A line AG at right angles to ^ J5 is measured 300 feet long, and the angle ACB is found by measurement to be 62° 30'.4. What is the distance from Ato B? Ans. 576.45 feet. 6. From the top of a lighthouse, 150 feet above the sea level, the angle of depression to a buoy was 12° 10'.2, and that to the shore, measured in the same vertical plane with the buoy, was 62° 14 '.8. Find the distance in feet of the buoy from the shore. Ans. Log. Tables, 616.60 ; Nat. Tables, 616.61 26 PLANE AND ANALYTICAL TRIGONOMETRY. 7. The angle of elevation to the top of the vertical wall of a tower is 20° 10 .4, and the angle of depression to the bottom is 10^ 11 '.0, the horizontal distance from the observer to the wall being 250 feet. Find the height of the wall. Ans. 136.802 feet. 8. We wish to make a ladder that would reach from a point 20 feet in front of a building to the fourth story, a height of 45 feet. Find the length of the ladder and the angle it would make with the ground in this position. Ans. 49.244 feet; 66° 2'.2. 9. The ridgepole of a roof is 15 feet above the center of the garret floor, and the garret is 40 feet wide. What is the inclination of the roof to a horizon- tal plane ? Ans. 36° 52 '.2. 10. A chord of a circle is 20 feet long, and the angle at the center subtended by it is 46° 43'.6. Find the radius of the circle. Ans. 25.217 feet. 11. The angle between two lines is 40° r2'.4, and a circle whose radius is 5730 feet is tangent to both lines. Find the distance from the point of tangency to the point of intersection of the two lines when the circle is in the smaller angle, and when it is in the larger angle formed by producing one of the lines. Ans. 15055 and 2097.2 feet. 12. The legs of a pair of dividers are set so that the angle between them is 80° 24'.4. What is the distance between the points, the legs being 6 inches long? Ans. 7.7460 inches. 13. An equilateral triangle is circumscribed about a circle whose radius is 10 inches. Find the perimeter of the triangle. Ans. GOVS inches. 14. A wedge measures 12 inches along the side, and its base is 2 inches wide. Find the angle at its vertex. Ans. 9^ 33'. 6. 15. The side of a regular decagon is 2.4304 feet. Find tlie radii of the inscribed and circumscribed circles. Ans. 3.7400 feet; 3.9325 feet. 16. The area of a regular octagon is 24 square feet. Find the radius of the inscribed circle and the length of one of the sides. Ans. 2.6912 feet ; 2.2295 feet. 17. The radius of the circumscribing circle of a regular dodecagon is 10 feet. Find the area of the dodecagon. Ans. 300.00 square feet. 18. A cord is stretched around two wheels with radii of 7 feet and 1 foot respectively, and with their centers 12 feet apart. Prove that the length of the cord is 12\/3 + 10 tt feet. 19. A cord is stretched around, and crossed between, two wheels whose radii are 5 feet and 1 foot respectively, their centers being 12 feet apart. Prove that the length of the cord is 12\/3 + 87r feet. 20. Find the radius and the length of an arc of 1° of the parallel of latitude at a place whose latitude is 42° 43'. 9, the earth being regarded as a sphere whose radius is 3963.3 miles. Ans. 2911.1 miles; 50.809 miles. 21. The altitude of a right circular cone is 4. 1436 feet, and the angle at its vertex is 20° 14'.2. Find its convex surface. Ans. 9.7780 square feet. RIGHT PLANE TRIANGLES. 27 22. The altitude of a right pyramid with a square base is 14.463 feet, and the sides of the base are each 4.703C feet. Find its slant height, its lateral edge, and the angle between a face of the pyramid and its base. Ans. 14.643 feet; 14.831 feet; 80° 45'. 5. 23. The base of a trapezoid measured 600.430 feet, and the angles at the ends of the base were found to be 62° 14'. 3 and 74° 18'. 6. Find the length of the other base, the altitude being 40 feet. Ans. 568.138 feet. 24. Find the length of the perpen- dicular from the vertex of the right angle of a triangle to the hypotenuse, the hypotenuse being 6.4603 inches long, and one of the angles of the tri- angle being 40° 40'. 4. Ans. 3.1934 inches. 25. A street-railway track is 10 feet from the curbstone (FB = HD = 10), Fig. 20. and in passing a corner where the street is deflected through an angle of 60°, the rail must be 4 feet from the corner (GC = i). Find the radius of the circular curve. Ans. 0C = 20-4v/3 2-^3 26. Before paying for a pavement, it was necessary to find the area shaded 28750 in Fig. 21. Prove that it is 1- 7500 square feet, the streets being 50 feet wide. ^ Fig. 21. 27. In the egg-shaped sewer (Fig. 22), C is the center of the arc ADB with a radius a ; / and J, of AF and BG respectively with the radii 3 a ; and IT, of FEG with the radius | a. Prove that its area is a2fj+ ^tan-i ^ + 9 tan-i?- 3^ = a2f^^ + §^tan-i?- 3^ = 4.59413 a2, \i5 4 6 4/\8 4 4/ where tan-i - is the angle whose tangent is ^ 3 3 28 PLANE AND ANALYTICAL TRIGONOMETRY. 28. A hill rises 1 foot vertically in a horizontal distance of 30 feet. What is the difference of elevation of two points that are 1000 feet apart, the distance being measured on the ground ? log tan a = 8.62288 - 10 CpL T' = 3.53611 log a' = 2.05899 S' = 6.46365 - 10 log sin o=: 8. 52264 -10 log 1000 = 3. log diff. of elev. = 1.52264 difE. of elev. = 33.315 feet. 29. The horizontal distance between the two extreme positions of the end of a pendulum 40 inches long is 4 inches. Through what angle does it swing ? Half-angle =2° 51 '.96. Ans. 5°43'.92. 30. The angular diameter of the moon is 31'.12, and its distance is 238840 miles. Find its diameter in miles. BAD = SV. 12, and ^C = 238 840. Ans. 2162.0 miles. 31. The equatorial horizontal parallax of the sun is 8". 8, and the radius of the earth is 3963.3 miles. Find the distance of the sun from the earth. BAC = 8". 8, and BC = 3963.3. Ans. 92 896 000 miles. 32. A circular chimney 100 feet high is 10 feet in diameter at the base, and 8 feet at the top. Find the angle at the vertex of the cone of which it is a frustum. Half-angle = 34'.376. Ans. 1°8'.752. Solve the following triangles, the first two elements being given 33. c = 0.02934, A = 31° 14'.2. 34. c = 4.6136, B = 47° 15'.6. 35. c = 436.53, A = 74° 10'.6. / 36. = 0.96724, B = 40° 40'.2. 37. = 110.97, a = 67.291. 38. = 1843.7, & = 618.42. 39. c = 8226.5: a = 814.33. 40. = 0.03672, 6=0.01296. ./ 41. = 4.8293, b = 0.31435. ^ = 58°45'.8; a = 0.015215; & = 0.025086. ^ = 42°44'.4; a = 3.1311 ; 6 = 3.3885. B=15°49'Ai a = 419.98; 6 = 119.03. A = 49° 19'.8 ; a = 0.73363 ; b = 0.63036. A = 37° 19'.8 ; ^ = 52° 40'.2 ; b = 88.236. A = 70° 24M ; 5 = 19° 35'.9 ; a = 1736.9. ^ = 81°50'.5; .5= 8° 9'.5 ; 6 = 116.74. A = 69° 19'.9 ; B = 20° 40M ; a = 0.034357. A = 86° 16'.1 ; ^ = 3° 43'.9 ; a = 4.8191. RKJHT PLANE TRIANGLES. 29 42. a = 43.148, 6 = 84.107. 43. a = 769.28, ft = 61.86. 44. a = 7642.5, ft =864.7. 45. a = 0.04326, ft = 0.54318. 46. a = 903.64, A = 22° lO'.S. 47. ft =0.47922, A = 62° 16' A. 48. a = 8.4642, i? = 30° 16'.4. 49. ft = 18.430, B = 65° 15'.6. .^ = 27° 9'.5; i? = 62°50'.5; c = 94.630. . ^ = 86° 6'.6; B= 3°54'.4; c = 761.06. . ^ = 83°32'.7; B= 6°27'.3; c = 7691.3. . A= 4°33'.2; I? = 85° 26'.8 ; c = 054489. •. 5 = 67°49'.7; ft = 2217.4 ; c = 2394.6. •. i? = 27°43'.6; a = 0.91176 ; c = 1.0300. : A = 59° 43'.6 ; ft = 4.9409 ; c = 9.80075. . A =2A° 44'.4 ; a = 8.4954 ; c = 20.299. Solve the isosceles triangles (Fig. 16) in the following examples, the first two elements being given : 50. a =57.906, ft =62.736. 51. a=3.4782, a=20°20'.6. 52. rt =99.674, /3=40°30'.4. 53. 6=0.96042, a = 70°10'.4. 54. ft = 1146.48, i3=80°36'.4. 55. a=87.904, j9=46.812. 56. 6=6.9044, p= 5.7806. 57. J9 = 18.478, a= 37° 19'.8. 58. i)=0.46424, /3=100°36'.8. .-. a= 67°12'.05; /8=66° 35'.9 ; p=48.673. .-. ^ = 139°18'.8; 6=6.5224; ;) = 1.209L .-. a= 69°44'.8; 6 = 69.008; .-. /3= 39°39'.2; a = 1.4158; a = 886.24; i3=115°38'.8 .-. a= 49°41'.8 .-. a= 32° 10 '.6 .-. a= 59° 9'.2; ^3= 61°41'.6 .-. ^ = 105°20'.4; a=30.471; .-. a= 39°41'.6; a = 0.72690; p = 93.610. p= 1.3319. jr)=675.87. 6 = 148.806. a =6.7330. 6=48.458. 6 = 1.11865. 4 CHAPTER III. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 30. Generation of Angles. — An angle may be considered as generated by a line revolving about a fixed point, the vertex ; thus OA revolving about in the direction a, to the position OB^ de- scribes the angle A OB. The side of the angle /rom which the revolution takes place is called the initial side, and that to ivhich the describing line moves is called the terminal side. The letters describing the initial side are Avritten first in the symbol of the angle, so that the angle A OB is one in which the motion is from OA to OB. J Fig. 24. 31. Direction of Measurement. — The revolving line can move from OA to OB either in the direction marked a or in that marked h. The former motion, contrary to that of the hands of a watch, is arbitrarily considered positive and the latter negative. Thus if the angle a;, between OA and OB, is 30°, the angle AOB is either + 30° or - 330°. Any angle has two measures less than 360°, one positive and the other negative, their numerical sum 90° being 360°. 32. Quadrants. — For convenience the measuring circle is divided into four parts called quadrants., as in the figure. An angle is in the first quad- rant when its value lies between 0° and 90°; in the second, between 90° and 180°; in the third, between 180° 30 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 81 and 270°; in the fourth, between 270° and 360°. Angles between 0° and — 90° are in the fourth quadrant ; between - 90° and - 180°, in the third ; between - 180° and - 270°, in the second ; between — 270° and — 360°, in the first. Also, an angle between zero and ^tt is in the first quad- rant ; between J ir and tt, in the second ; between ir and | tt, in the third ; and between | tt and 2 tt, in the fourth. 33. Complement and Supplement. — Two angles are said to be complementary Avhen their algebraic sum is 90°, as 60° and 30°, 120° and -30°, 260° and -170°; and supplementary when their algebraic sum is 180°, as 120° and 60°, 230° and - 50°, 300° and - 120°. Note. — In Fig. 2, ^ is the sine of B ; that is, it is the sine of the comple- h ment of A, and hence it is called the cosine of A. Since -|-7r corresponds to 90°, and tt to 180°, two angles are complementary when the algebraic sum of their circular meas- ures is ^ TT,. and supplementary when it is tt. 1. The complement of 200° is 90° - 200° = - 110°. 2. The complement of 90° + x is 90° - (90° + x) = -x. 3. The supplement of 2u0° is 180° - 200° = - 20°. 4. The supplement of 270° + x is 180° - (270° + x) = - 90° - x. 5. The complement of -^^ tt is i tt — y^^ tt = — | tt. 6. The supplement of | tt is tt — | tt = — |7r. Show that the complement of the first angle of each of the following pairs is equal to the second angle : 7. 145° and -55° ; 300° and -210° ; -70° and +160° ; -200° and +290°. 8. 180° - X and - 90° + x ; 270° - x and - 180° + x ; 360° - x and - 270° + X. 9. \ IT and \ir ; f tt and — tt ; rr — x and x — \ir ', lir + x and — ^ tt — x. Show that the supplement of the first angle of each of the following pairs is equal to the second angle : 10. 145° and 35° ; 225° and - 45° ; - 160° and 340° ; - 70° and 250°. 11. 270° - X and - 90° + x ; 90° + x and 90° - x ; x - 90° and 270° - x. 12. \ IT and | tt ; § tt and - | tt ; x — tt and 27r— x; |7r + x and — \-ir — x. 134. General Measure of an Angle. — The line OA may be brought into the position OB by revolving either through the 32 PLANE AXD ANALYTICAL TRIGONOMETRY. number of complete revolutions in either direction. The general measure of the angle A OB is then not x, but x+nZGO°^ where n is any whole number, positive or negative. The general circular measure of ""'- — '^ the angle whose circular measure ^'°' ^^* less than 2 tt is a: would be a; + 2 titt, since 2 tt corresponds to a complete revolution. 1. Show that 1000° is in the fourth quadrant.* 1000° = 720° + 280° = 2 X 360° + 280°, two complete revolutions and 280° beyond ; 280° lies in the fourth quadrant. 2. Show that — 3000° is in the third quadrant. - 3000° = - 2880° - 120° = 8(- 360°) - 120°, eight complete revolutions and 120° beyond in the negative direction ; — 120° lies in the third quadrant. 3. Show that - (8 w + |) is in the first quadrant. A -(8n + f)=2nx27r + j'^7r, 2n complete revolutions and -^^ ir beyond ; 2 -j3jj TT is in the first quadrant. 4. Show that 1500° is in the first quadrant, 2690° in the second, 2720° in the third, 2100° in the fourth. 5. Show that — 010° is in the second quadrant, - 1100° in the fourth, - 1400° in the first, - 1920° in the third. 6. Show that -(10n4-6^ is in the third quadrant, -(12n + 2^ in the IT ^ 3 second, t (8 w 4- 7) in the fourth, | tt (3 n + 2) in the third, 7. Show that f tt (10 n — i) is in the fourth quadrant, f tt (15 n — f ) in the third, f TT ( - 9 n - f ) in the third, ^ tt (10 w — 9) in the first. 8. Show that ^ (9 n + 1") will lie in the third or in the first quadrant, o according as n is odd or even. 9. Show that the general circular measure of 0° is 2 nir, and not wr. 10. Show that the general circular measure of 90° is (2 w + ^) ir ; of 180°, (2n + l)ir; of 270°, (2 n + |)7r. 11. If a; = 60°, show that one third of the general measure of x will be 20°, 140°, and 260°, the terminal side of the angle for all values of i x greater than 260° falling in one of these positions. We have, using the general measure, « + w 360°, a: = 60°, 420°, 780°, 1140°, 1500°, 1860°,-. .-. ia; = 20°, 140°, 260°, 380°, 500°, 620°,..- or I a; = 20°, 140°, 260°, 20°, 140°, 260°,... if we reduce the values oi ^x that are greater than 360° to others less than 360° by subtracting some multiple of 360°. * That is, show that when the angle is 1000° the terminal side will lie in the fourth quadrant. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 33 12. Jf X = 45°, show that ] x will be 15", 136°, 255°, three values. 13. If X = 20°, show that I x will be 5°, 95°, 185°, 275°, four values. 14. If X = 60°, show that I x will be 10°, 70°, 130°, 190°, 250°, 310°, six values. 1 r>i° 15. If X = m°, show that -x will have n values less than 3G0°, as — n n m° 360'^ n n wt° 720° n n rn^ (n- 1)360° n n 35. The definitions of the trigonometric ratios in Art. 8 are applicable only to angles less than 90°. We shall now con- sider the more general definitions, of which those in Art. 8 are special cases. 36. Map Drawing by Coordinates.* — Let ABCD be a field whose map is wanted. From any point in the field, measure the distances Oa, Oh^ Oc, and Od^ and also measure the dis- tances aA, bB, cO, and dD, at right angles to X'OX, Lay off on the paper a line X' X of indefinite length, and take on j it some point to represent I the point in the field. Lay off Oa according to some con- venient scale ; thus if Oa were _x 200 feet, and the scale were 20 feet to 1 inch, we would on the map make Oa 10 inches long. Then draw the line aA perpen- dicular to OX on the proper side of OX^ and lay off on it the distance corresponding to aA according to the same scale, thus locating the point A. The other points would be located in a similar manner. Since Oa and Oc are measured from in contrary direc- tions, and a A and cC are measured on opposite sides of X' X^ there is danger of laying them off in the wrong direction ; hence their directions must be carefully distinguished. 37. Coordinates. — The distance Oa, measured along X'OX, is called the abscissa of the point A ; aA^ measured parallel to * This is called the method of offsets. CBOCK. TRIG. — 3 Fig. 27. 84 PLANE AND ANALYTICAL TRIGONOMETRY. Fig. 28. Y'OY^ the ordinate of A; and the two distances Oa and aA^ the coordinates of A. The line X'OX is called the axis of abscissas ; the line Y'OY, the axis of ordinates; and the point 0, the origin of coordinates. The abscissa of a point is its distance from the axis of ordi- nates measured on a line parallel to the axis of abscissas. The ordinate of a point is its distance from the axis of abscissas measured on a line parallel to the axis of ordinates. The abscissa is positive when the point is on the right of the axis of ordinates, and negative when it is on the left; the ordi- nate is positive when the point is above the axis of abscissas, and negative when it is below. If we consider the abscissas as measured from Y'OY, and the ordinates from X'OX, they will be positive when measured to the right and upward respec- tively. Using the customary notation for directed lines,* Oc will represent a line measured from to /3. 17. cosx=-}. II. + fV2. — -2V2. -i\^. -3. + IV2. III. -f\/2. — + 2\/2. + iV2. -3. -|V2. 18. tanx=+^. I. + iV5. + f\/5. — + 2. + IV5. + V5. III. -^V5. -fV5. — + 2. -^>/5. -V5. 19. tan x= -2. II. + |V5. -|\/5. -h -V5. + 1V5. IV. -f\/5. + iV5. — -h + V5. -|Vg. * See Arts. 11 and 15. If sin x is positive, must be positive, since h is always positive, and the angle lies in quadrants I. and II. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 37 Quadrant, Sin. Cos. Tan. Cot. Sec. Cosec. cotx=+|. I. + f fl. + |. — + f- + !• III. -f. I. + |. — -i- _5 3* cot 2= -3. IT. +tVv1o. -^VTo. -f — -jVTo. + Vio. IV. -T^.VIo. +^vlo. -h — + ^VIo. ->/io. sec a; =+3. I. 4f>/2. +f + 2\/2. + iV2. — +IV2. IV. -fV2. +1. -2V2. -\V2. — -|v^. seca:= — ^. II. + !• -h -|. -I — +1- III. -f. -f. + t. + 1- — -l. cosecx=+-U. I. + tV + H- + tV +v. +f|. — II. +A. -H- -A- -V-. -ii- — coseca;=— y-. III. -^T. -if +iV +¥. -If. — IV. -/5- + lf -^i. -¥• + l|. — 21. 22. 23. 24. 25. 40. Trigonometric Functions. — One quc^ntity is said to be a function of anotlier when it depends upon the latter for its value. Thus, if 2/ = sin a;, ?/ is a functioii of x, since it depends upon X for its value, any change in the value of x producing a change in the value of y. The trigonometric functions are the sine, cosine, tangent, cotangent, secant, cosecant, versed sine, coversed sine, and suversed sine. The last three are defined by the eq.uations : The versed sine is vers a; = 1 - cos a? The coversed sine is covers a; = 1 - sin a; The suversed sine is suvers a; = 1 + cos a;. , (1) 41. Geometrical Representation of the Functions. — In Fig. 30 let the radius OB, of the circle described about the vertex of the angle AOB as a center, be unity, and let the angle AOY be equal to 90°. NM and FB are tangent to the circle at X and F respectively ; the triangles OAB, OXM, and OYB, are right-angled ; and the angle YB is equal to the given angle AOB. Then the trigonometric functions of the angle AOB are represented by the lines shown in the figure. For, in Figs. 2 and 29, B is any point on the terminal side OB of the angle AOB, and therefore we may choose the position of B so that OB, or A, shall be equal to unity. Comparing Fig. 30 with Figs. 2 and 29, and using the definitions in Arts. 8, 39, and 40, we see that 38 PLANE AND ANALYTICAL TRIGONOMETRY. sin AOB cos AOB tan AOB cot AOB sec AOB cosec AOB vers AOB covers AOB suvers AOB AB h OB • AB, a OA h OB -.OA, AB a OA XM OX = XM, a OA CB YD AB 00 or h OB a OA OM OX --0M, h OB OB OB YD. AB^ 00 ^ OY ^ ^^' 1 - cos AOB = OX -0A = AX. 1 - sin AOB = OY -00= CY. 1 + cos AOB = XO + 0A = X'A. The trigonometric functions are ratios^ — pure numbers, — and are represented by these lines in the circle whose radius is unity ; that is, they are actu- ally equal to the ratios of these lines to the radius. If, with a radius of unity and the vertex of the angle as the center, a circle be described and two tangents be drawn, one Avhere the initial side OA cuts the circle, and the other at a distance of + 90° from this point (at X and Y respec- tively), the trigonometric functions will be represented as follows : The sine of an angle will he the perpendicular distance from the point where the terminal side of thi» angle cuts the circle, to the initial side, produced if necessary ; positive when it is above, and negative when below, the initial side. Thus sin A OB = AB, sin AOJI = GH, sin A OK = aK, sin A OL = AL. AB and 6rl£, above X' OX, are positive, while GK and AL are nega- tive, being below X' OX. The sine is therefore positive when TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 39 the angle is in the first or second quadrant, and negative when it is in the third or fourth. The cosine will be the distance from the center to the foot * of the sine ; positive when measured to the rights and negative to the left, of the center. Thus cosAOB:=OA, cos AOI£= OG, cos A OK = 0G-, cos A 0L= OA. OA, measured to the right of the center, is positive, while 0(r, measured to the left, is negative. The cosine is therefore positive when the angle is in the first or fourth quadrant, and negative when it is in the second or third. The tajigent will be the distance along the line tangent to the circle at the point where the initial side cuts the circle, from this point to the point where this tangent is cut by the terminal side of the angle, produced if necessary ; positive when measured above, and negative when beloiv, the initial side. Thus tan ^0^ = XM, tan A 011= XJSr, tan A 0K= X^, tan AOL = XK XM, above X' OX, is positive, and XJSf, below X' OX, is negative. There- fore the tangent is positive when the angle is in the first or third quadrant,. and negative when it is in the second or fourth. ^ The cotangent will be the distance along the second tangent (FYD) from the point of tangency to the point ivhere this line is cut by the terminal side of the angle, produced if necessary ; posi- tive when measured to the right, and negative to the left, of the point of tangency. Thus cot A OB = YD, cot A OH = YF, cot AOK= YD, cot AOL = YF. YD, measured to the right, is positive, and YF, measured to the left, is negative. There- fore the cotangent is positive when the angle is in the first or third quadrant, and negative when it is in the second or fourth. Note, — The positive directions of measurement are above X'OX and to the right of TOY, and the negative are below X'OX and to the left of Y'OY. The secant ivill be the distance from the center along the ter- minal side of the angle, produced if necessary, to its point of intersection with the taiigent at the point of intersection of the initial side with the circle; positive when measured along the side itself, and negative when along the side produced. Thus sec AOB= OM, sec A 0H= OX, sec A 0K= OM, sec AOL= ON. * The foot of the sine is the point where the perpendicular line representin ; the sine cuts the initial side, produced if necessary. | % 40 PLANE AND ANALYTICAL TRIGONOMETRY. Since sec A OB and sec AOL are measured along the terminal side itself, they are positive. The terminal sides (^OH and OK} of the angles A OH and A OK must be produced in order that they may intersect the tangent line iV7l[f, and therefore ^ec A Off and sec A OK are negative. Hence the secant is positive when the angle is in the first or fourth quadrant, and nes^ative when it is in the second or third. The cosecant will he the distance from the center along the terminal side^ produced if necessary^ to its intersection with the second tangent^ FYD ; positive when measured along the side itself^ and negative when along the side produced. Thus GosQcAOB= OB, cosec ^Oir= OF, cosec A OK = OB, cosec A OL = OF, Since cosec A OB and cosec A OH are meas- ured along the terminal side itself, they are positive, while cosQc AOK and cosqgAOL, measured along the side pro- duced, are negative. Therefore the cosecant is positive when the angle is in the first or second quadrant, and negative when it is in the third or fourth. The versed sine (1 — cos x) will he the distance from the foot of the sine to the point where the initial side cuts the circle; always positive, hecause cosx can never he greater than the' radius, or unity. Thus vers A OB = AX, vers A Off = GX, Yeis AOK= ax, Yers AOL = AX. The coversed sine (1 — sin x) ivill he the distance from the point C or P, where a line drawn through the point of intersection of the terminal side and the circle parallel to the initial side cuts Y'OY, to the point Y; always positive, since sinx can never he greater than the radius, or unity. Thus covers A OB = CY, covers AOff= CY, covers AOK= PY, covers AOL = FY. The suversed sine (1 + oos x) will he the distance from the point yj , where the initial side produced cuts the circle, to the foot of the sine ; always positive, since cos x can never he alge- hraically less than minus unity. Thus suvers AOB = X^A, suvers A Off= Xa, suvers A 0K= X' a, suvers AOL = X'A. Note. — These lines represent th^ trigonometric functions, only when the radius of the circle is unity. If the radius differs from unity, the functions are equal to the lengths of these lines divided by the radius. 42. Changes in the Values of the Functions. — Let OX be the initial side of the angle, and let the terminal side first TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. 41 r Y cot D \ ^ . 1 C E\ A f / / cos ^ 1 X \ y^ OV ^ ven 1 ^ ■-^ P 1/ H coincide with OX^ and then, in revolving about 0, come into the positions OM, OY, OH, OX', OK, (9^^ ON, and OX, and let us consider the resulting changes in the values of the sine and of the tangent. The sine of 0°, the terminal side coinciding with OX, is zero. As the angle increases, the sine, being positive, also in- creases (sin A OB = AB}, until at 90° it is equal to the radius, or +l(sinJL07= OF). The sine then decreases {ain A Oil = G-H^, still being positive ; and at 180° it is zero, the ter- minal side coinciding with OX' . The sine then becomes negative, and decreases algebraically, in- creasing numerically (p\nAOK= (tK), until at 270° it is equal to the radius, or — 1 (sin ^ OF' = OF'). It then in- creases algebraically, decreasing numerically (sin^OX = ^X); and at 360° it again becomes zero. The tangent of 0° is zero ; the tangent then becomes posi- tive, and at 90° it is infinite, the terminal side being parallel to XM; then negative, and at 180° it is zero ; then positive, and at 270° it is infinite ; then negative, and at 360° it is zero. Just before the terminal side reaches the position F, the tangent is positive, and just after, it is negative ; therefore the tangent of 90° is ±qo, the upper sign being that of the function of an angle a little less than 90°. The table gives the values of the functions of 0°, 90°, 180°, 270°, and 360°, and their signs in quadrants I., II., III., and IV. : Fig. 31. 0°. I. 90°. II. lS()o. III. 270°. IV. 360°. sin. + + 1 p + _ -1 _ COS. + 1 + 9- — -1., — + + 1 tan. cot. CO + CO _ CO + + CO — CO sec. + 1 + CO.. - 1 — 00 + + 1 cosec. 00 + + 1 + QO - - 1 - QO ^' v^V 42 PLANE AND ANALYTICAL TRIGONOMETRY. 43. Limiting Values of the Functions. — The sine and cosine may have any value between +1 and —1, but they cannot have a value numerically greater than unity. Tlie tangent and cotangent may have any value between + c» and —00 ; that is, no matter what a number may be, there will always be some angle that will have that number as the value of its tangent, and another having it as its co- tangent. The secant and cosecant may have any value between -|-1 and + X), or — 1 and — oo ; but they cannot have a value numeri- cally less than unity. The versed sine, coversed sine, and suversed sine may have any value between zero and -f 2. Note. — In the first quadrant, all the functions are positive, and the sine, tangent, and secant increase as the angle increases ; while the cosine, cotangent, and cosecant decrease as the angle increases. Note. — The functions change signs only when they pass through the values zero and infinity. 44. Graphical Representation of the Functions. — Let the dis- tance OL represent 360°, so that 1° is represented hj -^\-qOL. At (7, such that OQ =^0L^ draw a line perpendicular to OL^ and D E \F 226' 270° 0° SO" 46" Wf 136" 160° IW Fig. 82. lay off on it any convenient distance 0 — — ^ - Vl + tan2 X - ^^ + ^*^'' ^ - ^^^^^ * . Vl — sin^ X cosx cotx Vcosec2x-l 1 1 Vl 4- tan2 X / i . ^^♦•2^ sec x cosec X = = — — = ■ = V 1 -f cot-* X = — - ■ sinx Vl -cos2x ^anx Vsec2 x - 1 If any one of the functions is given, the others may be found from these formulas. It is easier in general to find first the sine and cosine, and then to find the others. 48. Find the Unknown Functions in the Following : 1. tana;=— -, 2: being in the fourth quadrant. Compute the numerical values of the ratios by the^ method of Art. 15, and then select the proper signs for the func- tions in the fourth quadrant. Thus let = 3, sin 2:= — -, 5 4, . A = 5 cos x — -\- 5' cot 2; = --, sec a: = H- -, 4 cosec a: = — -• o 2. tan 2: = 2, a; being in the third quadrant. Then sma: tana: cos a: _1_ V5' Vl + tan2 X V5' Vi + tan2 x These convenient formulas may be easily remembered from Fig. 35. Knowing sin x and cos a:, we have cota: = - = 4-^; seca; = = _V5: tan X A cos x cosec X sma; z The radicals should be taken with the double sign. FUNCTIONS OF ONE ANGLE. 49 8. cot X = — % X being in the second quadrant. •. coseca; = ± Vl + cot2a;= + V5; sin 2: = =H — ; cosec X V6 cos x = ± Vl — sin^a; = ^ ; tan x = == — - ; V5 cot X 2 sec a; = =— -V5. cos X I 4. sec x = — -y-, X being in the third quadrant. . •. cos X = = — -— ; sin a: = ± V 1 — cos'^^ = ; sec a: 17 17 , sin a: , 15 . ,8 17 tana; = =-|-— -; cota; = +-— ; cosec a; = . cos a; 8 15 15 5. sinx = — I, X being in the third quadrant. . •• cos X = — f ; tan x = + | ; cot x = + | ; sec x = — | ; cosec id = — | . 6. cosx = + I, X being in the fourth quadrant. .-. sinx = — |V6; tanx = — ^VS; cotx = — |\/5; secx = +f; cosecx = — |V5. 7. tanx = — y\, X being in the second quadrant. . sin X = + t\ ; cos X = — {| ; cot x = — -^ ; sec x = — ^| ; cosec x = + ^?-. 8. cotx = + j\, X being in the third quadrant. . '. sin X = — II ; cos x = — ^^. 9. sec X = — II, X being in the second quadrant. . •. cos X = — If ; sin X = + y^y ; tan x = — -^-^. 10. cosec X = — V , X being in the fourth quadrant. If sin J^: If tan J^; •. sma ^ = -/t; ; CO .sx = +r_ where s = cos J^=- where s = cosi5= ; tan %■=- a^b^-c 2 ' ■^. 11. W^ -6)(s- 6c _^ show that Ms-a)_ > 6c a + 6 4- c 2 ' 12. =# -a)(s- s(s - c) ^ show that ks-c), ^ ^ ab 13. If sec 6 = a, show that sin 6 is imaginary if a is numerically less than unity. d = Vl - COS20 =a/1 ^ rrJl - 1 \ sec2^ ^ a2 Smt/=V1 -COS''£'=A/l ^r-^-v/i -— = — i. »2 /7 CROCK. TRIG.— 4 Vf^^^^^^'^^V '^ Of THE 50 PLANE AND ANALYTICAL TRIGONOMETRY. 14. If tan 6 = a, show that cosec 6 is real for all values of a. 16. If cos d = a, show that cosec 6 is imaginary when a is numerically greater than unity. 49. The Signs of the Functions are given by the formulas of Art. 46, so that it is necessary to remember only that the sine is positive in the first and second quadrants and the cosine in the first and fourth. Thus, in the second quadrant, sin a; -h , cos a: — tan X = = — = — ; cot X = = — = — ; cos a; — sma; + 1 + 1 + , . sec X = = — = — ; cosec x = - — = — = + . cos a; — sin a; + 50. Find the Values of the Following Expressions : 1. versa:tan3T-l ^^^^ tan a: = 4, x being in the third secic quadrant. Find the numerical values of vers x and sec x, and substitute. .-. cosa; = ^, seca; = — VTf, versa;=lH — . VlT + l .^ ^ Vr! 4VT7 + 4-V17 3VT7 + 4 -VI7 -17 17 • 2. SIP ^ sec X ^Yien versx = |, x in the fourth quadrant. Ans. + 16. cos X cosec X 8. tan a; - cot X ^^^^^ ^^^^^ x = - V5, x in the third quadrant. Ans. - f . tan X + cot X when cot a; = — ^, a; in the second quadrant. Ans. — 2. cosec X 4- cos x 5. sm a; + tan a; ^^^^^^ sec x = - f , x in the third quadrant. Ans. + ^%. cos X + vers x 6. sec X - vers x ^^^^ cot x = - 2, x in the second quadrant, sec X 4- vers x Ans 9 + 2V5 ^ 29 -^ 20V5 7. su^y^ + t^"'^^ when secx = - |, x in the second quadrant. Ans. ^Wz- cos-^ X + vers2 x 8. secx + sin X ^^^^ tan x = 2, x in the third quadrant. Ans. - ^ V5. 1 — cotx FUNCTIONS OF ONE ANGLE. 51 9. cosecx + seca ; ^^^^ sec a; = + VIO, x in the fourth quadrant. ^°^^^°«^ Ans. -20. 10. secx-coseca; ^^^^ cotx = - 2, x in the second quadrant. Ana. - 3. sec X + cosec x 11. versx- covers X ^^^^ sinx = - |, x in the fourth quadrant. sec x- cosec X ^,^^ _ ^^ 51. Change the Given Expression to Another containing only One Function : , 2 sec? X 4- sec^ x tan^ x — sec* x sec^ a; — 1 to contain only cosec x. It is best generally to change the expression to another con- taining only sin x and cos x^ and then to change this into one containing the proper function. 2 sin2 X 1 cos^ X cos* X cos* X _2 cos^ X + sin^ x — 1 1 -| cos2:c(l— cos^a;) COS^ 2J 2 — 2 sin^ 2; + sin''^ a; — 1 1 — sin^ x 1 o ; ! ; = ; ^ = — = COSCC^ X. (1 — sin^ x^ sin^ x (1 — sin^ x^ sin^ x sin^ x to contain only tan x. vers 2: — covers x sin^ a; — cos^ x . , , tan 2: . 1 = sm X + cos x= ± l-cosa;-l + sina: Vl + tan^a; VT+tan^^ where the signs used will depend upon the quadrant of x. The true result is ± — where the positive sign Vl + tan^ X corresponds to x in the first or fourth quadrant, and the nega- tive to X in the second or third. Use radicals as little as possible. 3. 1 — 2(1 — covers x)2 -\ ^~-^ — - to contain only cosx. Ans. cos*x. ^ (1 + t,an2 x)2 ^ 4 sec xcosecx-4sinx cosx ^^ ^^^^.^ ^^^^ ^.^ ^ ^,^^ (l-2sin2x)2^ sin X sec x sin^ x 5. (^ - covers x)^ cosec* x ^^ contain only tan x. ^ns. tan2 x + tan* x. (cosec- X — 1) cot-x « sec2 X — sec2 x sin* x(l + cot^ x) . . . , 6. — ; 7^—^ ^ to contain only cosec x. sni2 x cos2 X . Ans. *^^^^^*^ cosec2 X — 1 52 PLANE AND ANALYTICAL TRIGONOMETRY. 7. tan2^sec2^-sin2^cos2^ to contain only cot ^. Ans. l+'^cot2g+3cot*g ^ COt4^(l+COt2 0)2 8. ^ ~ ^" ^^' (cos* X — sin* x) to contain only sin x. Ans. (1—2 sin2 x)*. (l + tan'-ix)--^^ ^ ^ \ 9. — sec2 a sin2 a — ^^ contain only cosec a. ^ns. (tan a + 2 cot a)2 (2 cosec2 o - 1)2 - 10. ^'^'^^^"'^ to contain only sec 6. Ans. C^^^^'^--^)'. sin- e — cos-^ d sec2 ^ - 2 - 11. sec2^cosec2^+sec2^-cosec2^-l ^^ ^^^^.^ ^^^ ^ ^^^^ cot2^+2 tan2 d - cosec2 ^ + 1 1 -cot* d 52. Solution of Trigonometric Equations. — Transform the given equation into one containing only a single function (usually the sine or cosine), because in a single equation we must have only one unknown quantity. Then solve the equa- tion algebraically for this function as the unknown quantity. The corresponding angle may then be found from the tables. Test the angles by substitution in the given equation. I. sin ^ cos ^ = + J. .-. sin eVl - sin2 (9 = + J ; .-. sin2 (9(1 - sin2 (9) = J ; .-. sin*(9-sin2(9 + i = 0; .-. sin2(9 - 1 = ; .-. sin^ = ±Vj. .-. e might be 45°, 135°, 225°, or 315°. But the given equa- tion shows that the product of the sine and cosine must be positive, and hence that they must have the same sign. Both the sine and cosine are positive in the first quadrant, and nega- tive in the third, but they have contrary signs in the second and fourth quadrants. Hence the only admissible values of 6 are 45° and 225°. 2. tan ^ sec ^ = - \/2. .-. 5 = 225°, 315°. 3. cosec 5 = I tan 5. .-. 5 = 60^ 300°. 4. tan 5 + cot 5 = 2. .-. = 45°, 225°. 6, sec2 d + cosec2 6 = 4. .: d = 45°, 135°, 225°, 315°. 6. sin5=±\/3vers5. .-. 5 = 0°, 60°, 300°. 7. see + tan 5 = ± V3. .-. d = 30°, 150°. [300°, 330°. 8. sec2 + cot2 5 = J/ . .-. 5 = 30°, 60°, 120°, 150°, 210°, 240°, 9. sinx= + V3cosx. . •. x = 60°, 240°. ' 10. tan X = - 2\/3 cos X. .'. x = 240°, 300^ II. sin X cos X = - 1 V3. . •. x = 120°, 150°, 300°, 330° 12. sin 6 + cosec 9 = -^. .: $ = 210°, 330°. 13. 3 sin X = 2 cos2 x. .-. x = 30°, 150°. 14. sec X tan X = + 2 V3. . •. x = 60°, 120°. FUNCTIONS OF ONE ANGLE. 68 16. sec 6 vers 6 = \ — tan 6. 1 (l-C08^)=l-''"^ sin ^ = 2 cos ^ — 1 ; cos 6^ ^ cos^ .-. sin2^ = 4cos2^-4cos^ + l; .-. l-cos2^ = 4cos2^-4cos^ + l; . •. 5 cos2 ^ - 4 cos ^ = ; . •. cos ^ (5 cos ^ - 4) = ; . •. cos ^ = and 5 cos ^ — 4 = 0. (a) cos ^ = gives 6 = 90° or 270°. These values are re- jected for reasons involving the methods of the Differential Calculus. (J) cos ^=f gives sin ^= ± |, since this value of the cosine will allow the angle to lie either in the first or in the fourth quadrant. Transposing in the original equation, we have sec 6 vers + tan ^ — 1 = 0, and we test by substitution. For in the first quadrant, we have showing that 6 has a value in the first quadrant. For 6 in the fourth quadrant, we have not zero; and hence 6 does not have a value in the fourth quadrant. 16. sin X tan x = — t%. 17. vers x = 2 covers x. 18. sin X tan x = 2 cos x. 19. sec X cosec x = — 2. 20. cos X cot X = — f . 21. sin X cos X = — ^|. 22. tan x = — ^20 cos x. 23. sec X + tan x = 2. smx: 5» cos X = — i ; quadrants II. and III. •. cos X = I or — 1 ; first quadrant, and 180°. •. sin X = ± Vf , cos X = ± V-J ; four quadrants. •. sin x = ± ^\/2, cos X = T 1 V2 ; 135° and 316°. •. sin X = — f ; quadrants III. and IV. •. sin X = ± I or ± f ; quadrants II. and IV. 2 •. sin X = :: ; quadrants III. and IV. V5 •. tan X = + f ; first quadrant. 24. sec X tan 2 a: (1 — 2 cos x) = 0. The values of x are found by placing each factor equal to zero, and solving the resulting equations. Hence we have sec X = 0, tan 2 a; = 0, 1 — 2 cos x = 0, 64 PLANE AND ANALYTICAL TRIGONOMETRY. But sec a; = is impossible; tan 2 a; = gives 2 rr = 0° or 180°, and, using the general measures of the angles, x = 0°, 90°, 180°, 270°, the second and the last values being inadmissible. 1 — 2 cos a; = gives cos x = -J, and x = 60° or 300°. 25. tan ^ a; = 0. 26. vers 3 a; = 0. 27. sinxcosx(l+2 cosx)=0. 28. cos 2 X (3 - 4 cos^ x) = 0. 29. (l+tana;)(l-2sinic)=0. 30. tan X = — 2 sin X. 31. sin 2x vers3x = 0. x = 0°. X = 0°, 120°, 240°. X = 0°, 90°, 180°, 270°, 120°, 240°. [330°. X = 45°, 225°, 135°, 315°, 30°, 150°, 210°, X = 45°, 225°, 30°, 150°. X = 0°, 120°, 180°, 240°. X = 0°, 90°, 180°, 120°, 240°, 270°. 53. The Functions of an Angle Greater than 360° are the same as those of the angle less than 360°, found by increasing or diminishing the given angle by some multiple of 360° ; for the position of the terminal side would not be changed by these operations. Thus tan 1010° = tan (1010° - 720°) = tan 290° ; cos ( - 835°) = cos ( - 835° + 720°) = cos ( - 115°), or cos (- 835°)= cos (- 835° + 1080°)= cos 245°. 54. The Functions of 90° ± oc and of 270° ± a? are numeri- cally equal to the cof unctions of x, but may differ from them in signs. Let the arcs UB, EB, KJ, KM, and NP be equal, E T)^^ B / \ . / \ / \ \ r ^^. \ H i n r \ / V > /, J ^■--^ _^^M JiV Fig. 87. K Fitt. 36. the radii CB and RP each being unity. Then the right tri- angles PCB, FOB, LCJ, LCM, and SRP are equal, having FUNCTIONS OF ONE ANGLE. 55 the same hypotenuse (unity) and the angle x the same in each. Therefore FB = CI= LM= DF=nC= JL = SP, and CF=IB = HD = LQ^ MI = JH = RS. .'. sin( ^()°-x) = IB = OF=RS = + Go^x\ ] (4) cos( W-x)=CI =FB=SP = + &mx. ) ^^^ cos( dO''+x)=CH:=FI) = -I)F*=-SP=-8inx. ] ^ ^ sin (210°-x) = IfJ= CL = -L0*= -RS= - cos a;; ) .^. 008(270'' -x)= Cir=LJ = -JL* =-SP=-sinx.] ^ ^ sin (210° +x} = lM=OL = -LO* = -RS= -cosx; cos (270° + X) = CI = LM= SP = + sin x. Thus sin 100°= sin (90° + 10°)= + cos 10° ; cos 100°= cos (90° + 10°) =- sin 10°. sin 200° = sin (270° - 70°) = - cos 70° ; cos 200° = cos (270° - 70°)= - sin 70°. sin 300° = sin (270° + 30°) = - cos 30° ; cos 300° = cos (270° + 30°) = + sin 30°. 55. The Functions of i8o° ± y and of 360° — y are numeri- cally equal to the same functions of y, but may differ from them in signs. From Fig. 36, sin (180° - z/)= ^D = Zg = + sin y ; cos(180° -y)=CH =- HO* = -CI = - cos ?/. * ^^^ 8m(im° +y)=HJ = -JH* =-Z5 = -siny;) .^^ cos(180°-hi/)=(7^ =-^(7* = -6Y =-cos^/. ) ^^ sin(360°-j/) = 7ilZ=-if/* =-Zg = -siny;) ,„>. cos (360° -I/) =07" = + cos2/. ) ^^ Thus sin 100° = sin (180° - 80°) = + sin 80° ; cos 100° = cos (180° - 80°) = - cos 80°. sin 200° = sin (180° + 20°)= - sin 20° ; cos 200° = cos (180° + 20°) = - cos 20°. sin 300° = sin (360° - 60°)= - sin 60° ; cos 300° = cos (360° - 60°) = + cos 60 * See Art. 2. \o 66 PLANE AND ANALYTICAL TRIGONOMETRY. 56. The Functions of a Negative Angle are numerically equal to the same functions of an equal positive angle, but may differ from them in signs. B ^-^ ^ ? ^ B \ F / \ H \ r ^ t / I / \^ \ ) ^ L \ J J ^--^ — M K Fig. 38. sin (-?/) = IM= - MI"" = -IB\ = - sin «/ ; cos ( — 1/) = CT= + cosy. J Thus sin (a:-180°) = sin [-(180°-a:)] = - sin (180° -J)=- sin x. cos(a:-180°) = cos[-(180°-2;)] = + cos (180° — x)=— cos X, (1) 57. Summary. — Using the equations of Art. 46, , sin a; , cos a; tan X = , cot X = , cos X ^ sin X sec re cos a; cosec X sin a; and the results of Arts. T>4:^35, and 5Q, we have sin (90° — a;) = + cosx; tan (90° -x) = + cot x ; sec (90° — x) = + cosec x ; sin (90° + x) = + cos a; ; tan (90° + a:) = - cot X ; sec (90° + x) = — cosec x ; sin (180° - x) = +smx;/\ QannSIE^xJ sec (180° — x) = — secx; sin (180° + a;) = — sinx ; tan(180° + x) = +tanx; sec (180° + x) = — sec X ; sin (270° — x) = — cos X ; tan (270° - x) = + cot x ; sec (270° — x) = — cosec J ; sin (270° + x) = — cos X : tan(270° + x) = -cotx; sec (270° + x) = + cosec x ; sin (360° - x) = - sin x ; tan(360°-x) = -tanx; sec (360° — X) = + sec X ; sin ( — x) = — sin x ; tan ( — x) = — tan x ; sec(— x) = + secx; cos (90° — x) = + sin X ; cot (90° - x) = + tan X ; cosec (90° — x) = + sec x. cos (90° + x) = — sin X ; cot (90° + x) = - tanx; cosec (90° + x) = + sec x. cos (180° — x) = — cos X ; cot(180°-x) = -cotx; cosec (180° — x) = 4- cosec x. cos (180° + x) = — cosx ; cot(180° + x) = + cotx; cosec (180° + x) = — cosec x. cos (270° — x) =,^_sinjc ; cot (270° - X) = + tan x ; cosec (270° — x) = — sec x. cos (270° + x) = + sin x ; cot (270° + x) = - tan X ; cosec (270° -|- x) = — sec x. cos (360° - x) = + cos X ; cot(360°-x) = -cotx; cosec (360° — x) = — cosec x. cos ( — x) = + cos X ; cot (— x) = — cotx ; cosec ( — X ) — — cosec x. * See Art. 2. (1) (2) (3) (4) (5) (6) (7) (8) FUNCTIONS OF ONE ANGLE. 67 These formulas may be remembered from the three facts : (a) Whenever the angle is 90° ± x, or 270° ± x, the func- tions of the angle are numerically equal to the corresponding (7of unctions of x. (b) Whenever the angle is 180° ± x, 360° - a;, or - x, the functions of the angle are numerically equal to the same functions of x. ((?) The sign to be placed before the function of x is that of the original function when x is less than 90°. Thus sin (270° + x) =^-. cos x, since, when x < 90°, 270° + x will be in the fourth quadrant, and sin (270° + x) will therefore be negative. 58. General Method of Proof. — In Arts. 54, 55, and 56, both x and y were less than 90°, but the formulas in Art. 57 are true for all values of x. Sup- pose, for example, tbat we wish to prove the formulas for 270° 4- x when X is in the fourth quadrant, that is, when x is between 270° and 360°. Let KAUaJ=x; then AEaJKAEaJ= 270° + x. Let AEGKJ'^x. Then in the right triangles JCL and J' CL' the angles JCL and J' CL' are equal, each be- ing 360° — X ; therefore the triangles are equal, and CL = CL' and LJ= L'J' numerically, braically CL = - CL' and LJ= + L'J'. .-. sin(270° + 2:)=5'J^= CX = - OX' =-cosa:; cos (270° ■\-x)= CH= LJ = + L'J' = + sin a;. EXAMPLES. 1. From the preceding equations prove that Alge- (a) tan (- 1200°) = cot 30°. (&) sec 1000° = cosec 10°. (c) cos ( - 890°) = - cos 10°. {d) cot 1700° = cot 80°. (e) cosec ( - 1235°) = - sec 66°. (/) sin 1340° = - cos 10°. (gr) sin ( - 3000°) = - cos 30°. (/i) cos 1300° = -cos 40°. (0 tan 3200° = -tan 40°. ( j) cot ( - 1300°) = - cot 40°. (k) sec (- 2900°) = + sec 20°. (0 cosec 2420° = - sec 10°. 58 PLANE AND ANALYTICAL TRIGONOMETRY. 2. If tan ^ = - cot 140°, find the two values of 6 less than 360° . •. tan 5 = - cot (90° + 50°) - + tan 50°. .-. = 50^ 230°. 8. Find the values of 6 in the following equations : (a) sin ^ = + cos 220°. . •. 230°, 310°. (6) sin^ = + cos310°. .-. 40°, 140°. (c) sin^ = -cos210°. .-. 60°, 120°. (d) sin2 d = + cos2 200°. . •. 70°, 110°, 250°, 290°. (e) cos ^ = + sin 150°. . •. 60°, 300°. (/) cos ^ = + sin 250°. . •. 160°, 200°. {g) QOsd = - sin 170°. .-. 100°, 260°. Qi) cos ^ = - sin 275°. .-. 5°, 355°. (0 cos2 ^ :=: + sin2 100°. .'. 10°, 350°, 170°, 190°. (j) tan ^ = + cot 100°. . •. 170°, 350°. (A:) tan ^ = + cot 200°. . •. 70°, 250°. (0 tan^ = -cot230°. .-. 140°, 320° (?n) cot = + tan 260°. . •. 10°, 190°. (n) cot^ii:+tan345°. .-. 105°, 285°. (o) cot (? = - tan 245°. . •. 155°, 335° {p) cot ^ = - tan 305°. .-. 35°, 216°. (g) sec ^ = - cosec 100°. .-. 170°, 190°. (r) sec = + cosec 130°. . •. 40°, 320°. (s) sec ^ = + cosec 310° . •. 140°, 220°. (0 cosec ^ = + sec 315°. . •. 45°, 135°. (It) cosec ^ =: + sec 2.30°. . •. 220°, 320°. (v) cosec ^ = - sec 185°. . •. 85°, 95°. {w) cosec ^ = - sec 335°. . •. 245°, 295°. (x) cosec2 zr + sec2 250°. .-. 20°, 160°, 200°, 340°. {y) SQCd = - cosec 290°. . •. 20°, 340°. {z) sin ^ = - cos 300°. . •. 210°, 330°. 4. cos = sin 2 6. Show that one value of is 30°. 5. tan n = - cot 120°. Show that one value of 6 is 30° ^ n. 6. sec3e = cosec (n- 1)^. Show that one value of ^ is 90° -^(n + 2). 7. If cot 309° = - j%, find sin 219°. sin 219° = sin (180° + 39°) = - sin 39°. But cot 309° = cot (270° + 39°) = - tan 39° ; .-. tan 39° = + j%. ... sin39° = -^:^|?L==-4= = -^; .'. sin219° =-- 4< Vl +tan2 39° Vl64 Vil V41 8. If sin 21 7° = - j%, prove that tan 127° = - f- 9. If cos 125° = - a, prove that tan 325° = _^ - vl — a2 FUNCTIONS OF ONE ANGLE. 59 10. If cot 260° = + a, prove that cos 350° = + ^ vTTa2 11. If sec 340° = + a, prove that sin 110° = - , and tan 110° = — 12. If cos 300° = + «, prove that cot 120° = ^ 13. If sin 116^ = + a, prove that tan 205° sec 245° _ . VTIT^^ cosec 385° a 14. If cos 200° = - m, prove that tan 110° cosec 250° cot 290° = m 15. If cosec 185°= -m, prove that tan 355° tan 275° cos 175° = - ^^ — -. m 16. Show that cot ^ ( - x - 540°) = tan J x. cot K- X - 540°) = cot (- ^ X - 90°) = cot [ - (90° + 1 X)] = - cot (90° + i sc) = + tan ^ SB. 17. Show that sin (y — 90°) = — cos y. 18. Show that sin (y - 180°) = - sin y. - 19. Show that cos (y — 270°) = — sin y. 20. Show that sec (— x — 540°) = — sec x. 21. Show that tan (y - 360°) = + tan y. 22. Show that cos |(x - 270°) = + sin | x. [Note that 270° in the parenthe- sis is to be multiplied by i.] 23. Show that cos |( - 810° + a - 6) = - sin ^(a - 6). 24. Show that cosec |(x - 360°) = - sec ^ x. 25. Show that sec ^ ( - 900° - x) = - sec ^ x. 26. Show that tan |(360° + a - &) = + tan ^(a - 6). 27. Show that cos (180° — x) is equal to the sine of the complementary angle. Complement = 90° - (180° - x) = - (90° - x) ; sin [- (90° - x)] = - sin (90° -x)= - cos x. But cos (180° - x) = - cos x. . •. cos (180 - x) = sin [90°- (180°- X)]. q. e. d. 28. Show that cosec (270° — x) equals the secant of the complementary angle. 29. Show that tan (180° + x) equals the cotangent of the complementary angle. 30. Show that sec (270° + x) equals the cosecant of the complementary angle. 31. Show that cos (90° +x) equals the sine of the complementary angle. 32. Show that cot (360°— x) equals the tangent of the complementary angle. 33. Show that tan (270°+ a;) is equal to the negative of the tangent of the supplementary angle. Supplement = 180° - (270° + x) = - (90° + x) ; tan [- (90° + x)] = - tan (90° + x) = + cot x. But tan (270° + x) = - cot x. . •. tan (270° + x) = - tan [180° - (270° + x)]. q. e. d. 34. Show that cosec (180° + x) is equal to the cosecant of the supple- mentary angle. 60 PLANE AND ANALYTICAL TRIGONOMETRY. 35. Show that sin (360° — x) is equal to the sine of the supplementary angle. 36. Show that sec (90° + x) is equal to the negative of the secant of the supplementary angle. 37. Show that cos (270°— cc) is equal to the negative of the cosine of the supplementary angle. 38. Show that cot (270° + x) is equal to the negative of the cotangent of the supplementary angle. 59. The Trigonometric Tables. — The relations shown in Arts. 53 and 51 enable us to find the functions of any angle, although the tables contain only the sines, cosines, tangents, and cotangents of angles less than 45°. For, since sin (90° — x) = cos x, cos (90° — x^ = sin x, tan (90° -x) = cot x, cot (90° - x) = tan x, the tables are immediately extended to 90° by writing the proper degrees and minutes at the bottom and on the right of the page respectively. Then, since the value of any function of an angle greater than 90° can be found in terms of a function of an angle less than 90°, we can find the numerical value of the function from the tables. 1. Find from the tables the logarithmic functions of 580° 42'.4. 580° 42' A = 300° + 220° 42'.4. .-. sin 580° 42'.4 = sin 220° 42'.4 = sin (180° + 40° 42'.4) = - sin 40° 42'.4 ; .-. log sin 580° 42'.4 = 9.81437 n. cos 580° 42'.4 = - cos 40° 42'.4 tan580°42'.4 = + tan 40° 42'.4 cot 580° 42'.4 = + cot 40° 42'.4 .-. log cos 580° 42 '.4 = 9.87971 n. .'. log tan 580° 42'.4 = 9.93467. .-. los cot 580° 42'.4 = 0.06533. 2. Find from the tables the logarithmic functions of the following angles : Angle. log sin. log COS. log tan. log cot. 499° 29'.7. 9.81258. 9.88102 w. 9.93158 w. 0.06842 n. 597° S'.B. 9.92427 w. 9. 73449 «. 0.18978. 9.81022. 689° 27'.6. 9. 70598 w. 9.93514. 9.77084 w. 0.22916 n. 3. sin b = tan 250° 15'.5 cot 278° 17'.3 ; find 6 = 203° 57'.0 or 336° 3'.0. 4. cos /3 = cos 149° 27'.6 sin 216° 44'.0 ; find 8 = 58° 59^7 or 301° 0'.3. 6. tana = sin 319° 52'.0 -- cot 254° 30'.2 ; find a = 113° 16'.5 or 293° 16'.5. 6. cotc = cos216°44'.0-v-tan329°27'.6; find c = 36°21'.6 or 216°21'.6. FUNCTIONS OF ONE ANGLE. 61 60. Transform the First Member into the Second in the following examples. Usually it is best to change the given expression into one containing the sine and cosine, and then to change this into the required form. Any operation is admis- sible that does not change the value of the expression. Use radicals only when unavoidable. If the expression is factored, it is often advantageous to reduce each factor separately, not multiplying until it becomes necessary. tan X — sin x sec a? sin^ X 1 -f cos X sin a; — sin X tan X — ^\nx cos x _ sin a; (1 — cos a;) _ 1 — cos x sin^ x ~~ sin^ x cos x sin^ x cos x sin^ x 1 — cos X 1 sec X cos x(l — cos^ x^ cos x(l -\- cos a;) 1 + cos x 2. COS X tan a; + sin x cot x = sin oj + cos x. 3. (2 — vers x) vers x = sin^ x. . COSX f„„ « 4. = tan X. sin X cot2 X 5. (tan X + cot x) sin x cos x = 1. 6. (sec^x- l)(cosec2x — 1) = 1. 7. sec X cosec x (cos^ x — sin^ x) = cot x — tan x. 8. (sin X + cos x) (tan x + cot x) = sec x + cosec x. 9. cot X + ^^^^ = cosec x. 1 + cos X 10. sin X (sec x + cosec x) — cos x (sec x — cosec x) = sec x cosec x. 11. (cosec X- cot x)2 = ^ -cosx ^ 1 + cos X 12. (1 + tan2 x) (1 - cot'2 x) = sec^ x — cosec^ x. 13. tan X - cot X _ — 2 ^ [First change to an expression containing tan X + cot X cosec^ x only sin x, the reciprocal of cosec x.] 14. sec2 X cosec2 x — 2 = tan^ x + cot^ x. [Substitute for sec x and cosec x their values in terms of tan x and cot x respectively. ] ,K tan a + tan /3 .„„ ♦„„ o 15. ' — tan a tan p. cot a + cot j3 16. cot X - sec X cosec x (1 - 2 sin^ x) = tan x. [The expression reduces to sill X H- cos X.] 62 PLANE AND ANALYTICAL TRFGONOMETRY. 17. cosec X (sec a; — 1) — cot x (1 — cos x) = tan x — sin x. [Factor a^ sood as possible, and reduce each factor separately. ] 18. vers x (sec x + 1 ) + covers x (cosec x + 1) = sin x tan x + cos x cot x. j3^ vers x (1 + sec x) ^ covers x (1 + cosec x) ^ ^^^ ^ ^^^^^ ^ sin X cos X 20. sin2 X (tan2 x - 1) + cos^ x (cof-^ x - 1) = - 2 cos'^ x)^ seC^ x^ tan'^ X 21. sec X cosec x [vers x(vers x — 2) —covers x(covers x— 2)] =cot x— tan x. 22. cos* X — sin* x = cos x ( 1 — tan x) (sin x + cos x) . ^„ vers X (1 + cos x)— covers X (1 + sin x) tan*x — tan^x ^^, 23. ^^ r 5 ^ - — [Change to sec-* X cosec-^ x sec ^ x to an expression containing only sin x and cos x, and then substitute their values in terms of tan x. ] g, sec'^ X sin^^ x — cosec'^ x + cosec'^ x cos^ x _ . o ^ sec2 X sin2 x — cosec^ x cos^ x „c * o -9 - (sec2x + l)(sec2x- 1)2 25. tan2 x - sin2 x cos^ x = -^^ ^-^^ —- sec* X 2g cosxcotx-sinxtanx ^l_^3.^^^^3^ cosec X — sec X 2„ (sec X 4- cosec x)2 _ (1 + tan x)^ tan X + cot X tan x 2g_ 2 + 5il^i^il£2!i£ = sec2x + cosec2x. sin2 X cos2 X gg sec X + cosec x _ tan x + 1 _ 1 + cot x sec X — cosec x tan x — 1 1 — cot x „Q sin X — tan2 x covers x sin* x cosec X cot2 X (1 + sin x) cos2 x 81. sec2 X cosec2 x [vers x (vers x — 2) — covers x (covers x — 2) ] = cot2 X — tan2 X. 82. tan x + cot x = ^^^^ ^ "^ cosec2 j; ^ ^^^ .^ admissible to divide both sec X cosec x numerator and denominator by sin2 x cos2 x.] 88. tan2 a tan2 /3 - 1 = ?^5!^^^^ = ^H^^f=^. cos2 o cos2 /3 C0S2 a cos2 /3 g^ 1 - tan2 g tan2 /3 _ cos2 a - sin2 j3 _ cos2 ^ - sin2 g tan2 g tan2 /3 ~ sin2 ^ sin2 /3 ~ sin2 g sin2 ^ 85. sin2 X tan2 x + cos2 x cot^ x = tan2 x + cot2 x — 1. 36. sin2 X tan x + cos2 x cot x + 2 sin x cos x = sec x cosec x. 37. sec* X + tan* x = 1 + 2 sec2 x tan2 x. [It is admissible to add and sub- tract 2 sec2 X tan2 x. ] 38. (r cos 0)2 + (r sin sin 0)2 + (r sin cos 0)2 = r2. . •. r2 cos2 + r2 sin2 (sin2 e + cos2 0) = r2 (cos2 + sin2 0) =r2, since sin2 x + cos^ x = 1. FUNCTIONS OF ONE ANGLE. 63 89. (2 r sin a cos o)* + r^ (cos2 a - sin^ o)2 = r^. 40. (a sin 7)2 + (a cos 7 sin 5)^ -|- (a cos 7 cos 5)2 = a^. 41. (cos a cos & — sin a sin 6)2 + (sin a cos 6 + cos a sin 6)2 = 1. 42. (cos a cos 6 + sin a sin 6)2 + (sin a cos b — cos a sin 6)2 = l. 43. (x cos ^ - y sin ^)2 + (x sin ^ + ?/ cos dy = x2 + y2. ^ 1 4 tan2 X _ , (cos2 X - sin2 a;)2 (1 - tan2 x)2 ~ * 45. 1 (l-tan2x)2 ^^ 4 sin2 X cos2 x 4 tan2 x 46. (3 sin a cos2 a - sin^ a)2 + (cos^ a - 3 sin2 a cos a)2 = 1. 47. x^-hy^ + s^ = ?-2 when X = r cos a cos /3 + r cos i sin a sin /3, y = r cos I cos a sin /3 — r sin a cos /3, 2 = r sin t sin /S. CHAPTER V. RELATIONS BETWEEN FUNCTIONS OF SEVERAL ANGLES. 61. Sine and Cosine of the Sum of Two Angles. — Let x and y be the angles, each, as well as their sum, being less than 90°. QQ \s> perpendicular to OP, BC and BQ are perpendicular to OA^ and EC is parallel to OA^ the radius of the circle being unity. Then x-\-y = AOQ, the angle EQC=x^ 06'= cos i/, and CQ = sin y. sin (x-^y')=DQ = BO-\- EQ = OC^mBOC+ OQ cos EQC = cos y sin x + sin y cos x, or sin (a? + y) = sin a? cos ?/ + cos a? sin y. (1) cos (a: + 2/)= ^J^= OB-EO= OOcosBOO- CQsmEQO = cos y COS X — sin ^ sin x, cos (ac + y) = cos a? cos y - sin a? sin y. (2) or 1. sin 90° = sin (60° + 80°) = sin 60° cos 30° + cos 60" sin 30° ^\/8 V3 , 1. 1^1 2*2 2*2 2. cos 90° = cos (60° + 30°) = cos 60° cos 30° - sin 60° sin 30° = 1. V3 V3 , 1 ^ 2*2 2*2 3. If sin a = f , and sin /3 = y\, find sin (a + /3) and cos (a + /3) when a < 90°, and /3 < 90° A71S. sin (a + iS) = |f , cos (a + /S) = |f 64 J'UNCTIONS OF SEVERAL ANGLES. 66 4. If tana = f, and tan/3 = ,'j, find sin (o + /3) and cos(o + /3) when a < 90°, and /3 < 90°. Ans. sin (a + P)= *, cos (o + /3) = g. Note. — At a point A the angle of elevation DAB to the top of a vertical wall is o, and the angle of depression CAD to its base is /3. Find the height CB of the wrall, the horizontal distance AD being a feet. CB= CD -\- DB = a tan /3 + a tan a = a(tana + tan/3). (3) ^^/sina^sin^X \cosa COSjS/ _ sin g cos /3 + cos a sin /3 cos a cos /3 ^ sin (g + /3) cos g cos /3 Fig. 41. (4) Eq. (3) w^ould be solved by the use of the natural functions, while (4) is adapted to logarithmic computation. 62. Sine and Cosine of the Difference of Two Angles. — Let X and y be the angles, each being less than 90° and x being greater than t/. QC is perpendicu- lar to OP, BO and DQ are perpen- dicular to OA, and EQ is parallel to OA, the radius of the circle being unity. Then x — y = AOQ, EOQ=x, 0(7=cos?/,..and C§=siny. sin (x-y)=DQ= BO- EO = 0Os'mB0O- OQcosEOQ = cos ?/ sin X — sin «/ cos x, or siii(x-2/) = siu£ccos2/-cosa5sin2/. (1) cos (x - y)= OB = OB + EQ = 00 cos BOO + OQ sin EOQ = cos 1/ cos X + sin z/ sin x, or cos (pc-y)= cos a? cos y + sin x sin y, (2) In this proof we have assumed that x is the greater angle, but (1) and (2) are true when y is greater than x. To prove this, let y8 be greater than «. Then sin (a — /3) = sin [ — (/8 — «)] = — sin (yS — a), CROCK. TRIG. — 5 66 PLANE AND ANALYTICAL TRIGONOMETRY, and, developing sin (^ — a) by (1), sin (rt — yS) = — (siny8 cos a — cos ^ sin a) ss sin a cos /S — cos a sin ^. Also cos (a — /3) = cos [ — (/S — «)] = cos (/S — a) = cos )Q cos a H- sin /3 sin a. Q.E.D. Q.E.D. 1. sin 30° = sin (90° - 60°) = sin 90° cos 60° - cos 90° sin 60° 1.1-0.^ 2. cos 30° = cos (90° - 60°) = cos 90° cos 60° + sin 90° sin 60° = 0.1+1.^ = ^. 2 2 2 3. If sin a = j^j, and sin /3 = :f\, find sin (a - /3) and cos (a - /3) when o<90°,and/3<90°. Ans. sin (a - /S) = ^^i \ cos (a - /3) = |f f . 4. If tan a = f , and tan/3 = f, find sin (a — /3) and cos (a — /3) when a < 90°, and /3 < 90.° Ans. sin (a - /3) = ^V J cos (a - /3) = f f . Note. — At a pointy^ on a horizontal plane, the angle CAD to the top of a crag is 7, and a feet farther away in the same vertical plane (at 5), the angle CBD is 7'. ^,,„, .,, Find^C = a;. Ci> = ^C tan 7 = a; tan 7. CD = BCid^ny' =(a + a;) tan 7'. . •. X tan 7 = (a + a;) tan 7'. Fig. 43. ^ _ q tan 7^ tan 7 — tan 7' osiny C0S7 g sin 7^ cos 7 sin 7 cos 7' — cos 7 sin 7' sin (7 — 7') (3) (4) Eq. (3) would be solved by the use of the natural functions, while (4) is adapted to logarithmic computation. 63. General Proof of the Addition Formulas. — These formulas were shown in Art. 61 to be true when a;, y^ and x + y were each less than 90°. That they are true for all values of these angles may be shown by proving the special cases separately. Let us consider first the case when »< 90°, y < 90°, X + 2/ > 90° and < 180^^ FUNCTIONS OF SEVERAL ANGLES. 67 Let x = 90°-a, y = 90°-p; .-. x + y = 180° -(a + /3). .-. a = 90° - x, )8 = 90'' - y, a + i8 = 180° - (x + j/). .-. a < 90°, /3 < 90°, a + /3 < 90°, since a; + y > 90°. Then sin (a + /3) may be developed by (1), Art. 61, since the conditions of that article are satisfied. But sin (x + y) = sin [180° — (a + /3)] = sin (a -f |3) = sin a cos /3 + cos a sin /3 = sin (90° - x) cos (90° - y) + cos (90° - x) sin (90° - y) = cos X sin 2/ + sin 05 cos 2/. q.e.d. Also cos (x + y)~ cos [180° - (a + /3)] = - cos (a + ^) = - cos a cos j8 + sin a sin |3 = - cos (90° - x) cos (90° -y) + sin (90° - x) sin (90° - y) = — sin X sin y 4- cos X cos y. q.e.d. Hence the formulas are true for x < 90°, y < 90°, x + y < 180°. To illustrate the proof for any special case, let us take x in the second and y in the fourth quadrant. Place x = 90° + a, and y = 270° + /3, so that a and /3 are each less than 90°. Then sin (x + y)= sin [360° + (a + /3)] = sin (a + jS) = sin a cos /3 + cos a sin /3 =; sin (x - 90°) cos (y - 270°) + cos (x - 90°) sin (y - 270°) = (— cos x)(— sin ?/) + sin X cos y = cos X sin y + sin x cos y. q.e.d. Let the student prove the addition formulas in the following cases : 1. X in the first, and y in the third quadrant. 2. X in the second, and y in the second quadrant. 3. X in the second, and y in the third quadrant. 4. X in the third, and y in the third quadrant. 6. X in the third, and y in the fourth quadrant. 6. X in the fourth, and y in the fourth quadrant. 64. General Proof of the Subtraction Formulas. — These formulas were shown in Art. 62 to be true when x and y were each less than 90°, both for x>y and for x < y. That they are true 'for all values of the angles may be shown by proving the special cases separately. For illustration, let x be in the second, and y in the third quadrant. Place x = 90° + a, and y = 180° + /3, so that a < 90°, and ^ < 90°. Then sin(x-y)=sin[90°+a-(180°+/3)]=sin[-90° + (a-i3)] = -sin[90°-(a-i3)] = — cos (a — /3) = — COS a cos iS — sin a sin = - cos (X - 90°) cos (y - 180°) - sin (x - 90°) sin (y - 180°) = — sin X ( — cos 2/) — ( — cos x) ( — sin y) = sin X cos y — cos X sin ?/. q.e.d. Also cos {x-y)= cos [- 90° + (a - 0)] = cos [90° - (a - ^8)] = sin (a - 0) = sin a cos $ — cos a sin = sin (x - 90°) cos (y - 180°) - cos (x - 90°) sin (y - 180°) = (— cos x) (— cos y) — sin X (— sin y) = cos X cos y + sin X sin y. q.e.d. 68 PLANE AND ANALYTICAL TRIGONOMETRY. Let the student prove the subtraction formulas in the following cases ; 1. X in the fourth, and y in the first quadrant. 2. X in the fourth, and ?/ in the second quadrant. 3. X in the fourth, and y in the third quadrant. 4. X in the third, and y in the third quadrant. 5. a; in the third", and y in the fourth quadrant. 6. X in the second, and y in the fourth quadrant. 65. Tangent of the Sum and of the Difference of Two Angles. sin (x + ?/) sin x cos y + cos x sin y tan (a: + ?/) = t^~^ = ^ —• -^• ^ '^ cos {x + y) COS X COS ^ — sm a: sin i/ Divide both numerator and denominator by cos x cos y. sin X cos ^/ cos a: sin y sin a: sin y , , cos a: cos v cos a: cos y cos a; cos ?/ .-. tan (2:+?/)= ^^ ^ ^ = : -^; cos X COS y sin a; sin «/ ^ sin a; sm z/ cos X cos ?/ cos X cos y cos a; cos y In the same way, we may show that . , . tan 05 - tan y ^o\ 1. tan75°=tan(45°+30°) = ;^"^^°+\^"-l": =-^^^g±J^2+V3. ^ ' 1 — tan 45° tan SO"' i VS — 1 V3 2. tan 15-tan (45°-30°) = J^li5;;i5a^„= ^ =2^ = 2- V3. ^ ^ 1 + tan 4d° tan 30° 1 ^_|_ ^ V3 3. If sina = |f and sinj3 = |, find tan (a + 5) and tan (a-;8), when a<90° and B < 90°. Ans. tan (a + ;8) = - f | ; tan (a - )3) = f |. 66. Geometrical Proof. — In Fig. 44, let 0^1 = 1, AOB = x, BOC = y. Draw .5C perpendicular to OJ?, and CD parallel to OA; :. DBC = x^ DCE = x + y. Then tan (x + y)= AE = AB + BD + BE. FUNCTIONS OF SEVERAL ANGLES. 69 But BD = BC cos X = OB tan y cos x — sec x tan y cos x — tan y, and DE = CD tan (x + y) = BC sin x tan (x + y) = OZ? tan y sin x tan (x -f y) = sec X tan y sin x tan (x + y) = tan x tan y tan (x + y). . •. tan (x -f y) = tan x + tan y + tan x tan y tan (x + y). tan X + tan y /. tan (x + y) = 1 — tan X tan y Fig. 45. In Fig. 45, let 0.4 = 1, AOB = x, COB = y. Draw 50 perpendicular to OB, and Z>(7 parallel to OA ; .-. DBC = x, DCE=x-y. Then tan (.X - ?/) = ^E- =r ^i^ - i>5 - ^D. Rut DB = BC cos X — OB tan y cos x = sec x tan y cos x = tan y, and ^i) — DC tan (x — y) = BC sin x tan (x — y) = Oi? tan y sin x tan (x — y) = sec X tan y sin x tan (x — y) = tan x tan y tan (x — y). .-. tan (x — y) = tan x — tan y — tan x tan y tan (x — y). tan X — tan y tan (x — y) 1 + tan X tan y EXAMPLES. Find by inspection one value of x in Exs, (1-6) : 1. sin (w — l)a cos a + cos (n — l)a sin a = sin x. ^ws. x = wa. 2. cos (10^= + a) cos (10° - a) + sin (10° + a) sin (10°- a) = cos X. J.>iS. x = 2a. 3. sin(a-)8 + 10°)cos(i8-a + 10°)-cos(a-i3 + 10°)sin ()3-a + 10°)= sin x. Ans. X = 2(a — i8). 4. cos 45° cos (90° - a) - sin 45° sin (90° - a)= cos x. .4ns. x = 135° - o. 5. sin (90° + h, a) cos (90° - i a) + cos (90°+ i a) sin (90° - i a) = sin x. Ans. x= 180°. 6. cos(45°-a)cos(45°+a)-sin(45°-a)sin(45°+a)=cosx. ^ns.x = 90°. 7. Given the functions of 30° and 45°, find those of 75°. Ans. sin 75° = ^^+ ^ ; cos 75° = ^~^ ; tan 75° = ->^J- = 2 + V3. 2V2 2V2 Vs-i 70 PLANE AND ANALYTICAL TRIGONOMETRY. 8. Given the functions of 30° and 45°, find those of 15°. Ans. sin 15° = ^~ ^ ; cos 15° = ^^ ; tan 15° = 2 - VS. 2V^ 2\/2 9. If tan a = I and sin ;8 = }|, find the functions oi a + when a is in the third, and j8 in the second quadrant. Ans, sin (a + j8) = - f | ; cos (a + j8) = f f ; tan (a + 3) = - f f . 10. If cos a = — If and sin )8 = - y\-, find the functions of o — /3 when a is in the third, and j8 in the fourth quadrant. Ans. sin (a -)3) = -!§!; cos (a - )8) = - ||| ; tan (a - )8) = + f f f . 11. If cos o = f and sin ;8 = — |, find the functions of o + ;8 and ot a — when a is in the fourth, and in the third quadrant. Ans. sin (a + )8) = + /^ ; cos (a + 3) = - If ; tan (a + ;8) = - /^ ; sin (a - 3) = + 1 ; cos (a - i8) = ; tan (a - fi) = oo. Transform the first member into the secpnd (or last) in Exs. (12-32): 12. sin (a + ;8) sin (a ~ $)= sin2 a — sin^ $ = cos^ & — cos^ a. 13. cos (a + ;8) cos (a — )8) = cos^ a — sin^ $ = cos^ — sin^ o. 14. sin (60° + a) - sin a = sin (60° - o). 15. (r' cos v' — r cos vy + (r' sin v' — r sin u)2 = r^ + r'^ — 2 rr' cos (v' — v). 16. cos^ a + cos2 /3 — 2 cos o COS i8 cos w = sin2 w, when w = o + j8. [Place a = w — iS.] tPf 4- , tan sec a *„„ / , .\ 17. tan o H ■^-—. = tan (a + 0). cos a — tan

a — cos2 2a: (1 + cos 22:) 2' 1 + cos 2 a5 Also _ /l— cos 2 2: 1 — cos 22:_ /(I — cos 22:)2_ ^ 1 + cos 2x 1 — cos 2x ^ 1 — cos^ 2 x .*• tan 0? 1 - cos 2 05 • sin 2 a? (8) (4) (5) (fi) FUNCTIONS OF SEVERAL ANGLES. 73 Note. — The double sign is not used in (5) and (6), for sin 2 X 2 sin x cos x and 1 + cos 2 a; 2 cos-' x 1 — cos 2 X 2 sin^ x sin 2 X 2 sin x cos x = tan X, r= tan X. 1. To express the functions of 20° in terms of those of 40°, we have 2sin2 20° = 1 -cos 40°; 2cos2 20° = 1 + cos 40°; 1 -cos 40° tan2 20^ tan 20° = 1 + cos 40° sin 40° _ 1 - cos 40^ 1 + cos40°~ sin 40= 2. tan 2 ^ = — 2, 2 ^ being in the second quadrant. Find the functions of B. 1 o .-. cos 2 ^ = —, and sin 2 ^ = -f -=-. cos ^ = ± Y" ( 1 - -^) , from (2) ; .*. sin tan 5 = 2 V6 1 _ J_ Vs - 1 V5 from (5). Since 2 ^ is in the second quadrant, B may be either in the first or in the third quadrant ; hence sin 6 and cos 6 have the double sign, and tan B is positive. 3. Given the functions of 30°, find those of 15°. Ans. sin 15°^ ^- ^ cos 15= ^^ll±l- tan 15° = 2 -V3. 2V2 ■ 2V2 4. Given the functions of 45°, find those of 22J°. Ans. sin22i° = ^V2-V2; cos22|° =i V2 + \/2 ; tan22|° = V2 - 1. 70. Geometrical Proof. — Let the radius CA of the circle be unity. e^ ^P_ ^~6B'Ba _ABa ^Jba OP VOBToA ^ OA = J g-^ - CB ^ j \ -cos2x OB OB OP y/OB.OA ^0.\ =M-r-^ lOC+CB J\ +cos2x = \— ^ =\ 1 74 PLANE AND ANALYTICAL TRIGONOMETRY. BP BP sin 2 a; tana; = tana: OB 0C+ CB 1 + cos2a: BA ^ CA- CB ^ 1 -cos2x BP BP sin2x 71. Multiple Angles. — Suppose that we wish to express sin 3 X in terms of powers of sin x. sin 3 a: = sin (2 a; + ^) = sin 2 x cos x + cos 2 a: sin a: = 2 sin X cos^ a: + (1 — 2 sin^ x} sin x C^'^'Z ^ / j^ ^OyUji^J ^^ = 2 sin a; — 2 sin3 a: + sin a; — 2 sin^ a: ^ = 3«ina; — 4sin3,a^. Q.E.I. 1. Show that cos 3 x = 4 cos^ x — 3 cos x. 2. Show that tan3x=^^^^^-^^"'^. 1 -3tan2a; 3. Show that sin 4 x = 4 sin x cos a; — 8 sin^ x cos aj. [Use 4x = i2x + 2x.] 4. Show that cos 4 x = 1 — 8 sin^ x + 8 sin* x. E ou *u * * A 4tanx(l - tan2x) 6. Show that tan 4 x = -.^ \ — ^ • 1—6 tan2 X + tan* x 6. Show that sin 6 x = 5 sin x — 20 sin^ x + 16 sin^ x. [Use 5 x = 3 x + 2 x.] 7. Show that cos 5 x = 5 cos x — 20 cos^ x + 16 cos^ x. 8. Find the functions of 18°, of 36°, and of 72°. Place X = 18° ; then, since cos 54° = sin 36°, we have cos 3 X = sin 2 x. .•. 4 cos-5 X — 3 cos X = 2 sin X cos x. .'. cos X (4 cos2 X — 3 — 2 sin x) = 0. /. 1—4 sin2 X — 2 sin X = 0. .-. sinx = ^(— 1 ± Vo). .-. sin 18° = cos72° = K>/5 - 1) ; cos 18° = sin 72° = jVlO + 2V5. Hence sin 36° = 2 sin 18° cos 18° = ^VlO-2V5 ; cos36° = 1 -2sin218° = i(^^+ 1). 72. To change the Product of Functions of Angles into the Sum of Functions. — From Arts. 61 and 62, sin (a; + ^) = sin x cos y + cos x sin i/; /^ sin (x — y^= sin x cos y — cos x sin y. ^^^ .'. sin (a; H- «/) + sin (a; — ?/) = 2 sin a: cos «/, (1) and sin (x -\- y}— sin (a; — «/) = 2 cos x sin y. (2) FUNCTIONS OF SEVERAL ANGLES. 75 Also cos (a: + y) = cos x cos y — sin a: sin y\ ^ — cos (x — y)= cos X cos y + sin x sin y. -^ .*. cos (a; 4- y ) + cos (a; — ?/) = 2 cos a; cos y, (3) and cos (x + y)— cos (a: — ?/)= — 2 sin a; sin y . (4) Reversing (1), (2), (3), and (4), we have sina3COS2/ = ^sin(a!; + 2/)+^sitt(a5-2/), \ (5) cos ic sin 2/ = I sin (05 + 2/)-^ sin (05-1/). [ (6) cosxcosy = |cos(a5 + 2/)+ |cos(a5 -y). (7) sina5sin2/ = -|cos(a5 + i/) +|cos(a5- J/). (8) In applying these formulas, let x represent the larger angle. 1. sin 4 ^ cos 2 = ^ sin (4 ^ 4- 2 ^) + I sin (4 ^ - 2 ^), from (5), = 1 sin 6 ^ + 1 sin 2 d. 2. cos 6 d sin 2 ^ = ^ sin (6 ^ + 2 ^) - ^ sin (6 ^ - 2 6), from (6), = ^ sin 8 ^ - ^ sin 4 ^. 3. cos8^ cos2^ = ^coslO^ + ^cos6^, from (7). 4. sin 6 ^ sin 4 ^ = - I cos 10 ^ + ^ cos 2 6, from (8). 5. cos 2 sin 4 = i sin 6 ^ - i sin ( - 2 ^), from (6), = i sin 6 ^ + ^ sin 2 '24 ^ of z. 76. To prove that tan a? > £c> sin « when a; < — , a? being ex- 2 pressed in Circular Measure. — Let AOB = BOC = x^ the radius being unity. Evidently AT> SB, or tan x > sin x. Also, since the shortest distance from a point to a line is perpen- dicular to the line, SB < AB, or sin x&vcABO, since ABO is a convex polygon lying in the tri- At^ angle formed by a chord A with fig. 6i. the tangent lines TA and TO. Then 2 AT > arc ABO, or AT>a>TGAB, or tanic>a;. Hence tan x>x, and x > sin x. q.e.d. 77. To prove that sin a?, tan a?, and a? approach Equality as the Angle a? approaches Zero. — As the angle AOT decreases, When the angles are less than 90*^ 80 PLANE AXD ANALYTICAL TRIGONOMETRY. the points B and T approach A^ and hence approach each other. But SB sin X AT i'Awx cos a;. When the angle x approaches zero as its limit, cos x approaches unity as its limit. Hence -j-^, or , approaches unity as its limit, or sin x and tan x approach equality. The arc x is intermediate in value between sin x and tan x ; hence the three quantities approach equality as the angle be- comes smaller. That is, the three ratios sin X sin x tan x tan XX X approach unity as the angle approaches zero. Hence we may say that when the angle is small, its sine and its tangent are equal to the arc itself, and its cosine is equal to unity. The smaller the angle, the more nearly correct will be the assumption. 78. Development of sin a?, of cos «, and of tan a?. — Let us assume that sin X = a + hx -\- cx^ -\- ds? + es^ +f^ + ••• (1) is true for all values of x. Then it is true when x has the values 4- y ^^^ ~ V j hence sin^ = a-\-hy^-cf'-\- dy^ + ey^ -{-fy^ + ••• (2) and sm(— y}=a — by -{- cy^ — dy^ + ey^ —fy^ -\ (3) But sin y = — sin ( — ^), or sin ?/ + sin (—?/)= 0. Adding (2) and (3), 2 a + 2 £?/ + 2 e/ + ... = 0. (4) But (4) is true for all values of y, since (1) is true for all values of x. In order that all values of y may reduce the left member of (4) to zero, we must have a = 0, c = 0, e = 0, ..-• Hence (1) becomes sin x = hx -\- dx^ -\-fx^ + ••• (5) or ?HL£ = 6 + ^a;2+A*+- (6) X FUNCTIONS Or SEVERAL ANGLES. 81 But as X approaches zero, ^^^ approaches unity, and h + dx^ X -f/r*+ ... approaches h. Hence 1 = J, (7) and (5) becomes sin £c = a? + doc^ + fx^ + ••• (8) Again, let cos a: = ^ 4- J5a: + Cx^ ^ Bt? -\- Ex"^ + Fx^ + -.• (9) Since cos x = cos ( — x')^ we have A + Bx + Cx'^ + Dj^ + E3^ + F3^-[- ... = A-Bx-\-Cx'^- B^ H- J^;^;* - i<:r5 ^ ... (IQ) or 2j52: + 2i>ar^ + 2jP2:5+ ... = 0. (11) In order that this may be true for all values of x^ we must have ^ = 0, i> = 0, ^=0 ..., and (9) becomes cos a: = ^ + Cx^ + Ed" + - (12) But when a: = 0, (12) reduces to 1 = A, (IS) and hence (12) becomes cos a^ = 1 + Cic- + Bqc^ + ... (14) Substituting from (14) and (8) in the formula cos 2 a: = cos^ a: — sin^ a:, we have 1 + 4 Ca:2 _j_ iq ^^4 _^ ... ^ 1 ^(9 (7_ i)^2 + (2 ^ + 6^2 _ 2 d):»^ 4- ••• (15) Equating the coefficients of like powers of x^ 4(7=2(7-1, or 2(7+1=0. (16) lG^=2^+(72_2^, or WE-C^^ld^^. (17) Substituting from (14) and (8) in the formula sin 2 a: = 2 sin x cos a:, we have 2 a: + 8 cZa;3 4. 32 /.^;5 + ... ^ 2 a: + 2 ((7+ d^T? + 2(iE + Cd+f^7^+... (18) CROCK. TRIG. 6 82 PLANE AND ANALYTICAL TRIGONOMETRY. Equating the coefficients of like powers of a;, 4.d = C-\-d\ or 3c^-(7=0. (19) lQf=E + Cd+f, or 15f-U-Cd = 0. (20) From (16), 0=-^. (21) From (19), ^ = -1=-^. (22) From (17), U=+^ = +^ (23) From (20), / = +^=+^. (24) These values, substituted in (8) and (14), give ^inx = a^-^ + ^^.-^ ij^-^ (25) i_n cosaj= 1 g?^ . ag^ L2. LI Dividing (25) by (26), tan i» = 05 + 4 ^3 ^ ^ ar;5 _j. ... 15 (26) (27) In (25), (26), and (27), which are the required develop- ments, X must be' expressed in circular measure. 79. Computation of the Trigonometric Functions (First Method). — The functions may be computed by (25), (26), and (27), Art. 78. Thus, to find sin 20°, we place x=\'ir, the circular measure of 20°. log 7r8= 1.49145 col93 = 7.13727 -10 col6 = 9.22185 -10 log 7r6 = 2.4857 col96 = 5.2288 -10 C0II2O = 7.9208 - 10 log*^ - 5.6353 - 10 li .-.^=0.0000432 . X : sin = - = 0.34906 59 9 ^.= 0.00708 88 |3 0.34197 71 log ^ = 7.85057 - 10 .-. ^ = 0.0070888 ^=0.00004 32 20° = 0.34202 03 li In the tables, sin 20° = 0.34202. FUNCTIONS OF SEVERAL ANGLES. 83 80. Computation of the Trigonometric Functions (Second Method). — From (25), Art. 78, it may be shown that sin 1" = 0.00000 48481 36811 07637, while arc 1'' = 0.00000 48481 36811 09536. . •. arc 1'' - sin 1'' = 0.00000 00000 00000 02. Again, sin V = 0. 00029 08882 04563 42460, while arc 1' = 0.00029 08882 08665 72160. .-. arc r- sin 1^ = 0.00000 00000 04. Again, sin 1° = 0.01745 24064 37283 51282, while arc 1° = 0.01745 32925 19943 29577. .-. arc r- sin 1° = 0.00000 09. ' Also, from (26), Art. 78, cos V = 0.99999 99999 88 = 1 - 0.00000 00000 12. cos 1' = 0.99999 99576 92 = 1 - 0.00000 00423 08. cos 1° = 0.99984 76952 = 1 - 0.00015. In computing a set of five-place tables, we may assume sin 1' = arc V = 0.00029 08882 with an error of 5 x lO-^^, and cos 1' = 1 with an error of 4 x 10"^. Then sin 2^ = 2 sin 1' cos 1' ; cos 2' = cos^ 1^ — sin^ l^ sin 3' = sin 2' cos 1' + cos 2' sin 1' ; cos 3' = cos 2' cos 1' — sin 2' sin V. sin 4' = sin (3' + 1^ ; cos 4' = cos (3' + 1^, or sin 4' = 2 sin 2' cos 2' ; cos 4' = cos2 2' - sin2 2'. And so on. This method would be employed until the functions of all angles less than 30° had been computed. Then, since sin (30° + a:) = cos a; - sin (30° - rr), and cos (30° -h x)= cos (30° — x)— sin x, the functions of angles between 30° and 45° would be found by combining the functions already found. Thus, if 2; = 10°, we have sin 40° = cos 10° - sin 20°, and cos 40° = cos 20° - sin 10°. 84 PLANE AND ANALYTICAL TRIGONOMETRY. It is possible to compute independently the sine and cosine of 3°, 6°, 9°, ..., 39°, 42°, 45°. We have found in this chapter* the sine and cosine of 15°, of 18°, and of 36°, and we have 3° = 18°-15°, 6° = 3G°-30°, 9° = 45°~36°, 12° = 30°-18°, 21° = 36°-15°, 24° = 45°-21°, 2T° = 45°-18°, 33° = 18° + 15°, 39° = 45°- 6°, 42° = 45°- 3°. The values found from these relations would serve as checks upon the computation. The computations may also be checked by Euler's and Legendre's verification formulas : sin (36° + A)- sin (36° -A)- sin (72° + A)+ sin (72° - A) = sin A. cos (36° + ^) + cos (36° -A}- cos (72° + ^) - cos (72° - A) = cos^. 81. Approximate Assumptions. — It can be shown that tanV' - arc 1" = 0.00000 00000 00000 04 ; arc 1'' - sin 1" = 0.00000 00000 00000 02 ; tan 1" - sin 1" = 0.00000 00000 00000 06. Hence we may assume that siiil'' = taiil" = arcl". (1) In the whole circumference of a circle there are 1296000", so that the error due to placing arc V = sin 1" in finding the circumference of a circle with a radius of unity will be only 2^ units in the eleventh decimal place. In the computation of elliptic orbits there occurs the equation M= U — esinU, where M and U are expressed in circular measure. If M" is the number of seconds in the angle, M= M" SircV\ and ap- proximately M= M" sin 1^' and U = U" sin 1". Hence the equation may be written M" = E" ^sinJS'. sm 1" * Ex. 3, Art. 69, and Ex. 8, Art. 71. FUNCTIONS OF SEVERAL ANGLES. 86 Another assumption that is often made is that for small angles Sinn" = n sin 1". (2) The error introduced is for 1', w" = 60", error = + 0.00000 00000 04 ; for 1°, n" = 3600'S error = + 0.00000 09. Thus, if sin a = 0.4 sin 2°, we should have, since a must be small, a" sin 1" = 0.4 sin 2° or a" = ^liiHl^!. sm 1'^ 82. Transform the First Member into the Second (or last) in the following examples ; . cos a — sec a a <> ^ /^ t ^ IN 1. = 4 cos^ 1 a (cos2 i a — 1). sec a 2 V 2 J . The first member contains the angle a and the second J a ; hence we must change the angle. 1_ = cos2 a - 1 = (2 cos2 1 a _ 1)2 _ 1 cos a — cos a cos a ^4 gQg4 1 « - 4 cos2 i « = 4 cos2 J a (cos^ |^ « - 1). 2. cosec2a + cot2a = cota. 8. cosec 2 a - cot 2 a ^ ^^^, ^ cosec 2 a 4- cot 2 a 4. cot a — tan a = 2 cot 2 a. We may either reduce the expression as far as possible before changing the angle, or change the angle and then reduce. ^ ^ cos a sin a cos^ a — sin^ a cos 2 a o ^ 4- o (a) = — : = - — : — -— = 2 cot 2 a. sin a cos a sin « cos a ^ sin 2 « ,j. 1 + cos 2 « 1 - cos 2 « _ 2 cos 2 « _ g ^^^ g ^ sin 2 a sin 2 a sin 2 a Note. — Avoid radicals if possible. 6. sec a cosec o = 2 cosec 2 a. 8. cot J ^ 4- tan J ^ = 2 cosec 6. 6. (sin \d -\- cos J 0)2 = 1 -i- sin d. 9. sin x — 2 sin^ x = sin x cos 2 x. l-tan^^^^ 1^, ^(secg + sec2g)= l+tan^^ . l + tan2jv ^^ ^ (l-tan2je)2 S6 PLANE AND ANALYTICAL TRIGONOMETRY. . - 2 tan ^ V _ . 13. 1 + tan x tan J x = sec x. ^ 14. ^ (1 + tan i o)2 = i±^HLi^. opf»2 fl 1 + COS a 12. _§^1_^ = sec 2 ^. 2 — sec2 ^ 15. tan J a 4- 2 sin^ J o cot a = sin a. 16. «i"^a-tan^^)/ L_^_^ 1 \^sin2x. sec^x Vcosx — sinx cosxf sinx/ 17. (1 - tan2 d) sin cos ^ = cos 2 e-J- — cos 2 g + cos 2 ^ 18. l + ^^I^=sec2a. 20. ^-^^l±22l2J:l = cot^ 6. 1 — tan^ a sec ^ 4- cos ^ — 2 -g cosg— sing _ l— sin2g _ cos2g 21 * tan g ^ "^ ^^ ^ ^ — ^^" ^ cos^+sin^"" cos2g "~l4-sin2g* * 1 - tan J g~ 1 - sin^ 22. sec 2 a + tan 2 a + 1 = ^ 1 — tan o 23. (vTTsIna - Vl -sina)2 = 4sin2 Jo. t- 24. (Vl + sin a + VT— sino)2 =:4cos2 Ja. 25. 2 sin ^ - sin (^ - 5) - 4 sin A sin2 J 5 = sin (^ 4- -B). 26.t cos (36° + ^) + cos (36° - ^) - cos (72° + ^) - cos (72° -A)= cos ^. 27.t sin (36° + A)- sin (36° -A)- sin (72° + ^4) + sin (72° -A)z= sin ^. 28. ?ilL^±iHLl^ = cotJx. cos X — cos 2 X 29. 1 + cot2 ^v = sin V tan J v 30. tannv(l + cot2Av)8=-§ — sin^t) Q J sin a cos ^ g — 2 cos a sin ^ a _ „ ^^ga i 2 sin J a — sin a ^ 32 tan2 ^ x + cot2 j^ x __ _ 1 + cos2 x tan2 1 X — cot2 J X 2 cos x 33. Given tan J v =a/ "^ ^ tan J ^, show that '1 — e 1 — e cos ^ (1 + e) cos2 J V + (1 _ e) sin2 J v 1 - e* 34. tan (45° + ^) - tan (45° -A) =2 tan 2^. tan 45° + tan A tan 45° — tan A (a) 1 — tan 45° tan ^ 1 + tan 45° tan A _ 1 + tan A _ 1 — tan A _ 4 tan J. _ 2 tan 2 ^ 1 — tan A 1 + tan A 1 — tan^ A * After substituting, multiply both numerator and denominator by the quantity sin 0— 1 + cos d. t cos 36° = Kl + v^). FUNCTIONS OF SEVERAL ANGLES. 87 .,. 1 - cos(90° -f 2 A) 1 - co8(9Q° -2 A) ^ ^ sin (90° + 2 A} sin (90° - 2 ^) 1 + sin 2 ^ 1 - sin 2 ^ 2 sin 2 A o .. o a COS 2 A cos 2 ^ cos 2 A 35 tan (45° + M) + tan (45° - ^ A) _ ^^ tan (45° + M) - tan (45° - M) ~ 36. tan (45° + ^) - cot (46° + ^) = 2 tan 2 ^. 37. tan2 (45° + 6)+ cot2 (45° + ^) = 2 + 4 tan^ 2 $, 38. tan2 (450 + „) - cot2 (45° + o) = 4 tan 2 o sec 2 a. jg tan (45° + ^6) _ l + 8md tan (45° - ^ ^) 1 - sin tf' ^^40. tan^tan(45° + J^^)= ®^^^ 1 — sm ^ 41. cot (45° - J a) - tan (45° - J a) = 2 tan o. 42. tang(45"4-^a) = ^ + ^"^" . 1 — sin a 43. tan (45° + 6)+ tan (45° - ^) = 2 sec 2 d, 44. l-tan2(45°-^)^^.^^,^ iiK ^ />ico . 1 .l + tanArc l+sino; 45. tan (45° + J x) ^^ — 1 + tan2 (45° - 6) IJ 1 — tan ^x 1 — sin a; tan (45° + ^x) _ 1 + sin a; ***• 1 + cot2 (45° + ^x) - *°°^^ 1 - sinx* 47. sin (45° - J ^) + cos (45° - J e) = V2 cos J ^ = ^^^^ . VI - cos e CHAPTER VI. TRIGONOMETRIC EQUATIONS. 83. One Equation Containing Multiple Angles.* — Change the equation so that it shall contain a single angle, and then proceed as in Art. 52. 1. cos 3 a; = sin 1x ; find x. (See Ex. 8, Art. 71.) 4 cos^ x—K> cos a; = 2 sin x cos x . . * . cos 2: (1 — 4 sin^ 2: — 2 sin rr) - = 0. . • . cos a; = 0, giving x = 90° and 270° ; and 1 — 4 sin^ x — 2 sin x = 0, giving sin x = ^ ( V5 — 1) and sin a; = - ^ ( V5 + 1), or a: = 18°, 162°, 234°, 306°. 2. cos2 e + cos^ = - 1 ; find 6. Ans. 90°, 270°, 120°, 240°. 3. cot2 + tan = - f V3 ; find d. Ans. 150°, 330°, 120°, 300°. 4. C0S2.X + sinx ^ + 1 ; find x. Ans. 0°, 30°, 150°, 180°. 5. sin 3 X + sin 2 X = sin x ; find x. Ans. 0°, 180°, 60°, 300°. 6. tan 2 X = - 2 sin X ; find x. Ans. 0°, 60°, 180°, 300°. 7. tan2xtanx = + 1 ; find x. Ans. 30°, 150°, 210°, 330°. 8. tan2 X tan 2 X + 2 tan X = + V3 ; find x. Ans. 30°, 120°, 210°, 300°. 9. sin 4 2r - 2 sin 2 = ; find z. Ans. 0°, 90°, 180°, 270°. The equation may sometimes be solved by the use of the equations of Art. 73. 10. cos 3 a; — sin 2 a; = ; find x. cos 3 a; — sin 2 a; = sin (90° + 3 a;) — sin 2 a; = 2 cos (45° + f a;) sin (45° + 1 a:) = 0. cos (45°+ fa;) = gives 45°+ fa: = 90°, 270°, 450°, 630°, 810°, or x= 18°, 90°, 162°, 234°, 306°. sin (45° + ia:)= gives 45° + la; = 0°, 180°, or a; = -90° and 270'' . * See Art. 52 for the solution of equations wlien only one angle is involved. 88 TRIGONOMETRIC EQUATIONS. 89 11. cos 9 — cos 3 e = sin 2 fl ; find 6 by both methods. Ans. 0°, 30^ 90°, 150°, 180°, 270°. 12. sin 3 ^ + sin 2 ^ -f sin ^ = ; find by both methods. Ans. 0°, 90°, 120°, 180°, 240°, 270°. 13. cos2 = sin^; find d by both methods. Ans. 30°, 160°, 270°. 14. cos6^-cos3 + sin^ = O; findfl. ^ns. 0°, 180°, (2 n + J ± ^) -. 4 16. sin 5 + sin 3 ^ + 2 cos = ; find 6. Ans. 90°, 270°, (2 n + f ) -. 4 16. sin (60° -X)- sin (60° + x) = + J \/3 ; find x. Ans. 240°, 300°. 17. sin (30° + X) - cos (60° + a;) = - ^ V3 ; find x. Ans. 210°, 330°. 18. cos 4;? - cos 2 ;? = ; find 2. Ans. 0°, 60°, 120°, 180°, 240°, 300°. 84. Find r and from the Equations a and b being known. r sin «(» = a, r cos = &, (1) (2) (1)^(2) gives tan<^=:^. (3) From (1) and (2) r- ^ - h (4) s'm(f) 1. Find r and when loga=0.47141, and log 6=0.63927 n, r being positive. log (r sin 0)= log a = 0.47 141 (1) logsin0 = 9.74972 (5) logcos0 = 9.91758n (6) log (r cos , affect the logarithmic cosine less than the logarithmic sine. The angle is placed in the second quadrant, since r cos (f> is negative and r sin positive, r being considered positive. 2. Find r and

= 201° 30'.0 ; r = 79.497. 90 PLANE AND ANALYTICAL TRIGONOMETRY. 85. Find r, <|>, and 6 from the Equations r cos cos 9 = a, ] (1) r sin 4> cos = 6, (2) r a, h, and o being known sine =c, (3) (2)-^(l)gives tan cos ^ sin 1. Given loga = 0.46472, log6 = 0.72413 n, logc = 0.62817, find r, 0, and df d being numerically less than 90"^, and r being positive. log (r cos «^ cos e) = log a = 0.46472 (1) log cos (f) = (9.68314) Only as a check. (5) log sin 0= 9.94256 w (5) log (r sin cos d) =\ogb= 0.72413 n (2) (2) - (1) = log tan = 0.25941 n (3) = 298° 49'.4 (4) (2)-(5) = (l)-(5)=log(rcos^)= 0.78157 (6) log cos ^= 9.91291 (10) log sin d = (9.75951) Only as a check. (10) logc = log(rsin^)= 0.62817 (7) (7) - (6) = log tan ^ = 9.84660 (8) 0= 35° 5'. 1 (9) (6) - (10) = (7) -(10)= log r= 0.86866 (11) r= 7.3903 (12) The angle is placed in the fourth quadrant, since r cos 6 is positive, and therefore cos must be positive and sin negative, r cos cos being positive and r sin cos negative. 2. Given log a = 0.26903 n, log & = 0.32426, log c = 0.36903 w, find r, 0, and 0, r being positive and numerically less than 90°. Ans. = 131° 22'.0 ; = - 39° 45'. 6 ; r = 3.6572. 3. Given log a = 9.43942 w, log b = 9.40403 w, log c = 9.56700 w, find r, 0, and ^, r being positive and numerically less than 90°. Ans. = 222° 40'.1 ; d = - 44« 36'.4 ; r = 0.525425 or 0.52544. TRIGONOMETRIC JJQUATIONS. 91 86. Find ^ from the Equation a sin<|) + 6 cos 4» = c (1) by formulas adapted to logarithmic computation, a, J, and c being known. Let M be an auxiliary angle and m a positive constant, so that m cos M=b. J ^ -^ The angle M is always possible, for we have, by division, tanif=i (3) and since the tangent may have any value between + oo and — 00, there will always be some angle whose tangent is equal to Y* Also, squaring and adding Eqs. (2), we have m^ sin^ il[f + m^ cos^ M= m^= a^ -\- h\ or m = Va^ + b'^. (4) Therefore the assumptions in (2) are always possible, since M and m will be real quantities if a and h are real. Substituting (2) in (1), we have m sin M sin = c^ \ or m cos ((/> — Jf ) = c. (5) Hence, from (2) find M and m by the method of Art. 84 ; from (5) find <\) — iJf (two values < 360°), and thence find . 1. Find when 2 sin ^ — 3 cos = 1. Ans. M= 146° 18'.6 ;- = 220° 12'. 5, or 72° 24'. 7. 2. Find when 2 sin + 4 cos = — 3. Ans. ilf=26°33'.9; = 158°41'.8, or 254°26'.0. 87. Find <() from the Equation atan<|> + &cot<|> = c by formulas adapted to logarithmic computation, a, 5, and c being known. 92 PLANE AND ANALYTICAL TRIGONOMETRY. Substituting for tan and cot in terms of sin (f> and cos <^, we have, after reducing, (a — 6) cos 2(l> -\- c sin 2(f) = a -{- b. Let msinM=a — b^ m cosilff = c. \ .-. wsin (il[f 4- 2(/>) = a4- ^. 1. Find when 2 tan — cot = — 3. ^ns. ilf=135°; = 15°41'.O, 119M9'.0, 195°41'.0, 299° 19'.0. 2. Find when tan + 3 cot = — 2 VS. ^ns. M = 210° ; = 120° or 300°. 88. Find <}> from the Following Equations, a and a being known: (a) sin (<^ -f- «) = « sin . (1) Expanding, sin <^ cos a + cos <^ sin a = a sin <^. . • . sin (^ (a — cos a) = cos <^ sin a, , , sin a ,o\ .-. tan = (2) a — cos a Eq. (2) is not adapted to logarithmic computation. But from (1) we have sin (0 -f «) _ a sin<^ i and, by composition and division, sin (<^ + a) + sin <^ _ a + 1 sin ((/) + «)— sin <^ a — V and this, from the equations of Art. 73, becomes tan (<^ -f 1^ a) _ a 4- 1 tan^^a a — l' or tan (<^ + 1 a) = ^ "^ tan \ a. (3) Let tan/3 = a, and note that tan 45° = 1. ^ ^^ . ^ tan /3 + tan 45° ^ , ... tan(c/> + J«)=^-^^^^-^^^^^tanla sin(y8 + 45°)^ , = . \ri TF^ tan i a. sm (/S — 45°) 2 .-. tan(. . • . tan (<^ -f J «) = tan (45° — /S) cot J a, if tan /8 = a. ( + a) =3 a cos . .-. sin(<^ + a)=«sin(90° +<^); .-. tan(4'5° + + «)=acos(90° - <^); . • . tan (<^ 4- i a - 45°) = tan (45° - ff) cot (45° + J a), if tan = a. Note. — The equation a sin (0 + a) = a' sin (0 + a') and similar equations may be solved by expansion, the solution of the given equation being ,„ , a' sin a' — a sin a tan = -. a cos a — a' cos a' A solution adapted to logarithmic computation may be found by the method of this article, giving tan [0 + ^ (o + a')] = cot (j8 - 45°) tan K« - «')» i^ tan & = —- 89. Find from the Following Equations, a and a being known : (a) sin (<^ + a) sin = a. From (8), Art. 72, cos a — cos (2 <^ + a) = 2 a. . • . cos (2 <^ + a) = cos a — 2a. (1) Let tan /3 = ^-^. (2) sin a ^n , , ^ ' . a COS a COS j8 — sin a sin y3 .*. cos(2 + a) = cosa — sinatan)^= — ^• cos)8 ■ •■ coa^24> + a)= ^°^("+/\ (3) cosp ^ ^ (5) sin (a — <^) sin (\> = a. . • . cos (a — 2 <^) — cos a = 2 a ; . • . cos (a — 2 <^) = cos a + 2 a. ^ o JL\ cos(a — /3) ./. . o 2 a • o cos (« — 2 0) = ^^ — — ^-^ , if tan p = cos p sin a 94 PLANE AND ANALYTICAL TRIGONOMETRY. (c) sin (0 + a) cos cf) = a. . • . sin (2 + a) -t- sin « = 2 = a. . • . cos (2 (^ + a) + cos a = 2 a. , • . cos (2 + «)=« tan (f>. tan ( a — l' sin (2 <^ + ct) _ g + 1 sin a a — 1 Let tan/8 = a; (2) .-. ^ = cot(yS-45°), and sin (2 <^ + «) = cot (/3 - 45°) sin a. (3) Find /3 from (2) and 2 <^ + « from (3). (V) tan ((^ + a) = a cot . . • . cos (2 <^ + «) = tan (45° — yS) cos a, if tan = a. (c) cot (a — <^^= a cot <^. . • . sin (2 (/) — a) = tan (/3 — 45°) sin a, if tan /3 = a. (c?) cot ( = cot (45° — yS) sin 2 a, if tan fi = a. (e) tan ((^ + a) = a tan ( + «'). . • . sin (2 <^ + a + «') = cot (y8 - 45°) sin (a - «'), if tan yS = a. (/) cot ((/> + a) = a cot (<^ + a')- .• . sin (2 + a 4- "0 = cot (/S - 45°) sin (a' - a), if tan /S = a. (^) cot (<^ + a) = a tan ( + «')• . • . cos (2 (^ + a + a') = tan (y3 — 45°) cos (a — a'), if tan yS = a. (1) TRIGONOMETRIC EQUATIONS. 95 91. Find ^ from the Following Equations, a, a, and a' being known : * (a) tan (<^ -|- «) tan -}- a) sin = a cos ( + a) cos ^. From the equations of Art. 72, we have — cos (2 + a) + cos a = a cos (2 (f) -{• a) -\- a cos a; .-. cos (2 + «) = tan (45° — ;8) cos a. (V) tan (<^ + a) cot = «. . • . sin (2 — a)= a. . • . cos 2 <^ = tan (45° — y8) cos 2 a, if tan P=a. (d) tan (^ + a) cot (<^ + «') = a. . • . sin (2 + « + «0 = cot (y8 — 45°) sin (a — a'), if tan ^ = a. 92. Find r and from the Following Equations, a, h, a, and /9 being known : r sin (<^ + «) = a, I (1) rcos(<^ + /3)= J. J (2) sin (d) -f a) <* i • ^ j , \ ^ i , o\ . • . 5 sin <^ cos a-\-h cos <^ sin a = a cos <\> cos ^ — a sin <^ sin yS. . • . 5 sin (\> cos a 4- a sin <^ sin p = a cos <^ cos /S — b cos <^ sin a. . • . sin (f) (b cos a-{- a sin y8) = cos (a cos /3 — 5 sin a). . • . tan 6 = — r, (3) b cos a -\- a sin p and r = = (4) sin(<^ + «) cos( + y8) The quadrant of (j) will be determined by the sign assigned to r. * The method of Art. 90 may be used, since tan x = and cot x = • cot X tan X 96 PLANE AND ANALYTICAL TRIGONOMETRY. 1. If r sin (0 + a) = a, and r sin (0 + /3) = 6, show that 6 cos a — a cos/3 2. If r cos (0 + a) = a, and r cos (0 + /3) = 6, show that ♦«^ -J. rt cos /3 — ?> cos a tan = as>'\n3 — b sin a ;:i 93. Find r and <|) from the Following Equations, a, 5, a, and /3 being known, and the formulas derived being adapted to logarithmic computation : r sin (. From (3) and (4), we have a + b 0) (2) r sin [<^ + K« + ^)] rcos[<^4-K" + ^)] = 2cos-|-(te-y3)' 2 a — h 2sini(«-y8) J from which r and 4- | (« + y8) are found by the method of Art. 84. 1. If r cos (0 4- a) = a, and r cos (0 + /3) = &, show that a -\-h rcos[0+i(« + ^)] = rsin[0 4-Ka + '8)] = 2 cos I (a — ii)* h — a 2 sin \{a- fi) 2. If r sin (0 4- a) = a, and r cos (0 + y8) = &, show that by placing cos (0 + /3) = sin (90"^ + + 3) we may obtain a + & rsin[0 + 45° + Ha + )8)] = rcos[0 + 45° + Ka + i8)] 2 cos [45°- Ka-^)] h — a 2 sin [45°- K«-^)] 3. Find r and when rsin (0 + 100°) = 2, and r sin (0 + 200°) = 3, r being positive. Ans. = 290° 28'. 4 ; r = 3.9436. CHAPTER VII. OBLIQUE PLANE TRIANGLES. 94. It has been shown in Geometry that a triangle can be constructed when three elements, one being a side, are known. If the three angles only are given, there will be an infinite number of triangles satisfying the conditions of the problem, since the data determine the shape and not the size of the triangle. We also know that in any triangle (1) The sum of the three angles is 180°. (2) If one angle is 90°, the sum of the other two is 90°. (3) The greater side is opposite the greater angle, and conversely. (4) Any side is less than the sum of the other two. 95. The Sine Proportion. — The sides of a triangle are to each other as the sines of the opposite angles. In Fig. 52, p = a sin 7 ; p = c sin a. .*. a sin 7 = b, 1. Find a when c = 1.4345, b = 2.3671, and o = 112°43'.4. log tan x = I (log b + log c) + log 2 + log sin ^ a + col (b — c\. log a = log (6 — c) — log cos X. log b = 0.37422 logc = 0.15070 log (6 -c) = 9.96970 -logcosic = 9.46361 log 6c = 0.53092 log a = 0.50609 log a/6c = 0.26546 log 2 = 0.30103 logsin|a = 9.92041 col (&-c)= 0.03030 log tan ic = 0.51720 X = 73° 5'.6 a = 3.2069 2. Find 6 when a = 101.47, c = 99.367, jS = 47°48'.2. .-. x = 88° 31'. 17; 6=81.396. 3. Find a when 6 = 19.937, c = 62.475, a = 130° 9'.4. .-. x = 56°23'.7; a = 76.858. OBLIQUE TRIANGLES SOLVED BY RIGHT TRIANGLES. 103. Case I. Given a, a, y, — In Figs. 63 and 64, on the next page, draw DB perpendicular to AC Considering the first figure, in the triangle BDO we know a and 7, and we compute DB and BQ\ then in the triangle BDA we know DB and a, and we compute AD and C= 1.6868; i)J5 = 2.9929 ; ^D = 9.3650 ; c = 9.8315; & = 11.0518. 2. Solve the triangle when a = 54.376, 7 = 103° 3'.2, /3 = 40° 10'. 3. Ans. o = 36° 46'.5 ; c = 88.478 ; h = 58.592. 3. Solve the triangle when c = 230.47, a = 21° 32'.2, /3 = 36'^ 24'.4. Ans. 7 = 122° 3'.4 ; a = 99.825 ; h = 161.3975. 104. Case II. Given a, c, a. — In the right triangle ABB we know c and «, and we compute AD and DB; then in the triangle CBD we know DB and a, and we find DQ and 7; then 6 = ^2) + DC; /3 = 180°-(« + 7); b' = AD-DC; y = 180°-7; ^8' = 180° -(« + 7'). Two solutions are possible only when a is acute and a is less than - i)C = - 35.178 ; .41) = -9.373; 2)0 = 25.805. ^n.«?. a = 138° 31'.2 ; 7 = 17° 48'.6 ; )8 = 23° 40'.3. In this example D lies to the left of A. 110 PLANE AND ANALYTICAL TRIGONOMETRY. 2. Solve the triangle when a = 32.456, h - 41.724, c = 53.987. .-. AD- DC = 44.(507 ; AD = 43.1655 , DC = - 1.4415. Alls, a - 36° 54'.7 ; 7 = 92° 32'. 7 ; & - 50° 32'.6. 3. Solve the triangle when a = 0.14679, h - 0.10433, c = 0.04796. .-. ^2> - i>C = - 0.18448 ; ^Z) = - 0.040075 ; DC = + 0.144405. Ans. a = 146° 40'. 75 ; 7 = 10° 2l'.0 ; )8 = 22° 58'.25. 106. Case IV. Given 6, c, a. — In the triangle J.i>^, know- ing c and «, find AT) and BB. Then in the triangle BBC we know BB and BC =h — AB, so that we can compute a and 7. 1. Solve the triangle when b = 1143.7, c = 1822.4, a = 15°6'.4. .-. ylZ>= 1759.5; Z>^ = 474.96; DC = -615.8. Ans. 7 = 142° 21'.5 ; a = 777.68 ; 3 = 22° 32'. 1. The negative value of DC shows that D is to the right of C. 2. Solve the triangle when b = 19.937, c = 62.475, a = 130° 9'.4. .-. ^D = - 40.288; DC = 60.225. Ans. 7 = 38° 24'.5 ; i8 = 11° 26M ; a = 73.857, or 76.858. Note that a is obtuse. 8. Solve the triangle when a = 101.47, c = 99.367, j8 = 47°48'.2. Ans. 7 = 64° 44'.6 ; a = 67° 27'.2 : b = 81.394. AREAS OF TRIANGLES. 107. Given Two Sides and the Included Angle (6, c, a).— Represent the area by A. From geometry, in Fig. 67, A = \pb. But p = c sin a. .'. A = ^ 6c sin a, (1) or, the area of a triangle is equal to half the product of the two sides multiplied by the sine of their included angle. OBLIQUE PLANE TRIANGLES. HI 108. Given One Side and the Three Angles («», a, p, -y). — Substitute in (1), Art. 107, the value of c found from the sine proportion, _ ^ sin 7 sin P giving ^=f-^-^w|P- (1) 109. Given the Three Sides («, h, c).— We have A = \hc mna = hc sin J a cos J a. From (2) and (3), Art. 99, we have 110. Given Two Sides and the Angle Opposite One of them (6, c, P). — First find 7 by the formula c . sin 7 = - sin ^. Then a = 180° and A = ^bc sin a. EXAMPLES. 1. Find the area when b = 0.14367, c = 0.11412, = 42° 14'.6. 2. Find the area when a = 3.4356, o=17°43'.4, 7 = 60°36'.7. log 6 = 9.15737 log c = 9.05737 log sin a = 9.82755 col 2 = 9.69897 log ^=7.74126 A = 0.0055114 .-. 3 = 101° 40'.9. loga2 = 21oga= 1.07200 col 2= 9.69897 logsini8= 9.99091 log sin y = 9.94010 col sin a = 0.51652 log^= 1.21850 A = 16.539 3. Find the area when a = 0.0093146, b = 0.0176530, c = 0.0095768. 2s = 0.0365444 logs= 8.26179 5=0.0182722 \og(s-a)= 7.95219 s - a = 0.0089576 log (s-b)= 6.79183 s- 6 = 0.0006192 log(s-c)= 7.93929 s-c= 0.0086954 2 )10.94510-20 sum = 0.0365444 ^°S^ ^ 5.47255-10 a check. ■ ^= Q-QQQQ29686 112 PLANE AND ANALYTICAL TRIGONOMETRY. 4. Find the area when a = 9.4672, c = 14.433, o = 11° 14'.3. log c = 1. 15936 log a = 0.97622 log a = 0.97622 log sin = 9.28979 logc= 1.15936 logc = 1.15936 col a = 9.02378 col 2 = 9.69897 col 2 = 9.69897 log sin 7 = 9^47293 log sin ^= 9.67899 log sin /3^ = 9.02259 7= 17°17M log^= 1.51354 log ^1' = 0.85714 7' = 162° 42'.9 A = 32.624 A' = 7.1968 .-. /3= 151°28'.6 ■ /3' = 6° 2'.8 Note that log A and log A' can be found by adding log sin p and log sin /3' respectively to log a + log c -f col 2, a shorter method than that given in this example. 6. Find the area when a = 0.013456, b = 0.023678, a = 40° 31 '.4. Ans. 0.00010351. 6. Find the area when c = 43.145, o = 40° 40'.3, = 60° 30'.3. Ans. 538.19. 7. Find the area when a = 1.4142, b = 1.6735, c = 2.8533. A7is. 0.83826. 8. Find the area when a = 14.135, c = 23.647, a = 33° 17'.3. Ans. 164.61 or 91.948. 111. Illustrative Examples. — The hearing of a line is the angle it makes with the magnetic meridian, shown by the mag- netic needle. The letter indicating whether the line is meas- ured north or south of the point of beginning is written, then the number of degrees and minutes in the angle, and then the letter indicating whether the line lies to the east or to the west of the magnetic meridian. Thus, if the bearing of the line AB is S. 60° W., the line is measured from A to the west of south by an angle of 60°. The distances and the angles given in the examples are horizontal unless otherwise specified. 1. From a point on a horizontal plane the angle of elevation to the top of a crag is 40° 28'.6, and 4163.2 feet farther away in the same vertical plane the angle is 28° 50 '.4. Find the distances from the points to the top of the crag, and its height above the horizontal plane. BD = 13399 feet ; AD = 9956.2 feet ; CD = 6463.0 feet j BC = 11737 feet : AC = 7573.2 feet. OBLTQW: PLANK TRTANOLES. 113 3. A tower 160.43 feet high is situated at tlie top of a hill (Fig. C9) ; 600 feet down the hill the angle between the surface of the hill and a Jine to the top of the tower is 8° 40'.4. Find the distance to the top of the tower, and the inclination of the ground to a horizontal plane. .-. ^BC = 136°59'.7; ^C :- 726.60 feet ; DAB = 46° 59^.7. '>B Fig. 70. 3. To find the horizontal distance from a point A to an inaccessible point B (Fig. 70), the horizontal distance AC and the angles a and y were measured and found to be 1042.3 feet, 72°9'.4, and 14° 13'.7, respectively. • .-. AB = 256.69 feet ; CB =z 994.15 feet. 4. To find the distance between two points A and B not visible from each other (Fig. 71). — Select a third pohit C from which A and B are visible, and measure the distances (7^ = 444.38 feet, C5 = 222.76 feet, and the angle ^C^=17'' 17'.6. Ans. AB = 240.97 feet. --~i>B Fig. 71. Fig. 72. 5. To find the distance from a point A to another point JB, the latter being inaccessible and invisible from A (Fig. 72). — Select two points C and D so that C, J., and D shall be in the same straight line, A and B being visible both from C and from D. From measurement it is found that CA — 456.72 feet, AD = 490.74 feet, y = 71° 22'. 7, 5 = 36° 19'.4. .-. CB = 589.10 feet ; DB = 942.475 feet ; AB = 619.51, or 619.53 feet. CROCK. TRIG. 8 114 PLANE AND ANALYTICAL TRIGONOMETRY. 6. To find the elevation of the top of a church steeple D (Fig. 78) above the horizontal plane ACB, and the distances of the steeple from A and B. — Let the horizontal distance AB = 435.53 feet, the horizontal angles a = 140° 40'.2 and /3 = 10° 7'.6, and the vertical angles 7 = 32°45'.6 and 7' = 10° 7'.3. . •. AC= 156.95 feet ; BC = 565.74 feet ; CD = 100.99, or 101.00 feet. The agreement of the values of CD is a check upon the observed angles and upon the computations. 7. To find the elevation of the top of a church steeple D (Fig. 74) above the iYfo points A and B, not in the same horizontal plane, the inclined distance from A to B, and its angle of inclination 5 to a horizontal plane being measured, as well as the angles a, ^, 7, and 7', shown in the preceding example. — Let ^5=134.70 feet, 5=3°2'.7, a=43°14'.8, /3=63°17'.5, 7 = 56°36'.6, 7' = 62°17'.3. [First find the horizontal distance AF and the vertical distance FB in the right triangle AFB ; then solve the horizontal triangle AFC; and then find CD and FD from the right triangles ACD and BFD respectively.] . •. AF= 134.51 feet ; FB = 7.1553 feet ; FC = BE = 96.135 feet ; AC = 125.34 feet ; CD = 190.17 feet ; ED = 183.02 feet. Check: CD = FB + ED. Fig. 75. 8. To find the distance between two inaccessible points A and B. — Select two points C and D from which both A and B can be seen, and measure CD = 456.82 feet, a = 30° 40'.6, )8 =r 40° 14'.8, 7=35°16'.4, 5 = 56°47'.4. .-. .42) =449. 09 feet; ^C=274.41 feet ; jBJ9 = 398.66 feet ; 5C=616.66 feet ; AB = 405.57, or 405.58 feet. OBLIQUE PLANE TRIANGLES. 115 9. To find the distance between two inaccessible points A and being visible from only one accessible point C. — Select a point D from which A and C are visible, and another point E from which B and C are visible. From measurement CZ>=943.37 feet, C^=673.33 feet, a = 72°9'.3, /3 = 60°17'.9, 7 = 32° 14'.6, 8 = 67° 33'.9, e = 19° W.I. . '. CA = 1217.0 feet ; CB = 222.28 feet ; AB = 1035.8 feet. B, both 10. To find the distance between two inaccessible points A and B^ there being no accessible point from which both A and B are visible (Fig. 77). — Select the points C, 2>, E, and F so that A, (7, and E shall be visible from i>, and D, F, and B from E. Measure the angles a, /3, y, 5, e, and d, and the dis- tances CD, DE, and EF, Show how AB may be found from the data thus obtained. Ar, Fig. 77. Fig. 78. 11. Two points A and B, 8763.6 feet apart (Fig. 78), are situated at the sea level in the same north and south line ; a vessel is seen at C, and an hour later at D. The required quantities are AC, BC, AD, BD, CD, and the angle that CD makes with the north and south line, having measured BAC=: 120° 30'.6, BAD = 30° 14'.4, ABC = 40° 18'.8, ABD = 140° 28'.2. .-. AC= 17260 feet ; BC = 22985 feet ; AD = 34552.5 feet ; BD = 27340 feet ; ACD = 63° 14'.5 ; ADC = 26° 29'.3 ; BCD = 44° 3'.8 ; BDC = 35° 46'.8 ; CD=SSed6, 38697, or 38699 feet ; e=S60°-BAC-ACB-BCD=nG° 15'.0, or =ABD+BDA-{-ADC=176°U'.9. 12. In measuring the line from A to B, whose direction was known, it was necessary to pass an obstacle at F (Fig. 79). A distance CD =144.31 feet was measured, making an angle y = 19° 53'.4 with AB, and the angle 5 = 140° 10'. 3 116 PLANE AND ANALYTICAL TRIGONOMETRY. was laid off with the transit. It is required to find the distance DE to the line, the distance CE, and the angle e, in order that tlie line AC may be prolonged. Arts. CE = 27L06 feet ; DE = 143.98 feet ; e = 160° 3'.7. x:;;'^a G B Fig. 80. 13. Ill passing an obstacle at F it was necessary to use the broken line CDEG (Fig. 80). The distances CD and DE and the angles 7, 5, and e were measured. It is required to find the distance EG to the line AB, the dis- tance CG, and the angle 6, when CD = 100.37 feet, DE = 94.367 feet, 7 = 80°, 5 = 101°19'.8, ande = 110°. .-. DCE=S7°[>3'.3; DEC = iO° AG'.O ; CE = 150.67 feet ; EG = 108.46 feet ; CG' = 151.22 feet ; = 111°19'.8. 14. From the top of a lighthouse ABy 200 feet above the sea level, the angle of depression to a ship was 7 = 10° 14'.3 : an hour later it was 7' = 11° lO'.O; the horizontal angle between the directions of the ship at the two instants was a = 127° 14'.4. Find the distance sailed by the ship. .-. AC = 1107.3 feet ; AD = 1012.2 feet ; CD = 1899.3 feet. 15. A ladder 52 feet long is set 20 feet in front of an inclined buttress,' and reaches 46 feet up its face. Find the inclination of the face of the buttress. Ans. ABC = do°51'.S, or 95°51'.9. t^ ^ 16. The sides of a city block measured AB = 423.24, BC= 162.36, CD = 420.81, and DA = 160.62 feet, the first two sides being perpendicular to each other. Find the angles between the other sides. .-. ^0-^453.31 feet; BCA = C0° 0'.8; ^^C = 20°59'.2; ACD = 20° ^b'.O ; CAD = G8° 8'.8'; CZ>yl = 91° 6'.4 ; BCD = 89° 45'.S; BAD = S9° 8'.0. Fig. 83. OBLIQUE PLANE TRIANGLES. 117 17. A ship B is 12 miles S. 45° W. of a lighthouse A, and sails S. 50° E. to C, a distance of 15 miles. Find its distance from the lighthouse. B Ans. AC= 18.374, or 18.375 miles. ''^" Fig. 84. 18. In surveying a field a thick wood prevents the measurement of the angle ABD and of the distance BD. The angle ABC = 70° 14'.6 is measured, a line ^O is run 743.86 feet, the angle BCD is found to be 62° 14'.4, and the distance CD to be 912.82 feet. .-. CjB2) = 68°28'.1; CDB = i9°lT.5; BD = 868.34, 868.30, or 868.38 feet ; ^i?Z)z=138°42'.7. Fig. 85. 19. The distance OE and its bearing E'OE are required, the engineer having measured the distances a, b, c, d, and e and their respective bearings, N. 30° W., S. 60° E., N. 20° E., N. 40° W., and N. 50° E. OE' = OA' - B'A' JrB'C'+C>D>-{- D'E' = a cos 30° - h cos 60° + c cos 20° + d cos 40° -f e cos 50°. E'E = -AA'-\- B"B + C"C-DD"^E"E = - a sin 30° + b sin 60° + c sin 20° - d sin 40° + e sin 50°. Then OE cos E'OE = OE', OEsmE'OE = E'E; whence OE and E'OE can be found. Then the quadrant of E'OE fixes the direction of the line OE ; thus, if E'OE = 40°, the bearing is N. 40° E. ; if ^^0^"= 110°, the bearing is S. 70° E. ; if E'OE = 230°, the bearing is S. 50° W. ; if E'OE = 310°, the bearing is N. 50° W. 20. At a certain point the angles of elevation of the base of a vertical tower and of its top are a and /3 respectively, the height of the tower being h feet. Prove that the horizontal distance from the point to the tower is Fig. 86. 118 PLANE AND ANALYTICAL TRIGONOMETRY. j^cosa cos/3 cosec(i3 — a), and that the elevation of its top above the point is h cos a sin /3 cosec (/3 — a) . 21. At the top of a vertical tower whose height is h, the angles of depression to two points M and N in the same vertical plane with the tower were o and /3 respectively (/3 > a), the points being in the same horizontal plane with the base of the tower. Prove that the distance MN is h sin (/3 — a) cosec o cosec /3. 22. Two points M and iV in a horizontal plane are in the same vertical plane with a tower. The angle of elevation of the top of the tower from 31 is a, and from N it' is iS, /3 being greater than a. Prove that the horizontal distance of the tower from iV" is MN sin a cosjS cosec {^ — a). 23. Three points, A, B, and C, are in the same horizontal line, the distances AB and BG being a and b feet respectively (Fig. 87). The angles of elevation of the top of a tower measured at A, B, and C were a, /3, and y respectively. Find the elevation of the top of the tower above the horizontal plane through the points, and the horizontal dis- tances of the tower from the three points. Fig. 87. m = h cot a ; n = h cot ^; p = /i cot 7 ; m^ = a^-\-n^ -2 an cos ABD ; P' = b-2 + W2 + 2bn cos ABD ; a2 + n2- -m2 p^- 62 - w2 2 an 2bn > . h^ ab(a + b) a (cot2 7 - cot2 /3) + 6 (cot2 a - cot2 /3) 24. In Fig. 88 the distances a and & and the angles a, /3, and y are known, and the distance BG = x is required, ABGD being an inaccessible straight line. ^<^r;||v: FB a + X FG Fig. 88. sin a sin J. ' sin (a + jS) sin A ' FB a sin (a + /S) FG a + x sin a b FG . b + x _ FB sin 7 sin Z) ' sin (^ + 7) sin D FG b sin(j8 + 7). FB 6 + x sin 7 Multiplying (1) and (2), we have (a + x) (6 + x) sin a sin 7 = ab sin (a + jS) sin (^ + 7), from which x may be found, since the equation is a quadratic in x. (1) (2) OBLIQUE PLANE TRIANGLES. 119 a tower 25. Two points A and B in the same vertical plane with the top of are on a sidehill whose angle of inclination to a horizontal plane is 5, the inclined distance AB being a feet. The angles of elevation of the top of the tower were measured at A and B, and found to be a and )3. Prove that the horizontal distance of the top of the tower from B is a (cos S tan a — sin 5) cos a cos cosec (3 — a), and that the elevation of the top above B is a (cos 5 tan a — sin 5) cos a sin |3 cosec (/3 — a). 26. In a hydrographical survey, the distances between three points, A, Bj and C, on the shore having been determined, the observer in the boat P measures the angles 5 and e subtended by AB and BC. It is required to find the dis- tances of the boat from the three points. (1) Graphical Solution. — Construct on AB the segment of a circle APB that shall contain the measured angle 5, and on BC the segment of a circle BPC that shall contain the angle e. Their point of in- tersection P will be the position of the boat. There are four possible solutions, only one being shown in the figure. .' (2) Analytical Solution. — Let J.Z>CP » be the circle through A, C, and P. Then DAC= e, and DC A = 8. Hence in the tri- angle ADC we know one side AC and the three angles ; find AD and CD. In the tri- angle ABC we know the three sides; find ^ the three angles. In the triangle DAB we '^^^- ^^^ know two sides and the angle DAB = CAB — CAD ; find ABD. Then in the triangle ABP we know one side and the three angles ; find AP and BP. Also, compute DBC from the triangle DCB, and then BP and CP from the triangle BPC. The values of BP should agree. In the following examples find the last three elements, the first three being given: 27. a = 1.0431, /3 = 4°4'.4, 7 = 22°3'.6. .'. a = 153° 62'.0 ; b = 0.16822 ; c = 0.88942. 28. a = 103.37, a = 10° 11 '..3, /3 = 83°43'.6. 7-. M: & = 580.89: c = 583.02. 29. c = 74.344, a = 105° 6'. 7, /S = 60° 14'.4. .-. 7 = 14°38'.9; a = 283.82 ; 6 = 255.21. 30. c = 0.047365, p = 40° 7'. 7 y = 39° 41'. 9. .'. a = 100° 10'.4 ; a = 0.072990 ; b = 0.047792. 31. c = 4.4479, a = 11° 11'.3, y = 57° 37'.4. . •. /3 = 111° 11'.3 ; a = 1.0219 ; b = 4.9106. 120 PLANE AND ANALYTICAL TRIGONOMETRY. 82. b = 143.97, /3 = 30° 36'.8, y = 107° 15'.5. .•'. .-. a =42°7'.7; a = 189.64; c = 269.98. 33.6 = 10.467, c = 1.4321, /S = 114° 10'.3. .-. 7 = 7°10'.2; a = 58°39'.5; a = 9.79875. 34. a = 0.67375, b = 0.43213, a = 147° 11'.3. .-. i3 = 20°20'.2; 7 = 12°28'.5; c = 0.26858. 35. a = 1.4742, c = 0.97674, o = 25° 19'.9. . •. 7 = 16° 28'. 1 ; j3 = 138° 12'. ; 6 = 2. 2966. 36. a = 943.42, b = 647.15, a = 104° 6'.9, .-. /3 =41°42'.0; 7 = 34° 11'. 1; c = 646.59. 37. a = 0.10321, c = 0.047323, a = 45°9'.7. .-. 7 = 18°58'.4; /3 = 115°51'.9; 6 = 0.13097. 38. a = 4.4321, c = 5.4763, 7=100°11'.9. ' .-. o =52°48'.l; /3 = 27°0'.0; 6=2.5261. 39.0=23.111, 6 = 19.476, 7 = 47°16'.7. .-. ^=38°15'.0; a = 94°28'.3; a = 31.363. 40. a = 0.11111, = 0.12767, a = 23° 15'.6. . •. 7 = 26° 59'. 1 ; /3 = 129° 45'. 3 ; 6 = 0.21630 ; 7' = 153° 0'.9 ; /3' = 3° 43'.5 ; 6' = 0.018279. 4L 6 = 1.4326, c= 1.3671, 7 = 44° 17'.3. .-. /3= 47° 1'.9; a = 88°40'.8; a = 1.9574 ; /3' = 132° 58'. 1 ; a' = 2° 44'. 6 : a' = 0.093706. 42. a = 46.703, 6 = 57.147, a = 19° 17'. 7. .-. /3 = 23°50'.9; 7 = 136°51'.4; c = 96.652 ; /3' = 156° 9'.1 ; 7' = 4°33'.2 ; c' = 11.221. 43. a = 9.4327, c = 10.4751, a = 63° 17'.3. .-. 7 =82°45'.0; ^ = 33°57'.7; 6 = 5.8990; 7' = 97° 15'.0 ; ^' = 19° 27'.7 ; 6' = 3.5182. 44. a = 0.034337, c = 0.062774, a=9°6'.7. .: y= 16° 49'. 7 ; ^ = 154° 3'.6 ; 6 = 0.094846 ; 7' = 163°10'.3; /3' = 7°43'.0; 6' = 0.029115. 46. a = 0.79797, 6 = 0.46731, jS = 23° 19'.6. .-. a= 42°32'.5; 7 = 114° 7'.9 ; r.= 1.07705; a' = 137°27'.5; 7'= 19° 12'.9 ; c' = 0.38841. 46. a = 37.456, 6 = 43.987, c = 13.498. . •. i a = 26° 31'.0 ; ^ jS = 55° 7'.0 ; i 7 = 8° 22'.0. 47. a = 2.4568, 6 = 2.4743, c = 1.0047. .-. ia = 38°38'.0; i^ = 39°36'.7; i7 = ll°45'.3. 48. a = 47.474, 6=100.980, c = 93.929. .-. ia = 13°56'.8; |^ = 42°10'.2; i7 = 33°53'.0. 49. a =14.567, 6 = 9.4769, c = 11.113. .-. ia = 44°50'.9; i /3 = 20° 17'.5 ; i7 = 24°51'.5. OBLIQUE PLANE TRIANGLES. 121 50. rt = 2.1476, 6 = 1.9397, c = 3.4345. .-. ^0=17^22^8; i/3=15°29'.8; ^y = b7°T.3. 61. a = 115.03, b = 129.15, c = 112.06. .-. ia = 28°12'.9; J /3 = 34° 39'.2 ; J 7 = 27° 7'.9. 52. b = 113.47, c = 227.79, a = 19° 43'.4. .-. /3= 17°33'.8; 7 = 142°42'.8; a = 126.90 ; or log tan x = 9.68278 ; a = 126.89. 53. a = 99.416, c = 90.432, /3 = ll°7'.8. .-. a = 110°20'.4; 7 = 58°31'.8; 6 = 20.467; or log tan x = 0.31110 ; b = 20.467. 54. a = 1.4342, b = 9.7672 ; 7 = 109° 19'. 0. .-. o=7°31'.7; ^ = 63°8'.7; c = 10.330, or 10.331 ; or log tan x = 9. 86498 ; c = 10.331. 55. a = 1003.7, b = 943.67, 7 = 101° 19'.8. .-. a = 40°46'.9; i8 = 37°53'.3; c = 1506.7 ; or log tan x = 1.39930 ; c = 1506.7. 56. a = 222.76, 6 = 444.38, 7 = 17° 17'.6. .-. a = 15°57'.0; /S = 146°45'.4 ; c = 240.97; or log tan x = 9.63029 ; c = 240.97. 67. a = 363.24, 6 = 146.18, 7 = 68° 14'.4. .-. a = 88°2'.6 ; /3 = 23°43'.0 ; c = 337.55, or 337.56; or log tan x = 0.07590 ; c = 337.55. PART TWO. SPHERICAL TBIGOJfOMETBY. ^-if^Oo CHAPTER YIII. DEFINITIONS AND CONSTRUCTIONS. 112. Spherical Trigonometry treats of the relations between the face angles and the edge angles of a trihedral angle. An edge angle is the angle between two of the three planes forming the ^--^t^'^ rV trihedral angle ; it is measured by ct<<^^^^^ --^^^^^ ?l A the angle between the lines cut from >^>i5^^^ \ < j ry the two planes by a plane perpen- ^^^^^'^^' J^ dicular to the edge in which the two b^^^-J^ planes intersect. A face angle is the angle between two of the edges. 113. Representation of Trihedral Angles. — The relations between the elements of a trihedral angle are discussed by means of the spherical triangle formed by the intersections of the faces with a sphere described with any radius about the vertex as a center. The faces will cut arcs of great circles from the surface of the sphere, their angular measures being the same as those of the face angles; and the angles of the spherical triangle will correspond to the edge angles, each being measured by the angle between two lines lying in the planes of the faces and perpendicular to the line of intersection of the faces. 1^ 124 SPHERICAL TRIGONOMETRY. Hence, in the spherical triangle the sides correspond to the face angles, and the angles to the edge angles of the trihedral angle. The lengths of the sides in linear measure will depend upon the radius of the sphere, and are computed, when the radius is known, by the proportion 360°:a = 2Trr:«, (1) where a is the number of degrees in the arc, and I is its length. 114. Limitation of Values. — We shall consider only those triangles in which each element is less than 180°. In the gen- eral spherical triangle the sides and angles may have values greater than 180°, but in such a case it is always possible to substitute for the triangle, in the computations, another in which each element shall be less than 180°. 115. Definitions and Relations. — A great circle is cut from the surface of a sphere by a plane passing through its center; its radius is equal to the radius of the sphere. A %mall circle is cat from the surface by a plane not passing through the center; its radius is always less than the radius of the sphere. Two planes passing through the center will intersect in a diameter of the sphere, and the two corresponding great circles will intersect at the ends of this diame- ter. Hence any two great circles will intersect at two points 180° apart. To describe a great circle on a sphere, separate the points of a pair of compasses by a distance equal to the chord of 90°, or rV2, and describe an arc about any point. If any other ^ _ distance is used, a small circle will be Fig. 92. described. The point used as the center is called the fole of the great circle; its distance from all points on the great circle is evidently 90°. Any great circle passing through the pole of another great circle will be perpendicular to that great circle DEFINITIONS AND CONSTRUCTIONS. 125 Any two great circles drawn perpendicular to a third great circle will intersect in its pole. A great circle perpendicular to two great circles will pass through the poles of both, and its plane will be perpendicular to the diameter joining the points of intersection of the two great circles. The angle between two arcs of great circles is measured by the arc of a great circle described about the vertex as a pole, and limited by the sides, produced if necessary The shortest distance between two points on a sphere is the arc of the great circle passing through the points. 116. Constructions. — To find the pole of a given great cir- cle: from any two points on the circle as poles, describe arcs of great circles, and their intersection will be the point required. To draw a great circle through two points: find the pole as before, and describe the great circle. To draw a great circle through a given point perpendicular to a given great circle: from the point as a pole describe an arc of a great circle; its point of intersection with the given circle will be the pole of the required circle. Or, find the pole of the given circle, and then draw the great circle through this pole and the given point. To cut from a great circle an arc n° long: separate the points of the compasses by a distance equal to the chord of n^^ or 2 r sin ^ 7i°, place the points on the great circle, and the arc intercepted will be the one re- quired. To construct a great circle pass- ing through a given point and mak- ing a given angle with a given great circle: in Fig. 93, let ACB be the given great circle,* P its pole, F the given point, and a the given angle. With P as a pole, draw the small circle P'P" such that the angular FilTaT * The planes of the great circles ACB and CF, and of the small circle P'P"^ are perpendicular to the paper. 126 SPHERICAL TRIGONOMETRY. distance PP' = a; then the pole of the required great circle must be on this small circle. With jP as a pole, describe an arc of a great circle cutting the small circle P'P" in two points; these points will be the poles of two great circles through F, both of which satisfy the given conditions. Only the great circle OF, whose pole is P', is shown in the figure. To construct a great circle making a given angle with a given great circle, the point of intersection being given: from the given point as a pole describe a great circle, lay off on it from the given circle a distance equal in angular measure to the given angle, and pass a great circle through the point thus found and the given point of intersection. 117. Definitions. — A right spherical triangle is one which has one angle equal to 90°; a birectangular triangle lias two angles each equal to 90°; a trirectangular triangle has three angles each equal to 90°. A quadrantal triangle has one side equal to a quadrant, or 90°; a biquadrantal triangle has two sides each equal to a quadrant; a triquadrantal triangle has three sides each equal to a quadrant. A birectangular triangle is also biquadrantal, and a tri- rectangular triangle is also triquadrantal; and vice versd. 118. The Polar Triangle of any triangle is constructed by describing arcs of great cireles about the vertices of the origi- nal triangle as poles. Thus, about A, B, apd C as poles, describe the arcs B' C\ A!'Q\ and A! B\ respectively; that triangle is called the polar in which the vertices. ^ and A! , B and B\ C and C are on th'^ same side of BO, AO, and AB, respectively. The vertices of the polar triangle will be the poles of the sides of the original triangle, so that either triangle will be the polar of the other. The sides of a triangle are the supplements of the opposite angles of the polar, and the angles are the supplements of the DEFINITIONS AND CONSTRUCTIONS. 127 Fio. 95. opposite sides of the polar; a'= 180°— a, aJ — 180"— a. Thus, if the angles of a triangle be 120°, 80°, and 60°, the opposite sides of the polar will be 60°, 100°, and 120°. The polar of a quadrantal tri- angle is a right triangle, the angle in the polar opposite the quadrant being equal to the supplement of 90°; the polar of a biquadrantal tri- angle is birectangular; the polar of a triquadrantal triangle is trirectan- gular; and vice versd. The triquadrantal triangle is its own polar, each vertex being the pole of the opposite side. 119. In Any Spherical Triangle : (1) Each side must be an arc of a great circle. (2) Each side must be less than the sum of the other two. (3) The greater side is opposite the greater angle, and conversely. Equal sides are opposite equal angles. (4) The sum of the sides must be less than 360°. (5) The sum of the angles must be greater than 180° and (ess than 540°. 120. Construction of Triangles. — (1) Given the three sides, a, 5, c. — Draw an arc of a great circle and lay off on it an arc equal to one of the sides, as a. From the extremities of this arc as poles, with radii equal to the chords of h and c respec- tively,, describe arcs of small circles with the compasses, and find their point of intersection. Join this point and the extremities of a by arcs of great circles, and the triangle will be constructed. (2) Given the three angles, a, /3, 7. — Find the sides of the polar triangle, construct it, and then construct the given tri- angle by using the vertices of the polar as poles. (3) Given two sides and the included angle, a, 5, 7. — Draw an arc of a great circle, and lay off on it an arc equal to one of the sides, as a. Pass an arc of a great circle through one extremity of a, making the angle 7 with a, and lay off on it an arc equal to h. Join the extremities of a and h by an arc of a great circle, and the triangle will be constructed. v'^V^^' ouw^^^ s^ t>f CAi \rcS 128 SPHERICAL TRIGONOMETRY. (4) Given two angles and their included side, a, /3, o. — In the polar we know two sides and the included angle, and hence we can construct it by the method just given. Having the polar, we can then construct the required triangle. Or, draw a great circle and lay off on it an arc equal to e ; at the extremities of this arc, construct arcs of great circles making the angles a and y3 with c ; their point of intersection will be the third vertex. (5) Given two sides and the angie opposite one of them, a, 5, a. — Draw any great circle ADA\ and through any point on it, as A^ draw a great circle making the angle DA C= a with it. On this circle l:iy off from A the distance AC =h. With -^ ^ ^ ^ ^6^ as a pole, describe a small Fig. 96. • ^ ^ ^ Circle whose radius is equal to the chord of a, using the compasses ; pass arcs of great circles through C^ and the points B and B' where this small circle intersects the first great circle ADA'^ and the triangle will be constructed. There will be, in general, two points of intersection, and there may therefore be two triangles that will satisfy the con- ditions of the problem. Only those triangles can be taken in which each side is less than 180°, i.e. both B and B' must lie on the arc ADA' between A and A\ these points being 180° apart. If a is acute, as in Fig. 96, a must be greater than p and less than the shorter of the two distances CA and CA' (h and 180° - 6) in order that there may be two solutions. If a is obtuse, as in Fig. 97, CD' is the least and CD the greatest distance of C from ADA'D', DCD' being perpen- dicular to ADA'D'. Therefore a must be less than jt?, in ordei that the small circle may cui ■piQ, 97. ADA'D' ; a must also be greatei DEFINITIONS AND CONSTRUCTIONS. 129 than the longer of the two distances CA and CA' (6 and 180° — 6) in order that the two points of intersection may fall on the arc ABA'. The conditions, therefore, for two solutions are : a acute : a > p^ a < b^ a < 180° — b. a obtuse : a b^ a> 180° — b. Or, a must be intermediate in value between p and both b and 180° - b. If a is intermediate in value only between p and either b or 180° — 6, there will be one solution. If a is not intermediate in value between p and either b or 180° — 6, no solution will be possible, but ii p = a, there will be one solution — a right triangle. (6) Given -two angles and the side opposite one of them, «, /3, a. — In the polar triangle we know two sides and the angle opposite one of them, and we can construct it ; having the polar we can construct the required triangle. As the polar triangle may admit of two solutions, there may be two solutions of the problem. CROCK. TRIG. — 9 CHAPTER IX. GENERAL FORMULAS. Fia. 98. 121. The Cosine of Any Side of a Spherical Triangle is equal to the product of the cosines of the other two sides, increased hy the product of the sines of these two sides multiplied hy the cosine of their included angle. — Let the phxne BAC be per- pendicular to OA at any point A, and let J. (7, BC, and BA be its intersections with the faces of the trihedral angle. Then BA C — a, and AB and A are perpendicular to OA, i.e. OAB and OAO are triangles right- angled at A. In the triangle BA we have BC^ = AB^ -{- AC^- 2 AB' AC cos a. In the triangle BOO, BC'^= 0&-{- OC^-I OB' 00 cos a. Equating the values of BC^, and transposing, 2 OB ' O0cosa = OB^ - AB^ + 00^ - AO^ + 2AB' AOcosa. In the right triangles OAB and OAO, OB"- - AB"^ = 0A\ and 00^ - AO^ = OA^. .'. 2 OB' 00 cos a = OA^ + OA'^ + 2 AB ■ AC cos a; OB ' 00 cos a = OA^ + AB ■ AC cos a. or or cos a cos a OA 00 OA AC OB 00 AB OB cos a cos b cos c + sin 6 sin c cos a. 130 (1) GENERAL FORMULAS. 131 In this proof b and c are assumed to be less than 90°, while a and a may have any values less than 180°. The formula is true, however, when either b or c, or both b and c, exceed 90°. If, in the triangle represented by the full lines (Fig. 99), b is greater than 90°, then in the dotted triangle formed by completing the arcs of great circles, the two sides are 180° — 6, and c, both less than 90°, and the other side and its opposite angle are 180° — a, and 180° — a. Hence we can apply (1) to the dotted triangle, giving Fkj. 100. cos (180° - a)= cos (180° - b) cose + sin (180° - 6) sine cos (180° - o). .-. — cosa = — cos6 cose — sin6 sinecoso. .*. cos a = cos 6 cose + sin & sine cos a. q.e.d. If both b and c are greater than 90°, as in Fig. 100, then in the dotted triangle the two sides are 180° — b, and 180° — c, and the other side and its opposite angle are a and a. .-. cos a = cos (180° - b) cos (180° - c) + sin (180° - b) sin (180° - e) cos a = ( — cos &) ( — cos c) + sin b sin e cos a. .-. cos a = cos 6 cos c -f sin 6 sin c cos o. q.e.d. Therefore the formula is always true when each of the elements of the triangle is less than 180°. No assumption, then, has been made concerning any element that is not true for all the others. We may therefore change any angle to another, as a to yS, if at the same time we change the sides opposite, as a to 6, making also the reverse changes, 5 to a and ^ to a, in the formula ; for this is equivalent to changing the names arbitrarily assigned to the sides and angles. Thus, to permute (1) to find cos Eq. (1) is called the fundamental equation of spherical trig- onometry, since all the other formulas may be derived from it. 122. The Cosine of Any Angle of a Spherical Triangle is equal to the product of the sines of the other two angles multiplied by the cosine of their included side, diminished by the product of the cosines of the qther two angles. — We have cos a = cos b cos c -\- sin b sin c cos a. (1) Since the angles of the polar triangle are the supplements of the sides oppo- site in the original triangle, and vice versd, we have a = 180° -a', b = 180° - yS', c^ 180° - 7', a = 180° - a'. Substituting in (1), cos (180° - a') = cos (180° - yS') cos (180° - 7') + sin (180° - 13') sin (180° - 7') cos (180° - a'), (— cosy8')(— cos 7') 4- sinyS' sin 7' (~ cos a'). — cos yS' cos 7' -1- sin /3' sin Q = (9^ sin b sin 7. sin yS _ sin 7 sin 90° ; /3 < 90°; 90° ; a > 90° ; h< 90°. Second solution : « > 90° ; yS > 90° ; c<90°; a > 90° ; b> 90°. 1. a = 160^ 12'.2, a = 150° 37'. log sin a = 9.52979 log sin a = 9.69077 log sin c Fig. 109. log tan a = 9.55625 n + log cot a = 0.24942 n log sin b = 9.80567 b= 39''44M b' = 140° 15'.9 log cos a = 9.94020 n log cos a = 9.97354/1 9.83902 c'= 48°39M c = 136° 20' .9 log sin|3 = 9.96666 /3 = 67° 50' 2 /3' = 112° 9' 8 b = 40° 50', /3 = 62° 14'. .: a = 27° 3'.9 ; a = 38° 0'.4 ; c = 47° 38'.6 ; or a' = 152° 56'. 1 ; a' = 141° 59'.6 ; c' = 132°21'.4. • The triangle is supposed to be possible. The two solutions are identical when a = o. 142 SPHERICAL TRIGONOMETRY. 137. Special Cases. 1. C = 90°, a = 90^ 2. c = 90°, a = 90^. 3. a = 90°, /3 = 90°. 4. « = 90°, & = 90°. 6. a = 90°, i3 = 90°. 6. a = 20°, a = 20°. . a = 90° ; b and /3 indeterminate. . a = 90° ; b and /3 indeterminate. •. rt := 90° ; b = 90° ; c = 90°. . c = 90° ; a = 90° ; /3 = 90°. . c = 90° ; 6 = 90° ; a = 90°. ■. c = 90° ; & = 90° ; /3 = 90°. 138. Additional Examples. 1. rt = 40°42'.4, c = 63°20'. .-. 6=53°41'.9; a = 46° 52'.25 ; /3 = 64°24'.0. 2. a = 70°15'.5, a = 81°42'.7. .: b= 23°57'.0; ^= 25° 15'.7 ; c = 72° 1'.25 ; or &' = 156° 3'.0 ; ^' = 154° 44'.3 ; c' = 107° 58'.75. 3. 6 = 30°32'.4, a = 36° 44'. .-. a=20°46'.0; c = 36°21'.6; /3 = 58°59'.7. 4. c = 72° 10', a = 30° 43'. .'. a = 29° 5'.0 ; & = 69° 29'.0 ; j8 = 79° 41'.25. 6. = 106° 34- .2, /3 = 33°11'.7. .-. a = 121°23'.6; 6 = 29° ll'.O ; c = 117° 3'.0. 6. a = 28° 47', & = 110°27'.3. .-. c = 107°50'.2; a = 30° 23'.1 ; /3 = 100° 10'.9. 7. c = 54°12'.2, )S = 164°50'.4. .-. a = 99°0'.3; 6 = 167°45'.2; a = 126°45'.9. 8. a = 40° 8', i3 = 74°30'.2. .-. 6 = 66°43'.5; c = 72°25'.0; a = 42°32'.7. W 9. c = 102°30', a = 125°13'.4. .-. « = 127°8'.l; 6 = 68°49'.0; /3=72°49'.8. 10. a = 40°42'.4, ^ = 67°51'.0. .-. a = 35°4'.4; ?> = 54°42'.0; c = 61°46'.6. 11. 6 = 163°14'.2, c = 112°41'.8. . •. rt = m° 14'.1 ; a = 82° 45'.75 ; /3 = 161° 46'.9. 12. a = 120° 30'.2, b = 140° 12'. .-. c = 67°2'.8; a = 110° 39'.7 ; /3 = 135° 57'.7. 13. c = 50° 20'.2, /3 = 101° 29'.4. .-. rt = 166°29'.5; ?> = 131° 1'.7 ; a = 162''20'.1. 14. a = 82°4'.4, /3 = 8°22'.3. .-. a = 18°42'.2: c = 18° 53'.25 ; Zy = 2° 43' or 2° 44'. RIGHT SPHERICAL TRIANGLES. 143 15. a = 130°40'.7, c= 76° 31 '.6. .-. 6 = 112°33'.0; a = 128°26'.6; /3 = 107" 28'. 76. 16. 6 = 10°10'.2, /3= 16°40'.6. .: a= 39° 43'.9 ; c = 40° 48'. 1 ; o = 78° 0'.7 ; or a' = 140° 16M ; c' = 139° 11'.9 ; a' = 101° 59'.3. 17. 6 = 67°8'.3, a = 104°16'.2. .-. a = 106° 50'.8 ; c = 99° 2'.8 ; ^ = 58° 16'.4. 18. a = 20° 64', b - 6^° 26'.7. .-. c = m° 14'. 1 ; a = 22° 66'.5 ; /3 = 80° 19'. 2. 139. Isosceles Triangles. — If an arc of a great circle be drawn from the vertex perpendicular to the base, it will bisect both the base and the angle at the ver- tex, dividing the triangle into two equal right triangles that may be solved by Napier's rules. 1. a = 110°47'.3, y8=92°14'.6. .-. -l-y8 = 46°7^3. To find a: cosa= cot « cot JyS ; , cos« .-. cot a = Fig. 110. COt-l-yS To find J h : sin \h = sin a sin ^ ff. To find jt?: cos-J/S = tanj? cota; .-. tanjo = -^ — Check : sin J5 = tanjt? cot a. log cos a = 9.55013 n log sin a = 9.97076 log cos ^ /3 = 9.84081 log cot I /S = 9.98299 f log sin ^ /3 = 9.85783 - log cot a = 9.57936 n log cot a = 9.56714 n log sin ^b = 9.82859 log tanp = 0.26145 n 110° 15'.54 I b = 42° 22'.1 p = 118° 42'.6 ■ — • b = 84° 44'.2 2. a = 82° 26', /3 = 64° 42'. 3. & = 56°41', /3 = 112°44'.6. .-. a= 77°53'.6; \b= 31°32'.75. .-. a= 38°69'.6; a= 34°45'.6; or a' = 141° 0'.4 ; a' = 146° 14'.4. 144 SPHERICAL TRIGONOMETRY. 140. Quadrantal Triangles. — The polar of a quadrantal tri- angle is a right triangle whose angles are the supplements of the sides, and whose sides are the supplements of the angles, of the original triangle. ^90° We may therefore solve the polar by Napier's rules, and then find the ele- ments of the original triangle by taking the supplements of the ele- ments of the polar. A b' Fig. 111. 1. ^=90°, a= 23°14'.7, b=: 27°14'.6. .-. y = 90°, a' = 156°45'.3, yS' = 152° 45'.4 are the elements of the polar triangle.* To find c' : cos jS' = tan a' cot c' ; To find b' : sin a' = tan b' cot /3' ; To find a' : cos a' = cos a' sin ff. Check: cos a' = cot c' tan b'. cot c' tan b' ^ cos 13' tan a' _ sin a' ~cot/3'" log cote' = 0.31594 log tan 6' = 9.30795 n log cos a' ■= 9.62389 n c'= 64°12'.8 6'=168°30'.8 a' = 114°52'.4 .-. 7 = 115°47'.2 .: p= 11°29'.2 .: a= 65° 7'.6 log cos /3' = 9.94894 n log tan a' = 9.63300 n log sin a' = 9.59623 log cos a' = 9.96324 n log cot /3' = 0.28828 n + log sin /3' = 9.66065 c= 90°, 7 22'.7, I = 150° 47 150°26'.2 90' 90^ 90^ a = 121° 30', /3 = 112°16'.2. .: b = 108°51'.l: - 94°43'.5; jS = 99°36'.6. 123°30'.75; y= 102° 4'.7. a = 138°47'.8, 6 = 107°54'.9. .-. a = 142°15'.2; ^ = 117°50'.25; 7 = 111° 40'. 1. a = 112° 6'. 5, 7= 74° 30'. . •. 6 =._ 56° 39'.6 ; a = 116°46'.4; /3 = 53°36'.9. 6. c = 90°, a = 83° 20'.6, /3 = 77° 14'.3. .-. a= 83°30'.3; &= 77°19'.3; 7= 91°28'.0. 7. c= 90°, a= 94°22'.2, a = 108° 13'.3. .-. b= 14° 6'.2; /3= 13°25'.3; 7= 72° 17'.5; or 6' = 165° 53'.8 ; ^' = 166° 34'.7 ; 7' = 107° 42'.5. * Note that a', /3', and c' are not the parts of the right triangle, but their complements. RIGHT SPHERICAL TRIANGLES. 145 141. Quadrantal Triangles may also be solved by the use of Fig. 112, in which B^ one of the vertices adjacent to the quadrantal side, is the pole of the great circle MDGN. If the triangle has one side less than 90°, as BO in the tri- angle ABC, produce that side to B. In the triangle ACB, ABC =90", BAC = 90° - «, ACB = 180° - 7, AC=b, (7i>=90°-a, SindAB = l3 since AB = ABB. Therefore, if any two elements of ACB besides the quadrant are given, we know two elements of the right triangle ACB in addition to the right angle. Hence we could solve it by Napier's rules, thence obtaining the elements of ABC. If one side of the triangle is greater than 90°, as in BUF, then in the triangle GBF we have GF = a - 90°, GF = /3, FF = 6, FGF = 90°, GFF =a- 90°, and GFE = 7. If any two elements of BFF besides the quadrantal side are given, we then know two elements of the triangle GFF in addition to the right angle. Hence we could solve it by Napier's rules, thence finding the elements of BFF. Fig. 112. CROCK. TRIG. 10 CHAPTER XI. OBLIQUE SPHERICAL TRIANGLES. 142. To find an Angle, having given the Three Sides. (a) cos a = cos b cos c 4- sin 5 sin )cos J(w — v"), we have cos(y8+7)+cosa=2cos J (a + /3+7) cos J(y8 + 7 — '0 ♦ Compare with Art. OBLIQUE SPHERICAL TRIANGLES. 149 Let a-|-;9 + 7 = 2AS'; .-. /9 + 7 - « = 2aS' - 2a = ^(.S'- a). . *. cos (/S + 7) + COS a = 2 cos JS cos (aS' — a). Permuting, ^ sin p sin Y gin2 15= -cosScos(5-p) ^ sin a sin y sin'' - c = ~ *®'' '^ ^®* C-S^ - 7). (2) sin a sin P (c?)* Add each member of (1) to unity. ^ ^ cos a + cos yS cos 7 .-. l + cosa = lH : — ri—' sin p sin 7 _ cos a 4- cos /Q cos 7 + sin ^ sin 7 "" sin j3 sin 7 cos a -\- cos (yS — 7) sin /3 sin 7 2cos^^a = Applying the equation (Art. 73) cos u + cos V = 2 cos 1^ (t^ 4- v) cos J (m — v), we have cosa + cos(^ — 7) = 2 cos J(a + /Q — 7) cos J(a — /3 + 7). Let a + yS + 7 = 2AS'; .-. a + /3 - 7 = 2aS'- 27 = 2 (aS^ - 7); « - yg + 7 = 2aS'- 2/3 = 2(aS'- /3). . •. cos a + COS (yS — 7) = 2 cos (^S — /3) cos (aS' — 7). .. cos ^a- sinpsinv •r> , . o 1 , COS (5> - a) cos (S - y) Permuting, cos« ^ 6 = sinasinY ' «i cos (S - g) cos {S - P) ^^^^^= sinasinp (cT) Dividing sin^Ja by cos^i-a, we have -cosScos(S -g) cos(«-P)cos(;S-y)' (3) tan»^a Tj .• X oi^ - cos aS cos (*S - p) Permuting, tan«-6 = c^c^S - g) cos (^ - y) ^ tan^i^^ -cos^^cos(^-y) ^ 2 cos(-S'-g;cos(S-p) J (4) ♦ Compare with Art. 99. 150 SPHERICAL TRIGONOMETRY. We may write tan i a = cos {S - a} J -cosS' COS (S — a) cos (aS' — /3) cos (^S— 7)' >'C0S(^-a)C0S(>S-P)C0SC^'-Y)* ^^ .•. tan I a = JB cos (S - a). Permuting, tan 1 6 = U cos (5 - P), (6) tan I c = iJ cos (S - 7). Note. —Since the sum of the angles 2 S must be between 180° and 540°, S must be between 90° and 270°, so that COSTS' is always negative and hence — cos S is always positive. Note. — The center of the circumscribed circle of a spherical triangle is the point of intersection of the arcs of great circles perpendicular to the sides of the triangle at their middle points. .-. AN = NB; BL=LC; CM = MA. .-. 0AM = OCM; OANz= OBN; OCL = OBL. .'. 0AM + OAN-\- OCL= S, .'. OCL = S-{OAM-\-OAN) = S-a. In the right triangle OCL^ by Napier*s rules, cos OCL = tan LCcot OC. -.^ tanLC tania .-. tan 0C = 7vFrF = ttt — r cos OCL cos(/S^— a) S-a)\i COS /S cos (^ — a) cos (S — a) \cos («S'— /3)cos (S —7) 4 - cosS 'cos (6'-a)cos (S-p) C9S (0 cot ^ y' ~ — cos ^ (a' + ^')' Changing the signs and dropping the primes, tan I (tt + p) cos l(a-b) — ^-1 — = —i (4) cot ^7 COS ^ (a + 6) Eqs. (1), (2), (3), and (4) are called Napier's Analogies. 145. Gauss's Equations. — From (2) and (3), Art. 142, we have . - - -, sin (s — 6)^ /sin s sin (s — (?) sin (s — h') , sin i a cos i )8 = ^^ ^\ ■. ^. , ^ = ^ ^ cos i 7 ; ^ ^ sine ^ sin asm 6 sm^ ^ - .10 sin Cs — a')^ /sin s sin (s — - sin (»-<;)] _ sin 1 7 cos (g — ^ g) sin j- c "" sin ^ (? cos J g __ sin J 7 cos ^ (« + ft) ~~ cos ^ g .•. cos I c COS ^ (a + p) = sin |-y cos ^ (a + 6). (2) (3) sin K« - ^) = ^^ [sin (s - b) - sin (« - a)] _ cos 1 7 cos [8 — j^ (g + ^)] sin j (a — h') "~ sin 1^ c cos ^ g _ cos ^ 7 sin i(ct — h) ~" sin J (? .•. sm^csiii|(a-p) = cos|7siii|(a-6). (3) (4) cos^ (a — yS) = ^ . ^ ^ [sin s + sin (s — S'-^)= 9.54249 log tan i& =9.87569 S-y= 46°44'.0 log COS C^*- 7) =9-83594 J & = 36° 54'.6 3 .5= 305° 49'. 2 a check. col i?2 = 9.33359 logi22 = 0.66641 log i2 = 0.33320 log tan ^ c = 0. 16914 Jc =55°63'.l OBLIQUE SPHERICAL TRIANGLES. 157 In finding log tan J a, write log B on the margin of a slip of paper, place it above log cos (/i? — a), and write the sum opposite log tan J a; then find log tan J b and log tan J c in a similar manner. 2. a = 110° 36'. 4, /3 = 122° 8'. 7, y = 140° 20'.3. .-. Ja = 4r66'.3; J6 = 57°57'.5; Jc = 68°39'.4. 3. a = 120°60'.6, /3 = 78° 6'. 1, 7= 81° 12'.3. .-. Ja = 59°65'.2; i6=40°40M; Jc = 43°23'.4. 4. 0= 80° 20'. 2, /3= 73° 46'. 7, 7= 54° 8'. 5. .-. Ja = 32°23'.6; J6 = 30°63'.7; Jc = 24°l'.7. 6. a = 100°61'.3, /3= 80°47'.6, 7= 74°3'.3. .-. ia = 49°22'.4; J&=41°42'.5; ic = 37°41'.6. 150. Case III. Given Two Sides and the Included Angle (6, c, a). — By permuting (6) and (7), Art. 147, we have tanK^-7)=cotJ„?i^i|^, (1) tanK^ + 7)=cotJ-«5^jl|^. (2) Then /3 = K^ + 7)+K/8-7). Note that the larger angle must be opposite the larger side. To obtain a, we permute (4) and (5), Art. 147 : tan^a^tanK^-0;i:fg!g. (3) tan.a = tanK* + 0^^|ig±g. (4) The agreement of the values oi ^a found from (3) and (4) is a check upon the computation. The sine proportion may also be used as a check. Note. — In using these formulas, the larger side and the larger angle should be written first in the expressions b — c and /3 — 7. Thus for c > 6, (1) would be written * 1 ^ ON * 1 sin i (c - 6) tan i (7 - /3) = cot i a -^-i — — ix* ^^' ^^ ^ sm ^ (c + &) Eq. (1) may be read : " The tangent of half the difference of the required angles is equal to the cotangent of half the given angle, multiplied by the sine of half the difference of the given sides, ancf divided by the sine of half their sum." 158 SPPIERICAL TRIGONOMETRY. 1. Solve the triangle when h = 105" 14'.8, c = 43° 17'.2, a = 112° 6=105° 14'. 8 logcot^a = 9.82251 log cot J a = c= 43°17'.2 log sin i(6-c) =9.71159 col sin J (6 + c) =0.01658 J(6 + c)= 74°1C'.0 i(6-c)= 30°58'.8 Ja= 56°23'.7 log tan i (i3-7) =9.55068 K^-7)=19°33'.8 logtan|(&-c) =9.77843 log sin I (/3 + 7) =9.95566* col sin ^(P-y) =0.47514* log tan I a =0.20923 |a=58°17'.8 log tan 1(6+ c)= 0.55019 log cos ^ (^ + 7) =9.63322* col cos I (/3-7) =0.02582* logtan|a=0.20923 |a=58°17'.8 log cos J^ (6 — C): col cos ^ (6 + C): logtani(/3+7): H/3 + 7): K/3-7): 47'.4. : 9. 82251 9.93316 :0.56677 : 0.32244 -64° 32' .9 19° 33' .8 84° 6' .7 :44° 69' .1 2. a = 103° 44'. 7, 6 = 64° 12'.3, y = 98° 33'. 8. .-. K« + /S) = 82° 37'.0 ; K« - /S) = 16° 19'.0 ; a = 98° 56'.0 ; /3 = 66°18'.0; ^c = 51°45'.3. S. a = 156°12'.2, 6 = 112°48'.6, 7 = 76°32'.4. .-. K« + ^)=120°45'.6; K« - /3)= 33° 18'.5 ; a = 154°4M; )8 = 87°27'.l; Ac = 31° 54'. 4. 151. Case III. Second Method. Given 6, c, a, to find One Element only. (1) To find a only. Let cos a = cos b cos c + sin ^ sin c cos a. w sin M= sin c cos a, ' m cos M= cose. cos a = m (cos 5 cos ilff -f sin 5 sin JkT), cos a — m cos (5 — i^f ). (1) (2) (2) To find one angle only, /8 or 7. — From (6), Art. 124, sin a cot 7 = cot ^ sin 6 — cos b cos a. cot c sin b — cos ^ cos a cos .^ ^ sin = sin yS (1) sin a Find c by (4) and (5), Art. 147, tan J c = tan K« - ^) "^^ f ^^^ + ^> (2) tan J ^ = tan J (« + *) ^^'f^^ + ^> (3) 2 2 V ^^ COS ^ (« — yS) "" ^ Find 7 by (6) and (7), Art. 147, cot 1 7 = tan 1 (a - /3) ^^^f^^ + ^> (4) 2' 2v ^^sinj(a — 6) ^^ ^1 ^ 1^ , ON cos A (a +^) ,rx coti7-tanK« + ^) ^^3|^^_^j - (5) The agreement of the values of J we find D.4C= 180°- «= 103° 11'.6, ^CZ) = iV= - 17° 57'.5, giving iY the negative sign since it is exterior to the triangle. Hence a = 76° 48'.4 ; 7 = 3/-I- iV= 47° 19'.5. 2. a = 76°40'.4, 5 = 54°21'.3, c = 36°8'.7. .-. J(»w-w)=53°0'.38: w = 71°4'.73: n = - 34° 66'.03 ; /3 = 46°17'.3; Jlf=76°27'.0; iV = - 44° 48'.2 : 7 = 31°38'.8; a = 120°3'.6. 170 SPHERICAL TRIGONOMETRY. 3. a = 124° 34'.0, h = 6Q° 7'.2, c = 109° 43'.5. .-. J(m-w) = -76°37'.32; m = - 21°45'.57 ; n = + 131°29'.07 ; /3 = 74° 1'.7 ; ilf = - 26° 45'.6 ; a = 120° 2'.7 ; iV = + 124° 69'.2 ; 7 = 98° 13'.6. 4. a = 30°17'.6, 6=22°14'.4, c = 18°51'.8. .-. ?>i = 21° 14'.6 ; n = ~2°22'.8; /3 = 48°17M; J/=45°54'.8; a = 95° 50'.0 ; iV^ = - 6° 17'.9 ; 7 = 39° 36'.9. 5. a = 130°46'.0, b = 113°21'.4, c = 102° 16'.2. . •. i (m - «) = - 11° 8'.6 ; m = 39° 59'.5 ; n = 62° 16'.7 ; /3 = 136° 19'.25 ; Jf = 58° 3'.4 ; a = 145° 15'.9 ; i\r = 74° 37'.75 ; 7 = 132° 41'.2. 161. Case II. Given a, p, y, — ^Pply the method of Case I to the polar triangle, and thence find the elements of the original triangle. 1. a = 116° 19'.4, /3 = 83° 19'.2, 7 = 106° 10'.6. In the polar triangle, a' = 63° 40'.6, b' = 96° 40'.8, c' = 73° 49'.4. .-. J (m' - n') = - 66° 18M, m' =- 29° 23'.4, n' = + 103° 12'.8. The negative value of m' shows that B'C'D' is exterior to the triangle. From B'C'D' we find D'B'C = 180° - /3' = 73° 49'.2, D'C'B' = 3I'=- 33° 11'.8, giving M' the negative sign since it is exterior to the triangle. From A' CD' we find D'A'C = a' = 60° 4'.7, N' = + 101° 25'. 5. . •. p' = 106° lO'.S, 7' = M' + N'= 68° 13'.7. Passing from the polar to the original triangle, a = 119° 55'.3 ; 6 = 73° 49'.2 ; c = 111° 46'.3. 2. a = 110° 36'.4, /3 = 122° 8'.7, 7 = 140° 20'.3. . •. J (m' - n') = 29° 27'.90 ; m' = 49° 17'. 75 ; w' = - 9° 38'.05 ; /3' = 64° 4'.9 ; M' = 54° 5'.4 ; a' = 96° 7'.4 ; iY' = - 11° 24'.0 ; 7' = 42° 41'.4 ; .-. a = 83° 52'.6, b = 115° 55M, c = 137° 18'.6. 3. o = 120° 50'.6, /3 = 78° 6'.1, 7 = 81° 12'.3. . •. i (m' - w') = - 63° 33'.19 ; m' = - 14° 9'.34 ; n' = 112° 67'.04 ; /3' = 98° 39'. 7 ; M'=- 1G° 33'.0 ; a> = 60° 9'.6 ; N' = 109° 46'.0 ; 7' = 93° IS'.O ; .: a = 119° 50'.4, b = 81° 20'.3, c = 86° 47'.0. 4. a = 80° 20'.2, /3 = 73° 46'.7, 7 = 54° 8'.5. .-. i (m' - n') = 7° 15'.69 ; m' = 70° 11'.44 ; n' = 55° 40'.06 ; /3' = 118° 12'.7 ; M' = 72° 37'.5 ; a' = 115° 12'.8 ; N' =59°19'.l; 7' = 131° 56'.6 ; .'. a= 64° 47'.2, b = 61° 47'.3, c = 48° 3'.4. OBLIQUE SPHERICAL TRIANGLES. 171 6. a = 100° 61'.3, /3 = 80° 47'.C, 7 = 74° 3'.3. . •. J (m' - n') = - 83° 60'.76 ; m' = - 30° 62'.41 ; n' = 130° 49'.11 ; /3' = 96° 35'.0 ; 3P = - 31° 30'.0 ; a' = 81° 16M ; N' = 136° 6'.8 J 7' = 104° 36'.8 ; .: a = 98° 44'.9, b = 83° 25'.0, c = 76° 23'.2. 162. Case III. Given a, 6, -y. — From the end of one of the sides, as 5, let fall an arc of a great circle perpendicular to the other side. In the triangle J) AC we know b and 7 ; hence we find w, iV, and p by Napier's rules, considering p as of the same species as 7. Then m = a — n, being negative when n>a, showing that the triangle BAD is then exterior to the triangle BA C. Now in the triangle BAD we know DB and AD^ and we find = 162° 39'.9, j? = 120° 48'.86, 31 =:. 160° VJ. Then iV = a - i¥ = - 19° 18^5. . •. & = 119° 22'.5, ACD = 180° - 7 = 99° 45M, n = - 16° 44'.8, giving n the negative sign since it is exterior to the triangle. .-. 7 = 80° 14'.9, a = m-\-n = 145° 55M. 2. a = 104° 30'.7, /3 = 62° 52M, c = 56° 6'.4. . •. M = 42° 34'.8 ; iNr= 61° 55'.9 ; m = 34° 10'.2 ; p = 47° 37'. 5 ; b = 66° 46'.0 ; 7 = 53°30'.4 ; 7i = 54° 10'.7 ; a = 88° 20'.9. 8. a = 140° 24'.6, /3 = 12° 18'.6, c = 28° 7'.7. . •. ilf = 79° 6'.4 ; iNT = 61° 18'.2 ; w = 27° 34'.7 ; p = 5° 46M ; & = 11° 52'.9 ; 7 = 29° 13'.3 ; n = 10° 24'.3 ; a = 37° 59'.0. > 164. Case V. Given a, b, a. — Let fall an arc of a great circle from the intersection of a and J, ^^ perpendicular to c. In this case there ^^^^Vi\ will be two solutions if a is inter- y^ / 1 \a mediate in value between p and both X 7 'f \ b and 180° - h (Art. 120). ''^^O-— - ^/ \d ^ r ^^^ ^^® triangle ACD, knowing b "'^'^ ^ and «, find 771, M, and p by Napier's rules. Then in the triangle DOB^ knowing p and a, find DB^ DCB^ and BBC. Then in the tri- angle A CB we have c = AB = m-\-BB, y = ACB = lM-\- BOB, ^ = BBO; and in the triangle ACB' ^ c'=AB' = m-BB, y' =ACB' = M-BOB, j3' = 180'' -BBC. 1. a = 148° 34'.4, h = 142° 11'.6, a = 153° 17'.6. .-. p = 164° 0'.52, and there are two solutions. ->. = 34° 43'.5, M = 68° 19'.4. Also, DB = 27° 25M, DBC = 148° 6'.3, DCB = 62° 1'.8. .-. c = 62° 8'.6, 7 = 130° 21'.2, ^ = 148° 6'.3, and c' = 7° 18'.4, 7' = 6° 17'.6, j8' = 31° 53'.7. OBLIQUE SPHERICAL TRIANGLES. 173 2. a = 40° 20'.4, b = 20° 18'.2, a = 60° 44'.4. .'. p = 17° 37'.3 ; m = 10° IG'.O ; ^ = 30° 61 '.2 ; /3 = 27° 52'.9 ; DB - 36° 53 .7 ; DCB = 68° 2'.3 ; c = 47° 8'.7 ; 7 = 98° 63'.6. 3. a = 98° IC, & = 74° 38', a = 78° 40'. .: p = 70° 59'.25 ; m = 35° 34'.0 ; iW= 37° 6'.1 ; /3 = 72° 49'.25 ; DB = 116° 11'.4 ; DCB = 114° 66'.4 ; c = 151° 45'.4 ; 7 = 152° 2'.5. 165. Case VI. Given a, p, a. — Pass to the polar triangle, in which we shall know a', Z>^ and a', and solve by the method of Art. 164. There may be two solutions of the polar triangle, and therefore of the triangle itself. 1. a = 143° 17'.4, /3 = 70° 18'.4, a = 160° 40'.6. .-. a' = 36° 42'.6, h> = 109° 41'.6, a' = 19° 19'.4. • •. p' =z 18° 9'. 13, and there will be two solutions. M' = 96° 44'.3, m' = 110° 46'.3. Also D'B' = 32° 28'.25, D'C'B' = 63° 54'.9, D'B'C = 31° 24'.7. .'. ci' = 143° 14'.55, 7i' = 160° 39'.2, /3i' = 31° 24'.7, and ci" = 78° 18'.05, 71" = 32° 49'.4, jSi" = 148° 35'.3. Taking the supplements to obtain the elements of the original triangle, 7 = 36° 45'.45, c = 19° 20'.8, b = 148° 35'.3, and 7' = 101° 41'.95, c' = 147° 10'.6, b' = 31° 24'.7. 2. o = 117° 54'.4, p = 45° 8'.0, a = 76° 37'.5. .: p' = 136° 23'.8 ; 31' = 18° 37'.7 ; m' = 13° 5'.3 ; D'C'B' = 120° 17'.5 ; D'B'C = 128° 42'.1 ; D'B' = 130° 15'.9 ; 7' = 138° 55'.2 ; c' = 143° 21'.2 ; . •. 6 = 51° 17'.9 ; c = 41° 4'.8 ; 7 = 36° 38'.8. 3. a = 104° 40'.0, /3 = 80° 13'.6, a = 126° 50'.4. .-. p' = 52° 3'.8 ; 31' = 102° 46'.0 ; m' = 106° 1'.7 ; D'C'B' = 70° 22'. 9 ; D'B'C = 54° 36'.8 ; D'B' = 65° 40'.7 ; 7i' = 172° 8'.0 ; 71" = 32° 23'.1 ; d' = 171° 42'.4 ; ci" = 40° 21'.0 ; /3i' = 54° 36'.8 ; /3i" = 125° 23'.2. . •• c = 7° 51'.1 ; 7 = 8° 17'.6 ; b = 125° 23'.2 ; and c' = 147° 36'.9 ; 7' = 139° 39'.0 ; h' = 54° 36'. 8. CHAPTER XII. APPLICATIONS OF SPHERICAL TRIGONOMETRY. 166. To find the Shortest Distance between Two Points on the Surface of the Earth,* the earth being treated as a sphere. — North latitudes and west longitudes are considered positive. Let QQ' be the equator, P the north pole, A and B the two points, and PM the meridian from which the longitudes are measured. The longitude of A is MPC and that of B is MPD^ both being positive since they are measured west- ward. The latitudes are CA and DB^ the former being negative since it is measured southward. In the triangle APB the sides AP and BP are found by algebraically subtracting the latitudes from 90°, and the angle APB is the algebraic difference of the longitudes. Hence we know two sides and their included angle, so that we can solve the triangle, using the method of Art. 151 when the distance only is required, and that of Art. 150 when we wish to find all the elements. . 1. Find the shortest distance between New York, 40° 45'.4 N., 73° 58'. 4 W., and Rio Janeiro, 22° 54'.4 S., 43° 10'.4 W. .-. I?P=49° 14'.6, ^P=112° 54'.4, ^PS=30° 48'.0. Ans. .45=69° 48'. 2. 2. Find the shortest distance between New York, 40° 45'. 4 N., 73° 58'.4 W., and Paris, 48° 50'.2 N., 2°.20'.2 E. Ans. AB = 52° 2C'.8. * The shortest distance between two points on a sphere is the arc of the great circle passing through the points. 174 Fig. 126. APPLICATIONS OF SPHERICAL TRIGONOMETRY. 175 If the bearings of the great circle AB at A and B are required, it -will be necessary to find the angles PAB and PBA. 3. A ship sailed from Calcutta, 22° 34'.8 N., 88°27'.3E., on an arc of a great circle to Melbourne, 37° 48'.0 S., 144° 58'.0 E. Find the distance sailed and the bearings * at both points. Ans. At Calcutta, S. 41° 56'. 61 E. ; at Melbourne, S. 61° 21 '.47 E. ; dis- tance, 80°22'.4or80°22'.6. 4. A ship sailed from the Cape of Good Hope, 34° 22' S., 18° 29' E., on an arc of a great circle to Cape St. Roque, 5° 28' S., 36° 16' W. Find the distance sailed and the bearings * at both points. Ans. At G. H., N. 72° 28'.0 W. ; at S. R., N. 52° 16'.0 W. ; distance, 57° 20'.4. 6. A ship sailed from Bombay, I&° 56' N., 72° 53' E., on an arc of a great circle to the Cape of Good Hope, 34° 22' S., 18° 29' E. Find the distance sailed and the bearings * at both points. Ans. At Bombay, S. 44° 12'.8 W. ; at G. H., S. 53° 2'.6 W. ; distance, 74° 15'.2 or 74° 15'.4. 6. A ship sailed from Bombay, 18° 66' N., 72° 53' E., on an arc of a great circle for the Cape of Good Hope, 34° 22' S., 18° 29' E. Find the distance to the equator and the bearing* and longitude at the equator. [Use the triangle BDE; the angle PBA - 135° 47'.2 was found in Ex. 6.] Ans. S. 41° 16'. 1 W. ; distance, 25° 34'.5 ; longitude, 65° 21'.8 E. 7. From a point whose latitude is 17° N. and longitude 130° W. a ship sailed an arc of a great circle over a distance of 4150 miles, starting S. 64° 20' W. Find its latitude and longitude if the length of 1° is 69^ miles. Ans. Lat., 19° 40'.62 or 19° 40'.60 S. ; Long., 178° 20'.9 W. 167. Given the Lengths of the Three Edges of a Parallelo- piped that meet in a Point, and the Angles between them, to find the Surface and the Volume of the Parallelopiped. — Let OGr he the solid, AD the perpendicular from A to BOO, and hence AOD a plane per- pendicular to BOO. Let the angles and edges be BOC=:a, AOC=h, AOB = c, OA = l, 0B = m, 00 = n. Describe a sphere with a radius of unity about as a center, its intersections with the planes forming the figure marked by the primed letters. Fig. 127. The course of the ship. 176 SPHERICAL TRIGONOMETRY. Then the surface is ^=2 OBEC^- 2 OAFC+ 2 OBHA = 2 {mn sill a + In sin b + Im sin c). (1) In the triangle A'B'B', right-angled at D', we have sin B'A' = sin B'A' sin A'B'I)' ; . •. sin D'A' = sin 6> sin A'B'L'. But in the triangle A' B' C we know the three sides a, b^ c ; hence sin A'B'B' = 2 sin l A'B'B' cos ^ J.'^'i>' 2 sin a sin A= OA sin BOA 21 Vsin s sin (s — «) sin (s — b) sin (s — (7O = 90°; BEO==90\ Let a = length of an edge AB^ s = number of sides of each polygonal face, n = number of faces meeting at a vertex of the polyhedron, APPLICATIONS OF SPHERICAL TRIGONOMETRY. 177 iV^= number of faces of the polyhedron, E = edge angle CDE of the polyhedron. i Then CD = AD Got ACD = \a cot ^-^^' CO = CD tan CDO = CD tan -^ E, .'. CO = I acot^^ tan IE, (1) About as a center, with a unit radius, describe a sphere, and let its intersections with the three planes form the triangle A' CD'. Then A'C'D' = ACD = ^^; A'i>'(7' = 90°; C'^'D' = lM2!. s n By Napier's rules, cos C'A'D' = cos CD' sin A^ CD', 180° ^,j.f . 180° or cos = cos C^'-D' sin n s But cos CD' = cos COD = cos (90° - CDO} = sin CD0= sin J J57. 180° . 1 rr . 180° . •. cos = sm J E sin • 1 XT 180° 180° ,oN . •. sm -^ E — cos cosec (2) 7h 8 Then, if A is the area of a face, the volume is V=lCOxAxN= ^\Ma^ cot2 1?^ tan ^ E. (3) Find -1 -^from (2) and then Ffrom (3). 1. Dodecahedron, formed by 12 regular pentagons, 3 meeting at a vertex. . •. s = 5, n = 3, iV = 12. log cos 60° = 9.G9897 lo^ ^ = 39-Q4 log cosec 36° = 0.23078 ^24 " '' log sin I E = 9.92975 ^'^S cof^ 36° = 0.27748 log tan ^E=z 0.20896 0.88438 .-. F= 7.663 a*. 2. Tetrahedron, formed by 4 equilateral triangles, 3 meeting at a vertex. ■.-. s = 3, 71 = 3, iY=4. Ans. F= 0.1179 a^. 3. Cube, formed by 6 squares, 3 meeting at a vertex. .-. s = 4. n = 3, iV = 6. Ans. V=a\ CROCK. TRIG. — 12 178 SPHERICAL TRIGONOMETRY. 4. Octahedron, formed by 8 equilateral triangles, 4 meeting at a vertex .-. s = 3, « = 4, N=S. Ans. V=0A7Ua\ 6. Icosaliedron, formed by 20 equilateral triangles, 5 meeting at a vertex. .-. s = 3, n = 5, iV=20. Ans. F= 2.182 a^. 169. If from Any Point in a Trirectangular Triangle Arcs of Great Circles are drawn to the Vertices, COS'^ a + COS^ p + COS^ 7 = 1» where «, /3, and 7 are the arcs. — In Fig. 129, produce YP and ZP to D and U. In the right triangle PDX, sin PD = sin a sin PXD ; .-. cos /3 = sin ct sin PXi>. (1) In the right triangle PJSX, sinPU = sinasinPXU ; .-. cos 7 = sin a cos PXD. (2) Squaring (1) and (2), and adding, we have cos^ yS + cos^ 7 = sin^ «. . •. cos^ a + cos^ /3 + cos^ 7 = 1. Q.E.i). Fig. 130. 170. If from Any Two Points P and I*' in a Trirectangular Triangle Arcs of Great Circles are drawn to the Three Vertices, and if v is the Length of the Arc rr', prove that cos V = cos a cos a' + COS p cos p' + cos 7 cos y'. In the triangle PYP^ (Fig. 130), cos V = cos yS cos /3^ + sin yQ sin yS' cos PYP'. (1) But cos PZP^ = cos CZYPf - ZYP^. .-. cos P rP' = cos ZrP^ cos ZrP + sin ZZP' sin ZrP. (2) APPLICATIONS OF SPHERICAL TRIGONOMETRY. 179 (3) In ZYP, cos 7 = sin y3 cos ZYF.* In ZYF^, cosy = sinks' cos ZYF'.* In XYF, cos « = sin /9 cos XYF * = sin /3 sin ZYF. In XrP', cos a' = sin /S' cos XYF' * = sin /3' sin ZYF'. Substituting in (1) the values found from (2) and (3), cos V = cos /3 cos fi' + cos 7 cos y' -\- cos a cos a', q.e.d. This is the formula for the cosine of the angle between two lines in space, the angles made by them with tliree lines at right angles to each other being «, yS, 7, and a'^ 13', 7', respectively. 171. To find the Angle a' between the Chords of Two Sides of a Spherical Triangle, having given the Two Sides b and c, and the Angle a between them. — Let AB = c, AC = b^ the spherical angle BA C = «, and the plane angle BAC = «', being the center of the sphere. About ^ as a center de- scribe a sphere, and let its intersec- tions with the planes GAB, OAC, and BAG form the triangle BUF. Then I)F= GAB =90° FDE = « ; FF= BAC=a'. ^^^ ,31 . •. cosFF= cos BF cos BF + sin BF sin BF cos FBF. .-. cos «' = sin -|- 6 sin I" c? -f- cos I 5 cos -|- e cos «. (1) This formula is true for all values of 5, c, and «. When b and e are small, the correction that must be applied to a to obtain a' may be found from (1) as follows : Let p = b -\- c^ and q = b — c. Then, from Art. 72, cos a' = l cos J q — ^ cos -|-|) + -|- (cos \p + cos J q) cos a = — sin2 \q -{- sin2 |-p + (1 — sin^ -^ j; — sin^ ^ ^) cos a = (sin^ I j;> — sin^ -^ (^) (sin^ -i- « -f cos^ 1 a) -|- cos a — (Qi\i?\p + sin2 ^ g) (cos2 -|- « — sin^ | «) . . ♦. cos a' = cos a — 2 sin^ J e^' cos^ 1 a 4- 2 sin^ ^jo sin^ |- a. (2) *Eq. (2), Art. 12L l" 180 SPHERICAL TRIGONOMETRY. Let «' = «-{- 6^ where 6 is so small that we may place sin 6 = 6^ and cos 6 = 1. . '.' cos a' = cos a cos — sin a sin 0. .'. cos a' = cos a — sin a. (3) Comparing (2) and (3), 2 6 sin 1^ a cos |- « = 2 sin^ ^ ^ cos^ i « — 2 sin^ ^ jt? sin^ |^ «. . •. ^ = sin2 ^ ^ cot -|- a — sin^ ^p tan -J a. ' '' ^" ^ ^hTl^ ^'''' * ^ ''''^ 2 « - ^i^,- sinH^ tan ^ a, (4) since 6? = 6"' sin 1'^ (Art. 81). 172. The Angles of Elevation of Two Points, in the Direc- tions OA and OB, above a Horizontal Plane, and the Inclined Angle AOB, were measured with a Sextant. Find the Hori- zontal Angle between the Points, as seen from O. — Let OZ be the vertical line, Oah the horizontal plane; aOA = h^ and hOB=k the measured altitudes; and AOB=:c the inclined angle. Describe a sphere about as a center. Then in the triangle AZB, AZ = 90° -h, BZ= 90° - 7c, AB = c, and hence the required angle aOb = AZB may be computed, since we know the three sides of the triangle. When h and k are small, the correction to be applied to the measured value c to obtain a Ob may be found as follows :* From (2), Art. 121, Fig. 132. cos^Z^= c — sin h sin k cose hJc cos h cos k cos c — hk (Art. 78) 1 - K^' + ^') . cos AZB= cos c + 1(^2 4- F) cos c (l-J-A2)(l_l/,2) = (cos c - hk) [1 + i Qi^ + A;2)] . A^. (1) * Neglecting powers of h and k above the second. APPLICATIONS OF SPHERICAL TRIGONOMETRY. 181 Let 6 be the correction to c so that AZB = c + ^. . *. cos AZB = cos c cos 6 — sin c sin 6. . •. cos AZB = cos c — ^ sin c. (2) Comparing (1) and (2), (/,2 ^ ^2) (cos^ -|- ff - sin2 lc)-2hk (cos^ ^ g + sin^ i g) ~~ 4 sin ^ g cos ^ g .-. ^= i(^ + A:)2tanJg-|(A-/c)2cotl(7, (3) where ^, A, and A: are expressed in circular measure. To find e in seconds, let 6 = 6" sin 1", ^ = h" sin l'^ /c = k" sin 1^'. ... B'f ^\Qi'' +y'y&mV' t2in^c-\(h'' -h''y^mV' Qot I c. (4) SPHERICAL EXCESS. 173. Area of a Spherical Triangle. — From geometry we know that the areas of any two triangles are to each other as their spherical excesses, the spherical excess being the amount by which- the sum of the three angles exceeds 180°. We also knoAV that the area of the trirectangular triangle is -|- Trr^, and that its spherical excess is 90°. If A is the area of any triangle, and E its spherical excess expressed in degrees, we have Ai\'nr'^=Ei^{i\ (1) .-. A = E^^, (2) 180° ^ r and E = a'^' ' (3) 174. Lhuillier's Theorem. — We have ^V tan 1 xT_s^^i (« + ^ + 7-'^) 2cos^(ot + ^ + 7r-7) ^ - ^(jg 1 (^^ + ^ + ^ _ ^) 2 cos|(a + /S 4- TT - 7) _ sin -|- (« + /3) — sin j- (tt — 7) "" cos I (a + yS) + cos ^ (tt ~ 7)' from (6) and (7), Art. 72. ^ cos i (« + y8) 4- sm J 7 182 SPHERICAL TRIGOXOMETRY. Hence, from (1) and (2), Art. 145, substituting for sin^(a-f-/3) and cos ^ (a 4- yS), we have - ^ cos i (a — h)— cos i c cos X y tan i J5;= \^ — —j^ f ^-f-i- * cos ^ (a + 6) + cos J ) sin^ 1 (g - a) ^^ COS^ -1- 8 COS^ J (S — t?) sin J s cos J s sin J (s — c) cos ^ (s — sin ^ (s — a) cos J (s — «) sin |^ (s — ^) cos f) 1. . tan ^ -E'= Vtan |^ s tan |^ (s — a) tan |^ (s — ^) tan ^ (s — c?). Q.E.i. 175. Spherical Excess in Terms of Two Sides and their Included Angle. tan ^ J? =r ^""^ 2 (« + /^ + 7 - tt) ^ - cos-|- (« + ^ + 7) 2 cos 1 (a 4- ^ + 7 — tt) sin |- (a + /3 + 7) _ sin I (a + /3) sin j^ 7 — cos j- (a 4- /5) cos j- 7 "" siji ^ (a + 13} cos-|- 7 + cos ^ (« + fi) sin J 7 Substituting for sin |- (« + /3) and cos J (a + /3) from (1) and (2), Art. 145, ^ ^ _ sin ^ 7 cos I 7[cos J (a — h) — cos |^ (« + ^)] 2 "~ cos|^(rt — 6) cos2^7 4- cos J(a + 6) sin2l7 sin -|- 7 cos -|- 7 f 4- 2 sin J a sin ^ />! ~^[cos^(a — 6)4-cos^(a4-^)J + -2-[cos^(^a— 6) — cos-^(«4-^)]cos7 _ sin I" a sin J 5 sin 7 cos ^ a cos ^b -{- sin ^ a sin J 6 cos 7 ... tanA^=, tan^atan^ismy . ^.^.l. "* 1 4- tan J a tan J 6 cos 7 176. Approximate Value of the Spherical Excess, neglecting Powers above the Second. — Let the sides of the triangle be so APPLICATIONS OF SPHERICAL TRIGONOMETRY. 183 small that the powers of their circular measures higher than the second may be neglected. We have, from Art. 78, tana; = a; -I- Ja;3 4- •••, (1) where x is expressed in circular measure. Let the lengths of the sides be a, 6, and c when expressed in circular measure, and a\ h\ and c' in linear measure, r being the radius of the sphere. Then « = p ^ = p ^=7 (2) Placing these values of a, 6, and c for x in (1), and substituting in Lhuillier's theorem, we have, neglecting powers above the second, for, 1 7? ^V^^ s' — a' &' — V s' — c' ,oN where s' = ^(a' + 5' + c'). (4) .-. tani^ = -!-Vs'(s' - a')(«' - ^-OC^' - O- C^) 4 H Since \Il\s, small, we place its tangent equal to its arc. .-. \E=^ A^«'(«' - ^')(s' - ^')0' - O (6) 4H ^ = 1^, (7) where ^ is the area of the plane triangle whose sides are a', V ^ and c\ E being expressed in circular measure. To find the value of E in seconds of arc, divide both sides by sin 1''. = En = —A_. (8) E sin 1 f f r^ sin 1 " Hence, whenever the third powers of the circular measures of the sides can be neglected, the spherical excess is found by computing the area of the triangle, considering it as a plane triangle, and dividing the area by r^miV . 184 SPHERICAL TRIGONOMETRY. 177. Approximate Value of the Spherical Excess, neglecting Powers above the Fourth. — From Lliuillier's theorem, * L2r 24?-aJL 2r ^ 24r3 J fs' - b' (s' - hy ~\[ s' - c' (js' - gpn L 2r 24^3 JL 2r 24?-3 J where A^ = s'(.s' - a')(^' - ^00' - ^')- .4^ A^ 16 r* 192 r" 4r2V 24?-2 / ••• i^'s^^^l' =4^(,^-^ 247^ ^j- r2sinl"V 24 r2 J ^ This value exceeds that found in Art. 176 by A a^2 ^ yi 4. ^/2 r2sinl''* 24 r2 If a' = b' = c' = 100 miles, and r = 3963.3 miles, we obtain -^^— - = 56^^863; ^!-^tJ!l±^ ^qmOOS; ?-2 Sin 1'^ 24 H so that the correction to the value of U'' given by (8), Art. 176, is only 56'^863 X 0.00008 = 0'^003. 178. Legendre*s Theorem. — If the sides of a spherical tri- angle are veri/ small compared ivith the radius of the sphere, the angles of the plane triangle whose sides are of the same length as APPLICATIONS OF SPHERICAL TRIGONOMETRY. 185 those of the spherical triangle, are equal to the corresponding angles of the spherical triangle diminished hy one third of the spherical excels. — Let a', h\ and c' be the lengths of the sides of the spherical triangle expressed in linear measure, and a, 6, and c the lengths in circular measure. a' J h' c' .^^ r r r Let a be an angle of the spherical triangle and a' the corre- sponding angle of the plane triangle. We have cos a — cos b cos c cos a = (2) cosa = -V[(^2 + c2_^2) 2 be sin b sin c From Art. 78, cos a = 1 — I a2 -f 2^ a* — ••• sinb = b — -J- b^ + ... cos6 = 1 - -J62 + ^\b^ smc = c - i6'3 + ... cos (? = 1 — -J c2 + 2f . In the triangle SPS\ find P^S^^S' and PSS'. Then in the triangle S'PZ we know S^P, ZP, and PS'Z, and Ave find PZS' and ZPS'. In the triangle SPZ we know SP, ZP, and PSZ = 180° - PSS', and Ave find PZS and ZPS, The checks are PZS' = PZS, and S' PZ-SPZ=:a-«! . (2) S^P = 90° - S, S^P = 90° - a', S^PS^ = ar-a'; find PS^S^' and PS^'S-^^, these angles being the same as those at S and S' in the first case. Then from the tAvo triangles PS^Z and PS^'Z Ave find the angles PZS^ and PZS^', Avhich should bs identical, and also the angles S^PZ and S-^'PZ, Avhose dif- ference should be a — a'. "-•^^l Ir^ LOGARITHMItr AND iRIGONOiMETRIC TABLES FIVE DECIMAL PLACES EDITED BY C. W. CROCKETT PROFESSOR OF MATHEMATICS AND ASTRONOMY RENSSELAER POLYTECHNIC INSTITUTE J'i9^<^ NEW YORK •:• CINCINNATI .:• CHICAGO AMERICAN BOOK COM^^amv :92 CONTENTS. PAGH 3 Table I. Logarithms of Numbers S\T',S'\T",ioT2r-s' ...... 24 II. Logarithms of Trigonometric Functions . . 25 III. Natural Trigonometric Functions .... 71 IV. Lengths of Circular Arcs 9- V. Conversion of Logarithms Formulas q- Constants j^- Explanation of the Tables 10- NoTE. — The well-known tables of Gauss, Becker, and /■il^.Tr^t have been taken as the standards, the proof sheets have been read with gTCat care, and it is believed that the numl)er of errors cannot be iar°^e. The arrangement of the figures on the page is in accordance v'ith that adopted in the standard six and seven place tables. 'I'he natural tables were reduced from seven-place tables and compared with published five-place tables. For convenience in using the tables, the explanation ' ten placed after them instead of before them. \KIGHT, 1896, BY AMERIC-VN 3oOK ■ 3 \ f'^ >\ ( . UNIVERSITY ] I. COMMON LOGARITHMS OF NUMBERS ' • ' FROM I TO I 1000. N. Log. N. Log. N. Log. N. Log. N. Log. 2 3 * — 20 21 22 23 1.30 103 1.32 222 1.34242 1-36173 40 41 42 43 1.60206 60 61 62 63 1-77815 80 1.90309 O.OO CXXD 0.30 103 0.47712 1.61 278 1.62 325 1-63347 1-78533 1.79 239 1-79 934 81 82 1.90849 1.91 381 1.91 908 4-, 0.77815 24 25 26 1. 38 021 1-39 794 1.41 497 44 45 46 .1-64345 1.65321 1.66276 64 65 66 1.80 618 1.81 291 1.81 954 !4 8^^ 1.92428 1.92942 1.93450 7 8 9 10 i t 0.84510 0.90309 0,95 424 1. 00 000 r.04 139 1.07 918 1.11 394 27 28 29 30 31 11 1-43 136 1.44716 1.46 240 49 50 51 52 53 1.67 210 1.68 124 1.69020 67 68 69 70 71 72 73 1.82607 1-83251 1.83885 1.84 510 89 90 91 92 93 1-93952 ,1-54448 1-94539 M77I2 1.49- 1.50 i . 1.51851 1.69897 1.95 424 1.70757 1. 7 1 600 1.72428 1.85 126 1.85 733 1.86332 1.95904 1.96379 1.96848 H IS i6 1.14613 1.17609 1.20412 34 11 1-53 148 1.54407 1-55630 54 11 1-73239 1.74036 1.74 819 74 75 76 1.86923 1.87506 1.88081 94 96 1-97313 1.97772 1.98227 17 i8 1 20 1.23045 1-25 527 1.27875 1.30 103 37 38 39 40 1.56820 »-57 978 1.59 106 57 58 59 60 1-75 587 1-76343 1.77085 77 78 79 80 1.88649 1.89 209 1.89763 1.90309 97 98 99 100 1.98677 1.99 123 1.99564 1.60 206 1.77 815 2.00000 o I S'. ' 6.46 371 373 T. 37:^ 37 ^ 0° 0' - . I - i — 1 0" 60 20 S". T". 4-68 557 557 557 557 .^57 557 S'. T'.J N. L. 1 2 3 4 5 6 7 8 ' 9 P. P. ^ 6 "'■' f .. ,^^_ 366' 3.= iDO 00000 043 087 130 ^73 217 260 303 346 389 44 43 42 366 38s j 'Ol 432 475 578 561 604 /647 ♦072 689 732 775 817 366 38, i02 860 903 945 988 ♦030 *ii5|*i57 *i99 *242 J ^ . . ^ ,. „ ^•4 4- j 4--^ ' 8.8 8.6 8.4 13.2 12.9 12.6 17.6 17.216.8 366 • 366 3&'' 1 03 01284 703 326 745 3^8 787 410 828 452 870 494 912 536 ^11 578 995 620 ♦036 662 *078 2 3 4 366 X^i- \ 05 02 119 I bo 202 243 284 325. 366 407 449 490 5 6 22.0 21.5 2 f.o 26.425.825.2 366 3«n io6 531 572 612 653 694 735 776 816 857 898 366 3S7 ',07 938 979 ♦019 *o6o *IOO *i4i *i8i ♦222 *262 *302 7 30.8 30.1 2C.4 365 387 108 03342 383 423 463 503 543 583, 623 663 703 8 35.2 34.4 33.6 365 '365 I365 387 ^7 3S8 T09 III 743 J^2^ 822 862 902 941 9^1 *02I *o6o *IOO 9 39-6 38.7 37-8 04i39_ J79_ 218 258 "297^ 336 ~37^ 415 454 493 41 40 39 532 571 610 650 689 727 766 805 844 883 365 388 112 922 961 999 *038 *o77 *ii5 *^54 *192 ♦231 *269 ^ 4.1 4.0 3.0 8.2 8x3 7.:,, 365 38S "3 05 308 346 385 423 461 500 538 576 614 652 2 ■' 36S 389 114 690 729 767 805 843 881 918 956 994 ♦032 3 4 12.3 12.0 11.'^ 16.4 16.0 15.61 20.5 20.0 19.5 24.6 ? 1 n 23,4 1 365 389 115 06070 108 145 185 221 258 296 333 371 408 364 389 116 446 483 521 558 595 633 670 707 744 781 5 6 364 389 ri7 819 856 893 930 967 *oo4 ♦041 *o78 *ii5 *i5i 7 28.7 28.0 27.3 364 390 118 07 188 225 262 298 335 372 408 445 482 518 8 32.832.031.2 364 - 364 390 'I9 120 555_ 918 59t 954 628 6b4 700 737 *099 773 *i35 809 846 882 9 36.9 36.0 3^.1 990 ♦027 *o63 *i7i *207 *243 38 37 36 364 391 121 08 279 3H 350 386 422 458 493 529 565 600 363 39' J 22 636 672 707 743 778 8.4 849 884 920 955 I 3.» 3-7 3-^\ 363 39 ■ 123 991 *026 *o6i ♦096 *I32 *i67 ♦202 *237 *272 *307 2 7.6 7.4 7-2' 363 39 r 124 09342 377 412 447 482 517 552 587 621 656 3 11.4 11. 1 10.^' 15.2 14.8 14./ 19X) 18.5 18.0 22.8 22.2 2I.( 3<^3 392 125 691 726 760 795 830 864 899 934 368 ♦003 4 5 6 ,363 352 126 10037 072 106 140 ns 209 243 278 312 346 ,36, 392 127 380 415 449 483 517 551 585 619 653 687 7 26.6 25.9 25.2 I363 393 128 721 755 789 823 857 890 924 958 992 ♦025 8 ^0.4 29.6 28.8 362 362 393 393 394 129 I30 131, II 059 093 126 160 193 227 261 294 327 361 9 34-2 33-3 32.4' 394 727 428 461 494 528 561 594 628 661 694 35 34 33 760 793 826 860 893 926 9t;Q 992 ♦024 \lt.. 394 132 120^7 090 123 156 189 222 254 287 320 352 I 3-5 3-4 3-3 362 3;>4 ^^2, 385. 4j8 450 483 516 548 581 613 646 678 2 7.0 6.8 6.6 362 - 395 134 710 743 775 808 840 872 905 937 969 *OOI 3 10.5 10.2 9,9 14.0 13.6 13.2 17.5 17.0 16.5 21.0 204 19.8 24.5 23.8 23.1 28.0 27.2 26.4 ^i c -in 6 00-7 361 393 J 35 13033 066 oq8 130 162 194 226 258 290 322 4 361 361 395 396 136 U7 * 354 672 386 704 418 735 450 767 481 799 513 830 545 862 577 893 609 925 640 956 5 6 361 396 138 988 *oi9 *oqi ♦082 *ii4 *I45 *i76 *2o8 *239 ♦270 361 396 139 14301 333 364 ^1 426 457 489 520 551 582 Q 361 360 397. 397 140 141 613 922 644 953 675 983" 706 "737" *045 768 *076 799 *io6 829 *'37 860 ] 891 32 31 30 *oi4 *i68 *i98 i;6o 397 142 11; 229 259 290 320 351 381 412 442 473 503 I 3-2 3-t 3-0 \yo 393 143 ■^534 564 594 625 655 685 715 746 776 806 2 6.4 6.2 6.0 J360 398 144 836 866 897 927 957 987 *oi7 *047 *077 *io7 3 9.6 9.3 9.0 I360 398 ^45 16137 167 197 227 -2-56- 286 316 346 376 406 4 12.8 12.4 12.0 : 360- 399 146 435 465 49? 524 554 584 613 643 673 702 5 '^•° "il Tn 359 399 147 732 761 791 820 850 879 909 938 967 997 6 19.2 18.6 18.0 359 399 148 17026 056 085 114 143 ^73 202 231 260 289 7 8 22.4 21.7*21.0 25.6 24,8 24.0 359 359 400 400 149 ISO ' 3- 348 609 538 377 406 435 464 493 522 81 r 551 580 667 696 725 754 782 840 869 9 ^^'"^ -^i'y ^/-^ N. L. 1 1 2 3 4 5 6 7 8 9 P.P. S.' T.' II S." T." S." T." I' 646 373 373 0° i'= 60" 4.68557 557 0° 19'= 1 140" 4-68557 558,. 2 373 373 2 = 120 557 557 20=1200 557 558 21 =1260 5f7 50 22=1320 557 5ju ' '■' "^' :■ / J 3 = 180 ^K-j 557 i73 16 = 960 557 558 : " N>73 17 =IC20 557 558 c 23=1380 ■ • 557 558 ^3 t::\ 11 18 "=:I0S0 S"^/ 558 :r! =1440 557 558 11 , ,r. — f 1 »r» c^7 <;•,? i 5 =^ 500 557 558 1 ^ .(_ _,_ b . T'.' N~ L. 1. 2 1 3 1 4 5 1 6 7 8 9 p. p. 1 1 ■■'■ 400 40t 150 15' I7OO9 898 9:^13 667 OoOi 725 754 ♦041 *070 811 *«S99~ 840 *I27 869 *I56 29 9.^ i| 2.9 2! 5.8 3i 8.7 4!ii.6 1 i 955 984 1*013 401 IS2 18184 2K3 241 j/O , 298 327 355 384 412 441 401 ^53 469 498 526 55-J 1 583 611 <539 667 696 724 402 154 752 780 808 1 8:57 ! S65 893 921 , 949 977 *cx)5 358 402 155 19033 061 089 ii7 H5 173 j 201 229 257 28S 5 M-5 '4^' 617.4 r.,X il 1358 402 156 312 340 368 390 1 424 45' 1 479 507 535 562 i 358 403 157 590 ii% 645 673! 700 728 756 783 811 838 720.3 I i 357 403 i';S 86^. 893 921 948 976 *oo3 ♦030 *o5il ♦085 *II2 8 23.2 J 926.1 ^ ,. 27 iH , 357 151 357 404 40. 404 159 160 161 20 140 it>7 194 222 249 ^6 303 330 385 H'2 439 466 493 520 548 575 602 629 656 ' 683 710 '737 7631 790 817 844 '871 898 925 357 405 162 952 978 ♦005 *032 1*059 ♦085 *iU *i39 ♦165 *I92 I ^•7 -•'-■ 1 356 405 163 2Jl, 219 245 272 299 325 352 378 405 431 458 2 11 11 i 356 400 164 "484 5" 537 564 Sm 617 643 6(69 696 722 3 o» /" : 4 lp.8 ,io.4 1 513-5 J 30 , 6 16.2 i;.6 ' 7li3.9 18.2 ! 356 406 167 .748 775 801 ^-^^ 880 906 932 958 985 356 356 406 407 22 0ir 272 037 298 063 324 0S9 350 "5 376 141 401 }^ '94 220 479 246 505 427 453 355 407 168 531 557 il 608 634 660 686 712 737 763 821.6 20.8 i r 255 355 408 '403 408 169 I70 171 789 814 866 891 917 943 968 994 *oi9 9124.3 23.4 ! 25 23?4|_ 070. 096 121 147 172 198 223 249 274 '300 325 350 _ 376 401 426 452 477 502 52^8 \ 409 172 553 578 603 629 654 679 704 729 754 779 I 2. 409 ^n 805 830 851 880 905 930 9^ 980 *oo5 ♦030 2 .5- 410 174 24055 080 103 130 155 180 204 229 254 279 3 4 i 7 Tn TO 4 410 175 304 329 353 378 403 428 45^ 477 502 527 12 15... ; 1354 411 176 551 576 601 625 659 674 699 724 748 773 '353 411 177 797 822 846 895 920 944 969 993 ♦oi8 7 353 411 178 25042 666 091 115 139 164 188 212 237 261 8 20' \ 353 353 412 179 180 285 527 310 55J_ .314_ 575 358 382 406 431 _45J. 696 479 4JO 503 ,744 9 22 / 600 624 648 J72 1 353 413 181 768 792 816 840 864 888" 912 935 959 ;983 24 \ 352 413 182 26007 031. 055 079 102 126 150 174 198 221 I 2.4 . ;352 414 183 245 269' 293 316 340 364 387 411 435 4^8 2 4.8 ; '|3S» 414 184 482 505 529 553 576 600 623 647 670 694 3 7-2 . 9.6 1 I352 415 185 717 741 764 788 811 834 858 881 905 928 4 ■=T 415 415 186 187 951 27 184 I 975 207 998 231 *02I 254 *045 277 *o68 300 ♦091 323 *ii4 J46 *i38 370 *i6i 393 5j4Z.U ' 614.4 ^ } 7116.8- 1 f 8;i9.2 9i2i.6 22 . ,1 416 18S 416 439 462 485 508 531 554 577 600 623 ,3S. 350 350 416 417 417 189 190 191 646 669 692 -Zii 738 761 784 807 830 852. 875 898 921; 944 967 989 *OI2 *^35_ ♦058 *o8i 2^103 126 149 T^ 194 .217, 240 262 285 307 :-:;o 418 192 ^330 353 375 39S m .443 466 488 5" 533 i| 2.2 418 193 556 578 601 623 668 691 713 735 758 2J 4-4 419 194 780 803 825 8^7 i 870 892 914 937 959 981 i^ 203 3 6.6 419 195 29003 026 048 070 092 ^^5 137 '59 4 8.8 <^o 196 226 248 270 292 3M 336 358 380 403 425 ^ 197 667 469 491 513 535 5^57 579 601 623! 045 f J98 688 710 73-' 754 776 798 820 842 1 1863 7 '■.^■\ 817^ 9" . 199 '885 907 929 _95jj 973, 994 *oi6 *o38 *o6o "^m :200 30i«3 12? 146 i6S"' 190 2tl 233 255 "276 298 N. L. ! I 2 3 1 4 5 6 7 8 9 S.' T.' S." T." '' 0; 5 373 0° 2'= 120" 4-68557 557 .So" 4.6S , 373 373 3= iSc 557 557 c 4 c 240 557 558 29 :-.- 1 74p 30 := I Sod 31 -i860 3:'2 373 ' ' -3 25 =i:l\' -,58 26 =15 -,vS 32 = 1920 27 =16. .-58 33 = 1980 28 -16^ 3 58 34 = 2040 (, ■\ 1 N. L. 1 2. 3 4 5 i 6 7 18'^ 9 ^ P.P. 1 200 20 1 30103 125 146 168 190 211 233 255 :7( 298 5H 5(9. 01 1 •320, 341 363 384 406 428 449 471 492 202 535 557 578 600 621 643 664 68 s 707 728 ^ 203 750 771 792 814 835 856 878 899 920 942 3 4.4 4.2 6.6 6.3 8.8 8.4 204 963 984 *oo6 ♦027 *048 *o69 ♦091 *II2 *i33 *IS4 205 31175 197 218 239 260 281 302 323 34? 306 5 ii.o 10.5 13.2 12.6 2o6- •387 408 429 450 471 492 513 534 555 576 6 207 597 618 639 660 681 702 723 744 7<5? 78s 7 15.4 14.7 208 806 827 848 869 890 911 931 952 973 994 8 17.6 16.8 209 2ro 211 3201.5 035 056 077 098 118 139 160 181 201 9 19.8 18.9 222 428 243 263 284 305 325 346 366 387 408 on II 449 469 490 Sio 531 552 57^ 593 613 212 ^34 654 675 69s 71S 736 756 777 797 818 213 838 858 ^11- 899 919 940 960 980 *OOI *021 4.0 60 214 33041 062 082 102 122 143 163 183 203 224 4 8.0 215 •244 264 284 304 325 345 365 385 405 425 10.0 216 445 465 486 506 52b 546 56b 58b 606 626 12,0 217 ..646 666 686 706 726 746 766 786 806 826 7 14.0 2l8 ..846 866 885 905 925 945 965 985 *oo5 *025 8 16.0 219 220 221 34044 064 084 104 124 143 163 183 203 223 9 18.0 • 242 262 282 301 321 341 361 380 400 420 .0 II 439 459 479 498 518 537 557 577 S96 616 222 • 635 6S5 674 694 713 733 753 772 792 811 ^ 1.9 3-8 9.5 11.4 223 830 850 869 889 908 928 947 967 986 ♦005 3 4 224 35025 044 064 083 102 122 141 160 180 199 225 218 23« 257 276 295 315 334 353 372 392 226 411 430 449 468 488 507 52b 545 564 583. 227 603 622 641 660 679 698 717 736 755 774 7 133 22S 793 «I3 «32 851 870 889 908 927 946 965 8 15.2 229 230 -231 984 ♦003 *02I ♦040 *o59 ♦078 *097 *ii6 *i35 *i54 9 17.1 36173 lyz 211 229 248 267 286 305 324 342 ia II 361 380 399 418 43(? 455 474 493 511 530 3-e SA 7.2 9.0 10.8 232 549 S68 S86 60s 62^ 642 661 680 698 717 ^ 2S3 n^ 754 773 791 810 829 847 866 884 903 3 4 5 234 922 940 959 977 996 *oi4 *033 *osi ♦070 *oS8 235 37107 125 144 162 i8i 199 218 236 2S4 273 236 291 310 328 346 365 385 401 420 438 457 237 475 493 511 530 548 566 585 603 621 639 7 12.6 23^S 658 676 694 712 731 749 767 785 803 822 8. 14.4 , 239 240 241 840 858 876 894 912 931 949 967 985 *oo3 . 9 16.2 t 38021 202 039 220 057 238 075 o?3 112 310 —148 328 166 i 184 17 256 274 292 346 364 242 382 399 417 435 453 471 489 507 525 543 I 1.7 ;i 243 561 578 596 614 632 650 668 8^3 703 721 2 3-4 244 739 757 775 792 810 828 846 j •881 899 3 S.I 6.8 8.5 245 917 934 952 970 987 ♦005 *023 1 *04i ♦058 ^076 4 24b 39 ^-94 III 129 146 164 182 199 1 217 235 1 252 I 7 8 247 270 287 305 322' 340 358 375 i 393 410 j 428 11.9 i::.6 1 24S 445 463 480 498 515 533 550 568 585.} 602 1 249 I25O .^020 t\^7 655 672 690 '707 724 742 759 ! 777 . Q rc.^ 794 8;i I 829 846 863 88 1 898 ; 915 933 i 950 i N. L. '■ 1 2 ! 3 4 1 5 6 7 .,8 . 1 9 P.P. S.' T.' S." T. 0° '3^ 2160'' 4.68 557 55c, 2' 6.46373 373 0° 3'=-- I 80" 4.68 557 557 3 373 373 04=2 5 =^ 3 40 557 55^^ 00 557 558 0. 37 = 2220 557 559 20 372 2C 372 373 373 38=^2280 557 559 33 = 19 80 55/ '."'^ 39 = 2340 557 559 i 34 = 20 35 = 21 40 557, '■ 00 557 - r- 40 = 2400 557 559 - 41 =.2460 556 560 ; ■ ^ 36 = 21 60 557 v ^ = 252.) 556 sf / N . L. 1 ["2" 3 4 "5 6 7 .1 8 9 P. P. "] 250 251 39 794 811 S29 _846 8C.3 881 898 915 *o88 ♦100 950 ♦123 \ 18 967 " 98s^ *oo2 ♦019 *o37 *054" ♦071 252 40 140 '57 175 192 209 226 243 261 278 295 1 i.» 253 312 329 34^ 364 3^^ 398 4«5 432 449 466 2 3.^ 254 483 500 518 535 552 56.9 586 603 620 637 255 654 671 688 705 722 739 756 773 790 807 256 824 841 _§S8 875 892 909 926 943 960 970 (j 10 257 993 *oio ♦027 *044 ♦06 1 ♦078 *09? ♦in *I28 *M5 I 12 b 258 41 162 179 196 ■'212 229 246 263 280 296 3'3 14.4 259 260 261 330_^ 347 363 380 .J^2L 414 430 _447 4f>4 481 9 16.2 497* 514 531 547 _^64 581 5.97 614 6^1 647 17 664 681 697 714 731 747 764 780 797 814 262 830 847 863 880 896 913 929 946 963 979 I ^•7 ! 263 996 *OI2 ♦029 *046 *o62 ♦078 ♦095 ♦ill ♦127 ♦ 144 . 2 3-4 261 42 i6o - 177- -193 210 22^ 243 259 275 292. 308 3 4 '1 JO. 2, 261 325 341 357 374 390 406 423 439 455 472 266 488 504 521 537 553 570 586 602 619 635 I .267 651^ --^67 ♦' 684 1 700 716 732 749 765 •7S1 797 7 8 * J '^ 268 813 830 846 . 862 4^78 894 911 927 943 959 269 270 271 175_ 991 *oo8 *024 ♦040 *o56 *072 ♦088 ♦104 *120 9 I >• > 43 136 152 169 .185 201 217 233 249 265 1 281 .' 297 3^3 329 345 361 377 393 409 425 441 272 457 473 489 505 521 537 553 569 584 600 I I.O 273 616 632 648 664 680 696 712 727 743 759 2 g 8:0 9.6 274 775 791 807 823 838 854 870 . 886 902 917 3 275 933 949 965 981 996 *OI2 *028 ♦044 ♦059 ♦075 4 276 44091 iSZ. £22 138 »54 170 185 201 217 232 277 ;.f--248 264 279 295 ^1' 326 342 ^ 373 389 I 278 ,,0404 420 436 451 467 483 498 514 52.9 545 12.8 279 280 281 " ^560 576 592 607 623 638 654 669 685 700 I4.d 716 73^ 747 762 778 793 809 824 •840 _8J5. 15 il 871 886 902 917 932. 948 963 979 994 *OIO 282 45 0-5 040 056 071 086 102 117 ^33 148 163 ' 1.5 283 179^ 194 209 225 240 255 271: 286 3QI 317 2 30 284 > 332 347 362! 378 393 408 423 439 454 469 3 6.0 1 285 484 500 515 530 545 561 576 591 606 6.-' 4 286 637 652 667 682 697 712 728 743 758 7: /"•5 287 788 803 818 834 849 864 879 894 909 92a „ JP-S t_2.0 288 939 954 969 984 1 *OQO *ois ♦030 ♦045 ♦060 *o-/5 8 2S9 290 291 46 090 i05_ 120 135 150 165 180 195 210 22? '~^.'; 3^ 389 255 270 285^ 434 300 315 ^^ 345 ~359' _37 5^^ • ■^ 404 419 449 464 479 494 509 292 538 553 568 583 598 613 627 642 657 672. 293 687 702 716 731 746 761 776 790 805 820 294 835 850' 864 879 894 909 923 938 953 967 I 4.2 5.6 7.-0 ^ 4 295 982 997 *OI2 *026 *04i ^056 ♦070 ♦085 ♦100 *ii4 4 296 47129 144 159 173 188 , 202 2i7 232 246 2'- 297 "^276 290 305 319 334 349 3(>3 37S 392.! 4. 298 422 436 451 465 480 494 5c^9 524 5.v^ 299 300 567 582 596 611 625 _^| 654 669 68, 2'h 727 741 756 770 784 799 813 82^L^4_ N. L. 1 i 2 1 3 4 5 6 1 7 ^PT^ P.P. ' S.' T.' S." T." S." T." i 2' 646 37: 373 0° 4'= 240" 468557 558 0° 45'=: 2700" ■ " 3 37: 373 5 = 300 557 558 46 = 2760 . L . .« 25 372 ' 373 41 = 2460 ..'556 560 47 ~. 2820 26 372 373 42 = 2520 556 560 4S = 2880 '. :' ^ 27 372 374 43.= -'!5'8o 556 560 49 = 2940 y\. ' i 30 37 374 44 = 26^,0 556 560 50 .=: 3000 5..'' . ' _ 45 = ^700 556 sfo . fNr L. 1 2 3 4 5 6 7 1 8 1 9 P.P. 300 30 1 47712 857 727 741 756 770 784 799 813 828 842 972 986 871 885 900 914 929 943 958 302 48001 015 029 044 058 075 087 lOl II6^^ 130 15 303 144 159 173 187 202 216 230 2441 259, 273 304 287 302 316 330 344 359 373 387 401 416 ' 1-5 305 430 444 458 473 487 501 515 530 544 558 2 30 306 572 586 601 615 629 643 657 671 j 686 700 3 4-5 6.0 7-5 9.0 10.5 12,0 13-5 307 7H 728 742 756 770 785 799 813 827 841 4 5 5 308 «55 869 883 897 911 926 940 954 96^ 982 309 310 3" /l96_ *OIO *024 *038 ♦052 *o66 *o8o *094 *io8 ♦122 9 49 136 150 164 178 "192 206 220 234 248 262 276 290 304 31S 332 346 360 374 388 402 312 415 429 443 457 47.1 485 499 513 527 1 541 353 554 568 582 596 •iSio 624 638 651 665 1 679 3H 693 707 721 734- 748 762 776 790 803 817 14 315 831 845 859 872 ^k 900 914 927 1 941 955 316 969. 982 996 *OIO *024 *037 ♦051 *o65 *079 ♦092 I 1.4 317 50 106 120 133 147 161 174 188 202 1 215 229 2 2.8 318 " 243 256 270 284 297 3" •325 338 352 365 3 4.2 5-6 . 7.0 8.4 9.8 319 320 321 379 393 406 420 433 447 461 474 488 501 4 • 5 6 7 5J1. 529 542 556 569 583 596 610 623 __637; ^\ki 664 678 691 705 718 732 745 i 759 772 322 786 799 813 826 840 853 866 880 ! 893 907 8 II.-C 323 920 934 947 961 974 987 ♦ooi *oi4 *028 *04i 9 12.6 324 51^51 068 o&t- -095 108 121 131 148 162 175 325 188 202 215 228 242 255 268 282 295 428 308 326 322 335 348 362 375 388 402 415 441 327 455 468 481 495 508 521 534 548 561 574 13 32S 587 601 614 627 640 654 667 680 i 693 706 I 1.3 1 329 330 33i 720 733 746 759 772 786 799 812 825 957 838 2 3 4 2.6 3-9 7.8 9.1 851" 865 878 891 904 917 930 943 970 983 996 ♦009 *022 *035 *048 *o6i *o75 ! *o88 1 *ioi 332 52114 127 140 153 166 179 192 205 1 218 231 333 244 257 270 284 297 310 323 336' 349 362 7 334 375 388 401 414 427 440 453 466 479 492 8 10.4 335 504 517 530 543 556 569 582 595 608 621 9 II. 7 336 634 647 660 ^73 686 699 71.1 724 1 737 1 750 337 763 776 789 802 815 827 840 853 ■ 866 1 879 33^ 892 90? 917 930 943 5S6- ■0S4 969 982 ! 994 *oo7 339 340 341 53020 033 046 058 071 097 no: 122; 135 12 148 161 173 186 199 212 224 237 . 250 263 390 I 2 1.2 2.4 n 6.0 7.2 8.4 9.6 275 288 301 314 326 ^S 352 3641 377 342 403 415 428 441 453 466 479 491 ! 504 1 517 3 343 529 542 555 567 580 593 605 618 : 631 643 4 344 656 668 681 694 706 719 732 744 757 ! 769 5 5 345 .782 794 807 820 832 845 857 870 882 895 I 346 908 920 933 945 958 970 983 995 *oq8 ♦020 I 347 54033 045 058 070 083 095 108 12b 1 133 145 9 10.8 1 ,-;48 'i^ 170 183 195 208 220 233 245 i 258 : 270 : .349 350 283 _295 307 320 332 345 357 37^ 382 1 394 407 419 432 444 456 469 481 494 506 j 518 1 N. L. 1 2 3 4 '■'1 6 7 8 1 9 P. P. 1 S.' T.' S. / "Y ff S." T." I 3' 6.46 373 373 0° 5'= 3 00" 4.68557 558 0° 54'= 3240" 4 .68 556 561. '. 4 373 373 6- 3 60 557 558 55 = 3300 ■ 56 = 3360 556 561 1, ->.^ 372 374 5 -^ 30 00 55^ ) 561 372 374 5 5 1 =30 2 = 31 60 55^ 20 55^ ^ 561 ) 561 057= 3420 rS - -<«r, c6i 562 56i 5 3 = 3^ 80 55^ j 561 C) . 5 4 = 32 40 55^ > 561 _ N. L. 1 1 2 3 1 4 5 6 1 7 i 8 9 P. P. ll 350 351 54407 531 419 543 432 444 456 469 481 494 506 S18 555 568 580 593 605 617 642 352 353 -4ff 66j_ 790 '£- 814 704 827 839 851 "864" ^ 765> 888 13 354 900 913 9il 937 949 962 974 986 998 *OII 1 2 2:6 355 55 023 035 0^7 060 072 084 096 108 121 133 3 4 3-9 $•2 356 H5 157 169 182 194 206 218 230 242 255 357 267 279 ,291 3<^3 315 328 340 352 364 376 5 6.5 358 388 400 413 425 437 449 461 473 485 iH 6 7.8 359 360 361 5?9_ 522 534 546 558 570 582 594 606 618 7 8 9 9.1 10.4 11.7 630 642 654 666 678 691 703 715 727 739 751 763 775 787 799 811 823 835 847 859 362 871 883 895 907 919 931 943 955 967 979 363 991 ♦003 *oif ♦027 ♦038 ♦050 ♦062 ♦074 *o86 *098 ^ 364 56 no 122 134 146 158 170 182 194 205 217 12 365 -229^ 241 253 265 277 289 301 312 324 336 366 ■34^ 360 372 384 396 407 419 431 443 455 I 1.2 367 467 478 490 502 5H 526 538 549 561 573 2 24 3-6 4.8 6.0 f.2 8.4 368 583 597 608 620 632 644 656 667 679 691 3 369 370 371 703 714 726 738 750 761 773 785 797 808 4 1 820 832 844 855 867 879 891 902 • 914 ^26 937 949 961 972 984 996 ♦008 ♦019 ♦031 *043 S72 .57054 066 078 089 lor "3 i£4 136 148 159 8 9.6 373 171 183 194 206 217 229 2^1 252 264 276 9 10.8 374 287 299 310 322 334 345' 357 368 380 392 375 403 415 426 438 449 461 473 484 496 507 376 519 SiP 542 553 565 576 588 600 611 6^ i 377 634 646 657 669 680 692 703 715 726 1 11 378 749 761 772 784 795 807 818 830 841 .2 1 1.1 379. 380 381 864 978_ 875 8^ 8^8 910 921 933 944 955 907 2 3 4 2.2 3-3 4r4- 990 *o6i *oi3 *024 *035 ♦047 ♦058 ♦070 *o8i 58092 104 "5 127 138 149 161 172 184 ^^5 382 383 206 320 218 ■ 331 229 343 240 354- 252 365 263 377 274 388 286 297 309 422 1 I 1:1 399 410 384 433 444 456 467 478 490 501 512 524 535 385 546 557 569 580 591 602 614 625 636 647 9 9.9 386 -659 670 681 692 704 715 726 737 749 760 387 771 782 794 805 816 827 838 850 86i 872 i 388 883 894 906 917 928 939 950. 961 973 984 1 389 390 391 995 *oo6 *oi7 *028 ♦040 *o5i *o62 *073 *o84 *095 10 i 59 106 118 129 140 151 162 173 184 195 207 318 I 2 ,,0 2.0 ' 218 229 240 251 262 273 284 295 306 392 329 340 351 362 373 384 395 406 417 428 3 30 393 439 450 461 472 483 494 5C56 517 528 539 4 4.0 394 550 561 i 572 583 594 605 616 627 638 649 5 50 39'^ .660 671 682 693 704 715 726 737 748 759 6 6.0 396 770 780 791 802 813 824 835 846 857 868 7 7.0 397 879 890 901 912 923 934 945 956 966 977 8 8.0 398 988 999 *OIO *02I *032 *043 *054 *o65 ♦076 *o86 1 9.0 399 400 60097 108 119 130 141 152 163 173 184 195 i 206 j 217 228 239 249 260 ^71 282 j 293 304 N. L. 1 1 2 3 4 5 1 6 7 1 8 1 9 1 P.P. ij S.' T.' II S." T." S." T." j / 646373 373 0° 5'= 300" 4-68 557 55« 1° ^'-3660" .ir>s - ' 4 • -57 558 I 2 =^ 3720 ^ 35-- 39 -,7 558 3 - 3780 ■ ^ M5 562 4 - 3840 ' 40 35 562 I 5 ^^ 3QOO ft > 55 562 I 6 i: 3960 , |j I • ;55 562 I 7 T-^ 402c lo /- N, L. 1 1 2 3 1 4 1 5 6 7 1 8 9 P.P. 400 401 60 206 217 228 239 249 260 271 282 2^.3 304 314 325 336 347 358 369 379 390 401 412 402 423 433 444 455 466 477 487 498 509 520 403 531 541 552 563 574 584 595 606 617 627 404 638 649 660 670 681 692 703 713 724 735 405 . 746 756 767 778 788 799 810 821 'V3I 842 406 853 863 874 885 895 906 917 927 J 938 949 11 407 959 970 981 991 *002 *oi3 ♦023 *034 *04^ *0S5 I I.I 408 61066 077 087 098 109 119 130 140 151 162 2 2.2 409 410 411 172 183 194 204 215 225 236 247 257 26}? 4 3-3 4.4 11 278 289 300 310 321 33^ 342 ! 352 363 374 384 395 405 416 426 437 448 458 469 ! 479 412 490 500 5" 521 532 542 553 563 574 584 y 7-7 413 595 606 616 627 637 648 658 669 679 690 8 8.8 414 700 711 721 731 742 752 763 Z73 S78 784 794 9 9.9 415 805 815 826 836 847 857 868 888 899 416 909 920 930 941 951 962 972 982 993 *oo3 4'Z 62014 024 034 045 055 066 076 086 097 107 418 118 128 138 149 159 170 180 190 20 1 211 419 420 421 221 232 242 252 263 273 284 I 294 304 3'? 10 325 335 346 356 366 }77 387 397 408 418 428 439 449 459 469 480 490 j 500 511 S2I 422.. 531 542 5S2 562 572 583 593 1 603 613 624 ^ 1.0 423 634 644 651 665 675 685 696 1 706 716 726 2 2.0 424 737 747 757 7(>7 778 788 79S 808 818 829 3 4 3-0 4.0 5-0 6.0 425 839 849 859 870 880 890 900 910 921 931 426 941 951 961 972 982 992 *002 ♦012 *022 *033 427 63043 053 065 073 083 094 104 114 124 134 7 7.0 428 144 155 I6S 175 i8s 195 205 215 22s 230 8 8.0 429 430 431 246 256 266 276 286 296 306 3»7 •327 337 9 9.0 347 357 367 377 387 397 407 417 "5A8 428 5^ 438^ 538 448 4S8 468 478 488 498 508 432 548 558 S68 579 589 599 609 619 629 639 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 435 849 859 869 879 889 899 909 919 929 939 43t>. .--_^41 959 969 979 988 998 *oo8 *oi8 ♦028 *038 437 64048 058 068 078 088 098 108 118 128 ^37 9 438 H7 157 167 177 187 197 207 217 227- ^237 I °-9 439 440 441 246 256 266 276 286 296 306 316 326 335 2 3 •4 1.8 2.7 3.6 •4.5 5-4 6.3 345 355 365 375 385 395 404 414 424 434 444 454 464 473 483 493 503 5«3 523 532 442 542 552 S62 572 582 591 6oi 611 621 631 443 640 650 660 670 680 689 699 709 719 729 7 444 738 748 758 768 777 787 797 807 816 826 8 l-.l' 445 836 846 8s6 865 ^7^ 88^ 895 904 914 924 9 446 933 943 953 963 972 982 992 -*002 ♦on 1 *02I 447 65031 040 050 060 070 079 089 099 108 1 118 448 128 137 147 157 16.7 176 186 196 205 j 215 449 450 225 234 244 254 263 273 283 292 302 i 312 321 331 341 350 360 369 379 389 398 1 408 N. L. 1 ' 2 ! 3 1 4 5 1 6 1 7 8 9 1 P.P. S.' T.' S.' T." S/' T." 4' 646 373 373 0° 6'^ 360" 4 68557 558 I^' 9'= 4140" 4 •68555 563 5 373 373 7 = 420 8 = 480 557 557 558 558 I 10 = 4200 I 11= 4260 I 12 = 4320 554 563 554 564 554 564 40 42 372 372 375 375 I 6 = 3960 555 563 43 371 375 I 7 = 4020 55^ 563 I 13=4380 554 564 44 371 375 I 8 = 4080 553 563 I 14 = 44 ^.0 554 564 45 371 375 I 9 =r 4140 555 563 I ! ' 554 564 u 1 1 N. L. 1 k 3 4 5 6 7 8 9 P.P. 450 451 65321 418 _33i 427 _ 341 437 _35o 447 360 456 369 466 379 389 398 408 504 475 485 495 452 514 523 \:>?> 543 S52 562 571 581 591 600 453 610 619 029 639 048 658 667 677 686 696 454 706 7»5 72? 734 744 753 763 772 782 792 4S5 801 811 820 830 839 849 858 868 877 887 456 896 906 9.6 925 935 944 954 963 973 982 10 457 992 *OOI ♦oil *020 ♦030 *039 *049 ♦058 *o68 *o77 I I.O 458 66087 .. 096 106 "5 124 134 143 . >53 162 172 2 2.0 459 460 461 181 191 200 210 219 229 238 247 257 266 3 4 3.0 4.0 276 285 291 304 314 ,323 332 342 351 361 46^ 38a 389 398 408 417 427 436 445 455 462 -474 483 492 502 5'i 521 530 539 549 7 7.0 463 558 5^7 577 586 596 605 614 624 633 642 8 8.0 464 652 661 671 680 689 699 708 717 727 736 9 9.0 46 s 745 75? 764 773 783 792 801 811 820 829 466 839 848 857 867 876 885 894 904 913 922 467 932 941 9 so 960 969 978 987 997 *oo6 *oi5 468 67025 034 043 052 062 071 080 089 099 108 469 470 471 117 127 136 H5 154 164 173 182 191 201 9 210 219 228 237 247 256 265 274 284 293 302 3" 321 330 339 348 357 367 376 385 472 394 403 413 422 431 440 449 459 468 477 I 1 0.9 473 486 495 504 5M 523 532 541 550 560 569 2 ' " i.e 474 S78 S87 596 605 614 624 ^33 642 651 660 3 2.7 3.6 4.5 5.4 6.^ 475 669 679 688 697 706 715 724 733 742 7S2 4 476 761 770 779 788 797 806 815 825 834 843 I '477 8S2 861 870 879 888 ^tl 906 916 925 934 s^ ^47^ ■ 943 952 961 970 979 988 997 *oo6 *oi5 *024 8 7.2 ' 479 ^80 ^81 68034 043 052 061 070 079 088 097 106 "5 9 8.1 124 133 142 151 160 169 178 187 196 205 215 224 233 242 251 260 269 278 287 296 482 305 314 323 332 341 350 359 368 377 386 4S3 395 404 413 422 -431 440 449 458 467 476 484 485 494 502 5" 520 529 538 547 556. 565 .85 574 S83 592 601 610 619 628 637 646 655 486 664 673 681 690 699 708 717 726 735 744 8 487 753 762 771 780 789 797 806 815 824 833 4.S8 ' 842 8si 860 869 878 886 ^95 904 913 922 I 0.8 489 490 491 931 940 949 958 966 975 984 993 *002 *OII 2 1.6 3 24 4.3.2 5 |40 6 i 4.8 69020 028 037 046 055 064 073 082 099 099 108 117 126 135 144 152 161 170 179 188 492 197 205 214 223 232 241 249 258 267 276 493 285 294 302 3" 320 329 338 346 355 364 7 5-6 494 373 381 390 399 408 417 425 434 443 452 8 6.4 49 S 461 469 478 487 496 504 5^3 522 531 539 9 1 7-2 496 548 557 566 574 583 592 601 609 618 627 * 497 636 644 653 662 671 679 688 697 705 7H 40H 723 732 740 749 758 767 775 784 793 801 499 500 810 819 827 836 845 854 862 871 880 888 ! 897 906 914 923 932 940 949 958 966 975 N. |L. 1 1 2 I 3 4 5 6 7 8 1 9 P. P. S.' T.' S.' f Y." S.' T." 4' 646 373 373 0° 7'= A 20" 4.68 555 ' 558 1° 1 8' =4680" 4 .68554 565 5 37: ; 373 8=4 9=5 80 55/ 40 55/ ' 558 ' 558 I 19 =4740 I 20 =4800 I 21 =4860 554 565 554 5«^5 553 566 45 48 37' 37' 375 375 I 15 =45 00 55^ ^ 564 49 yn 376 I 16=45 60 55^ ^ 565 I 22 =4920 553 506 50 37] 376 I 17 =46 20' SSA ^ 565 I 23=4980 553 566 I 18 =46 80. 55^ ^ 565 I 24 = 5040 553 566 In. L. 1 1 2 3 4 5 6 1 7 ii 1 9 p.p. 500 69897 906 914 923 932 940 949 1 958 9'>6 1 975 984 992 *OOI *OIO *oi8 *027 ♦036 1 *044 *o,3 ♦062 502 70070 079 088 096'' 105 114 122 131 ub 148 503 ^57 165 174 183 191 200 209 217 226 234 ^04 243 252 260 269 278 286 29? 303 312 321 505 329 ^^ii 346 355 364 372 381 389 398 406 9 506 415 424 432 441 449 458 467 475 484 492 507 501 509 518 526 535 544 552 561 569 s78 I 0.9 1 508 586 603 612 621 629. 638 646 6S5 603 2 1.8 509 5fO 5" 672 680 689 697 706 7H 7?3 731 740 749 834 3 4 5 6 2.7 3-6 4.5 5-4 757 766 774 783 791 800 808 817 825 842 851 859 868 876 885 893 ,902 910 919 512 927 935 944 952 961 969 978 ,986 99? *oo3 7 6-3 513 71 012 020 029 037 046 054 063 071 079 088 8 7.2 514 096 105 lU 122 130 139 147 155 164 172 9 8.r 5^5 181 189 198 206 214 223 231 240 248 257 51(5 265 273 282 290 299 307 315 324 332 341 517 349 3S7 366 374 383 391 399 408 416 4'^^ 518 433 441 450 458 466 47? 483 492 500 508 520 521 517 525 533 542 550 559 5^7 575 584 592 675 B 600 609 617 625 634 642 650 659 667 684 692 700 709 717 725 734 742 750 759 522 767 775 784 792 800 809 817 82s 834 842 i ' 523 850- 858 867 875 883 892 900 908 917 925 2 3 4 2-1 3 '4 4.0 4.8 524 933 941 950 958 966 97? 983 991 999 *oo8 S2.S 2^016 024 032 041 049 057 066 074 082 090 i 526 099 107 "5 123 132 140 148 156 165 173 527 \i8i \263 189 198 206 214 222 230 239 247 2SS 7 5-6 S28 272 280 288 296 304 313 321 329 337 S 64 .; 529 530 1 "531 '346 428 354 362 370 378 387 39? 403 411 419 501 583 ^1 436 444 452 460 469 477 48? 493 509 .518 526 534 542 550 558 567 575 1 S32 591 599 607 616 624 632 640 648 656 06^ ! 533 673 681 689 697 705 713 722 730 738 746 1 534 535 -154:. \K35 762 ^43 770 -85^^ • Ul^ l%- 876 803 884 811 892 819 900 ^ol 1 , 536 916 925 933 941 949 957 965 973 981 989 537 997 *oo6 ♦014 *022 *030 ♦038 *046 *054 *062 *070 ' ! 53« 73078 086 094 102 III 119 127 135 143 151 1 539 540 541 159 167 175 183 191 199 207 215 223 231 \ ' s i 239 247 255 263 272 280 288 296 304 312 5 6 3 'v 4 ^ i 320 328 336 344 352 360 368. 376 ; 384 392 542 400 408 416 424 432 440 448 4S6 1 464 472 7 4 ^ 1 543 480 488 496 504 512 520 528 536 544 552 s : i 544 560 56^ 576 584 592 600 608 616 624 632 r 545 640 648vJ 656 664 672 679 687 695 703 7" 1 546 719 727 735 743 751 759 767 775 783' 791 547 799 807 815 823 830 838 846 854 ! Mi, .^70. S4H 878 886 894 L 902 910 918 926 933 1 9'4i ! 949 >549 550 957 965 973 981 989 997 *oo5 *oi3 *020 - >28 107 74036 044 052 060 068 076 084 1 092 ( 099 i N. L. 1 1 1 2 1 3 i 4 1 5 1 6 1 7 i 8 1 9 p.p. 1 1 S/ T/ S. II ^-n S." T." 1 rf 1 5 6.46 373 373 0° S'= 4S0" 4.68 55' I 558 jO 26' = 5 1 60" 4 .68553 56. 6 373 373 09= 540 55' 10 = 600 55 f 558 7 558 - J 27 =5J20 28 = 5280 29 = 5340 553 >' • 553 5 553 5 ■ 50 55 371 371 376 376 I 23 = 498c 55^ ^ S66 1 I 24 = 5040 55^ ^ =;% 30 = 5400 553 5"7 I 25 = 5100 55. 7, 06 31 = 54 &0 552 5'^S I 26 = 5160 55 3 537 32 = 55-0 552 568 U .. |L. "1 2 3 4 1 5 1 6 1 7 , -8 1 9 P. P. 1 650 74036 044 052 060 06S 076 0S4 092 099 107 186 Nsi "5 123 131 139 147 155 162 IfO 178 1 552 194 202 210 218 225 233 241 249 257 26^ 1 553 273 280 288 296 304 312 320 327 335 343 554 351 359 367 374 382 390 398 406 414 421 555 429 437 44? 453 401 46S 476 484 492 500 i 550 507 5^5 523 531 539 547 554 502 570 578 557 S86 593 601 609 6,7 624 632 640 648 6s6 i SS8 66,s 67. 679 687 695 702 710 718 726 733 559 560 S6i 741 749 757 764 772 780 788 796 803 811 8 819 827 834 842 850 858 865 873 881 889 896 904 912 920 927 93? 943 950 958 966 I 0.8 S62 974 981 989 997 *oo5 *OI2 *020 *028 *035 *043 2 1.6 563 75051 059 066 074 0S2 089 097 105 "3 120 3 2.4 S^H 128 136 143 151 159 166 174 182 189 197 4 3-2 5^5 205 213 220 228 236 243 25' 259 266 274 5 4.0 566 282 289 297 305 312 320 328 335 343 351 6 4.8 S67 3S8 366 374 381 389 397 404 412 420 427 7 8 5.6 6.4 568' 435 442 450 458 465 473 481 488 496 504 569 570 571 511 5>9 526 534 542 549 557 565 572 580 9 7.2 587 664 595 603 610 686 618 626 633 641 648 656 67. 679 694 702 709 717 724 732 572 740 747 755 762 770 778 78s 793 800 808 573 815 823 831 838 846 853 861 868 876 884 574 891 899 906 914 921 929 937 944 952 959 575 967 974 982 989 997 *oo5 *OI2 *020 ♦027 *o3? 576 76042 05J 057 of>? 072 080 087 095 103 no 577 118 125 133 140 148 155 163 170 178 185 S7« 193 200 208 215 223 230 238 245 253 260 579 580 S8i 268 275 283 290 298 305 3^3 320 328 335 7 343 350 358 365 373 380 388 395 403 410 418 425 433 440 448 455 462 470 477 4SI S«2 492 500 507 515 522 530 537 545 552 559 U.7 583 - 567 574 582 589 W 604 612 619 626 634 2 1:4 584 641 649 656 664 67! 678 686 693 701 708 3 4 ^ 5 6 28 585 K 716 723 730 738 745 753 760 .768 775" 849 78z f^ 586 -790 797 805 812 819 827 834 842 856 587 864 871 879 886 893 901 908 916 923 930 7 4.9 S88 938 945 953 960 967 975 982 989 997 *oo4 8 5-6 589 590 59 i 592 77012 019 026 034 041 048 056 06^ 070 078 9 6.3 .085 093 100 107 115 122 129 ^37 144 151 159 . 232 166 240 '^ 181 254 188 262 195 269 TA 210 217 291 225 298 283 ' 593 305 3^3 320 327 335 34^ 349 357 364 371 594 379 386 393 401 408 415 422 430 437 444 595 452 459 466 474 481 488 495 503 510 517 596 525 532 539 146 554 561 568 576 583 590 597 597 605 612 619 627 634 64! 648 656 663 598 670 677 685 692 699 706 714 721 728 735 599 600 743 7?o 757 764 772 779 786 793 801 80S 815 822 830 837 844 851 859 866 873 880 N. 1 L. 1 ! 2 3 4 5 '6 1 7 8 9 P .p. S.' TJ S." T." 6' 646 37: 373 0° 9'= 5 10 = 6 40" 4-68557 558 00 557 558 i°3 I 3 5'= 5700" 4.f 55 56 371 371 376 376 6 = 5760 I 31 =54 60 552 568 I 3 7 = 5820 57 371 377 I 32 = 55 20 552 568 A 3 8 = 5880 58 371 377 I 33 = 55 80 552 568 I 3 9 = 5940 59 37c 377 I 34 = 56 40 552 568 I 4 — 6000 60 37c > 377 I 35 = 57 00 552 569 '4 N. L. 0| 1 2 3 ! 4. 5 6 Me 9 P.P. 600 77815 S22 830 837 844 851 859 __866 1 g73_ 880 6oi 887 895" 902 909 916 924 931 93^ 945 952 602 960 967 974 981 988 906 1 *oo3 ♦oio *oi7 082 ^o§9 *025 603 78032 039 046 053 061 068 075 097 604 104 III n8 125 132 140 147 154 161 168 605 176 183 190 197 204 211 219 226 233 240 8 606 247 254 262 269 276 283 290 297 305 312 607 319 326 333 340 347 351 362 369 376 383 I 0.8 1.6 608 390 398 405 412 419 426 433 440 447 455 2 609 610 462 469 _47A 483 490 497 504 512 519 •526 3 4 5 6 24 533 540 547 554 561 569 576 583 590 597 3-2 4.0 4.8 6u 604 611 618 625 633 640 647 654 661 668 612 675 682 689 696 704 711 718 725 732 739 7 5.6 6.4 613 746 753 760 767 774 781 789 796 803 810 8 614 888 824 831 838 845 852 859 866 873 880 9 7.2 615 893 902 909 916 923 930 937 944 951 616 958 965 972 979 986 993 *ooo *oo7 ♦014 *02I 617 79029 036. 043 ojo 057 064 071 078 08I 092 618 099 106 i"^ 120; 127 134 141 148 155 162 ' ^119 169 176 183 190 197 204 211 2l3 225 232 1 026 239 246 253 260 267 274 ~isr 288 295 302 7 1 621 309 316 323 330 337 344 351 358 365 372 J 0.7 1.4 2.1 622 •379 386 393 400 407 414 421 428 43? 442 2 3 4 623 449 456 463 470 477. 484 491 498 505 5" 624 518 525 532 .539 546 553 560 567 574 581 ^.8 625 588 595 602 609 616 623 630 637 644 650 5 3-5 626 657 664 671 678 685 692 699 706 713 720 6 4.2 627 727- 734 741 748 P4 761 768 775 782 789 7 4-9^ 628 796 803 810 817 831 837 844 851 858 8 5-6 629 630 1 631 865 872 879 886 893 900 906 913 920 927 9 6.3 934. 941 i 948 955 962 "969" 975 982 "989" .996 80003 010 017 024 030 037 044 051 058 065 6^2 072 079 085 092 099 106 'o^- 120 127 134 633. 147 154 161 168 175 182 J 88 195 202 i 634 209 216 223 229 236 243 250 ' 257 264 271. ! 635 277 284 291 298 30? 3-12 ^i^ 325 332 339 636 346 353 359 366 373 380 387 393 400 407 6 0 j- ! I'.,; 171 178 184 191 198 204 211 ! 218 1 '3» 24? 251 258 ?65 271 278 1 285 .305 3" 3i8 325 33^ 338 345 351 ■ 1 i 2 i 3 I 4 5 ! 6 ! 7 1 8 j 9 P.P. ■r ■' S.' ' T." S." T." -;, 0° io'= ( )Oo" 4.6855 J 558 i°44'=r624o" . i-68 551 571 - II = ( j6o 55' 7 558 I 45 ==6300 " I 46 =6360 551 571 551 571 I 40 =6c xx) 55 f 570 I 41 =6c >6o 55 f 570 I 47 =6420 550 572 I 42 =6] 20 55 I 570 I 48 =6480 550 572 i I 43=6 I 44=62 80 55 540 55 I 570 I 571 I 49 =6540 550 572 1 N. L. 1 2 ^ 3 14' 5 '6^1 1 8 J 9 P.>. 750 87 506 512 518 523 1 529] 535 541 547 552 558 ' 751 5^ . 570 576 581', 5871 ';93 599 604 610 616 ■v2 622 628 633 (^39 64^ 6,Si 656 662 668 674 s.^ » 679 685 6^1 697 703 708 7H 720 726 73 » 754 737 743 749 754 760 766 772 777 783 789 755 79? 800 800 812 818 823 829 835 841 84b 75^^ 852 858 864 809 875 88 1 887 892 898 904 757 910 9«5 9211 978 927 933 938 944 950 955 961 7S8 967 973 984 990 996 *ooi ♦007 ♦013 *oi8 759 760 761 88024 030 036 041 047 053 058 116 064 ~121 070 076 R 081 087 093 098 104 no 127 ^33 190 138 144 150 156 161 167 »73 178 184 06 762 195 201 207 213 218 224 230 235 241 247 /> 1.2 1.8 1(^}> 252 258 264 270 275 281 287 292 298 304 3. 4 764 309 315 3?i 326 332 338 343 349 355 360 2.4 7^>5 366 372 377 383 389 395 400 406 412 4»7 5 3^0 7O6 423 429 434 440 446 45 » 457 4t)3 468 474 6 3-6 767 480 485 491 497 502 508, 513 519 525 530 7 4.2 768 S36 542 547 553 559 564 570 57^ 581 587 8 4.8 • 769 770 771 593 649 598 655 604 610 615 621 627 632 638 643 9 5-4 660 666 672 677 683 689 694 700 705 711 717 722 728 734 739 745 75O' 7S6 772 762 767 773 779 784 790 795 801 807 812 773 818 824 829 835 840 846 852 857 863 868 774 874 880 885 891 832. 902 908 913 919 925 775 930 936 941 947 953 958 964 969 975 981 776 986 992 997 *oo3 ♦009 *oi4 *020 *025 ♦031 *037 777 89042 048 053 059 064 070 076 081 087 092 77« 098 104 109 "5 120 126 131 ^Zl 143 148 779 780 781 154 159 .^^5 170 17b 182 187 193 198 204 ft 209 215 221 226 232 237 243 248 254 260 265 271 276 282 287 293 298 304 310 315 782 321 326 332 337 343 348 354 ?><^o^ 365 371 ' 0-5 I.O 1.5 2.0 l^Z 376 382 387 393 398 404 409 415 421 426 3 4 784 432 437 443 448 454 459 465 470 476 481 7^5 487 492 498 504 509 5'5 520 5^6 531 537 1 2.5 786 542 548 553 559 5^4 570 575 581 58b 592 30 787 597 603 609 614 620 625 631 636 642 647 7 3.5 788 653 658 664 669 675 680 686 691 697 702 8 4.0 789 790 791 70S 713 719 724 730 735 741 746 752 757 9 4.5 763 818 768 774 77^ 785 790 791 801 856 8g7 812 867 823 829 834 840 845- 851 862 792 873 878 883 889 894 900 905. 911 916 922 793 927 933 938 944 949 955 960 966 971 972 794 982 988 993 998 ♦004 *oo9 *oi5 *020 *Oi:6 *03i 795 90037 042 048 053 059 064 069 075 080 086 796 091 097 102 108 "3 119 124 129 135 140 797 146 ly 157 162 168 173 179 184 189 195 798 200 206 211 217 222 227. 233 238 244 249 799 800 255 260 -266 271 276 282 287 293 298 304 309 3H 320 325 32,^ 7,2,^^ 342 347 352 358 N. |L. 1 1 2 3 4 5 6 i 7. 1 8 j 9 P.P. S.' T/ S." T." S." T." 7' 646 373 373 0° 12'= 720" 4.6855^7 558 [3 = 780 5^7 558 [4 = 840 ^57 558 2° 8'= 7680" 4-68547 578 8 373 373 2 9=7740 547 578 2 10=7800 547 578 "211 =7860 547 579 75 80 369 369 380 380 2 5=7500 /S48 577 2 6=7560 ./ 548 577 7 =7620 / 548 577 2 12=7920 547 579 2 2 13=7980 547 579 2 8. .7680 . 547 578 2 14=8040 546 579 -.. L. 1 1 2 3 1 4 5 1 6' 1 7 8 1 9 PP. 800 8oi 90309 3M 320 32^ 380 :u^ 33(^ 342 347 352 ' 358 363 369 374 3^5 3'.- 396 401 407 ' 412 802 417 423 428 434 i 4J9 445 450 455 461 ] 466 803 472 477 482 488 493 499 504 509 5»5j 520 804 526 531 536 542 547 553 558 563 569! 574 805 .S8o 585 S90 596 601 607 612 617 623 628 806 634 639: 644 650 655 660 666 671 677 682 807 687 693 698 703 700 7U 720 725 730 736 808 741 747 7S2 757 763 768 m 779 784 789 809 810 811 795 800 806 811 816 822 832 838 843 6 849 854 859 865 870 87^ 881 886 891 1 897 1 902 907 913 918 Q24 929 934 940 945 950 812 956 961 966 972 977 982 .988 993 998 ♦004 I 0.6 «'3 91 009 014 020 025 030 036 041 046 052 057 2 1.2 814 062 068 073 078 0.84 089 094 100 los I 10 3 1.8 8.S 116 121 126 132 n7 142 148 J 53 158 164 4 2.4 816 169 174 180 185 190 196 201 2p6 212 217 ^ 30 3.6 4.2 817 222 228 233 238 243 1 249 .^4 259 265 270 I 818 275 281 286 291 297 302 307 312 3i8 323 48 819 820 821 328 334 339 344 350 35? 360 3^5 37 « 37^ 9 e;.4. 38 • 387 392 397 403 408 413 418 424 429 434 440 445 450 455 461 466 47» 477 482 822 487 492 498 503 .SOS 5H 519 524 529 535 «23 540 545 551 55(> 561 566 572 577 582 587 824 593 S98 603 609 614 619 624 630 635 640 82s 645 651 6s6 661 666 672 677 682 687 693 826 69S 703 709 714 719 ■724 730 735 740 745 827 751 7S6 761 819 772 777 782 787 793 79S 828 803 808 8m 824 829 8S4 840 84^ 850 829 830 831 855 861 866 871 876 882 887 892 897 903 95S "007 5 908 913 918 924 929 934 939 944 950 960 965 971 976 981 986 991 997 *,gQ^ I 05 I.O 8^2 92012 ot8 023 028 033 03S 044 049 054 059 2 ^33 065 070 075 080 085 091 090 lOI 106 III 3 ^•5 sh 117 122 127 132 137 143 148 153 158 163 4 2.0 83s 169 174 179 184 189 195 200 205 210 215 5 2.5 836 221 226 231 236 241 247 252 257 262 267 6 30 sm 273 278 283 288 293 298 304 309 3H 319 7 3-5 838 324 330 335 340 345 350 355 3^1 366 371 8 4.0 839 840 841 376 381 387 392 J97 402 407 412 418 423 9 45 428 433 438 443 449 454 459 464 469 474 480 48^ 490 495 Soo 505 5'* 516 521 526 842 531 536 S42 547 552 ",57 S62 5^7 572 578 843 583 588 593 598 603 609 614 619 624 629 844 634 639 645 650 6S5 660 665 670 675 681 845 686 691 696 701 700 711 716 722 727 ! 732 846 737 742 747 V 752 758 763 768 773 778 ! 783 847 788 793 799 804 800 814 819 824 829 ' 834 848 840 84s 850 855 860 865 870 875 88! 886 ' 849 850 891 890 901 906 911 916 921 927 932 I 937 942 947 952 957 1 962 967 973 978 983! 98^ N. L. 1 1 2 1 3 1 4 5 1 6 7 1 8 9 1 P.P. S.' T.' \ S." T." S.-" T." 8' 6.46 373 373 0-13'= 7S0" i.68557 558 2^ i6'=8i6o"4 .68 ^a6 580 9 373 373 14= 840 557 558 015= 900 557 558 2 2 ' 2 2 2 2 17 = 8220 18 = 8280 546 580 546 581 80 81 82 85 369 369 368 368 380 381 381 381 2 13^=7980 -\47 579 2 14 = 8040 5:,^ 579 2 15 =8100 54t 580 2 i6 = 8i'o 546] 5 So 19=8 20 = 8 21 =;:8 22=8 340 400 460 520 54 54 54 54 t> 5»» 5 582 5 582 5 582 N. L. 1 1 2 3 4 5 6 7 1 8 1 9 P.P. 850 8si 92942 947 _951 ♦003 957 *oo8 962 *oi3 967 *oi8 ♦024 978 983 988 993 998 ♦029 *034 *039 8s2 93 044 049 054 059 064 069 221 080 085 090 «53 095 100 105 no i'5 120 »25 13'. 130 141 8^4 146 151 156 161 166 171 176 181 .186 192 S55 197 202 207 212 217 222 227 232 237 242 a 856 247 252 258 263 268 273 278 283 288 293 06 I 2 8S7 298 303 308 313 318 323 328 334 339 344 2 858 349 354 359 364 369 374 379 384 389 394 3 4 5 6 I s 859 830 So I 399 404 409 414 420 425 430 433 440 445 2.4 3-0 3.6 430 455 460 46? 470 475 480 485 490 495 Soo 505 510 515 520 526 531 536 541 546 S62 551 556 S6i S66 571 576 581 586 591 59b 7 4.2 86,:; 601 bo6 611 616 621 626 631 636 641 646 8 4.8 S64 651 656 661 666 671 676 682 687 692 697 9 5-4 865 702 707 712 717 722 727 732 737 742 747 86b 752 757 762 767 772 777 782 787 792 797 867 802 807 812 817 822 827 832 837 842 847 868 852 857 862 867 872 877 882 887 892 897 869 870 871 902 95^ 94002 907 912 917 922 927 932 937 942 947 5 957 962 967 972 977 982 987 992 997 007 012 017 022 027 032 037 042 047 872 052 057 062 067 072 077 082 086 091 096 I 05 873 lOI 106 III 116 121 126 131 13(5 141 146 2 I.O 874 151 iS6 161 166 171 176 i8i 186 191 196 3 4 1-5 cf 87s 201 2Q6 211 216 221 226 231 236 240 245 2-5 30 3-5 876 250 255 260 265 270 275 280 285 290 295 877 300 305 310 315 320 325 330 335 340 34? ^ 878 349 354 359 364 3^9 374 379 384 389 394 8 4.0 879 880 881 399 404 409 414 419 424 429 433 438 443 493 9 4.5 448 498 453 458 463 468 473 478 483 488 503 507 512 517 522 527 5f 537 542 882 547 SS2 557 562 567 571 57^^ 581 586 591 883 596 601 606 611 616 621 626 630 ^35 640 884 645 650 655 660 665 670 673 680 685 689 88s 694 699 704 709 714 719 724 729 734 738 886 743 748 753 758 7^3 768 773 778 783 787 4 887 792 797 802 807 812 817 822 827 832 836 I 0.4 888 841 846 851 856 861 m 871 876 880 885 2 0.8 889 890 891 890 895 900 905 910 915 919 924 929 934 3 4 5 6 1.2 1.6 2.0 2.4 28 939 944 949 954 959 963 968 973 *022 978 983 988 993 998 *002 ♦007 *OI2 ♦017 ♦027 *032 892 95036 041 046 051 056 06 1 066 071 075 080 7 8 893 085 090 095 100 105 109 114 119 124 129 32 3.6 894 134 139 143 148 153 158 163 168 173 177 9 89s 182 187 192 197 202 207 211 216 221 226 896 231 236 240 245 250 255 260 265 270 274 897 279 284 289 294 299 303 308 3^3 318 323 898 328 332 337 342 347 352 357 3t>i 36b 371 899 900 37^ 381 386 390 395 400 405 410 415 419 424 429 434 439 444 448 453 458 463 468 n: 1 L. 1 1 ! . 2 1 3 4 5 1 6 7 1 8 1 9 1 P.P. S.' TJ 1 S." T." S." T." 8' 6.46 373 373 0° 14'= h >4o" A .68 557 558 2°25' = 87 00" . ^68545 583 9 373 373 15 - 9 00 557 558 2 26 =87 60 20 544 584 544 584 8S 368 381 2 21 =84 ^6o 545 582 " 2 27 =88 86 368 382 2 22 =81: 20 545 582 2 28=8S 80 544 584 89 368 382 2 23 =8c 80 545 583 2 29=89 40 544 585 r 368 383 2 24=;8e 2 25-=8; )40 700 545 583 545 583 2 30=9C 00 544 585 20 N. L. 0| 1 2 3 1 4 1 5 1 6 7 8 9 1 P.P. ~]\ 900 901 95424 429 434 439 444 448 453 458 463 468 472 477 482 487 492 497 501 506 5" 516 902 521 525 530 535 540 54,"; SSo 554 559 564 903 569 574 578 583 588 593 598 602 607 612 904 617 622 626 631 636 641 646 650 6S5 660 905 665 670 674 679 684 689 694 698 703 708 90b 713 718 722 727 732 737 742 746 751 756 907 761 766 770 775 780 785 789 794 799 804 908 809 813 818 823 828 832 837 842 847 8S2 909 910 911 856 861 866 871 875 880 885 890 895 899 6 1 904 909 914 918 923 928 933 938 942 947 952 957 961 966 971 976 980 985 990 995 0.5 912 999 ♦004 ♦009 *oi4 ♦019 ♦023 *028 *033 *os8 *042 2 I.O 913 96047 052 057 061 066 071 076 080 085 090 3 1-5 914 095 099 104 109 114 118 123 128 133 137 4 2.0 915 142 147 152 156 ibi 166 171 175 180 185 5 2-5 916 190 194 199 204 209 213 218 223 227 232 6 3.0 917 237 242 246 251 2S6 261 26s 270 275 280 I 3-5 918 284 289 294 298 303 308 313 317 322 327 4.0 919 920 921 332 33(^ 341 346 350 355 360 365 412 369 374 9 4.5 379 384 388 393 398 402 407 417 421 426 431 435 440 445 450 454 459 464 468 922 473 478 483 487 492 497 501 506 5" 515 923 520 525 530 534 539 544 548 553 558 562 924 567 572 577 581 S86 S9I 595 600 60s 609 925 614 619 624 628 633 638 642 647 6S2 656 926 661 666 670 675 680 685 689 694 699 703 927 708 713 717 722 727 731 736 741 745 750 928 755 759 764 769 774 778 783 788 792 797 929 930 931 802 811 816 820 825 830 834 881 839 844 4 -S48 453 858 862 867 872 876 886 890 895 900 904 909 914 918 923 928 932 937 932 942 946 951 956 960 965 970 974 979 984 I U.4 0.8 933 988 993 997 *002 ♦007 ♦oil *oi6 *02I ♦025 ♦030 3 4 934 97035 039 044 049 053 058 063 067 072 077 1.6 935 081 086 090 095 ICX) 104 109 114 118 123 5 6 936 128 132 137 142 146 151 155 160 165 169 2.4 937 174 179 183 188 192 197 202 206 211 216 7 2.8 93« 220 225 230 234 239 243 248 253 2S7 262 8 3-2 939 940 941 267 271 276 280 285 290 294 299 304 308 9 3-6 313 3'7 322 327 331 336 340 345 350 354 359 364 368 373 377 382 387 391 39^ 400 942 405 410 414 419 424 428 41^ 437 442 447 943 451 456 460 465 470 474 479 483 488 493 944 497 502 506 5" 516 520 525 529 534 539 945 543 548 552 SS7 S62 S66 571 575 580 585 946 589. 594 598 603 607 612 617 621 626 630 947 635 640 644 649 653 6s8 663 667 672 676 948 681 685 690 695 699 704 708 713 717 722 949 950 727 731 736 740 745 749 754 759 763 768 772 777 782 786 791 795 800 804 809 813 N. L. 1 1 2 1 3 1 4 5 1 6 1 7 1 8 1 9 1 P.P. S/ T.' S." T." S." T." 9' 6.46 373 373 0° I 5'- S 00" 4-68557 558 2° 3^ ^'=92 ^0" 4.68543 587 10 373 373 I 6= c )6o 557 558 2 3 2 3 5=93 3 = 93 30 543 587 So 543 587 90 368 383 2 1 = 9C )oo 544 585 91 368 383 2 -i I =9C )6o 544 585 2 3 7 =94 20 542 588 92 367 383 2 3 2=91 20 543 586 2 3 ^ = 94 So 542 588 4.0 542 588 94 367 383 2 3 3 = 91 80 543 586 2 3 :^=95 95 367 384 2 3 4 = 9: 'AO 543 587 N. L. 1 1 1 2 1 3 1 4 1 5 6 1 7 1 8 1 9 p.pr 950 97 772 777 782 786 791 1 795 800 804 809 813 95' 818 823 827 832 836 841 845 850 855 859 952 864 868 873 877 882 886 891 896 900 905 9S3 909 914 918 923 928 932 937 941 946 950 954 955 959 964 968 973 978 982 987 991 996 955 98000 •"^046 00^ 009 014 019 023 028 032 037 041 956 050 055 05,9 064 068 073 078 082 087 957 091 096 100 105 109 114 118 123 127 132 9S8 137 141 146 150 155 159 164 168 173 177 959 980 961 182 i85 191 195 200 204 254 214 218 223 5 227 272 232 236 241 245 250 _^59_ 304 263 308 268 "~3iT 277 281 286 290 295 299 962 3'8 322 327 331 336 340 345 349 354 358 I "•5 9^^ 964 408 367 412 372 417 376 421 381 426 385 430 390 435 394 439 399 444 403 448 2 3 4 5 6 I.O 1-5 96s 453 457 462 466 471 475 480 484 489 493 2-5 3.0 966 498 502 507 5" 516 520 525 529 534 538 967 543 547 552 556 561 565 570 574 579 583 7 3.5 968 S88 592 597 601 605 610 614 619 623 628 8 4.0 969 970 971 632 637 641 646 650 655 659 664 668 673 9 4.5 677 682 686 691 695 700 704 709 7'3 717 722 726 731 735 740 744 749 753 75^ 762 972 767 771 776 780 784 789 793 798 802 807 973 811 816 820 825 829 834 838 843 847 851 974 856 860 865 869 874 878 883 887 892 896 1 975 900 905 909 914 « 918 923 927 932 936 941 i 97(3 945 949 954 958 963 967 972 976 981 985 1 977 989 994 998 *oo3 *oo7 *OI2 *oi6 *02I ♦025 ♦029 1 978 99034 038 043 047 052 056 o6j 065 069 074 979 980 981 078 083 087 092 096 100 105 109 114 118 A 123 127 131 136 140 145 149 154 158 162 167 171 176 180 185 189 193 198 202 207 0.4 08 q82 211 216 220 224 229 233 238 242 247 251 983 255 260 264 269 273 277 282 286 291 295 3 4 1.2 984 300 304 308 3>3 317 322 326 330 335 339 1.6 98s 344 348 352 357 361 366 370 374 379 383 5 2.0 986 388 392 396 401 405 410 414 419 423 427 6 2.4 9S7 432 436 441 445 449 454 458 463 467 471 7 2.8 988 476 480 484 489 493 498 502 506 5" 515 8 3-2 989 990 991 520 524 528 533 537 542 546 550 555 559 9 3.6 564 568 572 577 581 ,585 590 594 599 603 • 607 612 616 621 62s 629 634 638 642 647 9^2 651 656 660 664 669 673 677 682 686 691 993 695 699 704 708 712 717 721 726 730 734 994 739 743 747 752 756 760 765 769 774 778 995 782 787 791 795 800 804 808 813 817 822 996 82a 830 835 839 •843 848 852 856 861 865 997 870 874 878 883 887 891 896 900 904 909 qq8 913 917 922 926 930 935 939 944 948 952 1 999 lOOC 957 961 965 970 974 978 983 987 991 996 00000 004 009 013 017 022 026 030 035 039 N. 1 L. 1 I 2 1 3 4 1 5 1 6 1 7 1 8 1 9 1 p.p. S.' T.' S." T.' S/' T." 9' 6.46373 373 0° 15'= 900" 4.68557 55^' 2° 41'= c )66o" 4.68542 589 10 373 373 16 = 960 557 55^ 2 42= c > 2 43= c )720 541 590 ?78o 541 590 ?840 541 590 95 367 384 98 367 384 17=1020 557 55^ 2 38=9480 542 58^ \ 2 44 = < 99 367 385 2 39 =9540 542 58^ \ 2 45 = < J906 541 591 100 366 385 2 40 =9600 542 58c J 2 46= < ^960 541 591 2 41 =9660 542 58c ) 2 47=i( D020 540 592 N. L. 1 2 3 1 4 5 6 7 8 9 1000 lOOI I002 1003 0000000 0434 0869 1303 1737 21 7 1 2605 3039 3473 3907 4341 8677 001 3009 4775 9111 3442 5208 9544 3875 5642 9977 4308 6076 ♦0411 4741 6510 *o844 5174 ^^943 *I277 5607 7377 *i7io 6039 7810 *2i43 6472 8244 ♦2576 6905 1004 1005 1006 7337 002 ibbi 5980 7770 2093 6411 8202 2525 6843 8635 2957 7275 9067 3389 7706 9499 3821 8138 9932 4253 8569 ♦0364 4685 9001 ♦0796 5116 9432 ♦1228 5548 9863 1007 1008 1009 iOlO lOI I 1012 1013 003 0295 4605 8912 0726 5036 9342 5467 9772 1588 5898 ♦0203 2019 6328 *o633 2451 6759 ♦1063 2882 7190 *I493 3313 7620 ♦1924 3744 8051 *2354 4174 8481 ♦2784 0043214 3644 4074 4504 4933 5363 5793 6223 6652 7082 7512 005 1805 6094 7941 2234 6523 8371 2663 6952 8800 3092 7380 9229 3521 7809 9659 3950 8238 *oo88 4379 8666 ♦0517 4808 9094 *0947 5237 9523 *i376 5666 9951 1014 1015 1016 006 0380 4660 8937 0808 5088 9365 1236 5516 9792 1664 5944 ♦0219 2092 6372 ♦0647 2521 6799 *io74 2949 7227 ♦1501 3377 7655 *I928 3805 8082 *2355 4233 8510 ♦2782 1017 1018 1019 1020 1021 1022 1023 0073210 7478 008 1 742 3^37 7904 2168 4064 8331 2594 4490 8757 3020 4917 9184 3446 5344 9610 3872 5771 *oo37 4298 6198 ♦0463 4724 6624 *o889 5150 7051 *i3i6 5576 6002 6427 6853 7279 7704 8130 8556 8981 9407 9832 0090257 4509 8756 0683 4934 9181 1 108 5359 9605 f533 .5784 ♦0030 1959 6208 *0454 2384 6633 ♦0878 2809 7058 ♦1303 3234 7483 *I727 3659 7907 *2I5I 4084 8332 *2575 1024 1025 1026 010 3000 7239 on 1474 3424 7662 1897 3848 8086 2320 4272 8510 2743 4696 8933 3166 5120 9357 3590 5544 9780- 4013 5967 *0204 4436 6391 *o627 4859 6815 ♦1050 5282 1027 1028 1029 1030 1031 1032 1033 5704 9931 012 4154 6127 *0354 4576 6550 ♦0776 4998 6973 *ii98 5420 7396 *I62I 5842 7818 *2043 6264 8241 ♦2465 6685 S664 ♦2887 7107 9086 *33»o 7529 9509 *3732 7951 8372 8794 9215 9637 *oo59 *048o *090i ♦1323 *I744 *2i65 0132587 6797 014 1003 3008 7218 1424 3429 7639 1844 3850 2264 4271 8480 2685 4692 8901 3105 5"3 9321 3525 5534 9742 3945 5955 ♦0162 4365 6376 *0583 4785 1034 1035 1036 5205 9403 015 3598 5625 9823 4017 6045 *0243 4436 6465 *o662 4855 6885 ♦1082 5274 7305 ♦1501 5693 7725 ♦1920 6ll2 8144 *2340 6531 8564 *2759 6950 8984 ♦3178 7369 »037 1038 1039 I040 104 1 1042 1043 7788 016 1974 6155 8206 2392 6573 8625 2810 6991 9044 3229 7409 9462 3647 7827 9881 4065 8245 ♦0300 4483 8663 ♦0718 4901 9080 *ii37 5319 9498 *i555 5737 9916 0170333 075 • 1168 1586 2003 2421 2838 3256 3673 4090 4507 ' 8677 0182843 4924 9094 3259 5342 95" 3676 5759 9927 4092 6176 *0344 4508 6593 *076i 4925 7010 *ii77 5341 7427 *i594 5757 7844 *20I0 6173 8260 ♦2427 6589 1044 1045 1046 7005 019 1 163 5317 7421 1578 5732 7837 1994 6147 8253 2410 6562 8669 2825 6977 9084 3240 7392 9500 3656 7807 9916 4071 8222 *0332 4486 8637 *0747 4902 9052 1047 1048 1049 1050 ,^67 020 3^r3 7755 4027 8169 ♦0296 4442 8583 ♦071 1 4856 8997 *II26 5270 941 1 *i540 5684 9824 *i955 6099 ♦0238 ♦2369 6513 ♦0652 ♦2784 6927 *io66 ♦3198 7341 •*I479 021 1893 2307 2720 3134 3547 3961 4374 4787 5201 5614 N. L. 1 2 3 4 5 6 7 8 9 S." T." S." T." 2° 46' = 9960 ' 4. 58541 591 2°5i' = 10260" 4.68 540 593 2 47 = 10020 540 592 2 52 = 10320 539 594 2 48 = 10080 540 592 2 53 = 10380 539 594 2 49 = 10140 540 592 2 54 = 10440 539 595 2 50 = 10200 540 593 2 55 = 10500 539 595 23 N. L. 1 1 2 3 1 4 5 6 1 7 1 8 9 1050 1051 1052 1053 021 1893 2307 2720 3»34 3547 3961 4374 4787 5201 5614 6027 0220157 4284 6440 0570 4696 6854 0983 5»09 7267 1396 5521 7680 1808 5933 8093 2221 6345 8506 2634 6758 8919 3046 7170 9332 3459 7582 9745 3871 7994 »054 1055 105b 8406 0232525 6639 8818 2936 7050 9230 3348 7462 9642 3759 7873 ♦0054 4171 8284 ♦0466 4582 8695 ♦0878 4994 9106 *1289 5405 9517 ♦1701 5817 9928 ♦2113 6228 *0339 1057 1058 1059 1060 1061 1062 1063 0240750 4^57 8960 1161 5267 9370 9780 1982 6088 ♦0190 2393 2804 6498 6909 ♦0600 *IOIO 3214 7319 ♦1419 3625 7729 *i829 4036 8139 ♦2239 4446 8549 *2649 025 3059 3468 3878 4288 4697 5^07 5516 9609 3698 7783 5926 6335 *0427 4515 8600 6744 *o836 4924 9008 7154 026 1245 5333 7563 1 654 5741 7972 2063 6150 8382 2472 6558 8791 2881 6967 9200 3289 7375 *ooi8 4107 8192 1064 1065 1066 9416 027 3496 7572 9824 3904 7979 *0233 4312 8387 ♦0641 47'9 8794 *i049 5127 9201 *i457 5535 9609 ♦1865 5942 *ooi6 ♦2273 6350 ♦0423 *268o 6757 ♦0830 *3o88 7165 *i237 1067 1068 1069 I070 1071 1072 1073 028 1644 5713 9777 2051 6119 *oi83 2458 6526 *059o 2865 6932 ♦0996 3272 7339 ♦1402 3679 7745 *i8o8 4086 8152 *22I4 4492 ♦2620 4899 8964 ♦3026 5306 9371 *3432 029 3838 4244 4649 5055 5461 5867 6272 6678 7084 7489 7«95 030 1948 5997 8300 2353 6402 8706 2758 6807 91U 7211 9516 3568 7616 9922 3973 8020 ♦0327 4378 8425 *0732 4783 8830 *ii38 5188 9234 *i543 5592 9638 1074 1075 1076 031 0043 4085 8123 0447 4489 8526 0851 4893 8930 1256 5296 9333 1660 5700 9737 2064 6104 *oi40 2468 6508 *0544 2872 6912 ♦0947 3277 7315 *i350 3681 7719 *i754 1077 1078 1079 1080 1081 1082 1083 0322157 6188 0330214 2560 6590 0617 2963 6993 1019 3367 7396 1422 3770 7799 1824 4173 8201 2226 4576 8604 2629 4979 9007 3031 5382 9409 3433 5785 9812 3835 4238 4640 5042 5444 5846 6248 6650 7052 *io68 5081 9091 7453 ♦1470 5482 9491 7855 8257 034 2273 6285 8659 2674 6686 9060 3075 7087 9462 3477 7487 9864 3878 7888 ♦0265 4279 8289 *o667 4680 8690 *i87i 5884 9892 1084 1085 1080 0350293 4297 8298 0693 4698 8698 1094 5098 9098 1495 5498 9498 1895 5898 9S98 2296 6298 *0297 2696 6698 *o697 3096 7098 *i097 3497 7498 ♦1496 3897 7898 *i896 1087 1088 1089 1090 1 09 1 1092 1093 036 2295 6289 0370279 2695 6688 0678 3094 7087 1076 3494 7486 1475 3893 7885 1874 4293 8284 2272 4692 8683 2671 5091 9082 3070 5491 9481 3468 5890 9880 3867 4265 4663 5062 5460 5858 6257 6655 *o635 4612 8585 7^3 *io33 5009 8982 7451 *i43i 5407 ■9379 7849 8248 038 2226 6202 8646 2624 6599 9944 3022 6996 9442 3419 7393 9839 3817 7791 *o237 4214 8188 ♦1829 5804 9776 1094 1095 1096 0390173 4141 8106 0570 4538 8502 0967 4934 8898 1364 5331 9294 1761 5727 9690 2158 6124 *oo86 2554 6520 ♦0482 2951 6917 *o878 3348 7313 *I274 3745 7709 *i670 1097 1098 1099 MOO 040 2066 6023 9977 2462 6419 ♦0372 2858 6814 ♦0767 3254 7210 *Il62 3650 7605 *i557 4045 8001 ♦1952 4441 8396 *2347 4837 8791 *2742 5232 9187 *3i37 5628 9582 *3532 041 3927 4322 4716 5"! •5506 5900 6295 6690 7084 7479 N. L. 1 2 3 4 5 6 7 8 9 S." T." S." T." 2° 55' = 10500" 4.( 38539 595 3°o'= 10800" 4.68538 597 2 56 = 10560 539 .595 3 I = 10860 537 598 2 57 = 10620 538 596 32= 10920 537 598 2 58 = 10680 538 596 3 3= 10980 537 599 2 59 = 10740 538 597 34=1 1040 537 599 24 3° / M. S'. T'. Sec. 1 S". T". ll 6.46 1 4.68 II I 180 353 412 10800 538 597 181 353 413 10860 537 598 2 182 352 413 10920 537 598 3 183 352 414 10980 537 599 4 184 352 414 1 1 040 537 599 5 185 352 41S moo 537 599 1 i 6 186 351 415 I II 60 536 600 7 187 351 415 1 1 220 536 600 8 188 35 1 416 1 1 280 536 601 9 II 189 351 416 1 1 340 536 601 190 350 417 1 1400 : 535 602 191 350 417 1 1460 1 535 602 12 192 350 418 1 1520 535 603 13 193 350 418 1 1580 535 603 14 194 350 419 1 1 640 534 604 IS 195 349 419 1 1 700 534 604 16 196 349 420 1 1 760 534 605 17 197 349 420 1 1 820 534 605 18 198 349 421 1 1 880 533 606 19 20 21 199 348 421 1 1940 533 606 200 348 422 12000 533 607 201 348 422 12060 533 607 22 202 348 423 1 21 20 532 608 23 203 347 423 12180 532 608 24 204 347 424 12240 532 609 25 20s 347 424 12300 532 bo9 26 206 347 425 12360 531 610 27 207 346 42s 12420 531 610 28 208 346 426 12480 531 611 29 30 31 209 34t) 426 12540 531 611 210 346 427 12600 530 612 211 345 427 12660 530 612 32 212 345 428 12720 530 613 33 213 345 428 12780 530 613 34 214 345 429 12840 529 614 35 215 344 1 429 12900 529 614 36 216 344 430 12960 529 615 37 217 344 430 13020 529 615 ^>i 218 344 431 13080 528 616 39 40 41 219 343 431 13140 13200 528 528 616 617 220 343 432 221 343 432 13260 528 ^'l 42 222 342 433 13320 527 618 43 223 342 434 13380 527 1 618 44 224 342 434 13440 527 1 619 45 22s 342 435 13500 526 j 620 46 226 341 435 13560 526 620 47 227 341 436 13620 526 621 48 228 341 i 436 13680 526 621 49 50 51 229 230 340 340 437 13740 525 622 437 13800 525 622 231 340 j 438 13860 525 i 623 II 52 232 340 1 439 13920 525 623 53 233 339 439 13980 524 624 54 234 339 440 14040 524 625 55 235 339 440 14100 524 i 625 II 56 236 33^ 441 141 60 523 626 57 237 338 441 14220 523 626 S8 238 33ii 442 14280 523 627 59 60 239 240 338 337 443 443- 14340 522 628 14400 522 1 628 II ' M. 1 S'. T'. 1 Sec. 1 S". T". 1 1 6.46 1 4.68 240 337 443 14400 522 ! 628 241 337 444 14460 522 629 2 242 337 444 14520 522 629 3 243 336 445 14580 521 630 4 244 336 446 14640 521 1 631 1 5 245 336 446 14700 521 1 631 1 6 246 33(> 447 14760 520 632 7 247 335 447 14820 520 632 8 248 1 335 : 448 14880 520 633 9 10 II 249 335 i 449 14940 520 634 250 i 334 1 449 251 i 334 ! 450 15000 519 519 634 635 15060 12 252 1 334 1 450 15120 1 519 63s 13 253 333 451 15180 j 518 636 H 254 333 452 15240 i 518 637 15 255 333 1 452 15300 518 637 16 256 332 453 15360 1 517 638 17 257 332 454 15420 1 517 638 18 258 332 454 15480 1 517 639 19 20 21 259 260 261 332 455 15540 1 516 640 331 456 15600 1 516 640 331 456 15660 516 641 22 262 331 457 15720 515 642 23 263 330 457 15780 515 642 24 264 330 458 15840 515 643 25 26s 330 459 15900 514 644 26 266 329 459 15960 5H 644 27 267 329 460 16020 514 645 28 268 329 461 16080 513 646 29 30 31 269 328 461 16140 513 646 270 328 1 462 16200 1 513 647 271 328 463 16260 1 512 648 32 272 327 463 16320 1 512 648 33 273 327 464 16380 1 512 649 34 274 327 465 16440 511 650 35 275 326 465 16500 511 650 36 276 326 ! 466 16560 1 511 651 37 277 326 i 467 16620 1 510 652 38 278 325 467 16680 510 652 39 40 41 279 280 325 325 468 16740 510 653 469 16800 i 509 654 281 324 ' 469 16860 1 509 654 42 282 324 470 16920 509 1 655 1 43 283 324 1 47» 16980 508 656 44 284 323 ! 472 17040 508 656 45 28s 323 j 472 17100 508 657 46 286 323 473 17160 507 658 47 287 i 322 474 17220 507 659 48 288 322 i 474 17280 507 659 49 50 51 289 321 ! 475 17340 506 b6o 290 321 j 476 1740G 506 i 661 291 1 321 1 477 17460 506 1 661 52 292 1 320 i 477 17520 505 j 662 53 293 320 478 17580 505 663 54 294 320 ! 479 17640 505 664 55 29 s 319 479 17700 504 bb4 56 296 319 480 17760 504 665 57 297 319 481 17820 503 666 58 298 318 482 17880 503 666 59 60 299 318 482 17940 503 667 300 317 483 18000 502 668 25 r 1 II. 1 THE LOGARITHMS • OF THE TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE. Formulas for the Use of the Auxiliaries 5 and T. 1. When a is in the Hrst five degrees of the quadrant : log sin a = log a' + S.' log a' = log sin a + cpl SJ log tan a = log a' + 7'.' = log tan a + cpl 7'.' log cot a = cpl log tan a. = cpl log cot a 4- cpl 7'.' log sin a = log a" + S." log a" = log sin a + cpl 5." log tan a = log a" + TJ' = log tan c + cpl 7"." log cot a = cpl log tan a. = cpl log cot a -1- cpl 7." 2. When a is in the last five degrees of the quadrant : log cos a = log(90° - a)' + SJ log(90° — a)' = log cos a + cpl SJ log cot a = log(90° - a)' + TJ = log cot a + cpl y\' log tan a = cpl log cot a. = cpl log tan a + cpl T.' log cos a 3= log(90° - a)" + 6'." log(90°-a)"= log cos a -f cpl 5." log cota = log(90° - a)" + 7V' = log cot a + cpl r." log tan a = cpl log cot a. = cpl log tan a + cpl 7'." a = 90° -(90° -a). / 26 0° // / L. Sin. 1 d. iCpl. S'. 1 Cpl. T'. L. Tan. jc. d. L. Cot. 1 L. Cos. o 646 373 '^0101 — — — 30103 17609 12494 — 0.00000 60 59 60 3-53 627 3.53627 6.46 373 3-53 627 0.00000 120 2 6.76476 17609 12494 3-53627 3-53627 6.76476 323 524 0.00 000 S8 1 80 3 6.94 085 353627 3-53627 6.94 085 3-05915 0.00 000 57 240 4 7.06579 9691 3-53 627 3.53 627 7.06579 9691 7918 66q4 2.93421 0.00 000 S6 300 5 7.16 270 3-53 627 3-53627 7.16 270 2.83 730 0.00 000 55 360 6 7.24 188 6694 3-53 627 353627 7.24 188 2.75812 ^ 0.00 000 54 420 7 7.30882 5800 3-53 627' 3-53627 7.30 882 5800 2.69 118 1 0.00 000 53 480 8 7.36682 3.53627 353627 7.36 682 2.63318 1 0.00 000 52 540 600 9 10 7.41 797 4576 4139 3779 3476 3218 2997 3-53 627 353627 7.41 797 4576 4139 3779 3476 3219 2oa6 2.58 203 0.00 000 51 50 49 746373 3-53627 3-53627 7-46 373 2-53 627 0.00000 660 750 5' 2 3-53627 3-53627 7.50512 2.49 488 0.00000 720 12 7.54291 •3-53 627 3-53627 7-54 291 2.45 709 o.coooo 48 780 13 7-,57 767 3-53627 3.53 627 7-57767 2.42 233 0.00 000 47 840 14 7.60 9S5 3-53628 3-53627 7.60 986 2.39014 0.00 000 46 900 15 7.63 982 2802 3-53 628 3-53627 7.63 982 2803 2633 2.36018 o.co 000 45 960 16 7.66 784 2633 353628 3-53 627 7.66 785 2.33215 0.00 000 44 1020 7 7.69417 2483 3-53 628 3-53 627 7.69418 2.30 582 9-99 999 43 1080 18 7.71 900 2348 2227 2119 3-53 628 3-53627 7.71 900 2348 2228 2119 2.28 100 9.99 999 42 1 140 19 20 21 7.74 248 3-53 628 353627 7-74 248 2.25 752 9-99 999 41 40 39 1200 1260 7-76 475 3.53628 353627 7.76476 2.23 524 9-99 999 7-78594 353628 3-53 627 7-78 595 2.21 405 9-99 999 1320 22 7.80615 3.53628 3.53627 7.80615 2.19385 9.99 999 38 1380 23 7-82 545 1930 1848 353628 3-53627 7.82546 1931 1848 2.17454 9.99 999 37 1440 24 7-84 393 3-53 628 3-53627 7-84 394 2.15 606 9 99 999 36 1500 2S 7.86 166 353628 3-53627 7.86 167 2.13833 9-99 999 35 1560 26 7.87 870 1639 353628 3-53627 7.87871 1639 2.12 129 9.99 999 34 1620 27 7.89 509 1579 1524 1472 1424 3.53 628 3-53626 7.89510 1579 1524 1473 1424 2.10490 9 99 999 V^ 1680 28 7.91 088 3-53 628 3-53.626 7.91 089 2.08911 9-99 999 32 1740 29 30 31 7.92 612 3-53 628 3-53626 7.92613 207387 9.99 998 31 30 29 1800 i860 7.94084 3-53 628 3-53626 7.94 086 2.05 914 9-99 998 7-95 508 3.53628 3-53626 7.95510 2.04 490 9-99 998 1920 32. 7.96887 1336 1297 3.53 628 3-53 626 7.96 889 1336 1297 2.03 1 1 1 9-99 998 28 1980 33 7.98 223 3.53 628 3-53 626 7.98 225 2.01 775 9-99 998 27 2040 34 7.99 520 3.53628 3-53 626 7.99 522 2.00 478 9-99 998 26 2100 3S 8.00 779 353628 3-53626 8.00 781 1.99 219 9.99 998 25 2160 36 8.02 002 1 190 1158 3-53 628 353626 8.02 004 1190 "59 1 128 1.97996 9-99 998 24 2220 37 8.03 192 3.53 628 3-53 626 803194 1.96806 9.99 997 23 2280 3« 8.04 350 3-53 628 3-53 626 804 353 1.95 647 9.99 997 22 2340 2400 39 40 41 S.05 478 1 100 1072 1046 353628 3.53 626 8.05 481 IIOO 1072 1.94 5 19 9-99 997 21 20 19 8.06 578 3-53 628 353625 8.06581 1.93419 9-99 997 2460 8.07 650 3-53 628 3-53 625 8.07 653 1-92347 9-99 997 2520 42 8.08 696 3-53 628 353625 8.08 700 1.91 300 9.99 997 18 2580 43 8.09718 999 976 3-53 629 353625 8.09 722 998 976 1.90 278 9.99 997 17 2640 44 8.10717 3-53 629 3-53625 8.10 720 1.89280 9.99 996 16 2700 4S 8. 1 1 693 3.53 629 3-53625 8. 1 1 696 1.88304 9-99 996 15 2760 46 8.12647 954 934 3.53629 3-53625 8.12651 955 934 1-87349 9 99 996 14 2820 47 8.13 581 3-53 629 3-53 625 8.13585 1.86 415 9.99 996 13 2880 48 8.14495 914 896 877 860 843 812 3-53 629 3-53625 8.14500 H'5 895 878 860 843 828 812 1.85 500 9.99 996 2940 49 50 SI 8.15391 3-53629 353624 8.15395 1.84605 9.99 996 11 10 9 3000 3060 8.16268 3-53 629 3-53 624 I 8.16273 1.83727 9.99 995 8.17128 3-53 629 3.53 624 8.17 133 1.82867 9-99 995 3120 S2 8.17971 3-53629 3-53 624 8.17976 1.82024 9-99 995 8 3180 53 8.18798 3-53629 3-53 624 8.18804 1.81 196 9-99 995 7 3240 S4 8.19 610 782 769 353629 3-53 624 8.19616 797 782 769 756 742 730 1.80384 9-99 995 6 3300 ss 8.20407 3-53 629 3-53 624 8.20413 1-79587 9-99 994 5 3360 56 8.21 189 3-53 629 3 53 624 8.21 195 1.78 805 9.99 994 4 3420 57 8.21 958 TCC 353629 3-53623 8.21 964 1.78036 9.99 994 3 3480 58 8.22713 ;r, 1 3-53 629 3-53623 8.22 720 1.77 280 9.99 994 2 3540 59 60 8.23456 730 3 53 630 3-53623 8.23 462 1.76538 9.99 994 I 3600 8.24 186 3.53 630 3-53 623 8.24 192 1.75 808 9-99 993 L. Cos. d. 1 1 L. Cot. c. d. L. Tan. L. Sin. / 89' 27 I L. Sin. d. I Cpi. s^ i cpl.T^ L. Tan. c d. L. Cot. L. Cos. 3600 3660 3720 3780 3840 3900 3960 4020 4080 4 140 4200- 4260 4320 4380 4440 4500 4560 4620 4680 4740 4800 "486^ 4920 4980 5040 5100 5160 5220 5280 5340 5400 5460 5520 5580 5640 5700 5760 5820 5880 5940 6000 6060 6120 6180 6240 6300 6360 6420 6480 6540 6600 ~66fe 6720 6780 6840 6900 6960 7020 7080 7140 7200 o I 2 3 4 5 6 7 8 9 iO II 12 13 H »5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 8.24 186 8.24 903 8.25 609 8.26 304 8.26 988 8.27661 8.28324 8.28977 8.29 621 8.30 25 ^ 8.30 879 8.31 495 8.32 103 8.32 702 8.33 292 8.33875 8.34 450 8.35018 8.35 578 8.36 131 8.36 678 8.37217 8.37 750 8.38 276 8.38 796 8.39310 8.39818 8.40 320 8.40816 8.41 307 8.41 792 8.42 272 8.42 746 8.43 216 8.43 680 8 44 139 8.44 594 8.45 044 8.45 489 8.45 930 .46 306 8.46 799 8.47 226 8.47 650 8.48 069 8.48 485 8.48 896 8 49 304 8.49 708 8.50 108 ■50 504 8.50 897 8.51 287 8.51673 852055 8.52434 8.52810 8.53 183 8-53 552 8.53919 ■54 282 717 706 695 684 673 663 653 644 634 624 616 608 599 590 583 575 568 560 553 547 539 533 526 520 5H 508 502 496 491 485 480 474 470 464 459 455 450 445 441 436 433 427 424 419 416 411 408 404 400 396 393 390 386 382 379 376 373 369 367 363 3:53 ^30 3-53 630 3-53 630 3-53 630 3.53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3.53 630 3-53 630 353631 3-53631 353631 353631 353631 353631 3-5363' 3-53631 3.53631 353631 3-53 632 3-53 632 3-53 632 353632 3-53632 3-53 632 353632 353632 3-53 632 3-53 632 3-53 633 3-53 633 3-53 633 353633 3-53633 3-53 633 3-53633 3-53 633 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3.53 634 3-53 634 3-53 634 3-53 635 3-53 63 5 3-53 635 3-53 635 3-53 635 3-53635 3-53635 3-53 635 3.53 636 3-53 636 3 53 636 3_5 3 6^6 3-53636 3i53 623 353623 3-53623 3-53 623 3.53 622 353622 353622 353622 353622 353622 3-53621 3-53621 3-53621 3-53621 353621 3.53 620 3.53 620 3.53 620 3-53 620 3-53 620 3-53620 3-53619 3-53619 3-53619 3-53619 3-53619 3-53618 3-53618 3-53618 3-53618 3-53617 3-53617 3-53617 3-53617 3-53617 3-53616 353616 3-5: 616 3-53616 3-53615 3-53615 3-53615 3-53615 3-53614 3-53614 3-53614 3-53614 3-53613 3-53613 3-53613 3-53613 3-53612 3-53612 3-53612 3-53611 3-53611 353611 3-53611 3-53610 3-53 610 •53610 8.24 192 8.24910 8.25616 8.26312 8.26 996 8.27 669 8.28 332 8.28 986 8.29 629 8.30 263 8:30888 8.31 505 8.32112 8.32 7" 8.33 302 8.33 886 8-34461 8.35 029 8.35 590 8.36 143 8.36689 8.37 229 8.37 762 8.38 289 8.38 809 8-39 323 8.39 832 8.40 334 8.40 830 8.41 321 8.41 807 8.42 287 8.42 762 8.43 232 8.43 696 8.44 156 8.44 61 1 8.45 061 8.45 507 8.45 948 8.46 385 8.46817 8.47 245 8.47 669 8.48 089 8.48 505 8.48917 8.49 325 8.49 729 8.50 130 8-50527 8.50 920 8.51 310 8.51 696 8.52079 8.52459 8.52831 8.53 208 8.53578 8.53 94 5 8.54 308 718 706 696 684 673 663 654 643 634 625 617 607 599 591 584 575 568 561 553 546 540 533 527 520 514 509 502 496 491 486 480 475 470 464 460 455 450 446 441 437 432 428 424 420 416 412 408 404 401 397 393 390 386 383 .380 376 373 370 367 363 75808 75090 74384 73688 73004 72331 71 668 71 014 70371 69737 9-99 993 69 112 68T95 67888 67289 66698 66 1 14 65539 64971 64 410 63857 9-99 993 9.99 993 9 99 993 9.99 992 9 99 992 9 99 992 9.99 992 9-99 992 9i99 99i 9.-99 99 y 9.99991 9.99 990 9.99 990 9.99 990 9.99 990 9.99 989 9.99 989 9 99 989 9-99 989 63311 62771 6:2238 61 711 61 191 60 677 60 168 59 666 59170 58679 9.99-9 88 9.99 988 9-99 988 9-99 987 9 99 987 9.99 987 9-99 986 999986 9.99 986 9-99 985 58193 57713 57238 56768 56304 55844 55389 54 939 54 493 54052 9-99 98 5 9-99 985 9-99 984 9.99 984 9.99 984 9.99 983 9-99 983 9.99 983 9.99 982 9.99 982 53615 9.99 982 53183 52755 52331 51 911 5M95 51083 50675 50271 49870 9.99 981 9.99 981 9.99981 9-99 980 9.99 980 9.99979 9.99979 9.99 979 9-99978 49 473 49080 48 690 48304 47921 47541 47 165 46 792 46 422 46055 9.99 9 78 9.99977 9.99977 9.99977 9-99 976 9-99 976 9-99 975 9-99 97l 9.99 974 9-99 974 45692 9-99 974 L. Cos. L. Cot. |c. d. I L. Tan. L. Sin. QQ° 28 2^ n / L. Sin. j d. LCpl. S'. Cpl. T'. 1 L.Tan. c. d. 1 L .. Cot. L. Cos. 7200 7260 I 8.54 282 360 357 355 351 1 3-53 636 3.53610 8.54 308 361 '• 45692 9-99 974 60 S9 8.54 642 3.53 636 3.53 609 8.54 669 301 45331 9-99 973 7320 2 8.54 999 3-53 637 3-53 609 8.55 027 :^^r i I 44 973 9.99 973 58 • 7380 3 8-55 354 3-53 637 3.53609 i, 8.55382 000 I 44 618 9.99972 57 7440 4 8-55 705 3.53637 3.53609 8.55734 352 J 44266 9.99972 S6 7500 S 8.56054 349 346 3.53 637 3-53 608 1 8.56083 349 , 346 , 43917 9.99971 55 7560 6 8.56 400 3-53 637 3-53 608 8.56 429 43571 9.99971 54 7620 7 8.56 743 343 3.53637 3-53 608 8.56773 344 1 T ,1 I i 43227 9.99 970 53 7680 8 8.57084 341 3-53 637 3-53 607 1 8.57 114 341 1 T 338 1 ; 42886 9-99 970 52 7740 7800 7860 9 10 II 8.57421 337 336 332 3-53 638 3-53 607 1 18.57 452 8.57 788 42548 9.99 969 51 50 49 8-57 757 3-53 638 3.53 607 42 212 9.99 969 8.58089 3.53 638 3-53 606 8.58 121 ZZZ 1 J 41 879 9.99 968 7920 12 8.58419 ZIP 328 325 3.53 638 3-53 606 8.58451 330 \ , 328 i 326 ^ 41 549 9.99 968 48 7980 13 8.58 747 3.53 638 3.53 606 8.58 779 41 221 9.99 967 47 8040 14 8.59072 3-53 638 3-53 605 8.59 105 40895 9.99 967 46 8100 15 8.59 391 323 320 318 316 3-53 639 3-53 605 8.59428 323 I 40572 9.99 967 45 8160 16 8.59715 3-53 639 3-53605 !| 8.59749 321 J 40251 9.99 966 44 8220 17 8.60 033 3-53 639 3.53 604 \ 8.60 068 319 , 316 ; 39932 9.99 966 43 8280 18 8.60 349 3-53 639 3.53 604 i 8.60 384 39616 9.99 965 42 J340 8400 ~876o 19 20 21 8.60 662 zn 3" 309 3.53 639 3-53 604 I 8.60 698 3H I 39302 9.99 964 41 40 39 8.60973 3.53 639 3-53 603 8.61 009 31 ' J 38991 9.99 964 8.61 282 3-53 640 3-53 603 i 8.61 319 310 35681 9.99 963 8520 22 8.61 589 j ^^/ 3-53 640 3.53 603 i 8.61 626 307 , 38 374 9-99 963 38 8580 23 8.61 894 ! -^^s 3-53 640 3-53 602 ; 8 61 931 305 I 38069 9.99 962 Zl 8640 24 8.62 196 ! 302 3-53 640 3.53 602 j 8.62 234 303 J 37 766 9 99 962 36 8700 25 8.62 497 298 296 3-53 640 3.53602 i 8.62 535 301 I 299 1 I .37465 9.99 961 35 8760 26 8.62 795 3-53 640 3-53 601 i 8.62 834 3716b 9.99961 34 8820 27 8.63 091 3-53 641 3.53601 i!. 8.63 131 297 1 I _ ! * 36869 9.99 960 33 8880 28 8.63 385 294 3.53641 3.53601 i 8.63426 \^yji\\ 36574 9.99 960 32 8940 9000 9060 29 30 31 8.63 678 293 3-53641 3.536001! 8.63718 ^9^ 1 I 36 282 9-99 959 31 30 29 8.63 968 290 288 287 284 283 281 3-53 641 353600 j ,8.64009 291 35991 9-99 959 8.64 256 3-53641 3.53599 8.64 298 287 ; 285 I 284 281 , 280 I 278 , 276 I 35702 9.99 958 9120 32 8.64 543 3.53 642 3-53 599 8.64 585 35415 9-99 958 28 9180 ZZ 8.64 827 3-53 642 3.53 599 1 8.64 870 35 130 9.99 957 27 9240 34 8.65 no 3-53 642 3-53 598 i 8.65 154 34846 999956 26 9300 35 8.65 391 3-53 642 3.53598 {j 8.65435 34565 9.99 956 25 9360 36 8.65 670 ^yy 3-53 642 3.53598 j 8.65 715 34285 9-99 955 24 9420 37 8.65 947 2y; 276 3.53 642 3-53 597 il 8.65993 34007 9.99 955 23 9480 38 8.66 223 3-53643 3-53 597 1 8.66 269 33731 9-99 954 22 9540 9600 9660 39 40 41 8.66497 2;4 272 270 269 3-53 643 3.53 596 i 8.66 543 274 I 33 457 9-99 954 21 20 19 8.66 709 3-53 643 353596 ': 8.66816 273 I 33 184 9.99 953 8.67 039 3-53 643 3.53596 \\ 8.67087 271 ^ 269 I 268 , 266 264 , 263 , 32913 9.99952 9720 42 8.67 308 3-53 643 3-53 595 ' 8.67356 32644 9.99952 18 9780 43 8-67 575 ^^7 8.67841 ^f 8.68 104 i 263 8.68 367 1 263 3-53 644 3-53 595 il 8.67624 32376 9-99951 17 9840 44 3-53 644 3-53 594 ;| 8.67890 32 no 9.99951 16 9900 45 3.53 644 3.53594:! 8.68154 31 846 9-99 950 15 9960 46 3.53 644 3-53 594 : 8.68417 31583 9 99 949 14 10020 47 8.68627 260 3.53 644 3-53 593 1 8.68 678 261 260 I 258 , 31322 9 99 949 13 looSo 48 8.68 886 259 3-53 645 3.53593 ji 8.68938 31 062 9.99 948 12 10140 10200 10260 49 50 51 8.69 144 256 254 3-53 645 3.53592 1! 8.69196 30804 9.99 948 II 10 9 8.69 400 3.53 645 3-53592 !| 8.69453 257 J 30 547 r 9-99 947 8.09 054 3-53 645 3.53592 !l 8.69708 255 J 30292 9.99946 10320 52 8.69907. 253 3-53 646 3.53591 8.69962 254 I 30038 9-99 946 8 10380 53 8.70159 252 3.53 646 3-53 591 ! 8.70214 252 1 29786 9-99 945 7 10440 54 8.70409 250 3-53 646 3-53590 1' 8.70465 1^0 ' 29535 9.99 944 6 10500 55 8.70658 249 3-53 646 3.53590 !i 8.70714 249 I 248 I 2(19286 9-99 944 5 10560 56 8.70 905 1 247 3.53 646 3-53589 ,; 8.70962 29038 9-99 943 4 10620 57 8.71 151 24b 3-53 647 3.53 589!! 8.71208 246 J 28792 9-99 942 3 to68o S8 8-71 395 244 3-53 647 .3-53589 li 8.71453 245 I 28547 9-99 942 2 10740 10800 59 60 8.71 638 243 3 53 647 3.53 588 ' 8.71 697 j 8.71 940 244 I 28. 303 9-99 941 I 8.71 880 .4. 3.53 647 3-53 588 243 J 28060 9.99 940 L. Cos. d. 1 1! L. Cot. c. d. 1 L .Tan. L. Sin. t 87° 29 L. Sin. L. Tan. c.d. L. Cot. L. Cos. P.P. 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O I 2 3 4 5 b 7 8 9 iO II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 3« 39 40 41 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 56 57 5« 59 60 8.94 030 8.94 '74 J^.94317 8.94 461 8.94 603 8.94 746 8.94 887 8.95 029 8.95 170 8^5310 8-95_4|o_ 8.95 5^9 8.95 728 8.95 867 8.96 005 8.96 143 8.96 280 8.96417 ^•96553 8.96 689 8.96 825 8.9b 960 8.97 095 8.97 229 8.97 363 8.97 49b 8.97 629 8.97 762 8.97 894 8.98 026 8.98157 8.98 288 8.98419 8.98 549 8.98 679 8.98 808 8.98937 8.99 066 8.99 194 8.99 322 8.99 450 8.99 577 8.99 704 8.99 830 8.99956 9.00 082 9.00 207 9.00 332 9.00456 9 00 581 9.00 704 9.00 828 9.00951 9.01 074 901 196 9.01 318 9.01 440 9 01 561 9.01 682 901 803 9.01 923 144 H3 144 142 143 141 142 141 140 140 139 139 139 138 138 ^37 137 136 136 136 135 135 134 134 133 133 133 132 132 131 131 131 130 130 129 129 129 128 128 128 127 127 126 126 126 125 125 124 125 123 124 123 123 122 122 122 121 121 121 120 8.94 195 8.94 340 8.94 485 8.94 030 8.94 773 8.94917 8.95 000 8.95 202 8.95 344 8.95 4«6 8-95 ^27 8.95 767 8.95 908 8.96 047 8.96 187 8.96 325 8.96 464 8.96 602 8.96 739 8.96 877 8.97013 8.97 150 8.97 285 8.97421 8-97 556 8.97 691 8.97825 8.97 959 8.98 092 8.98 225 8.98 358 8.98 490 8.98 622 8.98 753 8.98 884 8-99015 8.99 145 8.99 275 8.99 405 8.99 534 8.99 662 8.99 791 8.99919 9.00 046 9.C0 174 9.C0301 9.00427 9-00553 9.C0 b79 9.00 805 9.00 930 9.0 J 055 9.01 179 9.01 303 9.01 427 9 01 550 9.01 673 9.01 796 9.01 918 g. 02 040 9.02 ib2 M5 •45 145 H3 144 M3 142 142 142 141 140 141 139 140 138 139 138 137 ^3^ 136 137 135 136 135 135 134 134 133 ^33 133 132 132 131 131 131 130 130 130 129 128 129 128 127 128 127 126 126 126 126 125 125 124 124 124 123 123 123 122 122 122 1.05805 1 .05 660 105 5»? 1.05 370 1.05 227 1.05083 1 .04 940 1 .04 798 1.04 656 104373 9-99 834 60 9.99 ii33 9.99 832 9.99831 9.99 830 9.99 829 9.99 828 9.99 827 9.99 825 9.99 824 9 -99 823 1.04233 1 .04 092 1-03953 1.03 813 1-03673 I;03 53^ 1.03398 1.03 261 1.03 123 9.99 822 9.99 821 9.99 820 9.99819 9.99817 9.99 816 9.99815 9.99814 9-99813 1.02 987 9.99 812 1.02 850 1.02 715 1.02579 1.02444 1.02 309 1-02 175 1.02 041 1. 01 908 i.oi 775 9.99 810 9-99 809 9.99 808 9-99 807 9.99 806 9.99 804 9.99 803 9.99 802 9.99 801 I.OI 642 9.99 8CO I.OI 510 I.OI 378 I.OI 247 I.OI 116 1.00985 1.00855 1. 00 725 1. 00 595 1 .00 466 .00338 1 .00 209 1 .00 08 1 0.99 954 0.99 826 0.99 699 0.99 573 0.99 447 0.99321 0.99 195 0.99 070 0.98 945 0.98821 0.98 697 0.98 573 0.98 450 0.98 327 0.98 204 0.98 082 0.97 960 0.97 838 L. Cos. I d. I L. Cot, led. L. Tan. | L. Sin. | ■99 798 ■99 797 •99 796 ..99795 9-99 793 9.99 792 9.99 791 9.99 790 9-99 788 9-99 787 , -99 786 9-99 78I 999 783 ,-99782 9.99781 9.99 780 9-99 778 9.99 777 9-99 776 999 775 9.99 773 9-99 772 9.99771 9-99 769 9.99 768 9-99 767 9-99 765 9.99 764 •99 763 9.99 761 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 147 147 294 44.1 588 73 5 88.2 102.9 117. 6 132.3 146 146 29 2 438 58.4 73 o 87.6 102. 2 I16.8 I3I4 I 143 14.2 2 28.6 284 S 42.9 426 4 57-2 56.8 t^ 71-5 71.0 6 85.8 85.2 7 100. 1 99.4 8 "4 4 113. 6 9 128.7 127.8 13-9 •27.8 41.7 55.6 69.5 834 97-3 III. 2 125.1 135 13-5 27.0 40-5 54-0 67.5 94.5 108.0 131 I3-I 26.2 39-3 52.4 655 78.6 91 7 104.8 117.9 127 12.7 25-4 38.1 50.8 635 76 2 88.9 loi 6 114.3 123 24.6 369 49.2 615 73-8 86. 1 "•4 138 27 6 41.4 552 69 o 82.8 966 1 10.4 124 2 145 39.0 43 5 58.0 725 870 101.5 1160 130.5 14 I 28.2 423 564 70-5 84.6 98.7 112.8 126.9 137 137 27.4 41. 1 54.8 68.5 82.2 95 9 109.6 123 3 14-4 288 576 72 o 864 X00.8 115.3 129.6 140 14.0 280 42.0 56 o 70.0 84.0 980 112.0 126.0 136 13-6 27 2 40.8 54 4 68.0 81 6 95-2 108.8 134 133 132 13-4 26.8 40.2 . 53.6 67 o 80.4 93-8 107 2 120.6 130 130 26.0 390 52.0 65.0 78.0 91. o 104.0 117.0 126 12.6 25 2 37.8 50-4 63 o 75-6 88.2 100.8 II3-4 12.2 24.4 36.6 48.8 61.0 73-2 854 97.6 133 26.6 39 9 53-2 665 798 93 I 106 4 119.7 129 12.9 258 387 516 64.5 77-4 90-3 103 2 116.1 125 12.5 25 o 37 5 50 o 62 5 750 87.5 100 o 112.5 12. t 24 2 363 484 605 72.6 847 96.8 13-2 26.4 396 52.8 660 79.2 92 4 105 6 128 128 25 6 38.4 51 2 64.0 76.8 89.6 102 4 115-2 124 12.4 24.8 37 2 496 62 o 74-4 86 8 99.2 111.6 12 o 24.0 36 o 48.0 60.0 72.0 840 96.0 110.7 109.8 108 9 108.0 P. P. Qy1° 32 ■^ 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 2>7 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. 9.01 923 9.02 043 9.02 163 9.02 283 9.02 402 9.02 520 9.02639 9.02 757 9.02 874 9.02 992 903 109 9.03 226 9-03 342 9-03 458 903574 9.03 690 9.03 805 9.03 920 9.04 034 9.04 149 9.04 262 9.04 376 9.04 490 9.04 603 9.04715 9.04 828 9.04 940 9.05 052 9.05 164 9-05 27? 9-05 386 9-05 497 9.05 607 9.05 717 9.05 827 905 937 9.06 046 9.06155 9.06 264 9.06 372 9.06481 9.06 589 9.06 696 9.06 804 9.06 911 9.07018 9.07 124 9.07 231 907 337 9.07 442 9-07 548 9-07 653 9.07 758 9.07 863 9.07 968 9.08 072 9.08 176 9.08 280 9.08 383 9.08 486 9.08 589 L. Cos. i«^1 L. Tan. 9.02 162 9.02 283 9.02 404 9.02 525 9.02 645 9.02 766 9.02 885 9.03 005 9.03 124 9.03 242 903 361 9.03 479 9-03 597 9.03 714 9.03 832 9.03 948 9.04 065 9.04 181 9.04 297 9.04413 9.04 528 9.04643^ 9.04 758 9.04 873 9.04 987 9.05 lOI 9.05 214 9.05 328 9.05 441 9-05 553 9.05 666 9-05 778 9.05 890 9.06 002 9.06 113 9.06 224 906 335 1.06 445 9.06 556 9.06 666 9-o6 775 9.06 885 9.06 994 9.07 103 9.07 211 9.07 320 9.07 428 9.07 536 9.07 643 9.07751 9.07 858 9.07 964 9.08071 9.08177 9.08 283 9.08 389 9.08 495 9.08 600 9.08 705 9.08 810 9.08914 L. Cot. c.d. 21 21 21 20 21 19 20 19 i8 19 18 18 17 18 16 17 16 16 16 15 15 15 15 14 14 13 H 13 12 13 12 12 12 II II II 10 II 10 09 10 09 09 08 09 08 08 07 08 07 06 07 06 06 06 06 05 05 05 04 cTd! L. Cot. 0.97 838 0.97717 o 97 596 0.97 475 0-97 355 0.97 234 0.97115 0.96 995 0.96 876 0.96 758 0.96 639 0.96 521 0.96 403 o 96 286 0.96 168 0.96052 0-95 935 0.95 819 0.95 703 0-95 587 095472 0.95 357 0.95 242 0.95 127 0.95013 0.94 899 0.94 786 0.94 672 0.94 559 0.94 447 0-94 334 0.94 222 0.94 no 0.93 998 0.93 887 0.93 776 0.93 665 0.93 555 0.93 444 0.93 334 0.93 225 0.93115 0.93 006 0.92 897 0.92 789 0.92 680 0.92572 0.92 464 0.92 357 0.92 249 0.92 142 0.92 036 0.91 929 0.91 823 0.91 717 0.91 611 0.91 505 0.91 400 0.91 295 0.91 190 0.91 086 L. Tan. L. Cos. 9»9" 761 9.99 760 9-99 759 9.99 757 9-99 755 9-99 755 9-99 753 9-99 752 9-99751 9.99 749 9-99 748 9-99 747 9-99 745 9.99 744 9.99 742 9.99 741 9.99 740 9-99 738 9-99 737 9.99 73(> 9-99 734 9-99 1Z2> 9-99 73' 9-99 730 9.99 728 9.99 727 9.99 726 9.99 724 9-99 723 9 -99 72' 9.99 720 9.99718 9.99717 9.99 716 9.99714 9-99713 9.99711 9.99 710 9.99 708 9.99 707 ■99 705 9.99 704 9.99 702 9.99 701 9.99 699 9.99 698 9.99 696 9.99 695 9-99 693 9-99 692 9.99 690 9.99 689 9.99 687 9.99 686 9.99 684 9.99 683 9.99 681 9.99 680 9.99 678 9.99 677 ^99 675 L. Sin. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 ^^ 37 36 35 34 Z2> 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 I O P.P. 121 120 119 118 I 12. 1 12.0 j;:i 11.8 2 24.2 24.0 2^.6 3 36.3 36.0 35.7 ^S-4 4 4«.4 48.0 47-b 47.2 5 bo.5 60.0 59-5 59-0 b 72.6 72.0 71.4 70.8 7 «4-7 84.0 «S.^ 82.6 « 96.8 96.0 95.2 94.4 9 108.9 108.0 107. 1 106.2 117 116 115 114 I 11.7 11.6 "•5 II. 4 2 23.4 23.2 23.0 22.8 3 4 J?:s 34.8 46.4 34.5 46.0 34-2 4,S.6 5 6 7 58.5 70.2 81.9 58.0 69.6 81.2 69.0 80.5 79.8 8 93-6 92.8 92.0 91.2 9 105.3 104.4 103-5 102.6 "•3 22.6 33-9 45-2 56.5 67.8 79.1 90.4 101.7 109 10.9 21.8 32.7 43-6 54-5 65-4 76.3 87.2 22.4 33-6 44.8 56.0 67.2 78.4 89.6 100.8 108 10.8 21.6 32.4 43.2 54-0 648 75-6 86.4 97.2 22.2 33-3 44.4 55-5 66.6 77-7 107 10.7 21.4 32.1 42.8 53-5 64.2 74-9 85.6 96.3 22.0 33-0 44.0 55-0 66.0 77.0 106 10.6 21.2 31.8 42.4 53-0 63.6 742 84.8 95-4 105 1 I 10.5 2 I 21.0 3 j 31-5 4 i 42.0 5 j 52.5 63.0 73-5 84.0 94-5 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93-6 P. P. 103 10.3 20.6 309 41.2 51-5 61.8 72.1 82.4 92.7 • 83° 33 / L. Sin. d. L. Tan. c.d. L. Cot. L. Cos. P. P. I 9.08 589 103 9^08914 9.09019 105 0.91 086 9.99 b75 60 59 9.08 692 0.90981 9-99 674 2 9.08 795 I "3 9.09123 104 0.90 877 9-99 ^>72 58 105 104 103 3 9.08 897 9.09 227 104 0.90 773 9.99 670 57 I 10.5 10.4 10.3 4 9.08 999 9-09 330 103 0.90 670 9-99 (^('9 5^' 2 21.0 20.8 20.6 5 9.09 10 1 9-09 434 103 0.90 566 9.99 667 55 3 315 31.2 30-9 6 9.09 202 909 537 0.90 463 9.99 666 54 4 42.0 41.6 41.2 7 9.09 304 9.09 640 103 0.90 360 9.99 664 S3 5 52-5 52.0 5^-5 8 9.09 405 9.09 742 0.90 258 9.99 663 52 6 63.0 62.4 61.8 9 10 9.09 500 9.09 606 9^9707 100 lOI 9.09 845 9.09 947 9.10049 103 102 102 0.90 155 9.99 661 5^ 50 49 7 8 9 73-5 84.0 94-5 72.8 83.2 93-6 72.1 82.4 92.7 0.90053 9-99 659 0.89951 9.99 658 12 9.09 807 9.10 150 0.89 850 9-99 656 48 13 9.09 907 9.10 252 0.89 748 9-99 655 47 H 9.10006 99 9-10353 lOl 101 0.89 647 9-99 653 46 102 101 99 ^S 9.10 106 9.10454 0.89 546 9.99651 45 I 10.2 10.1 9-9 lb 9.10 205 99 9-10555 0.89 445 9-99 650 44 2 20.4 20.2 19.8 17 9.10304 99 98 9.10656 0.89 344 9.99 648 43 3 30.6 30.3 29.7 18 9.10 402 9.10756 0.89 244 9-99 647 42 4 40.8 40.4 39-6 19 20 21 22 9.10 501 99 98 98 98 98 9.10856 100 100 99 0.89 144 9-99 645 41 40 39 38 5 6 51.0 61.2 71.4 81.6 50-5 60.6 70.7 80.8 90.9 49-5 79-2 89.1 9.10599 9.10956 0.89 044 9.99 643 9.10697 9.10795 9.11 056 9" 155 0.88 944 0.88 845 9.99 642 9.99 640 23 9.10893 9.1 1 254 99 0.88 74b 9.99 638 37 9 y..^ 24 9.10990 97 9-II353 99 0.88 647 9.99 637 36 2S 9. 1 1 087 97 9.11452 99 0.88 548 999635 35 26 9.1 1 184 97 9-11551 99 98 98 98 98 97 98 97 97 96 0.88 449 9-99 (>33 34 98 97 96 27 9.11 281 9; 96 9. 1 1 649 0.88351 9.99 632 33 I 9-8 9-7 9.6 28 9-" 377 9.11 747 0.88 253 9-99 630 32 2 19.6 19.4 19.2 29 30 31 32 33 34 9. 1 1 474 9y 96 96 95 96 95 9.11 845 0.88155 9.99 629 31 30 29 28 27 26 3 4 5 6 7 8 9 29.4 39.2 49.0 58.8 68.6 29.1 38.8 48.5 58.2 67.9 77.6 87-3 28.8 38.4 48.0 57-6 67.2 76.8 86.4 9.11 570 9. 1 1 943 0.88057 9.99 627 9.11 666 9.1 1 761 9.11857 9.11952 9. 1 2 040 9.12 138 9.12235 9.12332 0.87 960 0.87 862 0.87 765 0.87 668 9.99 625 9.99 624 9.99 622 9.99 620 3S 9.12047 95 9.12428 0.87572 9.99 6x8 25. 36 9.12 142 95 9.12525 97 96 96 96 96 95 0.87 475 9.99617 24 37 9.12236 94 9.12 621 0.87 379 9.99615 23 95 94 93 3« 9-12331 95 9.12717 0.87 283 9.99613 22 39 9.12425 94 9.12813 0.87 187 9.99 612 21 I 9-5 9-4 9-3 40 41 9.12 519 94 93 9.12909 0.87091 9.99 610 20 19 2 3 4 5 6 19.0 28.5 38.0 47-5 57-0 18.8 28.2 37.6 47.0 56.4 18.6 27.9 37-2 46.5 55-8 9.12612 9.13004 0.86 996 9.99 608 42 43 9.12 706 9.12799 94 93. 93 9.13099 9-13 194 95 95 95 0.86 901 0.86 806 9.99 607 9.99 605 18 17 44 9.12 892 9.13289 0.86 711 9-99 603 16 7 66.S 65.8 65.1 4S 9.12985 93 9- 1 3 384 95 0.86616 9.99 601 IS 8 76.0 75-2 74-4 46 9.13078 93 9-13478 94 95 0.86 522 9.99 600 14 9 85.5 84.6 83-7 47 913 171 93 9-13573 0.86427 9-99 598 13 48 9.13263 92 9.13667 94 0.86 333 9-99 596 12 49 50 913355 92 92 92 9.13761 94 93 94 93 0.86 146 9-99 595 11 10 QQ 91 9.1 18 2 90 9-13447 9-13854 9-99 593 I 2 51 9-13539 9.13948 0.86052 9-99591 9 9-2 18.4 27.6 36.8 9.0 180 S2 9.13630 91 9. 14 041 0.85 959 9-99 589 8 3 4 27.3 36.4 27.0 36.0 53 9.13722 92 91 9.14134 93 93 93 0.85 866 9.99 588 7 54 9-13813 9.14227 0.85 773 9-99 586 6 S 46.0 45-5 45 -o 55 9.13904 91 9.14320 0.85 680 9-99 584 s 6 55-2 54-6 54.0 5^ 9-13994 90 9.14412 92 0.85 588 9.99 582 4 7 64.4 63-7 63.0 S7 9.14085 91 9.14504 92 0.85 496 9.99581 3 8 73.6 72.8 72.0 5« 914 175 90 9-14597 93 91 92 0.85 403 9-99 579 2 9 82.8 81.9 81.0 59 60 9.14266 91 90 9.14688 0.85 312 9-99 577 9.14356 9.14780 0.85 220 9-99 575 _ L. Cos. d. L. Cot. c.d.| L. Tan. L. Sin. / P.P. 1 82' 34 ' 1 L. Sin. 1 d. L. Tan. |c.d. | L. Cot. | L. Cos. | P. P. O I 9.14356 89 90 89 90 .89 8X 9.14 780 92 0.85 220 9-99 575 60 59 9-14 445 9.14872 0.85 128 9.99 574 2 9-14 535 9.14963 91 91 91 0.85 037 9.99 572 S8 92 91 90 3 9.14 624 9-15054 0.84 946 9.99 570 57 I 9.2 9.1 9.0 4 9.14714 9.15 145 0.84 855 9-99 568 56 2 18.4 18.2 18.0 5 9.14803 9.15 236 91 91 0.84 764 9-99 566 55 3 27.6 27.3 27.0 6 9.14 891 89 89 88 88 88 88 9.15327 0.84 673 9.99 565 54 4 36.8 36.4 36.0 7 9.14980 9.15 417 90 0.84 583 9.99 563 53 5 46.0 45.5 45.0 b 9.15069 9.15 508 91 0.84 492 9-99 561 52 6 55.2 54.6 540 9 10 II 915 157 9.15 598 90 90 89 90 89 0.84 402 9.99 559 51 50 49 7 8 9 64.4 63.7 63.0 73.6 72.8 72.0 82.8 81.9 81.0 9-15245 9.15 688 0.84 3 1 2 9.99 557 9-15333 9.15777 0.84 223 9-99 556 12 9.15421 87 88 87 87 87 87 86 86 87 86 9.15867 0.84 133 9.99 554 48 13 9.15508 9.15956 0.84 044 9.99552 47 H 9-15596 9.16046 90 89 89 88 89 88 88 88 88 0.83 954 9-99 550 46 89 88 »5 9.15683 9.16135 0.83 865 9.99 548 4S I 8.9 8.8 10 9.15770 9.16224 0.83 776 9.99 546 44 2 17.8 17.6 ^7 9-15857 9.16312 0.83 688 9-99 54? 43 3 26.7 26.4 i8 9-15944 9.16401 0.83 599 9-99 543 42 4 35-6 35.2 19 20 21 9.16 OJO 9.16489 0.83 51 1 9.99 541 41 40 39 7 8 44.5 44.0 53-4 52.8 62.3 61.6 71.2 70.4 80.1 79.2 9.16 116 9.16577 0.83 423 9.99 539 9.16 203 9.16 665 0.83 335 9.99 537 22 23 9.16 289 9.16374 85 86 85 86 85 85 85 84 85 84 84 84 84 83 84 S3 83 83 83 83 82 82 9-16753 9.16 841 88 87 88 87 87 87 86 87 86 86 0.83 247 0.83 159 9.99 535 9.99 533 38 37 9 24 9.16460 9.16928 0.83 072 9.99 532 36 25 9.16545 9.17016 0.82 984 9-99 530 3S 25 9-i6 631 9.17 103 0.82 897 9.99 528 34 87 86 85 27 9.16 716 9.17 190 0.82810 9-99 526 3^ I 8.7 8.6 8.5 28 9.16 801 9.17277 0.82 723 9.99 524 32 2 17.4 17.2 17.0 29 30 31 9.16886 9.17363 0.82 637 9.99 522 31 30 29 3 4 5 6 26.1 25.8 25.5 34.8 34.4 340 43.5 43.0 42.5 52.2 51.6 51.0 60.9 60.2 59.5 69.6 68.8 68.0 9.16970 9-17450 0.82 550 9.99 520 9-17055 9- '7 536 0.82 464 9.99518 32 9.17 139 9.17622 86 0.82 378 9-99517 28 7 ss 9.17223 9.17708 86 0.82 292 9.99515 27 8 34 9.17307 9.17794 86 0.82 206 9.99513 26 9 78.3 77-4 76.5 35 9-1739' 9.17880 85 86 85 85 85 85 84 85 84 84 84 84 83 84 83 83 ^3 83 82 82 82 82 82 0.82 120 9.99511 25 3b 9.17474 9.17965 0.82 035 9-99 509 24 37 9-17558 9.18051 0.81 949 9-99 507 23 84 83 3^ 9.17641 9.18136 0.81 864 9-99 505 22 39 40 41 42 43 9.17724 9.18 221 0.81 779 9.99 503 21 20 18 17 I " 2 3 4 5 6 0.4 0.3 16.8 16.6 25.2 24.9 33.6 33.2 42.0 41.5 50.4 49.8 9.17807 9.18 306 0.81 694 9-99 501 9.17890 9.17973 9.18055 9.18391 9.18475 9.18560 0.81 609 0.81 525 0.81 440 9.99 499 9.99 497 9-99 495 44 45 9.18137 9.18 220 82 81 82 82 81 81 81 81 81 9.18644 9.18728 0.81 356 0.81 272 9.99 494 9.99 492 16 IS I 58.8 58.1 67.2 66.4 4t) 9.18302 9.18812 0.81 188 9.99490 14 9 75-6 74-7 47 9.18383 9.18896 0.81 104 9.99 488 n 48 9.18465 9.18979 0.81 021 9 99 486 12 49 50 5' 9-18547 9.19063 0.80 937 9-99 484 II 10 9 fio. fii fin 9.18628 9.19 146 0.80 854 9.99482 I 2 8.2 8.1 8.0 16.4 16.2 16.0 24.6 24.3 24.0 32.8 32.4 32.0 9.18709 9.19229 0.80771 9.99 480 52 53 9.18790 9.18871 9.19312 9.19395 0.80 688 0.80 605 9.99 478 9.99 476 8 7 3 4 54 918952 Rj 9.19478 0.80 522 9.99 474 6 41.0 40.5 40.0 55 9-19033 80 80 80 80 80 919561 0.80 439 9.99472 5 6 49.2 48.6 48.0 5^ 9.F9113 9.19 643 0.80357 9.99 470 4 7 57-4 56.7 56.0 57 9.19 193 9.19725 0.80 275 9.99 468 3 8 65.6 64.8 64.0 • 5« 9 19273 9.19807 0.80 193 9.99 466 2 9 73.8 72-9 720 59 919353 9.19889 0.80 1 1 1 9.99 464 I 60 919433 9.19971 0.80 029 9.99 462 L. Cos. d. L. Cot. c.d. L. Tan. L. Sin. ' P. P. 1 sr 35 ' j L. Sin. 1 d. 1 L. Tan. c.d. 1 L. Cot. 1 L. Cos. 1 P.P. 1 9- 19 433 80 9.19971 82 81 82 81 ■ 81 81 81 81 80 81 80 80 80 80 80 0.80 029 9.99 462 60 I 9'95«3 9.20053 0.79947 9.99 4O0 59 2 3 9.19592 9.19672 79 80 9.20 134 9.20 216 0.79 866 0.79 784 9.99 458 9.99 456 58 57 J 82 8.2 81 8.1 80 8.0 4 9-'9 75i 79 9.20 297 0.79703 9-99 454 56 2 16.4 16.2 16.0 5 9.19830 79 9.20378 0.79 622 999452 55 3 24.6 24-3 24.0 6 9.19909 79 9.20459 0.79 541 9.99 450 54 4 32-8 32.4 32.0 7 9.19988 ;9 9.20 540 0.79 460 9.99 448 53 5 41.0 40.5 40.0 8 9.20067 V9 78 78 79 78 78 9.20621 0.79379 9.99 446 52 6 49-2 48.6 48.0 9 10 II 9.20 145 9.20 701 0.79 299 9.99 444 51 50 49 7 8 9 57-4 65.6 73-8 64.8 72.9 56.0 64.0 72.0 9.20223 9.20 302 9.20 782 0.79 218 9.99 442 9.20 802 0.79 138 9.99 440 12 9.20380 9.20942 0.79058 9.99 438 48 ^3 9.20458 9.21 022 0.78978 9-99 436 47 79 78 77 14 920 535 78 9.21 102 0.78 898 9-99 434 46 I 7.9 7.8 7-7 «S 9.20613 9.21 182 0.78818 9-99 432 45 2 15.8 15-b 15-4 i6 9.20691 9.21 261 79 80 0.78 739 9.99 429 44 3 237 23.4 23-1 I? 9.20768 n 9.21 341 0.78659 9.99 427 43 4 3i.t> 31.2 30.8 i8 9.20 845 'I'i 9.21 420 ;9 0.78 580 9-99 425 42 5 39.5 39-0 38.5 19 9.20922 ri 9.21 499 79 0.78501 9-99 423 41 6 47-4 46.8 54.6 62.4 70.2 46.2 20 21 9.20999 77 77 9.21 578 79 79 0.78422 9.99421 40 39 i 9 55-3 63.2 71. 1 53-9 61.6 69.3 9.21 076 9.21 657 0.78 343 9.99419 22 9-21 153 77 76 9-21 736 7^ 78 0.78:^64 9.99417 38 23 9.21 229 9.21 814 0.78 186 9-99415 37 24 9.21 306 76 76 76 76 9-21 893 79 7? 78 78 78 78 78 77 78 0.78 107 9-99413 36 76 75 74 2,S 9.21 382 9.21 971 0.78029 9.99411 35 I 7.6 7-5 7.4 26 9.21 458 9.22 049 0.77951 9-99 409 34 2 15-2 15.0 14.8 27 9-21 534 9.22127 0.77873 9.99 407 33 3 22.8 22.5 22.2 28 9.21 610 9.22 205 0.77 795 9.99 404 32 4 30.4 30.0 29.6 29 30 31 9.21 685 75 76 75 76 9.22 283 0.77717 9.99 402 31 30 29 I 38.0 45-6 53-2 60.8 37.5 45-0 60.0 37-0 66!6 9.21 761 9.22361 0.77 639 9.99 400 9.21 836 9.22438 0.77 562 9.99 398 32 9.21 912 9.22 516 0.77484 9.99 396 28 9 68.^ 67.5 ^2, 9.21 987 75 9.22 593 77 0.77 407 9-99 394 27 34 9.22062 Vb 9.22 670 77 0.77 330 9.99 392 26 73 72 71 35 9.22 137 75 9.22 747 77 0.77 253 9-99 390 25 3b 9.22 211 74 9.22 824 77 0.77 176 9.99388 24 I 7.3 7.2 7-1 37 9.22 286 75 9.22 901 77 76 0.77 099 9-99 385 23 2 14.6 14.4 14.2 38 9.22 36 F 75 9.22977 0.77 023 9-99 383 22 3 21.9 21.6 ^«-^ 39 40 41 9.22 435 74 74 74 923054 76 76 0.76 946 9-99 381 21 20 19 4 7 8 29.2 36.5 43-8 5I-I 58.4 28.8 36.0 43.2 50.4 57.6 64.8 28.4 42.6 497 56.8 639 9.22 509 9.23 130 0.76 870 9-99 379 9.22 583 9.23 200 0.76 794 9-99 377 42 9.22657 lA 9.23 283 76 0.76717 9.99 375 18 43 9.22731 74 9-23 359 0.76 641 9-99 372 17 9 65.7 44 9.22 805 74 923435 0.76 565 9-99 370 16 45 46 9.22878 9.22952 73 9.23 510 9.23 586 75 0.76 490 0.76414 9-99 368 9 99 366 ^5 14 74 t 47 9.23025 73 9.23 661 75 76 0.76 339 9-99 364 13 3 3 3 48 9.23098 73 923737 0.76 263 9.99 362 12 79 78 77 49 50 5« 9.23 171 9.23 244 9.23317 IS 9.23 812 lb 0.76 188 9-99 359 10 9 I 2 3 13.2 65!8 13.0 39.0 65.0 12.8 38.5 64.2 9.23 887 V5 75 0.76 113 9-99 357 9.23962 0.76038 9.99 355 52 9.23 390 73 9.24037 75 0.75 963 9-99 353 8 53 9.23462 72 9.24 112 IS 0.75 888 999351 7 3 3 3 54 9- 23 53? 73 9.24 186 74 0.75 814 9.99 348 6 1R 75 74 55 9.23 607 72 9.24261 75 0.75 739 9-99 346 5 I 5^ 9^.23 679 72 9.24 335 H 0.75 665 9-99 344 4 12.7 12.5 12.3 57 923752 73 9.24410 75 0.75 590 9.99 342 3 2 38.0 37-5 37-0 5« 9.23823 yi 9.24 484 74 075516 9.99 340 2 3 ^6-3 62.5 61.7 59 9.23895 72 72 9-24558 9.24632 ;4 74 0.75 442 9-99 337 I 60 9.23967 0.75 368 9-99 335 1 L.Cos. 1 d. L. Cot. |c.d.| L. Tan. L. Sin. 1 ' 1 P.P. 80^ 36 j / L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. 1 P.P. Il I 9-23967 72 71 71 72 71 71 9.24 632 74 73 74 73 74 73 0.75 368 9-99 335 2 60 59 9.24 039 9.24 70b 0.75 294 9.99 333 2 9.24 no 9.24 779 0.75 221 9-99 331 58 74 73 72 3 9.24 181 9-24 853 0-75 147 9-99 328 3 57 I 7.4 7-3 7-2 4 9-24 253 9.2492b 0.75 074 9-99 326 56 2 14.8 14.6 14.4 5 9.24 324 9.25 000 0.75 000 9.99 324 2 55 3 22.2 21.9 2i.b 6 9-24 395 9-25 073 0.74927 9-99 322 54 4 29.b 29.2 28.8 7 9.24 46b 7^ 70 71 70 71 70 70 70 70 70 9.25 14b 73 0.74 854 9-99 319 3 53 c 37.0 3b.5 3b.o 8 9.24 536 9.25219 73 73 73 72 73 72 73 72 72 0.74 781 9-99317 2 2 2 3 52 6 44.4 43-8 43-2 9 lO II 9.24 boy 9-25 292 0.74 708 9-99315 51 50 49 9 51.8 51. 1 50.4 59-2 584 57-6 bb.6 65.7 b4.8 9.24 b77 9.25 3^5 0.74 635 9-99 3'3 9.24 748 9-25 437 0.74 563 9.99310 12 9.24818 9-25510 0.74 490 9-99 308 48 13 9.24 888 9-25 582 0.74418 9.99 306 2 3 2 47 71 70 69 1 14 9.24958 9-25 655 074 345 9-99 304 46 I 7.1 7.0 b.9 15 9.25 028 9.25 727 0.74 273 9-99 301 45 2 14.2 14.0 13.8 lb 9.25 098 9-25 799 0.74 201 9-99 299 44 3 21.3 21.0 20.7 17 9.25 ib8 70 69 70 69 69 69 69 60 9.25871 72 72 71 72 71 72 71 71 71 0.74 129 9-99 297 2 3 43 4 28.4 28.0 27.b 18 9-25 237 9-25 943 0.74057 9.99 294 42 5 6 7 8 9 1 35-5 350 34-5 42.6 42.0 41.4 49.7 49-0 48.3 5b.8 56.0 55.2 bi.Q b^.o b2.i 19 20 21 925 307 9.26015 0.73 985 9.99 292 2 2 3 41 40 39 9-25 376 9.26 086 0.73914 9.99 290 9-25 445 9.26 158 0.73 842 9.99 288 22 9-25 5H 9.26 229 0.73 771 9.99 285 38 23 925 583 9.26 301 0.73 b99 9.99 283 2 37 68 67 66 24 9.25 b52 69 69 b8 69 b8 9-26372 0.73 628 9.99 281 3 36 25 9.25 721 9-26443 0-73 557 9.99 278 35 I b.8 b.7 6.b 2b 9.25 790 9-26514 0.73 48b 9-99 276 34 2 13.6 13.4 13.2 27 28 9 25 858 9.25927 9-26 585 9-26655 70 71 71 70 70 71 70 70 70 70- 69 r 69 70 69 69 69 69 69 69 69 68 0-73415 0-73 345 9.99 274 9.99 271 3 2 2 3 2 33 32 3 4 20.4 20.1 19.8 27.2 2b.8 2b.4 29 30 31 9-25 995 68 68 68 9.26 726 0.73 274 9.99 269 31 30 29 I 34-0 33-5 33-0 40.8 40.2 39.b 47. b 46.9 4b.2 54.4 53.b 52.8 9.2b ob3 9.26 797 0.73 203 9.99 267 9.2b 131 9.2b 8b7 0.73 ^33 9.99 2b4 32 9.2b 199 68 9-26937 0.73 ob3 9.99 2b2 28 9 bi.2 bo.3 59.4 33 9.26 2b7 68 9.27 008 0.72992 9.99 2bO 3 27 1 34 9-26335 68 9.27078 0.72922 9-99 257 2b 65 3 35 9.2b 403 67 68 67 67 67 67 67 67 67 66 67 6b 67 66 9.27 148 0.72852 9-99 255 3 2 2 25 , z: - „ _ 36 37 9.2b 470 926538 9.27 218 9.27288 0.72 782 0.72712 9-99 252 9-99 250 24 23 i "-3 'J-J > 1 13.0 0.6 5 i 19-5 0.9 [ 26.0 1.2 3H 9.2b bo5 9-27357 0.72 643 9-99 248 3 2 2 3 22 39 40 41 9,2b b72 9.27427 0.72 573 9-99 245 21 20 19 ) 32.5 1-5 ) 39.0 1.8 ' 45-5 2.1 9.2b 739 9-27 496 0.72 504 9-99 243 9.2b 80b 9.27 5bb 0.72434 9.99 241 42 9-26873 9.27 b35 0.72 3b5 9-99 238 18 I 52.0 2.4 43 9.2b 940 9-27 704 0.72 29b 9-99 236 3 2 17 ) 58-5 2-7 44 45 9.27007 9.27073 9-27773 9.27 842 0.72 227 0.72 158 9-99 233 9.99231 lb 15 1 4b 47 9.27 140 9.27 20b 9.27 911 9.27 980 0.72 089 0.72 020 9.99 229 9.99 22b 3 2 14 13 3 3 3 74 73 72 48 9.27.273 9.28049 0.71 951 9-99 224 3 2 2 12 j 49 50 9-27 339 66 66 9.28 117 69 68 0.71 883 9.99221 II 10 2 12.3 12.2 12.0 37.0 36.5 36.0 61.7 60.8 60.0 9.27 405 9.28 i8b 0.71 814 9.99219 51 9.27471 66 9.28 254 69 68 0.71 746 9.99217 3 2 9 3 52 9-27 537 65 66 9.28323 0.71 b77 9.99 214 • 8 53 9.27 b02 9.28 391 68 0.71 bo9 9.99212 3 7 3 3 3 3^ 54 9.27 6b8 66 9.28459 68 0.71 541 9-99 209 2 b 71 70 69 68 55 9-27 734 65 65 66 9.28527 68 0.71 473 9.99 207 3 2 2 5 11.811.7 11.511.31 35-5 35-034-5 34-Oi 59.258.357.5 5b.7 5^ 57 9.27 799 9.27 864 9-28 595 9-2§.6b2 67 68 0.71 405 0-71 338 9.99 204 9.99 202 4 3 I 2 5^ 9.27 930 65 65 9.28 730 68 0.71 270 9.99 200 3 2 2 3 59 60 9-27 995 9.28 obo 9.28 798 67 0,71 202 9 99 197 I 9.28 865 0.71 135 9-99 195 L. Cos. d. L. Cot. c. d. L.Tan. | L. Sin. d. f P. P. 79' 37 / L. Sin. i d. L.Tan. c. d. L. Cot. L. Cos. d. P. P. I 9.2S 060 9.28125 65 65 64 65 65 64 64 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 6? 9.28 865 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 0.71 135 9.99 195 3 60 59 928 933 0.71 067 9.99 192 2 9.28 190 9.29000 0.71 000 9.99 190 3 2 58 68 67 66 3 9.28 254 9.29067 0.70933 9-99 187 57 I 6.8 6.7 6.6 4 9.28319 9.29 134 0.70 866 9.99 185 3 56 2 13-6 13.4 13.2 ,s 9.28 384 9.29 201 0.70 799 9.99 182 55 3 20.4 20.1 19.8 6 9.28 448 9.29 268 0.70 732 9.99 180 3 2 3 2 3 2 54 4 27.2 26.8 26.4 7 9.28512 929 33^ 0.70 665 9.99177 53 5 340 33-5 330 8 9-28577 9.29 402 0.70 598 9.99175 52 6 40.8 40.2 39.6 9 iO II 9.28 b4i 9-28 705 9.28 769 9.29 468 0.70532 9-99 172 51 50 49 7 8 9 47.6 46,9 46.2 54.4 53.6 52.8 61.2 60.3 59.4 9.29 535 0.70 465 9.99170 9.29601 0.70 399 9-99 167 12 9.28 S33 9.29 668 0.70332 9-99 1 61 3 48 65 64 63 13 9.28896 9.29 734 0.70 266 9.99 162 47 H 9.28 960 9.29 800 0.70 200 9.99 160 3 46 I 6.5 6.4 6.3 ^5 9.29 024 9.29 866 66 0.70 134 9-99157 45 2 13.0 12.8 12.6 i6 9.29087 9.29 932 66 66 66 65 66 t^ 65 66 0.70 068 9-99 155 44 3 19.5 19.2 18.9 17 9.29 150 9.29 998 0.70002 9.99152 3 43 4 26.0 25.6 25.2 18 9.29214 9.30 064 0.69 936 9-99 150 42 7 8 9 32.5 32.0 31.5 39.0 38.4 37.8 45.5 44-8 44.1 52.0 51.2 50.4 t;8.t; i;7.6 1:6.7 19 20 21 9.29 277 9.30 130 0.69 870 9.99 147 3 2 3 41 40 39 9.29 340 930 195 0.69 805 9-99 145 9.29 403 9.30 261 0.69 739 9-99 142 22 9.29 466 9.30 326 0.69 674 9-99 140 3 2 38 23 9.29529 9-30391 0.69 609 9-99 137 37 62 61 60 24 9.29591 63 62 9-30457 65 65 65 64 65 64 65 64 64 65 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 0.69 543 9-99 135 3 36 25 9.29 654 9.30522 0.69 478 9-99 132 35 I b.2 b.i b.o 2b 9.29716 63 62 62 63 6'> 9.30 587 0.69413 9-99 130 3 3 2 3 34 2 12.4 12.2 12.0 18.6 18.3 18.0 24.8 24.4 24.0 31.0 30.5 30.0 37.2 36.6 36.0 43-4 42.7 420 49.6 48.8 48.0 27 28 -29 30 9 29 779 9.29 841 9.29 903 9.30652 9.30717 9.30 782 0.69 348 0.69 283 0.69 218 9.99 127 9-99 124 9.99 122 33 32 31 30 3 4 5 6 7 9.29 966 9.30 846 0.69 154 9-99 "9 31 9.30028 6-' 9.30 91 1 0.69 089 9.99 117 3 2 29 8 32 9.30 090 6t 9-30 975 0.69 025 9-99114 28 9 55-8 54.9 54.0 33 34 9.30151 9.30213 62 62 9.31 040 9.31 104 0.68 960 0.68 896 9.99 112 9-99 109 3 3 27 26 59 3 3.S 9-30 273 61 9.31 168 0.68 832 9.99 106 25 3^ 37 38 9-30 33^ 930 398 9-30459 62 61 6^ 9-31 233 9.31 297 9-31 361 0.68 767 0.68 703 0.68 639 9-99 104 9.99 lOI 9-99 099 3 2 3 3 2 3 24 23 22 I 2 3 4 5 6 7 5-9 0.3 1 1.8 0.6 17.7 0.9 23.6 1.2 29-5 1-5 35-4 1.8 41-3 2.1 39 40 41 9.30521 61 61 61 931 425 0.68 575 9.99 096 21 20 19 9.30 582 9.31 489 0.68 511 9-99 093 9-30 643 9-31 552 0,68 448 9-99091 42 9.30 704 61 9.31 616 0.68 384 9.99 088 18 8 47.2 2.4 43 9.30 765 61 9.31 679 0.68 321 9.99 086 3 3 n 9 53-1 2.7 44 45 9.30 826 9.30887 61 60 9-31 743 9.31 806 0.68 257 0.68 194 9.99 083 9.99 080 16 IS 4b 9-30 947 61 9.31 870 0.68 130 9.99078 3 3 14 3 3 3 47 9.31 008 60 9-31 933 0.68 067 999075 13 67 66 65 48 9.31 068 61 9.31 996 0.68 004 9.99072 12 49 50 51 9.31 129 60 61 60 9-32059 0.67 941 9.99 070 3 3 2 3 3 II 10 9 2 3 1 1.2 ii.o 10.8 33-5 33-0 325 55.8 55.0 54.2 9.31 189 9.32122 0.67 878 9.99 067 9.31 250 9.32 185 0.67815 9.99 064 52 9.31 310 60 9.32 248 0.67 752 9.99 062 8 53 931 370 60 9.32 311 0.67 689 9-99 059 7 3 3 3 , 54 931 430 60 9-32 373 63 62 63 62 62 0.67 627 9-99 056 6 64 63 62 55 9.31 490 59 60 60 59 60 932436 0.67 564 9-99 054 5 5t> 931 549 9-32498 0.67 502 9.99051 3 4 I 10.7 10.5 10.3 57 9.31 609 9.32561 0.67 439. 9.99 048 J 3 2 32.0 31.5 31.0 53-3 52.5 51-7 58 9.31 669 9-32 623 0.67 377 9-99 046 3 3 2 3 59 60 9.31 728 9-32685 62 0.6731? 9 99 043 I 9.31 788 9-32 747 0.67 253 9-99 040 L. Cos. d. L. Cot. c. d.| L.Tan. | L. Sin. d. f P.P. •7QC 38 / L. Sin. 1 d. L. Tan. c. d. L. Cot. L. Cos. d. P. P. 9.3 1 788 59 60 9-32 747 63 62 0.67 253 9.99 040 2 3 3 2 60 S9 931 847 9.32 810 0.67 190 9-99 038 2 9.31 907 59 59 59 59 59 9-32 872 61 0.67 128 9-99 035 58 63 62 61 3 9.31 966 932933 6-' 0,67 067 9-99032 57 I 6.3 6.2 6.1 4 9.32025 932995 6'> 0.67 005 9.99 030 3 3 56 2 12.6 124 12.2 5 9.32084 9-33057 ' 62 0.66 943 9.99027 55 3 18.9 18.6 18.3 6 9.32 143 9-33 "9 61 0.66 881 9.99 024 54 4 25.2 24.8 24.4 7 9.32 202 9-33 180 62 61 0.66 820 9.99 022 53 5 31.5 31.0 30.5 8 9.32261 59 58 59 59 58 58 59 58 58 58 58 58 58 58 57 58 57 58 57 57 58 57 57 57 56 57 57 57 56 9-33 242 0.66 758 9-99019 3 3 3 2 3 3 3 52 6 37-8 37-2 36-6 9 lO II 932319 9-33 303 62 61 61 0.66 697 9.99016 51 50 49 I 9 44-1 43-4 42.7 50.4 49.6 48.8 56-7 55-8 54-9 9.32 378 9-33 365 0.66 635 9.99013 9-32437 933426 0.66 574 9.99 on 12 9-32495 933487 61 0.66 513 9.99 008 48 60 59 13 932553 9-33 548 61 0.66452 9-99 005 47 14 9.32 612 9.33 609 6i 0.66391 9.99 002 46 I 6.0 5.9 15 9.32 670 9 33 670 61 0.66 330 9.99 000 3 45 2 12.0 11.8 lb 9.32 728 933731 61 61 60 0.66 269 9.98 997 44 3 18.0 17.7 17 9.32 786 9-33 792 0.66 208 9.98 994 3 43 4 24.0 23.6 18 9.32 844 933853 0.66 147 9.98991 3 2 3 3 42 5 6 7 8 9 30-0 29.5 19 20 21 9.32 902 9-33913 61 60 61 0.66087 9-98 989 41 40 39 36.0 35.4 42.0 41.3 48.0 47.2 54-0 53-1 9.32960 9-33 974 0.66 026 9.98 986 9.33018 9-34 034 0.65 966 9-98983 22 9-33075 934095 60 0.65 905 9.98 980 :> 38 23 9-33 ^33 9-34155 60 0.65 845 9.98978 3 3 3 2 3 3 3 3 2 3 3 37 58 57 24 9-33 190 9-34215 61 0.65 785 9-98975 36 25 9.33 248 9-34 276 60 65 724 9.98972 35 I 5-8 5-7 26 9-33 305 9-34336 60 0.65 664 9.98 969 34 2 1 1.6 1 1.4 27 933362 9-34 396 60 0.65 604 9.98 967 33 3 17.4 17.1 23.2 22.8 29.0 28.5 34-8 34.2 40.6 39.9 46.4 45.6 28 933420 934456 60 0-65 544 9.98 964 32 4 29 30 31 9-33 477 9.34516 60 59 60 0.65 484 9.98961 31 30 29 i 9 33 534 9-34 576 9-34 635 0.65 424 9.98958 933591 0.65 365 998955 32 9 33 647 9-34 695 60 0.65 305 9-98953 28 9 52.2 51.3 33 9-33 704 9-34 755 59 60 0.65 245 9 98 950 27 34 9-33 761 9-34814 0.65 186 9-98 947 26 56 55 3 35 9.33818 934874 59 0.65 126 9.98 944 3 25 36 9-33 874 9 34 933 0.65 067 9.98941 3 24 1 50 5-5 0.3 37 9-33931 56 57 56 56 56 56 56 56 55 56 55 56 55 56 55 55 55 9.34992 59 59 60 0.65 008 9.98938 3 23 2 3 4 7 II. 2 ii.o 0.6 16.8 16.5 0.9 22.4 220 1.2 280 27.5 1.5 33.6 33.0 1.8 39.2 38.5 2.1 3^ 9-33987 9-35051 0.64 949 9.98936 3 3 3 3 3 22 39 40 41 9-34 043 9-35 III 59 59 59 59 58 59 59 58 59 58 59 58 58 58 58 58 58 58 58 58 57 0.64 889 9-98 933 21 20 19 9.34 100 9-35 170 0.64 830 9 98 930 9.34 156 9-35 229 0.64 771 9.98927 42 9.34212 9-35 288 0.64 712 9 98 924 18 8 44-8 44.0 2.4 43 9-34 268 9-35 347 0.64 653 9.98 921 17 9 50.4 49.5 2.7 44 45 9-34 324 9-34 380 9-35 405 9-35 464 0-64 595 0.64 530 9.98919 9.98 916 3 16 15 46 9.34 436 9-35 523 0.64 477 9.98913 3 14 3 3 3 47 934491 9 35 581 0.64419 9.98910 3 13 62 61 60 48 9-34 547 9 35 640 0.64 360 9.98 907 J 12 49 50 51 9.34 602 9.35 698 0.64 302 9.98 904 3 3 II 10 9 I 2 3 10.3 10.2 lO.O 31.0 30.5 30.0 51.7 50.8 50.0 9.34 658 9-35 757 0.64 243 9.98901 9-34713 935815 0.64 185 9.98 898 52 9-34 769 9-35 873 0.64 127 9.98 896 8 53 9-34 824 9-35 931 0.64 069 9-98893 3 7 3 3 3 54 9-34879 9-35 989 0.64 01 1 9.98 890 3 3 6 59 58 57 55 9-34 934 9.36047 0.63 953 9.98 887 5 5^ 57 58 9-34 989 9-35 044 9.35 099" 55 55 55 55 55 9.36 105 9.36 163 9.36221 0-63 895 0.63837 0.63 779 9.98 884 9.98881 9.98 878 3 3 3 3 3 4 3 2 I 2 3 9.8 9.7 9.5 29.5 29.0 28.5 49.2 48.3 47.5 59 60 9-35 154 9.36 279 0.63721 9.98875 I 9-35 209 9-36 336 0.63 664 9.98 872 1 L. Cos. d. L. Cot. |c. d.| ' L.Tan. 1 L. Sin. | d. P.P. || WMO 39 L. Sin. L. Tan. c. d. L. Cot. L. Cos. P. P. O r 2 3 4 5 6 7 8 9 10 II 12 13 15 i6 17 i8 19 20 21 .22 23 24 25 26 27 28 .29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9-35209 9-35 263 9-35318 9-35 373 9-35427 9.35 481 9-35 536 9.35 590 9-35 644 9-35 ^98 9-35 752 9.35 806 9-35 860 9-35 9H 9-35 968 9.36022 9.36075 9.36 129 9.36 182 9-36 236 9.36 289 9-36 342 9-36 395 936449 9.36 502 9-36555 9.36 608 9 36 660 9-36713 9.36 766 1.36 819 9.36871 9.36 924 9.36 976 9.37028 9.37081 9-37 ^33 9-37 185 9-37 237 9-37 289 9-37 341 9-37 393 9-37 445 9-37 497 9-37 549 9.37 600 9-37652 9-37 703 9-37 755 9.37 806 9.37 858 9-37 909 9-37960 938011 9.38062 9-38113 9.38 164 9.38215 9.38 266 9-38317 9.38 368 9.36 336 9 36 394 9.36452 9-36 509 9.36 566 9.36 624 9.36681 936 738 9-36 795 9.36852 9.36 909 9.36 966 9.37023 9.37 080 9-37 ^37 9-37 193 9-37 250 9-37 306 9-37 363 937419 9-37 476 9.37 532 9-37 588 9-37 644 9.37 700 9.37 756 9-37812 9.37 868 9-37924 9-37 980 9-38 035 9.38091 9-38 147 9.38 202 9-38 257 9-38313 9.38 368 9-38423 9-38 479 938 534 9-38 5^ 9.38 644 9.38 699 9-38 754 9.38 808 9.38 863 9.38918 9.38972 9.39027 9-39 082 9-39 136 9.39 190 9-39 245 9.39 299 9-39 353 9-39 407 9.39461 9-39515 9-39 569 9-39 623 9-39 677 0.63 664 0.63 606 0.63 548 0.63491 0.63 434 0.63 376 0.63319 0.63 262 0.63 205 0.63 148 9 98 872 9.98 869" 9.98 867 9.98 864 9.98861 9.98 858 9-98855 9.98 852 9.98 849 9.98 846 0.63091 0.63 034 0.62977 0.62 920 0.62 863 0.62 807 0.62 750 0.62 694 0.62 637 0.62 581 9.98 843 9.98 840 9.98837 9.98 834 9-98831 9.98 828 9.98 825 9.98 822 9.98819 9.98816 0.62 524 9.98813 0.62 468 0.62 412 0.62 356 0.62 300 0.62 244 0.62 188 0.62 132 0.62 076 0.62 020 9.98 810 9.98 807 9.98 804 9.98 801 9.98 798 9.98 795 9.98 792 9-98 789 9.98 786 0.61 965 0.61 909 0.61 853 0.61 798 0.61 743 0.61 687 0.61 632 0.61 577 0.61 521 0.61 466 9-98 783 9.98 780 9-98 777 9.98 774 9-98 77^ 9.98 768 9-98 765 9.98 762 9-98 759 9-98 756 0.61 411 9-98 753 0.61 356 0.61 301 0.61 246 0.61 192 0.61 137 0.61 082 0.61 028 0.60973 0.60918 9-9^ 750 9.98 746 9-98 743 9.98 740 998 737 998 734 998 731 9.98 728 9 98 725 0.60 864 9.98 722 0.60810 0.60 755 0.60 701 0.60 647 0.60 593 0.60 539 0.60 485 0.60431 0.60 377 9.98719 9.98715 9.98712 9.98 709 9.98 706 9.98 703 9.98 700 9.98 697 9 98 694 0.60 323 9.98 690 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 n 10 9 8 7 6 5 4 3 2 I ■ O 58 57 56 5.8 11.6 5.7 5-6 1.4 II. 2 17.4 17.1 16.8 23.2 22.8 22.4 290 28.5 28.0 34.8 34.2 33.6 40.6 39.9 39.2 46.4 45.6 44.8 52.2 51.3 50.4 55 5-5 54 5.4 53 5-3 no 10.8 10.6 16.5 16.2 15.9 22.0 21.6 21.2 27.5 27.0 26.5 33.0 32.4 31.8 38.5 37-8 37.1 44.0 43.2 42.4 49.5 48.6 47.7 52 51 5-2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3-2 3-6 5.1 10.2 15-3 20.4 255 30.6 35-7 40.8 45.9 3 0.3 0.6 0.9 1.2 1-5 1.8 2.1 2.4 2-7 55 4 54 58 6.9 6.8 9.7 9.5 20.6 20.2 29.0 28.5 34.433.848.347.5 48.147.2 — — 56 55 54 9.3 9.2 9.0 28.0 27.5 27.0 46.7 45-8 45 o L. Cos. L. Cot. c. d L. Tan. 7A° L» Sin. P. P. 40 ! ' 1 L. Sin. |d. L. Tan. |c. d. L. Cot. L. Cos. d. 1 P.P. ■ 9.38 368 50 51 50 51 50 50 9-39677 1 54 54 53 0.60 323 9.98 690 3 60 54 63 9.38418 9.39731 0.60 269 9.98 687 2 9.38 469 9-39 78I 0.60 2 1 5 9.98 684 3 58 3 9.38519 9-39 838 0.60 162 9.98 681 3 57 I 54 5-3 4 9.38 570 9-39 892 54 53 0.60 108 9.98 678 3 S6 2 10.8 10.6 5 9.38 620 9-39 945 0.60055 9.98 675 3 S5 3 16.2 15.9 6 9.38 670 9.39 999 54 0.60001 9.98 671 4 54 4 21.6 21.2 7 9.38721 51 50 50 50 50 50 50 50 50 49 9.40 052 53 54 53 53 54 53 53 53 53 53 0.59 948 9.98 668 3 5S 5 6 27.0 26.5 324 31.8 37.8 37-1 43-2 424 48.6 d.7.7 8 9-38 771 9.40 106 0.59 894 9.98 665 3 52 7 8 9 9 lO II 9.38821 9.40 159 0.59 841 9.98 662 3 3 3 51 50 49 9.38871 9.40 212 0.59 788 9.98 659 9.38921 9.40 266 0.59 734 9.98 656 12 13 9-38971 9.39021 9.40319 9.40 372 0.59681 0.59 628 9.98 652 9.98 649 4 3 48 47 52 51 50 14 9.39071 9.40 425 0-59 575 9.98 646 3 46 I 5.2 5-1 50 15 9.39 121 9.40 478 0.59 522 9.98 643 3 45 2 10.4 10.2 lO.O 16 9.39170 940531 0.59 469 9.98 640 3 44 3 15.6 15.3 15.0 17 9.39 220 50 50 49 50 49 49 50 49 49 49 9.40 584 53 52 53 53 53 52 53 52 53 52 0.59416 9.98 636 4 A^ 4 20.8 20.4 20.0 26.0 25.5 25.0 31.2 30.6 30.0 364 35.7 35.0 41.6 40.8 40.0 46.8 4';.q 4s.O 18 9.39 270 9.40 636 0.59 364 9-98 633 J 42 5 6 7 8 9 19 20 21 9.39319 9.40 689 0.59 31 1 9.98 630 3 3 4 3 3 3 41 40 39 9.39 369 9.40 742 0.59 258 9.98 627 9.39418 940 795 0.59 205 9.98623 22 9-39 467 9.40 847 0.59153 9.98 620 38 23 9.39517 9.40 900 0.59 100 9.98617 37 49 48 47 24 9.39 566 9.40952 0.59 048 9.98 614 36 25 9.39 615 9.41 005 0.58 995 9.98 610 4 35 I i 4.9 4.0 4-7 26 9.39 664 9.41 057 0-58 943 9.98 607 3 34 2 9.8 9.6 94 27 9-39713 49 49 9.41 109 52 52 0.58891 q.98 604 3 ^^ 3 4 14.7 144 14.1 19.6 19.2 18.8 28 9.39 762 9.41 161 0.58 839 9.98601 3 32 29 30 9-39 81 1 49 49 49 49 48 49 48 49 48 49 48 49 48 48 48 48 9.41 214 53 52 52 52 52 52 52 52 51 52 52 51 52 51 52 51 0.58 786 9.98 597 4 3 3 31 30 I 7 24.5 24.0 23.5 29.4 28.8 28.2 34.3 33.6 32.9 9.39 860 9.41 266 0.58 734 9-98 594 31 9.39 909 9.41 318 0.58682 9.98 591 29 8 39.2 38.4 37.6 32 9.39 958 9.41 370 0.58 630 9.98 588 3 28 9 44.1 43.2 42.3 33 9.40 006 9.41 422 0.58578 9.98 584 \ 27 34 940055 9.41 474 0.58 526 9.98 581 3 26 4 3 35 9.40 103 9.41 526 0.58 474 9.98578 3 25 J 0.4 0.3 0.8 0.6 3t> 9.40152 941 578 0.58422 9.98 574 4 24 2 ^J 9.40 200 9.41 629 0.58 21^ 9.98568^ 3 3 4 3 ^ J 1.2 0.9 3^ 9.40 249 9.41 681 0.58319 22 4 1.6 1.2 39 40 41 9.40 297 941 733 0.58 267 9.98565 21 20 19 I 2.0 1.5 2.4 1.8 2.8 2.1 9.40 346 9.41 784 0.58216 9.98561 9.40 394 9.41 836 0.58 164 998558 42 9.40 442 9.41 887 0.58 113 998555 3 18 3.2 2.4 3-6 2.7 43 9.40 490 941 939 0.58061 9.98551 4 17 9 44 9.40 538 9.41 990 0.58010 0.57959 9.98 548 9.98 545 3 t6 45 9.40 586 48 48 48 48 48 47 48 48 47 48 47 48 47 47 t8 9.42 041 51 52 51 51 51 51 51 51 51 51 51 51 50 51 51 eri 1 3 15 40 9.40 634 9.42 093 0.57907 9.98 541 4 14 4444 47 9.40 682 9.42 144 0.57 856 998 538 3 13 54 53 52 51 48 9.40 730 9.42 195 0.57805 998 535 3 4 3 3 4 12 ° 6.8 6.6 6.5 6.4 2 20.2 19.9 19.5 19.1 .33-8 33-1 325 31.9 ^47.246.445.544.6 49 50 51 9.40 778 9.42 246 0-57 754 9.98 531 II 10 9 9.40 825 9.42 297 0.57 703 9.98 528 9.40 873 9.42 348 0.57652 998 525 52 9.40921 942 399 0.57 601 9.98521 8 53 54 9.409685 9.41 016^ 9.42 450 9.42 501 0.57 550 0.57499 9.98518 9-98515 3 3 7 6 A A A 1 55 9.41 063 9-42 552 0.57 448 9.98 511 4 3 3 4 3 4 5 54 53 52 51 5^ 9.41 III 9.42 603 0.57397 9.98 508 4 ° 9.0 8.8 8.7 8.5 57 9.41 158 942 653 0-57 347 9-98 505 3 27.0 26.5 26.0 25.5 5^ 9.41 205 9.42 704 0.57296 9.98 501 2 ^45.044.243.342.5 59 9.41 252 942 755 o-57'245 9.98498 I 3 60 9.41 300 9.42805 1 " 1 0.57195 1 9.98 494 : L. Cos. d. 1 L. Cot. |c. d. L. Tan. 1 L. Sin. | d.l '1 P.P. 'y/;° 41 / L. Sin'. d. L. Tan. c. d. L. Cot. L. Cos. d. P.P. 9.41 300 47 9.42 805 5« 50 0.57 195 9.98 494 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 3 4 3 4 4 3 4 I 3 4 3 4 3 60 59 61 50 49 941 347 9.42 856 0.57 144 9.98491 1 2 9.4' 394 47 9.42 906 0.57094 9.98 488 58 3 9.41 441 47 942 957 5^ 0.57 043 9.98 484 57 ^ 5-» 5-0 4-9 10.2 lo.o 9.8 15.3 15.0 14.7 20.4 20.0 19.6 25.5 25.0 24.5 30.6 30.0 29.4 i 4 9.41488 4/ 943 007 S^ 0.56 993 9.98481 56 2 3 4 ' ,S 9-41 535 47 47 46 47 46 47 46 943057 50 5» 50 50 50 50 50 50 50 50 50 49 50 50 49 50 49 50 49 50 49 49 49 50 49 49 49 0.56 943 9.98477 55 i 6 ! 7 9.41 582 9.41 628 943 loS 943 158 0.56 892 0.56 842 9.98 474 9.98471 54 53 1 ^ 9.41 675 943 208 0.56 792 9.98 467 52 7 35-7 35-0 34-3 1 9 1 10 II 9.41 722 943 258 0.56 742 9.98 464 51 30 49 8 9 40.8 40.0 39.2 45.9 45.0 44.1 9.41 7^« 9.41 815 943 308 0.5b b92 9.98 460 943 358 0.56 642 9.98457 12 9.41 8b I 943 408 0.56 592 9-98 453 48 48 47 46 13 15 9.41 908 941 954 9.42001 47 46 47 46 46 ^I 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 46 45 45 46 45 45 45 45 45 45 45 943 458 943 508 943 558 0.56 542 0.56492 0.56442 9.98 450 9.98 447 9.98 443 47 46 45 2 3 4 4.8 4.7 4.6 9.6 9.4 9.2 14.4 14.1 13.8 19.2 18.8 18.4 lb 9.42 047 943 607 0.56 393 9.98 440 44 17 i8 9.42 093 9.42 140 943 657 943 707 056 343 0.56 293 9.98 436 9-98433 43 42 24.0 23.5 23.0 28.8 28.2 27.6 19 20 21 9.42 186 942 232 943 756 0.56 244 9.98 429 41 40 39 7 8 9 33.6 32.9 32.2 38.4 37.6 36.8 43.2 42.3 41.4 943 80b 0.56 194 9.98 426 9.42278 943 855 0.5b 145 9.98422 22 9.42 324 943 905 0.56095 9.98419 38 1 23 9.42 370 943 954 0.56 046 9.98415 37 45 44 1 24 25 9.42416 9.42 461 9.44 004 944053 0.55 996 0-55 947 9.98412 9.98 409 36 35 2 4-5 4.4 9.0 8.8 2b 942 507 9.44 102 0.55 898 9.98 405 34 3 13.5 13-2 27 942 553 9.44151 0.55 849 9.98 402 33 4 18.0 17.6 28 942 599 9.44 201 0.55 799 9.98 398 32 5 22.5 22.0 29 30 31 9.42 b44 944 250 0.55 750 9-98 395 31 30 29 I 27.0 26.4 31.5 30.8 36.0 35-2 40.5 39-6 9.42 690 944 299 0-55 701 9.98391 942 735 944 348 0.55652 9.98 388 32 9.42 781 944 397 49 49 49 49 48 49 0.55 603 9.98 384 28 9 33 9.42 826 944 446 0.55 554 9.98381 27 34 9.42 872 944 495 0.55 505 9.98 377 26 4 3 3S 9.42917 944 544 0.55 456 9-98 373 25 I 0.4 0.3 36 9.42 962 944 592 0.55 408 9.98 370 24 2 0.8 0.6 37 943 008 944 641 0.55 359 9.98 366 2^ ' 1.2 0.9 3« 943 053 944 690 49 48 49 49 48 49 48 0.55310 998 363 22 4 1.6 1.2 39 40 41 943 09^ 944 738 0.55 262 9.98 359 21 20 19 1 9 2.0 1.5 2.4 1.8 2.8 2.1 3.2 2.4 3.6 2.7 943 143 944 787 0.55213 998356 943 i«« 944 836 0.55 164 9.98352 42 43 44 4S 943 233 943 27^ 943 323 943 367 9.44 884 9-44 933 9.44981 945 029 0.55 116 0.55 067 0.55019 0.54971 9 98 349 9-98 345 9.98 342 9.98 338 18 17 16 44 45 48 4 15 46 9.43412 945 078 49 48 48 48 49 48 48 48 48 48 48 47 48 48 48 0.54922 9.98 334, 4 H 4 4 4 4 47 943 457 45 45 44 45 44 45 44 45 44 44 44 45 44 44 945 126 0.54 874 998331 3 4 13 50 49 48 47 48 943 502 945 174 0.54 826 9.98327 12 6.2 6.1 6.0 5.9 18.8,18.418.017.6 31.2 30.6 30.0 29.4 43.842.942.041.1 49 50 SI 943 546 945 222 0.54 778 9.98 324 4 3 II 10 9 I 2 3 943 591 945 271 0.54 729 9.98 320 943 635 945 319 0.54681 9.98317 52 9.43 680 945 367 0.54 633 9.98313 4 4 3 4 3 4 4 8 4 1 53 943 724 945415 0.54 585 9.98 309 7 3 3 3 3 51 50 49 48 54 55 943 769 943813 945 463 945 511 0.54537 0.54489 9.98 306 9.98 302 6 5 5t> 943857 945 559 0.54441 9.98 299 4 8.5 8.3 8.2 8.0 57 943 901 945 606 0.54 394 9.98 295 3 25.5 25.0 24.5 24.0 5« 943 94t> 945 654 0.54 346 9.98 291 2 42.541.740.840.0 59 60 943 990 945 702 0.54 298 9.98 288 3 4 3 944 034 945 750 0.54 250 9.98 284 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. / P.P. 42 w L. Sin. L. Tan. Ic. d.l L. Cot. L. Cos. P. P. 7 8 9 10 II 12 13 H '5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9- 44 034 9.44 078 9.44 122 9.44 166 9.44 210 9-44 253 9.44 297 9-44 341^ 9-44 385 9-44 428 9.44 472 9.44510 9-44 559 9.44 602 9.44 646 9.44 689 9-44 733 9.44 776 9.44819 9.44 862 9-44 905 9.44 948 9.44 992 9-45 035 945077 9.45 120 9-45 '63 9 45 206 9-45 249 9-45 292 945334 945 377 945 4" 9 945 462 945 504 945 547 945 589 945 632 945 ^74 945716 945 758 945 801 945 843 945 885 945 927 945 969, 9,46011 946053 9.46095 9.46 136 ..^^46,178 9.46 220 9.46 262 946 3<^3 946 345 9.46 386 9.46428 9.46 469 9.46 511 946 552 946 594 44 44 44 44 43 44 44 44 43 44 44 43 43 44 43 44 43 43 43 43 43 44 43 42 43 43 43 43 43 42 43 42 43 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 41 42 -42 42 41 42 41 42 41 42 41 42 945 750 945 797 945 845 945 892 945 940 945 987 9.46 035 9.46082 9.46 130 9.46 177 9.46 224 9.46 271 9.46319 9.46 366 9.46413 9.46 460 9.46 507 946 554 9.46 601 9.46 648 946 694 9.46741 9.46 788 946 835 9.46881 9.46 928 946975 9.47021 9.47 068 947 »M 9.47 160 947 207 947 253 947 299 947 346 947 392 947 438 947 484 947 530 947 576 1.47622 9.47 668 9.47714 947 760 9.47 806 9.47852 947 897 947 943 947 989 948 035 9.48080 9.48 126 9.48 171 9,48217 9.48 262 9.48 307 948353 9.48 398 948 443 9.48 489 948 534 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 46 45 46 45 45 46 45 45 46 45 o>54 250 9.98 284 0.54 203 0-54155 0.54 108 0.54 060 0.54013 0.53 965 0.53.918 0.53 870 0-53 823 9.98281 9.98 277 9.98 273 9.98 270 9.98 266 9.98 262 9.98 259 9-98 255 9.98251 0.53 776 9.98 248 0.53 729 0.53 681 0.53 634 0-53 587 0.53 540 0.53 493 0.53 446 0.53 399 053352 9.98 244 9.98 240 9.98 237 9-98 233 9.98 229 9.98 226 9.98 222 9.98218 9.98215 0.53 306 0-53 259 0.53 212 0.53 165 0.53 119 0.53072 0.53025 0.52979 0.52932 0.52886 9.98 21 1 9.98 207 9.98 204 9.98 200 9.98 196 9.98 192 9.98 189 9.98 185 9.98 181 9.98177 0.52 840 0.52 793 0.52 747 0.52 701 0.52 654 0.52 608 0.52 562 0.52 516 0.52470 0.52424 9.98174 9.98 170 9.98 166 9.98 162 9.98 159 998 155 9-98151 9.98 147 9.98 144 9.98 140 0.52378 9.98 136 0.52 332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52057 0.52011 0.51965 9.98 132 9.98 129 125 121 117 113 no 106 9.98 102 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 0.51 693 0.51 647 0.51 602 0-51 557 0.51 511 9.98 098 ^^^094" 9.98 090 9.98087 9.98 083 9.98 079 9.98075 9.98071 9.98 067 9 98 063 0.51 466 9.98 060 60 59 58 57 56 55 54 53 52 5» 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 '7. 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I O 48 47 46 I 4.8 4-7 4-6 2 9.6 9.4 9.2 144 H-i 13-8 4 1,9.2 18.8 18.4 5 24.0 23.5 .23.0 6 28.8 28.2 27.6 7 33.6 32.9 32.2 8 38.4 37.6 36.8 9 43.2 42.3 41.4 45 44 43 I 4-5 44 4-3 2 9.0 8.8 8.6 3 13.5 13.2 12.9 4 18.0 17.6 17.2 5 22.5 22.0 21.5 27.0 26.4 25.8 31.5 30.8 30.1 36.0 35.2 34.4 40.5 39-6 38.7 42 41 4.2 8.4 12.6 16.8 21.0 25.2 29.4 33-6 37-8 0.4 0.8 1.2 1.6 2.0 24 2.8 3-2 3.6 4.1 8.2 12.3 16.4 20.5 24.6 28.7 32.S 36.9 0.3 0.6 0.9 1.2 1-5 1.8 2.1 2.4 2.7 4^ 48 4 47 46 6.0 5.9 5.8 5.6 18.0 17.6 17.2 16.9 300 29.4 28.8 28.1 42.0 41. 1 40.239.4 3^ 48 3_ 47 46 45 8.0 7.8 7.7 7.5 24.0 23.5 23.0 22.5 40.0 39.2 38.3 37.5 L. Cos. I d. L. Cot. |c. d.| L. Tan. L. Sin. P. P. 43- L. Sin. L. Tan. c. d< L. Cot. L. Cos d. P.P. 2 3 4 5 7 8 9 10 II 12 13 14 J5 i6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Z1 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 46 594 46633 46 676 46717 46758 46 800 46 841 46882 46923 46964 4700I 47045 47086 47127 47 168 47209 47249 47290 47330 47371 47 41 1 47452 47492 47 533 47 573 47613 47654 47694 47 734 47 774 .47814 •47 854 47 894 •47 934 •47 974 ,48014 ,48 054 .48094 ■48 "i^ZZ .48173 48213 48 252 48 292 48332 48371 48 41 1 48 450 48 490 48 529 48 568 48 507 48647 48686 48725 48764 48803 48842 48881 48920 48959 48998 948 534 948579 9.48 624 9.48 669 9.48714 948 759 9.48 804 9 48 849 9.48 894 948 939 9.48 984 9.49 029 949 073 9.49 118 949 163 949 207 949 252 949 296 949 341 949 385 949 430 949 474 949519 949 563 949 607 949 652 9.49 696 949 740 949 784 9.49 828 949 872 9.49916 949 960 9.50 004 9.50 048 9.50092 9.50 136 9.50 180 9.50 223 9.50 267 •50 9.50355 950398 9.50 442 9-50485 9-50 529 9-50572 9.50616 9-50659 950703 9-50 746 9.50 789 9-50 ^2,:^ 9.50876 9.50919 9.50 962 9.51005 9.51 048 9.51 092 9-51 135 9.51 178 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 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 0.51 466 0.51 421 0.51 376 0.51 33» 0.51 286 0.51 241 0.51 196 0.51 151 0.51 106 0.51 061 9.98 060 9.98 056 9.98052 9.98 048 9.98 044 9.98 040 9-98 036 9.98 032 9.98 029 9.98023 0.51 016 9.98021 0.50971 0.50927 0.50 882 0.50 837 0.50 793 0.50 748 0.50 704 0.50 659 0.50615 9.98017 9-98013 9.98 009 9.98005 9.98001 9-97 997 9-97 993 9.97 989 9.97 986 0- 50 570 9-97 982 0.50 526 0,50481 0.50437 0.50 393 0.50 348 0.50 304 0.50 260 0.50 216 0.50 172 9-97978 9-97 974 9.97 970 9-97 966 9-97 962 9-97 958 9-97 954 9 97 950 9.97 946 0.50 128 0.50 084 0.50 040 0.49 996 0.49 952 0.49 908 0.49 864 0.49 820 0.49 777 0-49 733 9.97 942 9-97 938 9-97 934 9-97 930 9.97926 9.97922 9.97918 9.97914 9.97910 9.97 906 0.49 689 9-97 902 0.49 645 0.49 602 049 558 0.495^5 0.49471 0.49 428 0.49 384 0.49 341 0.49 297 9.97 898 9.97 894 9-97 890 9.97 886 9.97 882 9.97 878 9-97 874 9 97 870 9.97 866 0-49 254 9.97 861 0.49 2 1 1 0.49 167 0.49 124 0.49081 0.49 038 0.48 995 0.48952 0.48 908 0.48 865 9-97857 9-97 853 9-97 849 9.97 845 9.97841 9-97 837 9-97 833 9.97 829 9.97 823 0.48 822 9.97 821 60 59 58 57 56 55 54 53 52 5^ 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 3?> 32 3^ 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I 45 44 43 I 4-5 2 9.0 3 U-5 4 18.0 5 22.5 6 27.0 7 3'S 8 36.0 9 40.5 4-4 8.8 13.2 17.6 4-3 8.6 12.9 17.2 22.0 21.5 26.4 25.8 30.8 30.1 35-2 344 39-6 38.7 42 41 40 I 4.2 4.1 4.0 2 8.4 8.2 8.0 3 12.6 12.3 12.0 4 16.8 16.4 16.0 S 21.0 20. s 20.0 6 25.2 24.6 24.0 7 29.4 28.7 28.0 8 33-6 32.8 32.0 9 37.8 36.9 36.0 39 5 4 3 3.9 0.5 0.4 0.3 7.8 i.o 0.8 0.6 1 1.7 1.5 1.2 0.9 15.6 2.0 1.6 1.2 19.5 2.5 2.0 1.5 234 3-0 2.4 1.8 273 '3-5 2.8 2.1 31.2 4.0 3.2 2.4 35.1 4.5 3.6 2.7 43 4-3 12.9 21.5 30.1 387 4^ 43 4^ 45 4 44 5.6 5-5 16.9 16.5 28.1 27.5 394 "38-5 3^ 45 3^ 44 54 7-5 7-3 i6.i 22,5 22.0 26.9 37-5 367 37.6 — — L. Cos. L. Cot. c. d. L. Tan L. Sin. P. P. 70° 44 / L. Sin. d. 1 L. Tan. c. d.i L. Cot. 1 L. Cos. d.l P.P. I 9.48 998 39 39 39 38 39 39 3^ 39 39 3ii 39 38 38 39 38 38 39 3^ ^^ 38 38 ^8 9.51 178 43 43 0.48 822 9.97 821 4 60 59 9-49 037 9.51 221 0.48 779 9.97817 2 949 076 9.51 264 0.48 736 9.97812 5 58 3 9-49 115 9.51 306 42 43 0.48 694 9.97 808 4 57 43 42 41 4 949 153 9-51 349 0.48 651 9.97 804 4 56 I 4-3 4-2 4-1 5 9.49 192 9.51 392 43 0.48 608 9.97 800 4 55 2 8.6 8.4 8.2 b 9.49 231 9-51 435 43 0.48 565 9-97 796 4 54 3 12.9 12.6 12.3 7 949 269 9-51478 43 0.48 522 9.97 792 4 5^ 4 17.2 16.8 16.4 8 949 308 9-51 520 42 0.48 480 9-97 788 4 52 5 6 21.5 21.0 20.5 25.8 25.2 24.6 30.1 29.4 28.7 344 33-6 32.8 ^8.7 ^7.8 ^6.0 9 10 949 347 9-51563 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 42 42 42 41 42 42 42 42 41 42 41 42 42 41 42 0.48 437 9-97 784 4 5 51 50 949 385 9.51 606 0.48 394 9.97 779 II 949 424 9.51648 0.48 352 9-97 775 4 4 49 9 12 9.49 462 9.51 691 0.48 309 9.97771 48 13 949 500 9-51 734 0.48 266 9-97 767 4 47 H 949 539 9.51 776 0.48 224 9-97 763 4 4 5 4 46 ^5 949 577 9.51 819 0.48 181 9-97 759 45 39 38 37 i6 17 9.49615 949 654 9.51 861 9.51903 0.48 139 0.48 097 9-97 754 9-97 750 44 43 I 2 3.9 3.8 3.7 7.8 7.6 7.4 i8 19 20 2l' 949 692 949 730 9.51 946 9.51988 0.48 054 0.48012 9.97 746 9.97 742 4 : 42 41 40 39 3 4 5 6 II. 7 11.4 II. I 15.6 15.2 14.8 19.5 19-0 18.5 23.4 22.8 22.2 949 70^ 9.52031 0.47 969 9-97 738 9.49 806 9.52073 0.47 927 9-97 734 22 949 844 J'' 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 37 37 9-52 115 0.47 885 9-97 729 5 38 7 27.3 26.6 25.9 23 9.49 882 9-52157 047 843 9-97 725 4 37 8 31.2 30.4 29.6 24 9.49 920 9.52 200 0.47 800 9.97 721 4 36 9 35.1 34.2 33.3 2,S 949 958 9-52 242 0.47 758 9.97717 4 35 26 949 996 9.52 284 0.47 716 9-97713 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 34 27 9-50034 9-52 326 0.47 674 9-97 708 33 36 5 4 28 9.50072 9-52 368 0.47 632 9.97 704 32 29 30 31 9.50 no 9.52410 0.47 590 9-97 700 31 30 29 2 3 4 3.0 0.5 0.4 1 7.2 i.o 0.8 10.8 1.5 1.2 14.4 2.0 1.6 18.0 2.5 2.0 21.6 3.0 2.4 9.50 148 9.52452 0.47 548 9 97 69^ 9.50 185 9-52494 0.47 506 9-97 691 32 9.50223 9-52536 0.47 464 9-97 687 28 33 9.50 261 9-52578 0.47 422 9.97 683 27 34 9.50 298 9.52 620 0.47 380 9.97 679 26 7 25.2 3.5 2.8 35 9-50336 9.52 661 047 339 9-97 674 25 8 28.8 4.0 3.2 3^ 9-50 374 9.52 703 0.47 297 9-97 670 24 9 32.4 4.5 3.6 37 9.50411 9-52 745 0.47 255 9.97 666 23 3« 39 40 41 9.50449 9-52787 9.52829 0.47 213 0.47 171 9.97 662 9-97657 22 21 20 19 9.50486 5 5 5 9-50523 9.52870 0.47 130 9-97 653 '9-50561 9.52912 0.47 088 9.97649 42 9-50 598 9-52953 0.47 047 9.97 645 18 A9. A9. A^ 43 9.50 635 9.52995 0.47 005 9.97 640 17 44 9.50 673 9-53037 0.46 963 9.97 636 16 4.3 4.2 4.1 45 9.50710 9-53078 0.46 922 9.97 632 15 2 12.9 12.6 12.3 46 9-50 747 9.53 120 0.46 880 9.97 628 4 H 21.5 21.0 20.5 47 48 9.50 784 9.50821 3' J 37 11 37 37 37 36 37 37 37 37 36 37 9-53161 9-53 202 41 41 42 41 42 41 41 41 42 41 41 41 41 41 0.46 839 046 798 9-97 623 9.97619 5 4 4 5 4 4 5 4 4 5 4 4 5 4 13 12 4 5 30.1 29.4 28.7 38-7 37-^ 36.9 49 50 SI 9.50 858 9-53 244 0.46 756 9.97615 II 10 9 ± A A 9.50 896 9-53 285 0.46715 9.97 610 9.50933 9-53 327 0.46 673 9.97 606 52 9.50970 9-53 368 0.46 632 9.97 602 8 53 54 9.51007 9.51043 9-53 409 9-53450 0.46 591 0.46 550 9-97 597 9-97 593 7 6 43 42 41 54 5-2 5-1 55 9.51 080 9-53492 0.46 508 9-97 589 5 ^ 16.1 15.8 15.4 5^' 9-51 "7 9-53 533 0,46 467 9-97 584 4 3 4 26.9 26.2 25.6 1 ^^7 9.51 154 9-53 574 0.46 426 9-97 580 3 37.6 36.8 35.9 58 9.51 191 9-53615 0.46 38^ 9-97 576 2 59 60 9.51 227 9.53 656 0.46 344 9-97571 9.51 264 9-53697 0.46 303 9-97 567 L. Cos. d. L. Cot. c. d.l L. Tan. | L. Sin. d. / p.p. 7V 45 r*^ L. SinTT d. L. Tan. !c. d. L. Cot. L. Cos. d. P.P. I 9.51 264 37 37 36 9-53 697 9.53 738 41 0.46 303 9.97 567 4 5 4 4 5 4 60 59 9.51 301 0.46 262 9.97 563 2 3 9-5133^ 9-51374 9-53 779 9.53 820 4* 41 41 41 41 41 41 40 41 .41 40 41 41 40 41 40 41 40 41 40 41 0.46 221 0.46 180 9.97 558 9.97 554 58 57 41 40 39 4 9.51 411 37 36 9.53861 0.46 139 9.97 550 56 I 4.1 4.0 3.9 8.2 8.0 7.8 5 9-51447 9-53 902 0.46 098 9-97 545 55 2 7 9.51 484 9.51 520 37 36 37 9.53 943 9-53 984 0.46057 0.46016 9.97541 9.97 536 54 53 3 4 5 12.3 12.0 II. 7 16.4 16.0 15.6 20.5 20.0 19.5 24.6 24.0 23.4 28.7 28.0 27.3 32.8 32.0 31.2 36.9 36.0 35.1 8 9.51 557 9.54025 0.45 975 9.97 532 4 5 4 4 5 52 6 9 10 II 9.51 593 9.51 629 36 37 "•A 9-54065 0-45 935 9-97 528 5* 50 49 I 9 9.54 »o6 0.45 894 9-97 523 9.51 666 9-54 147 0.45 853 9-97 5^9 12 9.51 702 ^6 9-54 187. 0.45 813 9-97 5'5 48 13 9.51 738 36 37 9.54 228 0.45 772 9.97510 47 14 9.51 774 9.54 269 0.45 731 9-97 506 4 5 4 I 46 37 36 35 15 9.51 811 9-54 309 0.45 691 9-97 501 45 lb 9.51847 36 36 36 36 36 ^6 9.54 350 0.45 650 9-97 497 44 I 3-7 3-6 3-5 17 9.51883 9.54 390 0.45 610 9.97492 43 2 7.4 7.2 7.0 18 9.51 919 9-54431 0.45 569 9-97488 4 42 3 11. 1 10.8 10.5 19 20 9.51 955 9.54471 0-45 529 9.97 484 4 4 5 4 5 4 4 5 4 5 4 5 41 40 4 I 14.8 14.4 14.0 18.5 18.0 17.5 9-51 991 9.54512 0.45 488 9.97 479 21 9.52027 9-54 552 0.45 448 9.97475 39 7 25.9 25.2 24.5 22 9.52063 36 36 ^t 36 35 36 36 36 35 :56 9-54 593 40 40 0.45 407 9.97 470 38 8 29.6 28 8 28.0 23 9.52099 9-54 633 0.45 367 9.97 466 37 9 33-3 32.4 31-5 24 9-52135 954673 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 40 40 39 0.45 327 9.97461 3b 25 9.52 171 9.54 7H 0.45 286 9-97 457 35 26 9.52 207 9-54 754 0.45 246 9-97 453 34 27 9.52 242 9-54 794 0.45 206 9.97 448 33 34 5 4 28 9.52 278 9.54835 0.45 165 9-97 444 32 I 3-4 0.5 0.4 29 30 31 9-52314 9.54 875 0.45 125 9.97 439 31 30 29 2 3 4 6.8 i.o 0.8 10.2 1.5 1.2 13.6 2.0 1.6 9-52350 9.54915 045 085 9.97 435 9-52 385 9-54 955 0.45 04? 9-97 430 32 9.52421 0" 35 36 35 36 35 36 35 36 3^ lb 9-54 995 0.45 005 9-97426 4 5 4 I 5 4 5 4 5 4 5 28 5 17.0 2.5 2.0 33 34 35 36 37 9-52456 9-52492 9-52527 9-52563 9-52 598 9-55035 9.55075 9-55 "5 9.55155 9.55 195 0.44 965 0.44 925 0.44 885 0.44 8-45 0.44 805 9.97421 9.97417 9.97412 9-97 408 9-97 403 27 26 25 24 23 6 I 9 20.4 3.0 2.4 23-8 3-5 2.8 27.2 4.0 3.2 30.6 4.5 3.6 3« 39 40 41 42 9.52634 9.52 669 9-55 235 9-55 275 0-44 765 0.44 725 9-97 399 9-97 394 22 21 20 19 18 5 5 5 41 40 39 9-52 705 9-55315 0.44 685 9-97 390 9.52 740 9-52 775 9.55 355 9-55 395 0.44 645 0.44 605 9.97 385 9.97 381 43 9-52811 9-55 434 0.44 566 9.97 376 17 44 9.52846 35 35 35 9-55 474 40 40 40 39 0.44 526 9.97 372 4 5 4 s 16 4.1 4.0 3.9 12.3- 12.0 II. 7 20.5 20.0 19.5 28.7 28.0 27.3 45 46 9.52881 9.52916 9-55 5H 9-55 554 0.44 486 0.44 446 9-97 367 9.97 363 15 H 2 3 47 9-52951 35. 35 35 36 34 35 35 35 35 35 35 34 35 9-55 593 40 40 39 40 39 40 39 40 39 40 39 39 40 0.44 407 9.97 358 5 4 5 4 5 13 4 36.9 36.0 35.1 48 9-52986 9-55 633 0.44 367 9.97 353 12 5 49 50 51 9.53021 9-55 673 0.44 327 9.97 349 11 10 9 4 4 4 9.53056 9.55 712 0.44 288 9-97 344 9.53092 9.55 752 0.44 248 9.97 340 52 9.53126 9-55 791 0.44 209 9.97 335 8 53 9.53 161 9-55 831 0.44 169 9-97 331 4 5 4 5 5 7 41 40 39 54 9.53 196 9-55 870 0.44 130 9.97 326 6 5.1 5.0 4.9 55 9.53 231' 9-55910 0.44 090 9-97 322 S 15.4 15.0 14.6 S^ 9.53 266 9.55 949 0.44051 9-97317 4 3 4 25.6 25.0 24.4 5-7 9.53 301 9-55 989 0.44011 9.97312 3 35-9 35-0 34-1 5« 9.53 336 9.56028 0.43 972 9-97 308 4 5 4 2 59 60 9.53 370 9-56 067 9.56 107 0.43 933 9-97 303 9.53 405 0.43 893 9.97 299 1 L. Cos. 1 d. 1 L. Cot. |c. d.| L. Tan. | L. Sin. 1 d. / 1 P.P. 7n' 46 20' L. Sin. L. Tan. c. d L. Cot. L. Cos. P.P. 9 10 II 12 13 14 15 16 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 30 37 3^^ 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.53 405 9-53 440 9-53 475 9-53 509 9-53 544 9-53578 9-53613 9-53 647 9.53682 953716 9.53751 9-53 785 9.53819 953854 9-53 888 953922 9-53 957 9-53 99_i 9-54025 9-54 059 9-54093 9.54127 9-54 161 9-54 195 9.54 229 9-54 263 9-54 297 9-54 33' 9-54 365 9.54 399 9-54 433 9.54 466 9-54 500 9-54 534 9-54 567 9.54601 9-54 635 9.54 668 9.54 702 9-54 735 9-54 769 9.54 802 9.54 836 9.54 869 9-54 903 9.54936 9-54 969 955003 9-55 036 9-55 069 9-55 '02 9-55 '36 9-55 169 9.55 202 9-55 235 9.55 268 9-55 301 9-55 334 9-55 367 9-55 400 9-55 433 9.56 107 9.56 146 9.56 185 9.56 224 9.56 264 9-56 303 9.56 342 9-56381 9.56 420 9-56459 9-56 498 9-56537 9-56576 956615 9-56 654 9.56 693 9-56 732 9.56771 9.56 810 9.56 849 9-56 887 9.56926 9-56 965 9.57004 9.57042 9.57081 9.57 120 9-57 158 9-57 »97 9-57 235 9-57 274 9-57 3'2 9-57351 9-57 389 9.57428 9.57466 9-57 504 9-57 543 9-57581 9.57619 9-57658 9.57 696 9-57 734 9-57 772 9.57810 9-57 849 9.57887 9.57925 9.57 963 9.58001 58039 9.58077 9.58 115 9-58153 9.58 191 9-58 229 9.58 267 9-58 304 958342 9.58 380 9.58418 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 38 39 39 38 39 38 39 38 •39 38 39 38 38 39 38 38 39 38 38 38 38 39 38 38 38 38 38 38 38 38 38 38 38 37 38 38 0.43 893 9-97 299 0.43 854 9.97 294 0.43815 9.97 289 0.43 776 9-97 285 0-43 736 9.97 280 0.43 697 9 97 276 0.43 658 9.97 271 0.43 619 9.97 266 0.43 580 9.97 262 0.43 541 9-97 257 0.43 502 9-97 252 0.43 463 9.97 248 0.43 424 9.97 243 0.43 385 9.97 238 0.43 346 9-97 234 0.43 307 9.97 229 0.43 268 9.97 224 0.43 229 9.97 220 0.43 190 9.97215 0.43 151 9.97 210 0.43 I '3 9.97 206 0.43 074 0.43 035 0.42 996 0.42958 042-919 0.42 880 0.42 842 0.42 803 0.42 765 9.97 201 9-97 196 9-97 192 9.97 187 9.97 182 9-97 178 9-97 173 9.97 168 9-97 ^63 0.42 726 9-97 159 0.42 688 0.42 649 0.42 611 042572 0.42 534 0.42 496 0.42 457 0.42419 0.42 381 9-97 154 9-97 149 9 97 145 9-97 140 9-97 135 9.97 130 9.97 126 9.97 121 9.97 116 0.42 342 9 97" 0.42 304 0.42 266 0.42 228 0.42 190 0.42 151 0.42 113 0.42 075 0.42037 0.41 999 9.97 107 9 97 102 9.97 097 9.97 092 9-97 087 9-97 083 9.97 078 997073 9.97 068 0.41 961 9.97 063 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 9-97 059 997054 9.97 049 9-97 044 9-97 039 9-97035 9-97 030 9-97 025 9.97 020 0.41 582 I 9.97015 L. Tan. [ L. Sin. fift° 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41' 40 39 3^ 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 40 39 4.0 8.0 12.0 16.0 20.0 7.6 II.4 15.2 19.0 3-9 78 11.7 15.6 19-5 24.0 23.4 22.8 28.0 27.3 26.6 32.0 31.2 30.4 36.0 35.1 34.2 37 35 34 3-7 7-4 II. I 14.8 18.5 3.4 6.8 10.2 13.6 17.0 3.5 7-0 10.5 14.0 17-5 22.2 21.0 20.4 25.9 24.5 23.8 29.6 28.0 27.2 33-3 31.5 30.6 33 I 3.3 0.5 0.4 2 6.6 I.O 0.8 3 9-9 1.5 1.2 4 13.2 2.0 1.6 5 16.S 2.5 2.0 6 19.8 30 2.4 7 23.1 .3-5 2.8 8 26.4 4.0 3-2 9 29.7 4.5 3-6 AAA 40 39 38 4.0 3-9 3-8 12.0 II. 7 11.4 20.0 19.5 19.0 28.0 27.3 26.6 36.0 35.1 34.2 AAA 37 39 38 3-7 4-9 [I.I 146 4-8 [4.2 18.5 24.4 23.« 25-9 34-1 33.2 33-3 — — L. Cos. d. L. Cot. c. d. P. P. 47 O I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. 9-55 43: 9-55 4^^ 9.55 499 9.55 532 9-55 564 9-S5 597 9-55 630 9-55 663 9-55 695 9-55 728 9-55 761 9-55 793 9.55 826 9-55 858 9-55891 955923 9-55 956 9 55 988 9.56021 956053 9-56 085 9.56 118 9.56 150 9.56 182 9.56215 9.56 247 9.56 279 9563" 9-56 343 9.56375 9.56 408 9-56 440 9.56472 9.56 504 9-56536 9.56 568 9-56 599 9.56631 9.56 663 9-56 695 9-56 727 956 759 9-56 790 9.56822 9.56854 9.56 886 9.56917 9.56 949 9.56 980 9.57012 9-57 044 9-57075 9.57 107 9-57 138 9.57 169 9.57 201 957232 9.57 264 9-57 295' 9-57 326 9-57 358 L. Cos. I d. L. Tan. c. d. L. Cot. 9.58418 958455 958493 9.58531 9.58 569 9.58606 9:58 644 9.58681 9.58719 9-58757 9. 58 794 9.58 832 9-58 869 9.58907 958 944 9-58981 9.59019 9.59056 9-59 094 9-59 131 9.59 168 9.59 205 9-59 243 9.59 280 9.59317 9-59 354 9-59 391 9-59 429 9.59466 9-59 503 9-59 540 9-59 577 9.59614 9.59651 9.59 688 9-59 725 9-59 762 9-59 799 9-59 835 9.59872 9-59 909 9-59 946 9.59 983 9.60019 9.60056 9.60 093 9.60 130 9.60 166 9.60 203 9.60 240 9.60 276 9.60313 9.60349 9.60 386 9.60 422 9.60459 9-60 495 9.60 532 9.60 568 9.60 605 9.60 641 L. Cot. 37 38 38 38 37 3^ 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 Q.41 582 0.41 545 0.41 507 0.41 469 0.41 431 0.41 394 0.41 356 0.41 319 0.41 281 0.41 243 0.41 206 0.41 168 0.41 131 0.41 093 0.41 056 0.41 019 0.40 981 0.40 944 0.40 906 0.40 869 0.40 832 0.40 795 0.40 757 0.40 720 0.40 683 0.40 646 0.40 609 0.40571 0.40 534 0.40 497 0.40 460 0.40 423 0.40 386 0.40 349 0.40312 0.40 275 0,40 238 0.40 201 0.40 165 0.40 1 28 0.40091 0.40 054 0.40017 0.39 981 0.39 944 0.39 907 0.39 870 0.39 834 0.39 797 0.39 760 0.39 724 0.39 687 0.39651 0.39614 0.39 578 0.39 541 0.39 50? 0.39 468 039 432 0.39 395 0.39 359 c. d.| L. Tan. L. Cos. 9.97015 9.97 010 9-97 005 9.97 001 9.96 996 9.96991 9.96 986 9.96981 9.96 976 9.96971 9.96 966 9.96 962 9-96957 9.96952 9-96 947 9.96 942 9-96 937 9-96 932 9.96927 9.96922 9.96917 9.96 912 9-96 907 9.96 903 9.96 898 9.96 893 9.96 888 9.96 883 9.96 878 996 873 9.96 868 9.96 863 9.96 858 996853 9.96 848 9.96 843 9.96 838 9.96 833 9.96 828 9.96823 9.96818 9.96813 9.96 808 9.96 803 9.96 798 996 793 9.96 788 9.96 783 9.96 778 9.96 772 .96 767 9.96 762 996757 9-96 752 9.96 747 9.96 742 996 737 9.96 732 9.96 727 9 96 722 996 7^7 L. Sin. vr 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 P. P. 38 37 3-8 7.6 11.4 15.2 190 3.7 3-6 7-4 7-2 II. I 10.8 14.8 14.4 18.5 18.0 22.8 22.2 21.6 26.6 25.9 25.2 30.4 29.6 28.8 34.2 33-3 32.4 33 32 31 3-3 6.6 9-9 13.2 16.5 19.8 23.1 3.2 6.4 9.6 12.8 16.0 19.2 22.4 21.7 26.4 25.6 24.8 29.7 28.8 27.9 9-3 12.4 J5-5 18.6 I 0.6 0-5 0.4 2 1.2 I.O 0.8 3 1.8 1-5 1.2 4 2.4 2.0 1.6 5 30 2-5 2.0 6 3-6 30 2.4 7 4.2 3-5 2.8 8 4.8 4.0 3-2 9 5-4 4.5 3.6 7 6 2 5 3 4 4 3 5 2 I 6 37 3.1 9-2 15-4 21.6 27.8 33-9 3.6 10.8 18.0 25.2 32-4 5 38 3.8 U.4 19.0 26.6 34.2 38 4.8 14.2 23-8 33-2 37 3-7 II. I 18.5 25-9 33-3 37 4.6 139 23.1 32.4 P. P. 68° 48 1 L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. P.P. I 9-57358 31 31 31 31 32 31 31 9.60.641 36 36 I 36 36 36 36 3.6 36 ^i 36 36 ^i 36 36 36 36 36 36 35 36 36 36 35 36 ^% 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35 35 36 35 35 36 35 0-39 359 9.96717 6 5 60 59 9-57 389 9.60 677 0.39 323 9.96711' 2 9.57420 9.60714 0.39 286 9.96 706 58 37 36 35 3 9-57451 9.60 750 0.39 250 9.96 701 5 5 5 5 5 5 6 5 5 57 I 3-7 3-6 3-5 4 9.57 482 9.60 786 0.39214 9.96 696 56 2 7-4 7-2 7.0 S 9-57 5H 9.60 823 0.39177 9.96 691 55 3 II. I 10.8 10.5 6 9-57 545 9.60 859 0.39 141 9.96 686 54 4 14.8 14.4 14.0 7 9-57576 9.60 895 0.39 105 9.96 681 S3 5 18.5 18.0 17.5 8 9.57 607 31 9.60931 0.39 069 9.96 676 52 6 22.2 21.6 21.0 9 10 II 9-57 638 31 31 31 31 9.60 967 0.39 033 9.96 670 51 50 49 7 8 9 25.9 25.2 24.5 29.6 28.8 28.0 33-3 32.4 31-5 9.57 669 9.61 004 0.38 996 9.96 665 9-57 700 9.61 040 0.38 960 9.96 660 12 9-57 731 9.61 076 0.38 924 9.96655 5 48 13 9-57 762 31 31 31 31 30 31 31 31 30 9.61 112 0.38 888 9.96 650 5 5 5 6 5 5 5 5 6 47 H 9-57 793 9.61 148 0.38 852 9.96 645 46 32 31 30 15 16 9-57 824 9-57855 9.61 184 9.61 220 0.38816 0.38 780 9.96 640 9.96 634 45 44 I 2 3.2 3-1 30 6.4 6.2 6.0 17 9-57 §85 9.61 256 0.38 744 9.96 629 43 3 9.6 9-3 9.0 18 9.57916 9.61 292 0.38 708 9.96 624 42 4 12.8 12.4 12.0 19 20 21 9-57 947 9.61 328 0.38 672 9.96 619 41 40 39 5 6 7 16.0 15.5 15.0 19.2 18.6 18.0 22.4 21.7 21.0 9-57978 9.61 364 0.38 636 9.96 614 9.58008 9.61 400 0.38 600 9.96 608 22 9.58 039 31 9.61 436 0.38 564 9.96 603 5 38 8 25.6 24.8 24.0 23 9.58070 31 31 30 31 9.61 472 0.38 528 9.96 598 5 37- 9 28.8 27.9 27.0 24 9.58 lOI 9.61 508 0.38 492 9-96 593 5 36 2S 9-58 131 9.61 544 0.38 456 9.96 588 \ 5 5 5 5 6 5 5 35 ■ 25 9.58 162 9.61 579 0.38421 9.96 582 34 29 6 5 27 9.58 192 30 31 9.61 615 0.38385 9.96577 33 2.9 0.6 0.3 5.8 1.2 I.O 8.7 1.8 1.5 1 1.6 2.4 2.0 14.5 3.0 2.5 28 9.58223 9.61 651 0.38 349 9.96572 32 2 29 30 31 9.58253 .5° 31 30 31 9.61 687 0.38313 9.96 567 31 30 29 3 4 9.58 284 9.61 722 0.38 278 9.96 562 9.58 3'4 9.61 758 0.38 242 9-96 556 32 958 345 9.61 794 0.38 206 9-96551 28 6 17.4 3.6 3.0 Zl 9-58375 S^ 9.61 830 0.38 170 9-96 546 27 7 20.3 4-2 3-5 34 9.58 406 31 9.61 865 0.38135 9.96 541 5 6 5 5 5 6 5 5 6 5 5 \ 26 8 23.2 4.8 4.0 3S 9-58 436 30 31 30 30 9.61 901 0.38 099 9.96 535 25 9 26.1 5.4 4.5 36 9.58467 9.61 936 0.38 064 9.96 530 24 ^^ 9-58497 9-58527 9.61 972 9.62 008 0.38 028 0.37 992 9.96 525 9.96 520 23 22 39 40 41 9.58 557 30 31 30 9.62 043 0.37957 9.96514 21 20 19 6 6 36 35 9-58 588 9.62079 0.37921 9-96 509 9.58618 9.62 114' 0.37 886 9.96 504 42 9.58 648 30 9.62 150 0.37 850 9.96498 18 3.0 2.9 9.0 8.8 15.0 14.6 43 44 9-58678 9.58 709 30 31 9.62 185 9.62 221 0.37815 0.37 779 9-96 493 9.96 488 17 16 I 2 4S 9-58 739 30 30 9.62 256 0.37 744 9.96483 15 3 21.0 20.4 46 9.58 769 9.62 292 0.37 708 9.96477 5 14 4 27.0 26.2 47 9-58 799 30 9.62 327 0.37 673 9.96472 13 330 32.1 48 9.58829 30 9.62 362 0.37 638 9.96467 12 49 50 5'i 9.58859 30 30 30 9.62 398 0.37 602 9.96 461 5 5 II 10 9 5 5 5 9.58889 9.62 433 0.37 567 9-96456 9.58919 9.62 468 0.37 532 9.96451 S2 9-58 949 30 30 9.62 504 0.37 496 9-96 445 5 8 37 36 35 53 9-58979 9.62 539 0.37461 9.96 440 7 54 55 9-59 009 9-59 039 3*^ 30 9.62574 9.62 609 3b % 35 35 35 35 0.37 426 0.37391 9-96 435 9.96429 5 6 5 5 6 5 5 6 5 I 2 3-7 3.6 3.5 I I.I 10.8 10.5 18.5 18.0 17.5 1 25.9 25.2 24.5 33-3 32.4 31.5 5^ 57 58 9.59 069 9-59 098 9.59 128 29 30 9.62 645 9.62 680 9.62715 0.37 355 0.37 320 0.37 285 9.96 424 9.96419 9.96413 4 3 2 3 4 5 59 60 9-59 158 30 30 9.62 750 0.37 250 9.96 408 I 9-59 188 9.62 785 0.37215 9-96 403 L. Cos. d. L. Cot. |c. d. L. Tan. L. Sin. d. / P.P. 67° 49 L. Sin. L. Tan. c. d L. Cot. L. Cos. d. P.P. 2 3 4 5 6 7 8 9 10 II 12 13 H 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 3B 39 40 41 42 43 44 45 46 47 48 49 50 51 52 5.3 54 55 56 57 58 59 60 9.59 188 9.59218 9.59 247 9-59277 9-59 307 9-59 33i^ 9-59 366 9-59 396 9-59425 9.59 451 9-5 9 484 9-59 5 H 9-59 543 9-59 573 9.59 602 9-59 ^32 9.59651 9.59 690 9.59 720 9.59 749 9-59 778 9.59 808 9-59 837 9-59 866 9-59 895 9-59 924 9-59 954 9-59983 9.60012 9.60041 9.60070 9.60 099 9.60 128 9.60 157 9.60 186 9.60 215 9.60 244 9.60 273 9.60 302 9.60331 9.60359 9.60 388 9.60417 9.60 446 9.60 474 9.60 503 9.60 532 9.60 561 9.60 589 9.60618 9.60 646 9.60 675 9.60 704 9.60 732 9.60 761 9.60 789 9.60818 9.60 846 9.60 875 9.60 903 9.60931 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 9.62 785 9.62 820 9.62855 9.62 890 9.62 926 9.62 961 9.62 996 9.63 031 9.63 066 9.63 lOI 9»fa3 135 9.63 170 9.63 205 9.63 240 9.63 275 9.63310 9-63 34l 9-63 379 9.63414 9-63 449 9.63 484 9.63519 9.63 553 9.63 588 9.63 623 9.63 657 9.63 692 9.63 726 9.63 761 9.63 796 9.63 830 9.63 865 9.63 899 9-63 934 9.63 968 9.64 003 9.64 037 9.64 072 9.64 106 9.64 140 9-64175 9.64 209 9.64 243 9.64 278 9.64312 9.64 346 9.64 381 9.64415 9.64 449 9.64 483 9.64517 9.64552 9.64 586 9.64 620 9.64 654 9.64 688 9.64 722 9.64 756 9.64 790 9.64 824 9.64858 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 0.37215 9.96 403 0.37 180 0.37 M5 0.37 no 0.37 074 0.37 039 0.37 004 0.36 969 0.36 934 0.36 899 9.96 397 9.96 3^2 9.96 387 9.96 381 9.96 376 9.96 370 9-96 365 9.96 360 9-96 354 o.36't>65 9.96 349 0.36 830 0.36 795 0.36 760 0.36 725 0.36 690 0.36655 0.36 621 0.36 586 0-36551 9-96 343 9.96 33^ 9-96 333 9.96327 9.96 322 9.96316 9.96 311 9.96 305 9.96 300 036510 0.36481 0.36 447 0.36412 0.36 377 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 0.36 170 0.36135 0.36 lOI 0.36 066 0.36032 0-35 997 0.35 963 0.35 928 0.35 894 0.35 860 0.35 825 0.35 791 0.35 757 0.35 722 0.35 688 0.35 654 0.35 619 0.35 585 0-35551 0-35517 0-35 483 0.35 448 0.35414 0.35 380 0.35 346 0.35312 0.35 278 0.35 244 0.35 210 0.35 176 0.35 H2 9- 96 294 9.96 289 9.96 284 9.96 278 9.96 273 9.96 267 9.96 262 9.96 256 9.96 251 9.96 245 9.96 240 9.96 234 9.96 229 9.96 223 9.96 2l8 9.96 212 9.96 207 9.96 201 9.96 196 9.96 190 9-96 185 9.96179 9.96174 9.96 168 9.96 162 9.96157 9-96151 9.96 146 9.96 140 9-9 6 135 9.96 129 9.96 123 9.96 118 9.96 112 9.96 107 9.96 lOI 9.96 095 9.96 090 9.96 084 9^96 079 9-96 073 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 3S 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I O 35 34 2 3.6 3.5 7.2 7.0 3-4 6.8 3 10.8 10.5 10.2 4 5 6 144 M-o 18.0 17.5 21.6 21.0 13.6 17.0 204 7 8 25.2 24.5 28.8 28.0 23.8 27.2 9 324 3' -5 30.6 I 2 3-0 6.0- 2.9 S.8 3 4 9.0 12.0 . 8.7 11.6 5 6 7 15.0 18.0 21.0 14-5 17.4 20.3 8 9 24.0 27.0 23.2 26.1 6 0.6 1.2 1.8 24 3-0 3-6 4.2 4.8 5.4 29 28 2.8 5.6 84 II. 2 14.0 16.8 19.6 22.4 25.2 0.5 i.o 1-5 2.0 2.5 3-0 3.5 4.0 4-5 6 36 35 2.9 8.8 3.0 9.0 5.5 5.5 15.0 14.6 14.2 21.0 204 19.8 27.0 26.2 25.5 33.0 32.1 31.2 35 3.5 10.5 17-5 24-5 31-5 34 34 10.2 17.0 23.8 30.6 I L. Cos. i d. I L. Cot, c. d. L. Tan. L. Sin. d. I P. P. 50 L. Sin. d. L. Tan. |c. d. L. Cot. L. Cos. d. P.P. O I 2 3 4 5 6 7 8 9 10 II 12 13 H ^5 i6 17 18 19 20 21 22 ^3 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.60931 9.60 960 9.60 988 9.61 016 9.61 045 9.61 073 9,61 lOI 9.61 129 9.61 158 9.61 186 9.61 214 9.61 242 9.61 270 9.61 298 9.61 326 9.61 354 9.61 382 9.61 411 9.61 438 9.61 466 9.61 494 9.61 522 9.61 550 9.61 578 9.61 606 9.61 634 9,61 662 9.61 689 9.61 717 9.61 745 9.61 773 9.61 800 9.61 828 9.61 856 9.61 883 9.61 911 9.61 939 9.61 966 9.61 994 9.62021 9.62 049 9.62076 9.62 104 9.62 131 9.62 159 9.62 186 9.62 214 9.62 241 9.62 268 9.62 296 9.62 323 9.62 350 9.62 377 9.62 405 9.62 432 9.62459 9.62 486 9.62513 9.62 541 9.62 568 9-62 595 L. Cos. 9.64 9.64 892 9.64 926 9.64 960 9.64 994 9.65 028 9.65 062 9.65 096 9.65 130 9.65 164 9.65 197 9.65 231 9.65 265 9.65 299 9.65 333 9.65 366 9.65 400 9-65 434 9.65 467 9.65501 9-65 535 9.65 568 9.65 602 9.65 636 9.65 669 9.65 703 9.65 736 9.65 770 9.65 803 9-65 837 9.65 870 9.65 904 9-65 937 9.65 971 9.66 004 9.66038 9.66071 9.66 104 9.66138 9.66 171 (.66 204 9.66 238 9.66 271 9.66 304 9-66 337 9.66371 9.66 404 9.66437 9.66 470 9-66 503 9-66 537 9.66 570 9.66 603 9.66 636 9.66 669 9.66 702 9-66 735 9.66 768 9.66 801 9.66 834 9.66 867 L. Cot. 34 34 34 34 34 34 34 34 34 33 34 34 34 34 33 34 34 33 34 34 33 34 34 33 34 33 34 33 34 33 34 33 34 33 34 33 33 34 33 33 34 33 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 cTd 0.35 142 9.96073 0.35 108 0.35 074 0.35 040 0.35 006 0.34972 0.34 938 0.34 904 0.34 870 0.34 836 9.96067 9.96062 9.96056 9.96050 9.96 045 9.96039 9.96 034 -9.96 028 9.96 022 0.34 803 9.96017 0.34 769 0-34 735 0.34 70 i 0.34 667 0.34 634 0.34 600 0.34 566 0.34 533 0.34 499 9.96 01 1 9.96 005 9.96 000 9.95 994 9.95 988 9.95 982 9.95 977 9-95971 9-95 965 0.34 465 9.95 960 0.34 432 0.34 398 0.34 364 0.34331 0.34 297 0.34 264 0.34 230 0.34 197 0.34 163 9-95 954 9.95 948 9-95 942 9-95 937 9-95 931 9-95 925 9.95 920 9-95 9H 9.95 908 0.34 130 9.95 902 0.34 096 0.34 063 0.34 029 0.33 996 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 9-95 897 9.95 891 9.95 885 9.95 879 •9-95 873 9.95 868 9.95 862 9-95 856 9-95 850 0.33 796 9-95 844 0.33 762 0.33 729 0.33 696 0.33 663 0.33 629 0.33 596 0.33 563 0.33 530 0.33 497 9-95 839 9-95 833 9.95 827 9.95821 9-95 815 9.95 810 9.95 804 9-95 798 9-95 792 0.33 46: •95 786 0.33 430 0.33 397 0.33 364 0-33 33^ 0.33 298 0.33 265 0.33 232 0.33 199 0.33 166 9.95 780 9-95 775 9-95 769 9-95 763 9-95 757 9-95751 9-95 745 9-95 739 9-95 733 0-33 ^33 L. Tan. 9-95 728 L. Sin. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 34 3-4 6.8 10.2 13-6 17.0 20.4 23-8 27.2 30.6 33 3-3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 29 28 27 I 2.9 2.8 2.7 2 5.8 5.6 S-4 3 8.7 8.4 8.1 4 11.6 II. 2 10.8 s 14.5 14.0 1.^.5 6 17.4 16.8 16.2 7 20.3 19.6 18.9 8 23.2 22.4 21.6 9 26.1 25.2 24.3 6 0.6 1.2 1.8 2.4 30 3-6 4.2 4.8 5-4 0-5 i.o 1-5 2.0 2-5 30 3-5 4.0 4-5 6^ 34 2.8 8.5 33 2.5 8.2 14.2 13.8 19.8 19.2 _5^ 34 3-4 10.2 17.0 2^.s 25.5 24.8 30.6 31.2 30.2 — p.p. a^° 5' O I 2 3 4 5 6 7 8 9 10 II 12 13 H 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L. Sin. 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 9.62 730 9.62 757 9.62 784 9.62 81 1 9.62 838 9.62 86g 9.62 892 9.62918 9.62 945 9.62972 9.62 999 9.63 026 9.63052 9.63 079 9.63 106 9-63 133 9.63 159 9.63 186 9.63213 9.63 239 9.63 266 9.63 292 963319 9-63 345 9-63 372 9-63 398 9.63 425 9.63451 9.63 478 9-63 504 9-63 531 9-63557 963 583 9.63 610 9.63 636 9.63 662 9.63 689 9-63715 9-63 741 9.63 767 9-63 794 9.63 820 9.63 846 9.63 872 9.63 898 •63 924 9-63 950 9.63 976 9.64 002 9.64028 9.64054 9.64 080 9.64 106 9.64 132 9.64 158 9.64 184 L. Tan. 9.66 867 9.66900 9-66 933 9.66 966 9.66 999 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 9.67 196 9.67 229 9.67 262 9.67 295 9.67 327 9-67 360 9-67 393 9.67 426 9.67 458 9.67 491 9.67 524 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 9.67 850 9.67 882 9.67915 9-67 947 9.67 980 9,68012 9.68 044 9.68077 9,68 109 9.68 142 9.68 174 9.68 206 9-68 239 9.68 271 9.68 303 9.68336 9.68 368 9.68 400 9.68 432 9.68 465 9.68 497 9.68 529 9.68 561 9-68 593 9.68 626 9.68 658 9.68 690 9.68 722 9.68 754 9.68 786 9.68818 c. d 32, 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 32 33 32 33 32 32 33 32 33 32 32 33 32 32 33 32 32 32 33 32 32 32 32 33 32 32 32 32 32 32 L. Cot. ^13J33^ 0.33 too 0.33067 0.33 034 0.33001 0.32 968 0.32 935 0.32 902 0.32 869 0.32837 0.32 804 0.32771 0.32 738 0.32 705 0.32 673 0.32 640 0.32 607 0.32 574 0.32 542 0-32 509 0.32 476 0.32 444 0.32 411 0.32 378 0.32 346 0.32313 0.32 281 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 0.32085 0.32053 0.32 020 0^1 988 0.31 956 0.31 923 0.31 891 0.31 858 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 0.31 664 0.31 632 0.31 600 0.31 568 0-31 535 0.31 503 0.31 471 0.31 439 0.31 407 0.31 374 0.31 342 0.31 310 0.31 278 0.31 246 0.31 214 0.31 182 L. Cos. 995 728 9-95 722 9-95 716 9-95 710 9.95 704 9.95 698 9.95 692 9.95 686 9.95 680 9-95 674 9-95 668 9.95 663 9.95 657 9-95 651 9-95 64I 9-95 639 9-95 633 9-95 627 9.95 621 995615 9.95 609 9.95 603 9-95 597 9-95 591 9.95 585 9-95 579 9-95 573 9-95 567 9-95 561 9-95 555 9-95 549 9-95 543 9-95 537 9-95 531 9-95 525 9-95519 9-95513 9-95 507 9-95 500 9-95 494 9.95 488 9.95 482 9-95 476 9-95 470 9-95 464 9-95 458 9-95 452 9-95 446 9.95 440 9-95 434 9-95 427 9-95 421 9-95415 9.95 409 9-95 403 9-95 397 9-95 391 9.95 384 9-95 378 9-95 372 9-95 366 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 3^ 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I O P. P. 33 32 3-2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 27 26 I .3.3 2 6.6 3 9-9 4 13.2 5 16.5 6 19.8 7 23.1 8 26.4 9 29.7 2-7 5-4 8.1 10.8 13-5 16.2 18.9 21.6 243 2.6 5-2 7-8 10.4 13.0 15.6 18.2 20.8 23-4 I 0.7 0.6 0.5 2 1.4 1.2 I.O 3 2.1 1.8 1.5 4 2.8 2.4 2.0 5 3.5 3-0 2-5 6 4.2 3.6 3-0 7 4-9 4.2 3-5 8 5.6 4-8 4.0 9 6.3 5-4 4-5 32 2-3 6.9 11.4 16.0 20.^ 25.1 29.7 32 2.7 8.0 13-3 18.7 24.0 29-3 33 3-3 9-9 16.5 23.1 29.7 L. Cos. L. Cot. c. d. L. Tan- fid." L. Sin. I d. P.P. / L. Sin. d. L.Tan. c. d. L. Cot. 1 L. Cos. d.| P. P. 1 I 9.64 184 26 26 26 9.68818 32 0.31 182 9.95 366 6 6 60 59 9.64 210 9.68 850 0.31 150 9.95 360 2 9.64 236 9.68 882 32 32 0.31 118 9-95 354 58 3 9.64 2b2 26 25 26 9.68914 0.31 086 9-95 348 57 4 9.64 288 9.68946 32. 32 10 0.31 054 9-95 341 7 5 56 32 31 5 964313 9.68978 0.31022 9-95 335 6 55 I 3-2 3.1 1 6 9-64 339 26 26 26 25 26 26 9.69 OIO 32 32 32 32 32 32 32 32 32 31 0.30 990 9-95 329 6 6 7 6 6 6 54 2 6.4 6.2 7 9.64 365 9.69 042 0.30 958 9-95 323 53 3 9-6 9-3 8 9.64391 9.69 074 0.30 926 9-95317 52 4 12.8 12.4 9 10 II 9.64 4P7 9.69 106 0.30 894 9-95 310 51 50 49 ' 5 6 7 8 16.0 15.5 19.2 18.6 22.4 21.7 25.6 24.8 9.64 442 9.69 1.38 0.30 862 9-95 304 9.64 468 9.69 170 0.30 830 9.95 298 12 9.64 494 9.69 202 0.30 798 9.95 292 A 48 9 28.8 2-7.0 13 9.64519 25 26 26 9-69 234 0.30 766 9.95 286 47 14 9-64 545 9.69 266 0.30 734 9.95 279 7 46 IS 9-64571 9.69 298 0.30 702 9-95 273 6 6 45 i6 9.64 596 25 26 11 25 26 9.69 329 0.30671 9.95 267 44 17 9.64622 9.69 361 32 0.30 639 9-95 261 43 26 25 24 18 9.64 647 9-69 393 C 0.30 607 9-95 254 7 t 6 42 I 2.6 2.5 2.4 1 19 20 21 9.64 673 9.69 425 32 31 0-30 575 9-95 248 41 40 39 2 3 4 5.2 5.0 4.8 7-8 7-5 7-2 10.4 lO.O 9.6 9.64 698 9-69 457 '^.30 543 9.95 242 9.64 724 9.69 488 0.30 5 1 2 9.95 236 22 9.64 749 25 26 25 26 9.69 520 32 32 32 31 32 32 31 32 32 31 32 31 32 32 31 32 31 •32 31 32 31 32 0.30 480 9.95 229 7 6 6 6 3^ 7 8 13.0 12.5 12.0 1 15.6 15.0 14.4 18.2 17.5 16.8 20.8 20.0 19.2 ] 23 24 9-64 775 9.64 800 9.69 552 9.69 584 0.30 448 0.30416 9.95 223 9-95217 37 36 2.S 9.64826 9.69615 0.30385 9.95 211 7 6 35 g 22.A 22. C 21.6 1 25 9.64851 26 9.69 647 0.30 353 9-95 204 34 27 9.64877 9.69 679 0.30321 9.95 198 33 28 9.64 902 25 9.69710 0.30 290 9.95 192 32 29 9.64927 ^5 9.69 742 0.30 258 9.95 185 7 6 31 30 31 9.64 953 25 9.69 774 0.30 226 9-95 179 I 7 A 30 29 7 6 9.64 978 9.69 805 0.30 195 9-95 173 1 0.7 0.6 32 9.65 003 'I 9.69 837 0.30 163 9.95 167 28 > 1.4 1.2 33 9.65 029 9.69 868 0.30132 9-95 160 27 1 1 2.1 1.8 34 9-65 054 25 9.69 900 0.30.100 9-95 154 6 26 ^ 2.8 2.4 35 9.65 079 25 9.69932 0.30 068 9-95 148 7 6 6 25 ) 3-5 3-0 36 9.65 104 25 26 25 9.69 963 0.30037 9-95 141 24 ) 4.2 3.6 37 9.65 130 9-69 995 0.30 005 9-95 135 23 f 4-9 4-2 I 5.6 4.8 ) 6.3 5.4 18 9-65 155 9.70026 0.29 974 9-95 129 22 39 40 41 9.65 180 25 25 25 25 26 9.70058 0.29 942 9.95 122 7 6 6 7 6 21 20 19 9.65 205 9.70 089 0.29 911 9.95 116 9.65 230 9.70121 0.29 879 9-95 "o 42 9-65 255 9.65 281 9.65 306 9.70152 9.70 184 9.70215 0.29 848 0.29 816 0.29 785 9.95 103 9-95 097 9.95 090 18 17 16 43 44 25 25 25 31 32 31 31 32 31 32 31 31 32 7 6 4S 9-65 33i 9.70 247 0.29 753 9-95 084 6 15 46 9-65 356 9.70278 0.29 722 9-95 078 7 6 H 47 9.65 381 25 25 i 9.70 309 0.29 691 9.95071 13 7 7 6 48 49 50 51 9.65 406 9-65431 9.70341 9.70372 0.29 659 0.29 628 9.95 065 9-95 059 6 7 6 7 12 II 10 9 32 31 32 9.65450 9.70 404 0.29 596 9-95052 I 2 2.3 2.2 2.7 6.9 6.6 8.0 9.65 481 9-70 435 0.29 565 9-95 046 52 9.65 506 25 9.70466 0.29 534 9-95 039 8 11.4 II. 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Cos. I d. I L. Cot. Ic. d.| L. Tan. | L. Sin. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 i 30 29 28 30 6.0 9.0 12.0 15.0 2.9 5.8 8.7 11.6 14-5 18.0 17.4 21.0 20.3 24,0 23.2 22.4 27.0 26.1 25.2 2.8 5-6 8.4 II. 2 14.0 16.8 19.6 22 21 2.2 4-4 6.6 8.8 II.O 13.2 15.4 17.6 19.8 0.8 1.6 2.4 3-2 4.0 4.8 5-6 6-4 7-2 2.1 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 0.7 1-4 2.1 2.8 3-5 4.2 4.9 5-6 6.3 30 29 28 2.1 2.1 2.0 6.4 6.2 6.0 10.7 10.4 1 0.0 15.0 14.5 14.0 19.3 18.6 18.0 23.6 22.8 22.0 27.9 26.9 26.0 P. P. 57 1 L. Sin. ["d. L. Tan. c. d. L. Cot. L. Cos. d. P. P. I 9.71 184 9.71 205 21 1 9.77877 977 906 29 29 28 0.22 123 9-93 307 8 60 59 0.22 094 9.93 299 2 9.71 226 2 1 977 935 0.22 ob5 9.93291 7 8 7 8 58 3 9.71 247 977 9t>3 29 28 29 '>8 0.22037 9.93 284 57 4 9.71 268 21 9.77992 0.22008 9.93 276 56 29 28 5 9.71 289 9.78020 0.21 980 9.93 2b9 55 I 2.9 2.8 6 9.71 310 9.78049 0.21 951 9.93 2bl 8 54 2 5.8 5.6 7 9-71 331 21 21 20 21 21 21 21 9.78077 29 29 28 29 28 29 28 29 98 0.21 923 9-93 253 7 8 8 7 8 8 7 8 53 3 8.7 8-4 8 971 352 9.78 lob 0.21 894 9.93 24b 52 4 11.6 II. 2 9 10 II 971 IIZ 978 135 0.21 8b5 9-93 238 51 50 49 I 7 8 9 14.5 14.0 17.4 16.8 20.3 19.6 971 393 9.71 414 9.78 163 0.21 837 9-93 230 9.78 192 0.21 808 9-93 223 12 13 971 435 971456 9.78 220 9.78 249 0.21 780 0.21 751 9-93215 9.93 207 48 47 23.2 22.4 26.1 25.2 14 9.71 477 9.78277 0.21 723 9.93 200 46 15 9.71498 '> I 9.78 30b 0.21 b94 9.93 192 c 45 lb 971 519 978 334 0.2 1 (^iib 9.93 184 44 17 971 539 20 978 2>^Z 29 2S 0.21 b37 9-93177 7 8 43 21 20 18 9.71 5bo 21 9.78391 28 0.21 bo9 9.93 ib9 8 42 I 2.1 2,0 19 20 21 9-^ 581 21 20 978419 29 28 29 28 0.21 581 9.93 ibi 7 8 8 7 8 8 41 40 39 2 3 4 4.2 4.0 b.3 bo 8.4 8.0 9.71 602 978448 0.21 552 9-93154 9.71 b22 9.7847b o.5i 524 9.93 146 22 971 643 978 505 0.21 495 9.93 138 :<^ 5 10.5 lO.O 23 9.71 bb4 978533 0.21 4b7 9-93131 31 b 12. b 12.0 24 971 685 20 9.78 5b2 29 0% 0.21 438 9.93 123 36 I 14.7 140 ib.8 ibo 2S 9.71 705 9.78 590 28 29 28 0.21 410 9-93115 35 9 18.9 18.0 26 27 9.71 72b 9.71 747 21 9.78 bi8 9.78 b47 0.21 382 0.21 353 9.93 108 9.93 100 7 8 8 34 2,3 28 9.71 7b7 ■7 T 978675 29 28 28 29 28 0.21 325 9.93 092 8 7 8 8 8 32 29 30 31 971 788 21 20 9.78 704 0.21 29b 9.93 084 31 30 29 8 7 9.71 809 9.78 732 0.21 2b8 9-93077 9.71 829 9.78 7bo 0.21 240 993069 I i 0.8 0.7 32 9.71 850 20 9.78 789 0.21 211 9-93061 28 2 |i.6 I.; 33 9.71 870 21 978817 28 0.21 183 993053 7 8 8 27 3 1 2.4 2.1 34 9.71 891 -70 9.78 845 29 28 0.21 155 9-93 046 26 4 13-2 2.8 35 9.71 911 9.78 874 0.21 12b 9-93 038 25 5 4.0 3-5 37 9.71 932 9.71952 20 9.78 902 9.78930 28 29 28 0.21 098 0.21 070 9-93 030 9.93022 8 8 24 23 I 4.8 4.2 5-6 4-9 6.4 5.6 3« 971 973 21 978 959 0.21 041 9-93014 7 8 8 8 22 39 40 41 9.71 994 20 20 21 978 987 28 28 29 28 28 0.21 013 9-93 007 21 20 19 9 1 /"^ ^-j 9.72014 9.79015 0.20 985 9-92 999 9.72034 979043 0.20957 9.92991 42 43 9.72055 9.72075 20 21 9.79072 9.79 100 0.20 928 0.20 900 9-92 983 9.92976 7 8 18 17 44 9.72 096 20 9.79 128 28 0.20872 9.92 968 8 16 45 9.72 lib 979 156 29 0.20844 9.92 9bo 8 15 4b 9.72 137 20 979 185 0.20 815 9.92952 8 14 8 8 8 47 9.72157 20 9.79213 -'S 0.20 787 • 9.92 944 8 13 30 29 28 48 9.72177 9.79241 28 0.20 759 9-92936 7 8 8 8 12 49 50 51 9.72 198 20 20 21 979 269 28 29 0.20731 9.92 929 II 10 9 I 2 3 1.9 1.8 1.8 5-6 5'4 5-2 9.4 9.1 8.8 13.1 12.7 12.2 ib.9 ib.3 15.8 20.b 19.9 19.2 9.72 218 979 297 0.20 703 9.92921 9.72238 979 326 0.20 b74 9.92913 52 9.72259 20 979 354 ->« 0.20 b4b 9.92 905 s 8 4 53 9.72 279 20 979 382 ?8 0.20 bi 8 9-92 897 8 7 5 b 54 9.72 299 21 9.79410 28 0.20 590 9.92 889 8 b 24.4 23.b 22.8 55 9.72 320 20 979438 28 0.20 5b2 9.92 881 7 8 8 5 . 8 28.1 27.2 2b. 2 5^^ 57 9.72 340 9.72 3bo 20 979466 979 495 29 28 0.20 534 0.20 505 9.92 874 9.92 8bb 4 3 5« 9.72381 20 979523 ->« 0.20477 9.92 858 s 2 59 60 9.72401 20 979551 28 0.20 449 9 92 850 8 972421 979 579 0.20421 9.92 842 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. / P.P. .^»^ 58 / L. Sin. d. L. Tan. 0. d. L. Cot. L. Cos. d. P.P. I 9.72421 20 979 579 28 28 28 0.20421 9.92 842 8 8 60 59 9.72441 979 607 0.20 393 9.92 834 2 9.72461 979 635 0.20 365 9.92 826 s 58 3 9.72 482 20 20 979 663 28 28 0.20 337 9.92818 8 I 57 4 9.72 502 979 691 0.20 309 9.92 810 56 29 28 27 5 9.72522 20 9.79719 28 0.20 281 9.92 803 55 I 2.9 2.8 2.7 6 9.72542 20 20 979 747 29 28 0.20 253 9-92 795 « 54 2 5.8 5.6 5.4 7 9.72562 979 776 0.20 224 9.92 787 8 53 3 8.7 8.4 8.1 ■ 8 9.72582 979 804 0.20 196 9.92 779 52 4 11.6 11.2 10.8 9 10 II 9.72 602 20 21 20 9.79832 28 28 0.20 168 9.92 771 8 8 8 51 50 49 5 6 14.5 14.0 13.5 17.4 16.8 16.2 20.3 19.6 18.9 23.2 22.4 21.6 9.72 622 9.79 860 0.20 140 9.92 763 9.72 643 9.79 888 0.20 112 992 755 12 9.72 663 20 9.79916 ->« 0.20084 9.92 747 8 48 9 26.1 21;. 2 2^.1 9.72 683 9.72 703 20 20 979 944 979 972 28 28 0.2005b 0.20028 992 739 9.92 731 8 8 47 46 15 9.72 723 9.80000 28 0.20000 9.92 723 8 45 16 972 743 9.80 028 28 28 28 28 28 0.19972 9.92715 8 8 8 8 8 8 8 8 8 44 17 9.72 763 20 -20 20 20 9.80 056 0.19 944 9.92 707 43 21 20 19 iS 9-72 783 9.80 084 0.19916 9.92 699 42 I 2.1 2.0 1.9 19 20 21 9.80 112 0.19888 9.92691 41 40 39 2 3 4 4.2 4.0 3.8 6.3 6.0 5.7 8.4 8.0 7.6 9.72 823 9.80 140 0.19 860 9.92 683 9.72 843 9.80 168 0.19832 9.92 675 22 23 24 9.72863 972 883 9.72 902 20 19 20 9.80 195 9.80 223 9.80251 27 28 28 28 0.19 805 0.19777 0.19 749 9.92 667 9.92 659 9.92651 38 37 36 5 6 I 10.5 lo.o 9.5 12.6 12.0 H.4 14.7 14.0 13.3 16.8 16.0 15.2 18.9 18.0 I7.I 25 9.72922 9.80 279 -pR 0.19 721 9.92 643 8 8 8 8 8 8 8 8 35 9 26 9.72942 9.80 307 28 28 08 0.19693 9-92 635 34 27 9.72 962 9-8o 335 0.19 665 9.92627 33 28 9.72982 9.80 363 0.19637 9.92 619 32 29 30 31 9.73002 20 19 9.80391 28 28 27 28 0.19609 9.92 611 31 30 29 9 8 7 9.73022 9.80419 0.19581 9.92 603 9.73041 9.80447 0.19553 9.92 595 T 0.9 0.8 0.7 32 9.73061 9.80 474 0.19 526 9.92 587 28 2 1.8 1.6 1.4 33 34 9.73081 9.73 lOI 20 26 19 9.80 502 9.80 530 28 28 28 0.19498 0.19470 9-92 579 9.92571 8 8 8 27 26 3 4 2.7 2.4 2.1 3.6 3.2 2.8 35 9.73 121 9.80558 0.19442 9.92 563 25 5 4-5 4.0 3.5 3^ 973 140 9.80 586 28 28 0.19 414 9-92 555 24 6 5.4 4.8 4.2 37 973 160 9.80614 0.19 386 9.92 546 9 8 8 8 8 8 8 23 7 6.3 5.6 4.9 38 973 180 9.80 642 0.19358 9.92538 22 8 7.2 6.4 5.6 39 40 41 973 200 19 20 20 19 9.80 669 2y 28 28 28 28 0.19 331 9.92 530 21 20 19 9 8.1 7.2 6.3 9.73219 9.80 697 0.19 303 9.92 522 973 239 9.80 725 0.19275 9.92514 42 43 44 973259 9.73278 973 298 9-8o 753 9.80 781 9.80 808 0.19 247 0.19 219 0,19 192 9.92 506 9.92 498 9.92 490 18 17 16 20 27 28 28 8 8 9 8 8 45 973318 19 9.80 836 0.19 164 9.92482 15 4b 973 337 9.80 864 28 27 0.19 136 9.92473 14 47 973 357 9.80 892 0.19 108 9.92 465 13 8 8 7 48 973 377 19 20 9.80919 0.19 081 9.92457 s 12 sa 2S 28 49. 973 396 9.80 947 -^8 0.19053- _2:S2^9 '8- 11 50 51 52 53 54 55 9.73416 19 20 19 20 19 20 9.80975 28 '27 28 28 0.19025 9.92441 8 8 9 8 8 8 10 9 8 7 6 5 I 2 3 4 1.8 1.8 2.0 5.4 5.2 6.0 9.1 8.8 lo.o 12.7 12.2 14,0 16.3 15.8 18.0 19.9 19.2 22.0 23.6 22.8 26.0 973 435 973 455 973 474 973 494 973 5'3 9.81 003 9.81 030 9.81 058 9.81 086 9.81 113 0.18997 0.18970 0.18942 0.18914 0.18 8S7 992433 9.92425 9.92416 9.92 408 9.92 400 5^ 973 533 19 9.81 141 28 0.18859 9.92 392 8 4 8 27.2 26.2 — 57 973552 9.81 169 0.18 831 9.92 384 8 9 8 3 58 973572 19 20 9.81 196 27 0.18804 9.92376 2 59 60 973591 9.81 224 28 0.18 776 9.92 367 I 973 611 9.81 252 0.18748 992 359 L. 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Cos. L. Cot. c. d. L. Tan. L. Sin. I d. P. P. Kl^ 39' 65 t L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. P.P. I 9.79 887 16 15 16 9.90 837 26 26 25 26 26 26 26 25 26 26 26 26 0.09 163 9.89 050 10 10 60 59 9.79 903 9.90 863 0.09 137 9.89 040 2 9.79918 9.90 889 0.09 1 1 1 9.89 030 58 3 9-79 934 16 11 9.90914 0.09 086 9.89 020 57 26 26 4 9-79 950 9.90 940 0.09 060 9.89 009 56 5 979965 9.90 966 0.09 034 9.88999 ir» 55 I 2.0 2,5 6 9.79 981 9.90 992 0.09 008 9.88 989 54 2 5.2 5.0 7.8 7.5 10.4 1 0.0 13.0 12.5 15.6 15.0 18.2 17.5 20.8 20.0 7 9.79 996 ^5 16 \i 15 16 9.91 018 0.08 982 9.88 978 10 II S3 3 8 9.80012 9.91 043 0.08957 9.88 968 52 4 7 8 9 io II 9.80027 9.91 069 0.08931 9.88 958 51 50 49 9.80 04.? 9.91 095 0.08 905 9.88 948 9.80 058 9.91 121 0.08 879 9.88937 12 9.80 074 15 t6 9.91 147 25 96 0.08 853 9.88927 ir> 48 9 23.4 22.1: 13 9.80 089 9.91 172 0.08 828 9.88917 10 47 14 9.80 105 \l 9.91 198 26 0.08 802 9.88 906 46 ^5 9.80 1 20 9.91 224 26 0.08 776 9.88 896 45 16 9.80 136 15 15 16 9.91 250 26 0.08 750 9.88 886 10 44 16 15 17 9.80 151 9.91 276 25 26 0.08 724 9.88875 43 I 1.6 1.5 18 9.80 ibb 9.91 301 0.08 699 9.88 865 42 2 3-2 3-0 19 20 21 9.80 182 15 16 9.91 327 26 26 '4 0.08 673 9.88 855 II 10 41 40 ^9 3 4 4.8 4.5 6.4 6.0 8.0 7.5 9.80 197 9.91 353 0.08 647 9.88 844 9.80213 9-91 379 0.08 621 9.88 834 22 9.80 228 15 16 9.91 404 0.08 596 9.88 824 11 10 38 6 9.6 9.0 23 24 9.80 244 9.80 259 15 9.91 430 9.91 456 26 26 25 26 0.08 570 0.08 544 9.88813 9.88 803 37 36 7 8 9 1 1.2 10.5 12.8 12.0 14.4 13-5 25 9.80 274 15 9.91 482 0.08518 9.88 793 35 2b 9.80 290 9.91 507 0.08 493 9.88 782 34 27 9.80 305 9.91 533 26 0.08 467 9.88 772 33 28 9.80 320 9.91 559 26 0.08441 9.88 761 10 10 II 10 32 11 in 29 30 31 9.80336 15 15 16 9-91 585 25 26 26 0.08 4 1 5 9.88751 31 30 29 I 2 3 4 5 6 1.1 I.O 2.2 2.0 3-3 3.0 4.4 4.0 5-5 5-0 6.6 6.0 9.80351 9.91 610 0.08 390 9.88 741 9.80 36b 9.91 636 0.08 364 9.88 730 32 9.80 382 15 15 16 15 15 9.91 662 26 0.08 338 9.88 720 28 33 34 9.80 397 9.80412 9.91 688 9-91 713 25 26 26 26 0.08 3 1 2 0.08 287 9.88 709 9.88 699 10 27 26 35 9.80 428 9-91 739 0.08 261 9.88 688 25 7 7.7 7.0 3^ 9.80 443 9.91 765 0.08 235 9.88 678 24 8 8.8 8.0 37 9.80458 9.91 791 0.08 209 9.88 668 23 9 9.9 9.0 3« 9.80 473 15 16 15 15 9.91 816 11 26 25 26 26 26 0.08 184 9.88 657 ^' 22 39 40 41 9.80 489 9.91 842 0.08 158 9.88 647 II 21 20 19 9.80 504 9.91 868 0.08 132 9.88 636 9.80519 9.91 893 0.08 107 9.88 626 42 9-8o 534 9.91 919 0.08081 9.88615 ^^ 18 43 9.80 550 15 9-9 1 945 0.08 055 9.88 605 M 17 44 9.80565 9.91 971 0.08 029 9.88 594 16 45 9.80 580 ^5 15 15 15 16 9.91 996 25 26 26 0.08 004 9.88 584 lu 15 11 11 4b 9-8o 595 9.92022 0.07 978 9-88573 ^' H 26 25 47 9.80 610 9.92 048 0.07 952 9.88563 13 1.2 I.I 48 9.80 625 9.92073 0.07 927 9.88552 " 12 I 3.5 3.4 49 50 51 9.80 641 15 15 15 15 15 15 15 16 15 15 15 9.92 099 26 25 26 26 0.07 901 9.88 542 10 II 10 9 2 3 4 S.s 8!o 10.6 10.2 9.80 656 9.92 125 0.07 875 9.88531 9.80671 9.92 150 0.07 850 9.88521 52 9.80 686 9.92176 0.07 824 9.88510 8 13.0 12.5 53 9.80 701 9.92 202 0.07 798 9.88499 7 I 15.4 14.8 54 9.80716 9.92227 25 26 26 0.07 773 9.88 489 6 17.7 17.0 55 9.80 731 9-92 253 0.07 747 9.88478 5 9 22.5 21.6 24.8 23.9 5^ 9.80 746 9.92 279 0.07 721 9.88 468 4 10 57 9.80 762 9.92 304 25 76 0.07 696 9.88457 10 3 II 5^ 9.80 777 9.92 330 26 25 0.07 670 9.88447 2 59 60 9.80 792 9.92 356 0.07 644 9.88436 II 9.80 807 9.92381 0.07 619 9.88 425 L. Cos. d. L. Cot. c. d. L. Tan. L. Sin. d. / P.P. 66 ' L. Sin. 1 d. L. Tan. |c. d.| L. Cot. | L. Cos. 1 d.| 1 P.P. 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Cos. 1 d. 1 L.Cot. |c. d.j L.Tan. | L. Sin. | d. | ' | P.P. I] /IQ° ^\^ 67 L. Sin. L. Tan. c. d. L. Cot. L. Cos. P. P. 2 3 4 5 6 7 8 9 10 II 12 ^3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9,81 694 9.81 709 9.81 723 9.81 738 9.81 752 9.81 757 9.81 781 981 796 9.81 810 9.81 825 9-8 i 839 9.81 854 9.8I 868 9.81 882 9.81 897 9.81 911 9.81 926 9.81 940 9-8i 955 9.81 969 9.81 983 9.81 998 9.82012 9.82026 9.82041 9.82 055 9.82 069 9 82 084 9.82 098 9.82 112 9.82 126 9.82 141 9.82 155 9.82 169 9.82 184 9.82 198 9.82 212 9.82 226 9.82 240 9.82 255 9.82 2b9 9.82 283 9.82 297 9.82 31 1 9.82 326 9.82 340 9.82 354 9 82 368 9.82382 9.82 9.82 410 9.82 424 9.82439 9.82453 9.82 467 9.82 48_^ 9,82 495 9.82 509 9.82 523 9-82537 9-82551 9.93916 9-93 942 9-93 967 9-93 993 9.94018 9-94 044 9.94 069 9-94 095 9.94 120 9.94 146 9.94 171 9-94 197 9.94 222 9.94 248 9-94 273 9-94 299 9-94 324 9-94 350 9-94 375 9.94 401 9 -94 426 9.94452 9-94 477 9-94 503 9.94528 9-94 554 9.94 579 9.94 604 9.94 630 9-94 655 9.94681 9-94 706 9-94 732 9-94 757 9-94 783 9.94 808 9.94 834 9-94 859 9.94 884 9.94910 9-94 935 9.94 9b I 9.94 986 9.95 012 9-95 037 9.95 062 9.95 088 9-95 "3 9 95 139 9-95164 9.95 190 995215 9.95 240 9.95 266 9.95 291 9-95317 9-95 342 9-95 368 9-95 393 9.95 418 9-95 444 0.06 084 0.06 058 0.06 033 0.06 007 0.05 982 0.05 956, 0.05931 0.05 905 0.05 880 0.05 854 9-87 778 9.87 767 9.87 756 9-87 745 9.87 734 9.87 723 9.87712 9,87 701 9.87 690 9-87 679 0.05 829 9.87 668 0.05 803 0.05 778 0.05 752 0.05 727 0.05 701 0.05 676 0.05 650 0.05 625 0.05 599 9.87 657 9.87 646 9-87 635 9,87 624 9.87613 9.87 601 9.87 590 9-87579 9.87 568 0.05 574 9-87557 0.05 548 0.05 523 0.05 497 0.05 472 0.05 446 0.05 421 0.05 396 0.05 370 0-05 345 9.87 546 9-87 535 9.87 524 9-87513 9-87 501 9.87 490 9.87 479 9.87 468 9-87457 005319 9-87 446 0.05 294 0.05 268 0.05 243 0.05 217 0,05 192 0.05 166 0.05 141 0.05 116 0.05 090 9-87 434 9-87 423 9.87412 9.87 401 9.87 390 9-87 378 9.87 367 9-87 356 9-87 345 0.05 065 9-87 334 0.05 039 0.05 014 0.04 988 0.04 963 0.04 938 0.04 912 0.04 887 0.04 86 1 0.04 836 9.87322 9.87 31 1 9-87 300 9.87 288 9.87 277 9.87 266 9-87 255 9-87 243 9.87 232 0.04 8ro 9.87 221 0.04 785 0.04 760 0.04 734 0.04 709 0.04 683 0.04 658 0.04 632 0.04 607 0.04 582 9.87 209 9.87 198 9.87187 9-87175 9.87 164 987 153 9.87 141 9-87 130 9.87 119 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15- 14 13 12 II 10 9 8 7 6 5 4 3 2 I 26 25 2.6 2.S 5-2 50 7.8 7-5 10.4 1 0.0 13.0 12.S 15.6 15.0 18.2 17-5 20.8 20.0 23-4 22.5 15 14 I 1-5 1.4 2 30 2.8 3 4-5 4.2 4 6.0 S.6 5 7-5 7.0 6 9.0 8.4 7 10.5 9.8 8 12.0 II. 2 9 13-5 12.6 12 11 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 5-5 6.6 7-7 8.8 9-9 12 12 11 26 25 25 i.o 3-1 5-2 7-3 9.4 11.5 »3-5 16.2 15.6 17.0 18.4 17.7 19.3 20.6 19.8 21.6 22.8 21.9 239 24.9 24.0 — I.I 3-2 5-4 7.6 9.8 11.9 14.1 3-4 5-7 8.0 10.2 '2-5 14.8 0.04 556 9.87 107 L.Cos. I d. L. Cot. |c. d.| L. Tan. L. Sin. I d. P.P. o8 # L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. d. p.p i[ o I 9.82551 H 14 14 9-95 444 25 26 0.04 556 9.87 107 II 60 59 9.82 565 9.95 469 0.04531 9.87 096 2 9.82 579 9.95 495 25 0.04 505 9.87085 1 2 58 3 9.82 593 9.95 520 0.04 480 9.87073 57 26 25 4 9.82 607 14 14 9-95 545 25 ?6 0.04 455 9.87 062 12 56 2 6 2-5 5 9.82621 9-95571 25 26 25 25 26 25 0.04 429 9.87050 T T 55 6 7 9-82635 9.S2 649 14 995 596 9-95 622 0.04 404 0.04 378 9.87 039 9.87028 II 54 53 3 4 10.4 5.0 7-5 1 0.0 a 9.82 663 ^4 14 14 H 9-95 647 0.04 353 9.87016 I I 52 5 6 7 8 I'?.0 12.5 15.0 '7-5 20.0 9 \o II 9.82677 9.95 672 0.04 328 9.87005 12 II 51 50 49 15.6 18.2 [ 20.8 9.82691 9.95 698 0.04 302 9.86993 9.82 705 9-95 723 0.04 277 9.86982 12 9.82719 ^4 H 9-95 748 0.04 252 9.86970 1 1 48 9 I23.4 22.5 13 9-82 733 9-95 774 25 ■2 ft 0.04 226 9-86 959 47 H 9.82 747 ^4 14 14 9-95 799 0.04 201 9-86947 1 1 46 :i 9.82761 9.82 775 9.95 825 9-95 850 25 0.04 1 75 0.04 150 9.86936 9.86924 12 II 45 44 14 13 17 9.82 788 ^3 9-95 875 25 26 0.04 125 9.86913 43 ' 'A ^•3 i8 9.82 802 ^4 9.95 901 0.04 099 9.86 902 12 1 2 42 2 2.6 19 20 21 9.82816 14 14 14 9-95 926 11 25 25 26 0.04 074 9.86 890 41 40 39 3 4 I 4.2 . 5-6 7.0 8.4 3-9 5-2 6.5 7.8 9.1 9.82 830 9-95 952 0.04 048 9.86879 9.82844 9-95 977 0.04 023 9.86 867 22 9.82858 H 9.96002 0.03 998 9.86855 38 7 9.8 23 9.82872 H 9.96028 0.03 972 9.86 844 37 8 II. 2 10.4 24 9.82 885 13 9.96053 25 0.03 947 9.86832 T T 36 9 12.6 11.7 2S 9.82 899 M 9.96078 25 26 0.03 922 9.S6821 35 26 9.82913 H 9.96 104 0.03 896 9.86 809 34 27 9.82927 14 9.96129 25 0.03871 9.86 798 33 12 11 28 9.82941 H 9.96 155 0.03 845 9.86 786 I I 32 29 30 31 9.82955 H 13 H 9.96 180 25 25 26 0.03 820 9.86775 12 II 31 30 29 2 3 1.2 2.4 .3-6 4-8 6.0 7.2 8.4 I.I 2.2 3-3 9.82 968 9,96 205 0.03 795 9-86 763 9.82982 9.96 231 0.03 769 9.86752 32 9.82 996 H 9.96 256 25 0.03 744 9.86 740 28 4 4-4 33 - 34 9.83010 9.83023 13 9.96 281 9.96 307 25 26 25 0.03719 0.03 693 9.86 728 9J6^7- II 27 _26. I 7 8 5-5 6.6 1-1 1t. 9-83037 14 996332 0.03 668 9.86 705 25 9-6 10.8 8.8 ^83051 14 9.96357 25 26 0.03 643 9.86 694 24 9 9-9 37 9.$3o65 9-96 383 0.03617 9.86682 \: 23 38 9.83 078 ii 9.96 408 25 0.03 592 9.86 670 \2 , T 22 39 40 41 9.83 092 14 14 14 9-96433 26 25 26 0.03 567 9.86 659 12 21 20 19 9.83 106 9-96 459 0.03 541 9.86647 9.83 120 9-96 484 0.03 5 1 6 9.86635 42 9-83 ^33 13 9.96510 0.03 490 986624 12 18 43 9-83 H7 14 9-96 535 25 0.03 465 9.86 612 17 12 11 11 44 9.83 161 H 9.96 560 25 26 0.03 440 9.86 600 16 26 26 25 4S 9.83174 ^3 9.96 586 0.03414 9.86 589 15 o| 46 9.83 188 14 9.96 611 25 0.03 389 9-86577 14 li I.I I 2 I.I 47 9.83 202 M 9.96 636 25 26 25 25 26 25 25 26 0.03 364 9.86565 13 2j 3-2 '3 5 3-4 48 983215 M 9.96 662 0.03 338 986554 12 3 5-4 5-9 5.7 7.6 8.3 8.0 9.8 10.6 10.2 1.9 13.0 12.5 4.1 15.4 14.8 49 50 51 9.83 229 14 13 14 9.96 687 0.03313 9-86 542 12 12 II 10 9 4; |i 9.83 242 9.96712 0.03 288 9-86 530 9.83 256 9.96 738 0.03 262 9.86518 52 9.83 270 14 9.96 763 0.03 237 9.86 507 1 2 8 Zii 6.2 17.7 17.0 53 9-83 283 13 9.96 788 0.03 212 9-86 495 12 T f 7 8.4 20.1 19.3 54 9.83 297 14 9.96814 0.03 186 9.86 483 6 in^ 0.6 22 5 21.6 55 9.83310 13 9-96 839 25 0.03 161 9.86472 5 l^ 2.8 24 8 23.9 56 9-83 324- H 9.96 864 25 26 0.03 136 9.86460 T 4 12^ 4.9 - — 57 9-83 338 14 9.96 890 0.03 no 9.86448 3 5« 9-83351 13 9.96915 25 0.03 085 9.86436 2 59 60 9-83 365 14 13 9-96 940 11 0.03 060 9.86425 12 9.83 378 9.96966 0.03 034 9.86413 L. Cos. d. L. Cot. c. d.i L.Tan. L. Sin. d. / P. P i\y^o 43° 6S > / L. Sin. d. 1 L.Tan. |c. d.| L. Cot. | L. Cos. | d. | | P.P. I 9.83 378 14 9.96 966 25 11 0.03 034 9-86413 12 12 60 59 9.83 392 9.96991 0.03 009 9.86 401 2 9.83 405 9.83419 9.97016 9.97 042 0.02 984 0.02 958 9-86 389 9.86377 12 58 57 26 25 4 9-83432 U 9.97 067 25 25 26 0.02 933 9.86 366 1 2 56 I 2.6 2-5 5 9.83 446 14 13 13 14 13 13 14 9.97 092 0.02 908 9-86 354 12 55 5:s 10.4 13.0 15.6 18.2 20.8 23.4 5-0 6 7 9-83 459 983 473 9.97 "8 9-97 143 25 25 25 26 25 25 26 0.02 882 0.02 857 9-86 342 9.86330 12 12 54 53 3 4 5 6 7-5 1 0.0 12.5 15.0 17-5 20.0 22.5 8 9.83486 9.97 168 0.02 832 9.86318 52 9 10 II .9.83 500 9.97 IQ3 0.02 807 9.86 306 II 12 12 5* 50 49 I 9 983513 9.83 527 9.97 219 0.02 781 9.86 295 9.97 244 0.02 750 9.86 283 12 9.83 540 9.97 269 0.02 731 9.86271 12 48 ^3 9-83 554 9.97 295 25 11 0.02 705 9.86 259 1 2 47 14 9-83 567 13 14 9.97 320 0.02 680 9.86 247 46 ^S 9.83 581 9-97 345 0.02 655 9.86 235 45 14 13 lb 983 594 ^3 9-97 37^ 25 25 0.02 629 9.86 223 44 I 1.4 1-3 17 9.83 608 14 13 13 14 13 9.97 396 0.02 604 9.86 211 I T 43 2 2.8 2.6 i8 9.83621 9.97 421 0.02 579 9.86 200 I 2 42 3 4.2 3-9 19 20 21 983 634 9-97 447 25 25 26 0.02 553 9.86 188 12 12 T '? 4J 40 39 4 5.6 7.0 8.4 9.8 5-2 6-5 7.8 9.83 648 9.97 472 0.02 528 9.86 176 9.83601 9-97 497 0.02 503 9.86 164 22 9.83 674 13 9.97 523 0.02 477 9.86 152 38 'J 8 9 9-1 23 24 9.83 688 9.83 701 '4 13 9-97 548 9-97 573 25 25 25 26 25 25 26 25 25 9fi 0.02452 0.02 427 9.86 140 9.86128 12 37 36 12.6 10.4 11.7 25 9-.83 7I5 H 9.97 598 0.02 402 9.86 116 T 35 26 9.83 728 M 9.97 624 0.02 376 9.86 104 12 12 34 27 9.83 741 13 9.97 649 0.02 351 9.86092 33 12 11 28 9-83 755 H 9.97 674 0.02 326 9.86 080 32 29 30 31 9.83 768 13 13 14 13 9.97 700 0.02 300 9.86 068 12 12 1 2 31 30 29 2 3 4 2.4 3-6 4.8 2.2 3-3 4-4 9.83 781 9-97 725 0.02 275 9.86 056 983 795 9-97 750 0.02 250 9.86 044 32 9.83 808 9.97 776 25 0.02 224 9.86032 T -7 28 5 6.0 5-5 33 9.83821 13 9.97 801 0.02 199 9.86 020 27 6 7.2 6.6 34 9-83 834 M 9.97 826 25 0.02 174 9.86 008 26 7 8.4 7-7 3S 9.83 848 14 9-97851 11 0.02 149 9.85 996 25 8 9-6 8.8 36 9.83 861 U 9.97877 0.02 123 9-85 984 12 24 9 10.8 9-9 37 9.83 874 13 9.97 902 25 0.02 098 9.85 972 23 3i^ 9.83 887 U 9.97 927 25 26 25 25 26 0.02 073 9-85 960 12 12 12 22 39 40 41 9.83901 13 73 9-97 953 0.02 047 9-85 948 21 20 19 9.83914 9-97 978 0.02 022 9.85 936 9.83927 9.98 003 o.oi 997 9.85 924 42 9.83 940 M 9.98 029 o.oi 971 9.85912 18 43 9.83 954 A4 9.98 054 25 O.OI 946 9-85 900 17 13 13 12 44 9.83 967 U 9.98079 25 O.OI 921 9.85 888 16 26 i 25 25 4S 9.83 9S0 M 9.98 104 11 25 O.OI 896 9.85 S76 12 vs I.O I .0 1.0 46 9-83 993 13 9.98 130 O.OI 870 9.85 864 14 1 3.0 2 •9 3-1 47 9.84 006 ^3 9-98 155 O.OI 843 9.85851 13 12 13 2 5-0 A .8 5.2 48 9.84020 M 9.98 180 11 25 25 O.OI 820 9-85 839 12 3 7.0 6 -7 7-3 49 50 51 984033 13 13 9.9S 206 O.OI 794 9-85 827 12 12 II 10 9 4 I 7 9.0 8 1 1.0 IC 13.0 12 .7 9.4 .6 11.5 -5 13.5 9.84046 9.98231 O.OI 769 9.85 815 9.84059 9.98 256 O.OI 744 9.85 803 52 9.84072 13 9.98281 26 O.OI 719 9-85 791 8 8 15.0 14.4 15.0 S3 9.84085 M 9.98 307 O.OI 693 985 779 7 Q 17.0 16.3 17.7 19.0 18.3 19.8 21.0 20.2 21.9 23.0 22.1 24.0 54 55 9.84098 9.84 112 14 13 9-98 332 ^9.98357 ^5 25 26 O.OI 668 O.OI 643 9.85 766 9-85 754 12 1 2 6 5 10 II 5^ 9.84125 9-98383 25 O.OI 617 9.85 742 4 12 25.0 2^ l-.o — 57 9.84 138 13 9.98 408 O.OI 592 9.85 730 3 13 58 9.84151 13 998433 25 O.OI 567 9.85718 I- 2 59 60 9.84 164 13 13 998458 il O.OI 542 9.85 706 13 I 9.841/7 9-98 484 O.OI 516 9.85 693 L. Cos. 1 d. L. Cot. |c. d. 1 L. Tan. L. Sin. 1 d. t P.P. Aa^ 70 ' L. Sin. d. L. Tan. c. d. L. Cot. L. Cos. 1 d. P. P. I 9.84177 13 13 13 13 13 13 9.98484 25 25 26 o.oi 516 9.85 693 12 60 59 9,84 190 9.98 509 0.0 1 491 9.85 681 2 9.84 203 9-98 534 O.OI 466 9.85 669 S8 26 25 14 3 9.84216 9.98 560 25 25 25 26 O.OI 440 9.85 657 T 57 I 2.6 2.5 14 4 9.84 229 9.98 585 O.OI 415 9.85 645 13 56 2 5.2 5.0 2.8 S 9.84 242 9.98 010 O.OI 390 9.85 632 55 3 7.8 7.5 42 6 9-^4 255 998 635 O.OI 365 9.85 620 54 4 0.4 lo.o 5.6 7 9.84 269 14 9.98661 O.OI 339 9.85 608 5S 5 3-0 12.5 7.0 8 9.84 282 13 9.98 686 25 O.OI 314 9-85 596 52 6 5.6 15.0 8.4 9 10 11 9.84 295 9.64 308 9.84321 13 13 13 13 9.98 711 9-98 737 2^ 25 25 25 26 O.OI 289 9.85 583 ^3 12 12 12 13 51 50 49 7 15.2 17.5 9.5 8 20.8 20.0 II. 2 9 23.4 22.5 I2.b O.OI 263 9-85 571 9.98 762 O.OI 238 9-85 559 12 9-^4 334 9.98 787 O.OI 213 9.85 547 48 13 9-^4 347 9.98 812 O.OI 188 9-85 534 47 1.^ 19. H 9.84 360 9.98 838 25 25 25 26 O.OI 162 9.85 522 12 13 12 1 2 46 IS 9.^4 373 12 I -1 9.98 863 O.OI 137 9.85510 45 1.3 1.2 2.6 2.4 3-9 3.6 5.2 4.8 i6 9.84 3«5 9.98 888 O.OI 112 9-85 497 44 3 I? 9.84 398 I ■> 9.98913 O.OI 087 9.85 485 43 i8 9.84 41 1 13 13 9-98 939 25 25 26 25 25 25 26 O.OI 061 9-85 473 42 5 6 I (x.S 6.0 7.8 7.2 9.1 8.4 10.4 9.6 19 20 21 9.84 424 9-^4 437 9.84450 9.98 964 9.98 989 O.OI 036 9.85 460 13 12 12 13 12 12 41 40 39 O.OI on 9.85 448 9.99015 0.00985 9-85 436 22 9.84 463 13 13 9.99 040 0.00 960 9.85 423 38 9 11.7 10.8 23 9.84476 9.99 065 0.00 935 9-85411 37 24 9.84489 9.84 502 9.84515 9.99 090 9.99 116 9.99 141 0.00910 0.00 884 0.00 859 9-85 399 9-85 i86-. 9:35-374 36 25 26 13 25 % 26 25 25 «- 4 27 9.84 528 ^j 9.99 166 0.00 834 9.85361 13 ZZ 13 13 28 9.84 540 13 13 13 13 13 13 12 13 9.99 191 0.00 809 9-85 349 32 29 30 31 9-84 553 9.99217 0.00 783 9-85 337 13 12 13 12 13 31 30 29 26 25 9.84 566 9.99 242 0.00 758 9-85 324 I I.O I.O 3.0 2.9 9M 579 9.99 267 0.00 733 9.85312 32 9.84 592 9.99 293 25 25 25 26 0.00 707 9.85 299 28 2 5.0 4.8 Zi 9.84 605 9.99318 0.00 682 9.85287 27 3 7.0 6.7 34 9.84618 9-99 343 0.00 657 9-85 274 26 4 9.0 8.7 3S 9.84 630 9.99 368 0.00 632 9.85 262 12 25 I I.O 10.6 36 9.84 643 9-99 394 0.00 606 9-85 250 24 I 13.0 12.5 37 9.84 656 13 9.99419 25 0.00 581 9-85 237 13 23 15.0 14.4 38 9.84 669 13 9.99 444 25 0.00 556 9.85 225 22 9 17.0 16.3 19.0 18.3 21.0 20.2 23.0 22.1 25.0 24.0 39 40 41 42 9.84 682 13 12 13 13 13 9.99 469 26 25 25 25 26 0.00 531 9.85212 13 12 13 12 13 21 20 18 10 II 12 13 9.84 694 9.99 495 0.00 505 9.85 200 9.84 707 9.84 720 9.99 520 9-99 545 0.00 480 0.00 455 9.85 187 9.85 175 43 9-84 733 9-99 570 0.00 430 9.85 162 17 44 9-84 745 13 13 13 12 9.99 596 25 25 26 25 25 25 26 25 25 0.00 404 9.85 150 13 12 13 12 16 45 9.84 758 9.99 521 0.00 379 985 137 15 12 12 46 9.84771 9.99 646 0.00 354 9-85 125 14 26 25 47 9.84 784 9.99 672 0.00 328 9.85 112 13 I.I I.O 48 9.84 796 9.99 697 0.00 303 9.85 100 12 I 3-2 3-1 49 50 51 9.84 809 13 13 12 13 9.99 722 0.00 278 9.85 087 13 12 13 1 2 II 10 9 2 3 4 5 6 5-4 5-2 7-6 7-3 9-8 9-4 9.84 822 9-99 747 0.00 253 9.85 074 9.84835 9-99 773 0.00 227 9.85 062 52 9.84847 9.99 798 0.00 202 9.85 049 8 11.9 11.5 53 9.84 860 9.99 823 0.00 177 9.85 037 7 14.1 13-5 54 9.84873 13 12 9.99848 25 26 0.00 152 9.85 024 13 12 6 16.2 15.6 18.4 17.7 20.6 19.8 22.8 21.9 24.9 24.0 55 9.84 885 9.99 874 0.00 126 9.85 012 5 9 56 9.84898 9.84 91 1 13 13 9.99 899 9.99 924 25 25 "I 0.00 10 1 0.00 076 9.84 999 9.84 986 13 4 3 10 II 5« 9.84 923 13 13 9.99 949 0.00051 9.84 974 13 12 2 12 59 60 9.84 936 9-99 975 25 0.00 025 9.84961 I 9.84 949 0.00 000 0.00 000 9.84 949 L. Cos. d. L. Cot. |c. d. L. Tan. L. Sin. d. ' P.P. ; 46' n III. NATURAL TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE. 72 , 0° 1 ^ 1 t N.Sin. |N.Tan.|N.Cot. N.Cos.l I .00000 .00000 00 1. 0000 60 59 029 029 3437-7 000 2 058 058 1718.9 000 58 3 087 087 "45-9 000 57 4 116 116 85944 000 56 5 .00145 .00145 687.55 1. 0000 55 6 175 175 572.96 000 54 7 204 204 491. II 000 53 8 233 233 429.72 000 52 9 10 II 262 262 381.97 000 51 50 49 .00291 .00291 343.77 1. 0000 320 320 312.52 -99999 12 349 349 286.48 999 48 13 37« 378 264.44 999 47 14 407 407 245-55 999 46 IS .00436 .00436 229.18 .99999 4S 16 465 465 214.86 999 44 ■ 17 495 495 202.22 999 43 18 524 524 190.98 999 42 19 20 21 553 553 180.93 998 41 40 39 .00582 .00582 171.89 .99998 611 611 163.70 998 22 640 640 156.26 998 38 23 669 669 149.47 998 37 24 698 698 143.24 998 36 25 .00727 .00727 137-51 ■99997 3S 26 756 756 132.22 997 34 27 785 785 127.32 997 997 33 28 814 81S 122.77 32 29 30 31 844 844 118.54 996 31 30 29 .00873 .00873 "4-59 .99996 902 902 110.89 996 32 931 931 107.43 996 28 33 9bo 960 104.17 995 27 34 .00989 .00989 lOI.II 995 26 35 .01018 .01018 98.218 .99995 25 36 047 047 95.489 995 24 37 076 076 92.908 994 23 38 105 105 90.463 994 22 39 40 41 134 135 88.144 994 21 20 19 .01164 .01164 85.940 .99993 193 193 83.844 993 42 222 222 81.847 993 18 43 251 251 79.943 992 17 44 280 280 78,126 992 16 4S .01309 .01309 76.390 .99991 15 46 33^ 338 74.729 991 11,4 47 367 367 73-139 991 13 48 396 396 71.615 990 12 49 50 SI 425 425 70.153 990 II 10 9 .01454 •01455 68.750 .99989 483 484 67.402 989 S2 513 513 66.105 989 8 53 542 542 64.858 988 7 S4 571 571 63-657 988 6 ss .01600 .01600 62.499 -99987- -^ 56 629 629 61.383 987 4 S7 658 658 60.306 986 3 sB 687 687 S9.266 986 2 59 60 716 716 58.261 985 I •01745 .01746 57.290 .99985 N. Cos. N.Cot.lN.Tan. N.Sin.j ' i ^ 1 ' iN.Sin. N.Tan.|N.Cot. N.Cos I •01745 .01746 57-290 .99985 60 S9 774 775 56.351 984 2 803 804 55^442 984 S8 3 832 833 54^561 983 57 4 862 862 53-709 983 S6 5 .01891 .01S91 52.882 .99982 SS 6 920 920 52.081 982 54 7 949 949 51^303 981 S3 8 .01978 .01978 50.549 980 S2. 9 10 II .02007 .02007 49.816 980 51 50 49 .02036 .02036 49.104 .99979 065 066 48.412 979 12 094 095 47.740 978 48 13 123 124 47-085 977 47 14 152 153 46.449 977 46 15 .02181 .02182 45-829 .99976 4S 16 211 211 45.226 976 44 17 240 240 44.639 97S 43 18 269 269 44.066 974 42 19 20 21 298 298 43-508 974 41 40 39 -02327 .02328 42.964 •99973 356 357 42.433 972 22 385 386 41.916 972 38 23 414 415 41.411 971 37 24 443 444 40.917 970 36 25 .02472 -02473 40.436 .99969 3S 26 501 502 39.965 969 34 ■ 27 530 531 39-506 968 33 28 560 560 39.057 967 32 29 30 31 589 589 38.618 966 31 30 29 .02618 .02619 38.188 .99966 647 648 37.769 965 32 676 677 37.358 964 28 33 705 706 36.956 963 27 34 734 735 36.563 963 26 35 .02763 .02764 36.178 .99962 2S 36 792 793 35.801 961 24 37 821 822 35-431 960 23 3^ 850 851 35-070 959 22 39 40 41 879 881 34.715 959 21 20 19 .02908 .02910 34.368 •99958 938 939 34.027 957 42 967 968 33.694 956 18 43 .02996 .02997 33.366 955 17 44 .03025 .03026 33.045 954 16 45 .03054 -03055 32.730 •99953- IS 46 083 084 32.421 952 14 47 112 114 32.118 952 13 48 141 143 31.821 951 12 49 50 SI 170 172 31.528 950 II 10 9 .03199 .03201 31.242 .99949 228 230 30.960 948 S2 257 259 .30.683 947 8 53 286 288 30.412 946 7 S4 316 317 30.145 •945 6 SS ■03345 .03346 29.882 .99944 5 56 374 376 29.024 943 4 57 403 405 29.371 942 3 S8 432 434 29.122 941 2 59 60 461 463 28.877 940 .03490 -03492 28.636 •99939 1 N.Cos.|N.Cot. N.Tan. N. Sin. ' 89° 88' 2° A 1 N. Sin. N. Tan. In. Cot. N.Cos. I .03490 519 .03492 28.636 •399 .99939 938 60 59 5ii 2 54Ji 550 28.166 937 58 3 577 579 27-93r 936 "57 4 606 609 .712 935 S6 5 •03635 .03638 27,490 •99934 55 bb4 667 .271 933 54 7 693 696 27.057 932 S3 8 723 725 26.845 931 52 9 10 11 752 754 .637 930 49 .03781 .03783 26.432 .230 •99929 927 810 812 12 «39 842 26.031 926 48 '3 8b8 871 25-835 925 47 '4 897 900 .642 924 46 '5 .03926 .03929 25452 •99923 45 lb 955 958 .264 922 44 17 .03984 .03987 25.080 921 43 18 .04013 .04016 24.898 919 42 19 20 21 042 046 .719 918 41 40 39 .04071 .04075 24.542 .99917 100 104 .368 916 22 129 ^33 .196 915 ?>^ 23 159 162 24.02b 913 37 I24 188 191 23.859 912 36 25 .04217 .04220 23^695 .99911 35 2b 246 250 •532 910 34 l'2 275 279 •372 909 ?>?> ,28 304 308 .214 907 32 29 30 31 33Z 337 23.058 906 31 30 29 .043b2 .04366 22,904 .99905 391 395 •752^ 904 32 420 424 .602 902 28 33 449 454 •454 901 27 34 478 483 .308 900 26 35 .04507 .04512 22.164 .99898 2S 3b 536 541 22.022 897 24 37 565 570 21.881 896- 23 3« 594 599 •743 894 22 39 40 41 623 628 .606 893 21 20 19 .04653 .04658 21.470 .99892 682 687 •337 890 42 711 716 .205 889 18 43 749 745 21.075 888 17 44 769 774 20.946 886 16 45 .04798 .04803 20.819 •9988s 15 4b 827 833 .693 883 H 47 856 862 .569 882 13 48 885 891 .446 881 12 49 50 51 914 920 •325 ! 879 II 10 9 •04943 .04949 .04978 20.206 1 .99878 .04972 20.087 87b 52 .05001 .05007 19.970 87.5 8 53 030 037 .855 873 7 54 059 066 •740 872 6 55 .05088 •05095 i9-627 .99870 5 5t> 117. 124 .516 869 4 ^57 146 1 153 .405 867 3 5^ 175 182 .296 866 2 59 60 205 212 .188 864 ii .05234 1 .05241 19.081 .99863 .N.Cos.|n. Cot.|N.Tan.| N. Sin.) ' J\\ 8 T ) ^ 3° 73 / N. Sin. N. Tan. N. Cot. N.Cos. .05234 .05241 19.081 .99863 60 I 263 270 18.976 861 59 2 292 299 .871 860 S8 3 321 328 .768 858 57 4 350 357 .666 857 56 S •05379 .05387 18.564 •99855 55 6 408 416 .464 854 54 7 437 445 .366 852 53 8 466 474 .268 851 52 9 10 II 495 503 .171 ^849 .99847 846 5' 50 49 05524 •05533 18.075 17.980 553 562 12 582 591 .886 844 48 13 611 620 .793 842 47 14 640 649 .702 841 46 15 .05669 •05678 17.611 •99839 45 16 698 708 .521 838 44 17 727 737 .431 836 43 18 7S6 766 .343 834 42 19 785 795 .256 833 41 20 21 .05814 .05824 17.169 17.084 .99831 40 39 844 854 829 22 873 883 16.999 827 38 23 902 912 •915 826 37 24 931 941 .832 824 36 25 .05960 •05970 16.750 .99822 35 26 .05989. .05999 .668 821 34 27 .06018 .06029 •587 819 33 28 047 058 •507 817 32 29 30 31 076 087 .428 815 31 30 29 .06105 .06 u6 16.350 .99813 134 145 .272 812 32 163 175 .195 810 28 33 192 204 .119 808 27 34 221 233 16.043 806 26. 35 .06250 .06262 15.969 .99804 25 36 279 291 .895 803 24 37 308 321 .821 801 23 38 337 350 .748 799 22 39 40 41 366 379 .676 797 19 .06395 .06408 15.605 .99795 424 438 •534 793 42 453 467 •464 792 18 43 482 - 496 .394 790 17 44 5" 525 •325 788 16 4S .06540 •06554 15^257 .99786 IS 46 569 584 .189 784 14 47 598 613 .122 782 13 48 b27 642 15056 780 12 49 50 51 656 671 14.990 778 11 iO 9 .06685 .06700 14.924 .99776 7»4 • 730 .860 774 52 743 759 .795 772 8 53 773 788 .732 770 7 54 802 817 .669 768 6 5S .06831 .06847 14.606 .99766 5 56 860 876 •544 764 4 57 889 90s .482 762 3 ■.58 918 934 .421 760 2 59 60 947 963 .361 758 .06976 .06993 14.301 .99756 N.Cos. N.Cot. N.Tan.lN.Sin. 1 86° 74 t 4° w\ / N.Sin.jN.Tan. N. Cot. N.Cos I .06976 .06993 14.301 •99756 60 59 .07005 .07022 .241 754 2 034 051 .182 752 58 3 063 080 .124 750 57 4 092 no .065 748 56 5 .07121 .07139 14.008 .99746 55 b 150 168 13-951 744 54 7 179 197 .894 742 53 8 208 227 .838 740 52 9 lO 237 256 .782 738 51 50 49 .07266 .07285 13.727 •99736 295 3'4 .672 734 12 324 344 .6.7 731 48 13 353 373 .563 729 47 H 382 402 .510 727 46 '5 .07411 .07431 13457 .99725 45 10 440 461 .404 723 44 17 469 490 •352 721 43 i8 498 519 .300 719 42 19 20 21 527 548 .248 716 41 40 39 .07556 .07578 13-197 .99714 585 607 .146 712 22 614 636 .096 710 38 23 643 665 13.046 708 37 24 672 695 12.996 705 36 2.S .07701 .07724 12.947 •99703 35 26 730 753 .898 701 34 27 759 782 .850 699 33 28 788 812 .801 696 32 29 30 31 817 841 •754 694 31- 30 29 .07846 .07870 12.706 .99692 875 899 •659 689 32 904 929 .612 687 28 33 933 958 .566 685 27 34 962 .07987 .520 683 26 3S .07991 .08017 12.474 .99680 25 3^ .08020 046 .429 678 24 37 049 075 .384 676 23 3« 078 104 .339 673 22 39 40 41 107 134 .295 671 21 20 19 .08136 .08163 12.251 .99668 165 192 .207 666 42 194 221 .163 664 18 43 223 251 .129 66 1 17 44 252 280 .077 659 16 45 .08281 .08309 12.035 •99657 15 4b 310 339 11.992 654 14 47 339 368 .950 652 13 48 368 397 .909 649 12 49 50 SI 397 427 .867 647 11 10 9 .08426 .08456 11.826 .99644 455 485 .785 642 52 484 5H •745 639 8 53 513 544 .705 637 7 S4 542 573 .664 635 6 55 .08571 .08602 11.625 .99632 5 56 600 632 •585 630 4 57 629 661 .546 627 3 58 658 690 .507 625 2 59 60 687 720 .468 622 1 .08716 .08749 11.430 .99619 N. Cos. N. Cot. N.Tan.iN.Sin.| .n 1 N.Sin iN.Tan. N.Cot. 1 N.Cos I .08716 .08749 11.430 .99619 60 59 745 778 .392 617 2 774 807 •354 614 58 3 803 837 .316 612 57 4 831 866 •279 609 S6 5 .08860 .0S89S 11.242 .99607 55 6 889 925 .205 604 54 7 918 954 .168 602 S3' 8 947 .08983 .132 599 52 9 10 11 .08976 .09013 .095 5,6 51 50 49 .09005 .09042 11.059 •99594 034 071 11.024 591 12 063 lOI 10.988 588 48 13 092 130 •953 586 47 14 121 159 .918 583 46 15 .09150 .09189 10.883 •99580 45 16 179 218 .848 578 44 17 208 247 .814 575 43 18 237 277 .780 572 42 19 20 21 266 306 .746 570 41 40 .39 09295 •09335 10.712 .99567 ' 324 365 .678 564 22 353 394 .641 562 38 23 3^^ 423 .612 559 37 24 411 453 .579 556 36 25 .09440 .09482 10.546 •99553 35 26 469 5" .5M 551 34 27 498 541 .481 548 33 28 52? 570 .449 545 32 29 30 31 556 600 .417 542 31 30 29 •09585 .09629 10.385 .99540 614 658 .354 537 32 642 688 .322 534 28 33 671 717 .291 531 27 34 700 746 .260 528 26 35 .09729 .09776 10.229 .99526 25 36 758 805 .199 523 24 37 787 834 .168 520 23 38 816 864 .1.S8 517 22 39 40 41 845 893 .108 5H 21 20 19 .09874 .09923 10.078 •995" 903 952 .048 508 42 932 .09981 10.019 506 18 43 961 .10011 9.9893 503 17 44 .09990 040 .9601 500 16 45 .10019 .10069 9.9310 .99497 15 46 048 099 .9021 494 14 47 077 128 .8734 491 13 48 106 158 .8448 4S8 12 49 50 51 135 187 .8164 485 11 io 9 .10164 .10216 9.78^2 .99482 192 246 .7601 479 52 221 275 .7322 476 8 53 250 305 .7044 473 7 54 279 334 .6768 470 6 55 .10308 .10363 9-6493 .99467 S 56 337 393 .6220 464 4 57 366 422 .5949 461 3 58 •395 452 •5679 458 2 55 60 424 481 .5411 455 1 .10453 .10510 9.5144 .99452 C_. N.Cos.'N. Cot.JN.Tan. N. Sin.| / 86' 84' 6° •* T 75 / N.Sin. N.Tan.jN.Cot. N.Cos. / N.Sin. N.Tan. N. Cot. N.Cos. ! I .10453 482 .10510 9-5 '44 •99452 60 S9 .12187 .12278 308 8.1443 .1248 •99255 251 60 59 540 .4878 449 216 2 511 569 .4614 446 58 2 245 338 •'Sl^ 248 58 3 540 599 .4352 443 57 3 274 367 .0860 244 57 4 569 628 .4090 440 S6 4 302 397 .0667 240 56 5 .10597 .10657 9.3831 •99437 ss 5 .12331 .12426 8.0476 •99237 55 6 626 687 •3572 434 54 b 360 456 .0285 233 54 7 6SS 716 •3315 431 S3 7 389 485 8.0095 230 53 8 684 746 .3060 428 52 8 418 515 7.990b 226 52 9 10 II 713 775 .2806 424 51 50 49 9 iO II 447 544 .9718 222 .99219 51 50 49 .10742 .10805 834 9.2553 .99421 .12476 •12574 79530 771 .2302 418 504 603 •9344 215 12 800 863 .2052 415 48 12 533 633 .9158 211 48 13 829 893 .1803 412 47 13 562 662 .8973 208 47 H 858 922 .1555 409 46 14 591 692 •8789 204 46 ^S .10887 .10952 9.1309 .99406 4S rs .12620 .12722 7.8606 .99200 45 lb 916 .10981 .1065 402 44 16 649 751 .8424 197 44 ^7 94^ .11011 .0821 399 43 '7 678 781 :8243 193 43 i8 .10973 040 •0579 396 42 18 706 810 .8062 189 42 '19 20 21 .11002 070 .0338 393 41 40 39 19 20 21 735 840 .7882 186. 41 40 19 .11031 .11099 128 9.0098 .99390 .12764 .12869 7.7704 .99182 obo 8.9860 386 793 899 •7525 178 22 089 158 .9623 383 38 22 822 929 •7348 17^ 38 23 118 187 •9387 380 37 23 851 958 .7171 171 37 24 147 217 .9152 377 36 24 880 .12988 .6996 167 36 25 .11176 .11246 8.8919 •99374 3S 2S .12908 .13017 7.6821 .99163 3S 2b 205 276 .8686. 370 34 26 937 047 .6647 160 34 27 234 305 •8455 367 33 27 966 076 .6473 IS6 33 28 263 335 •8225 • 364 .32 28 •12995 106 •6301 152 32 29 30 31 291 364 .7996 360 31 30 29 29 30 31 .13024 136 .6129 148 31 30 29 .11320 •I 1394 8.7769 .99357 •13053 •13165 7.5958 .99144 349 423 .7542 354 081 195 .5787 141 32 37« 452 •7317 351 28 32 no 224 .5618 137 28 33 407 482 .7093 347 27 33 139 254 •5449 ^33 27 34 436 5" .6870 344 26 34 168 284 .5281 129 26 35 .11465- .11541 8.6648 .99341 25 35 •13197 '^3313 7^5"3 •99125 2S 3^ 494 570 .6427 337 24 36 226 343 .4947 122 24 37 523 600 .6208 334 23 37 254 372 .4781 118 23 3ii 552 629 •5989 33^ 22 38 283 402 .4615 114 22 39 40 41 580 659 •5772 327 21 20 19 39 40 41 312 432 •445' no 21 20 19 .11609 .11688 8-5555 .99324 .13341 .13461 7.4287 .99106 638 718 .5340 320 370 491 .4124 102 42 667 747 .5126 317 18 42 399 521 .3962 098 18 43 696 777 .4913 314 '7 43 427 550 .3800 094 17 44 725 806 .4701 310 16 44 456 580 •3639 091 16 45 •"754 .11836 8,4490 •99307 IS 4S .13485 .13609 7^3479 .99087 IS 4b .7«3 865 .4280 303 14 46 514 639 .33 '9 0S3 14 ^Z 812 895 .4071 300 13 47 543 669 .3160 079 13 48 840 924 •3863 297 12 48 572 698 .3002 075 12 49 50 51 869 954 .3656 293 10 9 49 50 SI 600 728 .2844 071 11 10 9 .11898 .11983 8.34I0 .99290 .13629 •13758 7.2687 .99067 927 .12013 •3245 286 658 787 .2531 063 52. 956 042 .3041 283 8 S2 687 817 .2375 059 8 53 .11985 072 .2838 279 7 53 716 846 .2220 055 7 54 .12014 101 .2636 276 6 54 744 876 .2066 051 6 55 .12043 .12131 8.2434 .99272 S SS .13773 .13906 7.1912 •99047 S 5^ 071 160 .2234 269 4 56 802 935 .1759 043 4 ^l 100 190 .2035 265 3 S7 831 965 .1607 039 3 5^ 129 219 •1837 262 2 S8 860 .1399? .1455 03? 2 59 60 158 249 .1640 258 I 59 60 889 .14024 .1304 031 I .12187 .12278 8.1443 •99255 .13917 .14054 7."54 .99027 N.Cos. N. Cot. N.Tan. N.Sin. ' N.Cos. N.Cot.lN.Tan. N.Sin.l ' || 83° 82° 76 8° r N. Sin. N.TanJN.Cot. N.Cos. I .13917 .14054 7-1154 .99027 60 59 946 084 .1004 023 2 •1397? 113 .0855 019 58 3 .14004 143 .0706 015 57 4 033 173 .0558 on 56 S ,14061 .14202 7.0410 .99006 55 6 090 232 ,0264 .99002 54 7 "3 262 7.0117 .98998 53 8 148 291 6.9972 994 52 9 10 II 177 321 •9827 990 51 50 49 .14205 .1435' 6.9682 .98986 234 381 .9538 982 12 263 410 •9395 978 48 13 292 440 .9252 973 47 H 320 470 .9110 969 46 IS •14349 .14499 6.8969 .98965 45 i6 378 529 .8828 961 44 17 407. 559 .8687 957 43 18 436 588 .8548 953 42 19 20 21 464 618 .8408 948 41 40 39 .14493 .14648 6.8269 •98944 522 678 .8131 940 22 551 707 .7994 936 38 23 580 737 .7856 931 37 24 608 767 .7720 927 36 2S .14637 .14796 6.7584 •98923 35 26 bb^ 826 .7448 919 34 27 695 It .7313 914 ZZ 28 723 886 .7179 910 32 29 30 SI 752 915 •7045 906 31 30 29 .14781 .14945 6.6912 .98902 810 •14975 .6779 897 32 838 .15005 .6646 893 28 33 867 034 .6514 889 27 34 896 064 .6383 884 26 3S .14925 .15094 6.6252 .98880 25 36 954 124 .6122 876 24 37 .14982 153 .5992 871 23 38 .15011 183 .5863 867 22 39 40 41 040 213 .5734 863 21 20 19 .15069 •15243 6.5606 .98858 097 272 •5478 854 42 126 302 .5350 849 18 43 155 332 .5223 845 17 44 184 362 .5097 841 16 4S .15212 •15391 6.4971 .98836 15 46 241 421 .4846 832 14 47 270 451 .4721 827 13 48 299 481 .4596 823 12 49 50 SI 327 511 .4472 8i8 II 10 9 ■ '5356 •15540 6.4348 .98814 385 570 .4225 809 S2 414 600 .4103 80s 8 53 442 630 .3980 800 7 S4 471 660 •3859 796 6 ss .15500 .15689 6.3737 .98791 5 56 529 719 •3617 787 4 S7 557 749 .3496 782 3 S8 586 779 •3376 778 2 59 60 615 809 .3257 773 I •15643 • 15838 6.3138 .98769 N.Cos. N.Cot.lN.Tan. N. Sin. / ' N.Sin. N.Tan. N.Cot. N.Cos. I .15643 .15838 6.3138 .98769 60 S9 672 868 .3019 764 2 701 898 .2901 760 S8 3 730 928 •2783 755 57 4 758 958 .2666 75» S6 5 .15787 .15988 6.2549 .98746 SS 6 816 .16017 •2432 741 54 7 845 047 .2316 737 S3 8 873 077 .2200 732 S2 9 10 II 902 107 .2085 728 51 50 49 .15931 .16137 6.1970 •98723 959 167 .1856 718 12 .15988 196 .1742 714 48 13 .16017 226 .1628 709 47 14 046 256 .1515 704 46 IS .16074 .16286 6.1402 .98700 45 16 103 316 .1290 695 44 17 132 346 .1178 690 4S 18 160 376 .1066 686 42 19 20 21 189 405 •0955 681 41 40 39 .16218 .16435 6.0844 .98676 246 465 .0734 671 22 275 495 .0624 667 38 23 304 525 .0514 662 37 24 zzz 5SS •0405 i 657 S6 25 .16361 •16585 6.0296 .98652 35 26 390 615 .0188 648 34 27 419 645 6.0080 643 33 28 447 674 5^9972 638 32 29 30 31 476 704 .9865 633 31 30 29 .16505 •16734 5-9758 .98629 533 764 .9651 624 32 562 794 .9545 619 28 33 591 824 •9439 614 27 34 620 854 .9333 609 26 3S .16648 .16884 5.9228 .98604 25 36 677 914 .9124 600 24 37 706 944 .9019 595 23 38 734 .16974 .8915 590 22 39 40 41 763 .17004 .8811 585 21 20 19 .16792 .17033 5.8708 1 .98580 820 063 •8605 1 575 42 849 093 .8502 570 18 43 878 123 .8400 1 565 17 44 906 I S3 .8298 i 561 16 45 .16935 .17183 58197 .98556 15 46 964 213 .8095 551 14 47 .16992 243 .7994 546 13 48 .17021 273 .7894 541 12 49 50 SI 050 303 .7794 536 II 10 1 9 .17078 •17333 5. 7594}. 985 3 1 107 363 .7594 = 526 52 1.36 393 .7495 521 8 53 164 42-3 .7396 516 7 S4 193 453 .7297 511 6 SS .17222 •17483 57199 .98506 5 56 250 513 .7101 501 4 S7 279 543 .7004 496 3 S8 308 573 .6906 491 2 i 59 60 336 603 .6809 486 .17365 •17633 5.6713 .98481 i N.Cos. N.Cct.JN.Tan. N.Sin. 'J 8r 80' 10' » N. Sin IN.Tan.jN.Cot N.Cos I .17365 •17633 663 5^67i3 .6617 .98481 60 59 393 476 2 422 693 .6521 471 58 3 451 723 .6425 466 57 4 • 479 753 .6329 461 S6 5 .17508 •17783 5^6234 •98455 55 b 537 813 .6140 450 54 7 565 843 .6045 445 53 8 594 873 .5951 440 52 9 lO II 623 903 •5857 435 51 50 49 .17651 .17933 5.5764 .98430 680 963 •5671 425 12 708 •17993 .5578 420 48 13 737 .18023 •5485 414 47 H 766 053 .5393 409 46 15 .17794 .18083 55301 .98404 45 lb 823 113 .5209 399 44 17 852 143 .5118 394 43 18 880 173 .5026 389 42 19 20 21 909 203 •4936 383 41 40 39 .17937 .18233 5-4845 •98378 966 263 •4755 373 22 .17995 293 .4665 368 38 23 .18023 323 •4575 362 37 24 052 353 .4486 357 36 25 .18081 .18384 5.4397 -98352 35 2b 109 414 .4308 347 34 27 138 444 .4219 341 33 28 166 474 •4131 336 32 29 30 31 ^95 504 .4043 33^ 31 30 29 .18224 •18534 5.3955 •98325 252 564 .3868 320 32 281 594 •3781 315 28 33 309 624 •3694 310 27 34 338 6S4 .3607 304 26 35 .18367 .18684 5^3521 .98299 25 3^^ 395 7'4 •3435 294 24 37 424 745 •3349 288 23 35^ 452 775 .3263 283 22 39 40 41 481 805 .3178 277 21 20 19 .18509 .18835 5.3093 1 .98272 538 865 .3008 267 42 567 895 .2924 261 18 43 595 925 .2839 25b 17 44 624 955 •2755 250 16 45 .18652 .18986 5.2672 .98245 IS '40 681 .19016 .2588 240 14 i 47 710 046 .2505 234 13 - 48 738 076 .2422 229 12 ^9 767 106 •2339 223 II 10 9 50 •18795 .19136 5-2257 .98218 ^I 824 166 .2174 212 , ,2 ^5^ 197 .2092 207 8 63 881 227 .20II 201 7 54 910 257 .1929 196 6 55 .18938 .19287 5.1848 .98190 5 5^ 967 .317 •1.767 185 4 57 .18995 347 .1686 179 3 158 .19024 378 .1606 174 2 052 408 .1526 168 ^0 .19081 .1943SI 5.1446 .98163 ;.cos. N^^CotJ IH.Tan.'N.Sin.l ' | ir 77 / N.Sin. N.Tan.N.Cot.|N.Cos I .19081 .19438 5.1446 .98163 60 S9 109 468 .1366 157 2 138 498 .1286 152 58 3 167 529 .1207 146 57 4 195 559 .1128 140 S6 5 .19224 .19589 5.1049 •98135 55 6 252 619 .0970 129 54 7 281 649 .0892 124 53 8 309 680 .0814 118 52 9 10 II 338 710 •0736 112 51 50 49 .19366 .19740 5.0658 .9«io7 395 770 •0581 lOI 12 423 801 •0504 096 48 13 452 831 .0427 090 47 H 481 861 •0350 084 46 15 .19509 .19891 50273 .98079 45 16 538 921 .0197 073 44 17 566 952 .0121 067 43 18 595 .19982 5.0045 o6i 42 19 20 21 623 .20012 4.9969 056 41 40 39 .19652 .20042 4.9894 .98050 680 073 •9819 044 22 709 103 •9744 039 38 23 737 ^33 .9669 033 37 24 766 164 .9594 027 36 25 .19794 .20194 4.9520 .98021 35 26 823 224 .9446 016 34 27 851 254 •9372 010 33 28 880 285 .9298 .98004 32 29 30 31 908 315 •20345 •9225 .97998 31 30 29 •19937 4.9152 .97992 965 376 .9078 987 32 .19994 406 .9006 981 28 33 .20022 436 •8933 975 27 34 051 466 .8860 969 26 35 .20079 .20497 4.8788 •97963 25 36 108 527 .8716 958 24 37 136 557 .8644 952 23 38 165 588 •8573 946 22 39 40 41 193 618 •8501 940 21 20 19 .20222 .20648 4.8430 .97934 250 679 •8359 928 42 279 709 .8288 922 18 43 307 739 .8218 916 17 44 336 770 .8147 910 16 45 .20364 .20800 4.8077 •97905 15 46 393 830 .8007 899 14 47 421 861 •7937 893 13 48 450 891 .7867 887 12 49 50 51 478 921 1 -7798 881 II 10 9 •20507 .20952 4.7729 •97875 869 535 .20982 .7659 52 563 .21013 •7591 863 8 53 592 043 .7522 857 7 54 620 073 •7453 8s. 6 55 .20649 .21104 4.7385 •97845 S 56 677 134 ■73^ 839 4 57 706 164 .7249 833 3 58 734 195 .7181 2^7 2 59 60 763 225 .7114 821 .20791 .21256 4.7046 .97815 N.Cos.l N.Cot.lN.Tan., N.Sin. 1 / 7Q^. 7QO 78 1 2° 13° 1 N.Sin. N.Tan. N. Cot. N.Cos. 1 r N.Sin. N.Tan. N. Cot. N.Cos. .20791 .21256 4.7046 .978i5 60 59 I .22495 .23087 4.3311 •97437 60 59 820 286 .6979 809 523 117 .3257 430 2 848 3it) .6912 803 58 2 552 148 .3200 424 58 3 «77 347 .6845 797 57 3 580 179 '3^43 417 57 4 905 377 .6779 791 56 4 608 209 .3086 411 56 S .20933 .21408 4.6712 .97784 55 5 .22637 .23240 4.3029 .97404 55 962 438 .6646 778 54 6 665 271 .2972 398 54 7 .20990 469 .6580 772 53 7 693 301 .2916 391 53 8 .21019 499 .6514 766 S2 8 722 332 .2859 384 52 9 10 11 047 529 .6448 760 5i 50 49 9 10 II 750 363 .2803 378 51 50 [ 49 .21076 .21560 4.6382 .97754 .22778 807 .23393 4.2747 •97371 104 590 .6317 748 424 .2691 365 12 1.32 621 .6252 742 48 12 835 455 .2635 358 48 13 161 651 .6187 735 47 13 863 485 .2580 351 47 '4 189 682 .6122 729 46 14 892 516 .2524 345 46 IS .21218 .21712 4.6057 •97723 45 15 .22920 •23547 4.2468 .97338 45 i6 246 743 .5993 717 44 16 948 578 .2413 331 44 17 275 773 .5928 711 43 17 .22977 608 .2358 325 43 18 303 804 .5864 705 42 18 .23005 639 •2303 318 42 19 20 21 331 834 .5800 698 41 40 39 19 20 21 033 670 .2248 311 41 40 39 .21360 .21864 4.5736 .97692 .23062 .23700 4.2193 .97304 388 89^ •5673 686 090 731 .2.39 298 22 417 925 .5609 680 38 22 118 762 .2084 291 S8 23 445 956 •5546 673 37 23 146 793 .2030 284 37 24 474 .21986 •5483 667 36 24 175 ?'3 .1976 278 36 2S .21502 .22017 4.5420 .97661 35 25 .23203 .23854 4.1922 .97271 35 26 530 047 •5357 655 34 26 231 885 .1868 264 34 27 559 078 •5294 648 33 27 260 916 .1814 257 33 28 587 108 .5232 642 32 28 288 946 -1 760 251 32 29 30 31 616 139 .5169 636 31 30 29 29 30 31 316 •23977 .1706 244 31 30 29 .21644 .22169 4.5107 .97630 •23345 .24008 4-1653 •97237 672 200 .5045 623 373 039 .1600 230 32 701 231 .4983 617 28 32 401 069 •1547 223 28 33 729 261 .4922 611 27 33 429 100 •1493 217 27 S4 7S8 292 .4860 604 26 34 458 131 .1441 210 26 3S .21786 .22322 4-4799 .97598 25 35 •23486 .24162 4.1388 •97203 25 3^ 814 353 •4737 592 24 36 5H 193 .1335 196 24 37 843 383 .4676 585 23 37 542 223 .1282 189 23 3« 871 414 .4615 579 22 38 571 254 .1230 182 22 39 40 41 899 444 •4555 573 21 20 19 39 40 41 599 285 .1178 176 21 20 19 .21928 .22475 4.4494 .97566 .23627 .24316 4.1126 .97169 956 505 .4434 560 656 347 .1074 162 42 .21985 53^ •4373 553 18 42 684 377 .1022 155 18 43 .22013 567 •4313 547 17 43 712 408 .0970 148 17 44 041 597 •4253 541 16 44 740 439 .0918 141 16 4S .22070 .22628 4.4194 .97534 15 45 .23769 .24470 4.0867 •97134 15 46 098 658 •4134 528 14 46 797 501 .0815 127 14 47 126 689 •4075 521 13 47 825 532 .0764 120 13 48 1^5 719 .4015 51S 12 48 853 562 .0713 "3 12 49 50 51 183 750 •3956 508 II 10 9 49 50 51 882 593 .0662 . 106 II 10 9 .22212 .22781 4-3897 .97502 .23910 .24624 4.061 1 .97100 240 811 •3838 496 938 6,^5 .0560 ""^oi S2 268 842 .3779 489 8 52 966 686 .0509 086 8 53 297 872 •3721 483 7 53 .23995 717 .0459 079 7 S4 325 903 .3662 476 6 54 .24023 747 .0408 072 6 SS •22353 •22934 4.3604 .97470 5 55 .2405 1 .24778 40358 .97065 5 56 382 964 •3546 463 4 56 079 809 .0308 058 4 S7 410 •22995 .3488 457 3 57 108 840 .0257 051 3 S8 438 .23026 .3430 450 2 58 136 871 .0207 044 2 59 80 467 056 •3372 444 I 59 i60 164 902 .0158 037 I 22495 .23087 4-33'5 •97437 .24192 •2A933 4.0108 .97030 °i N.Cos. N. Cot. N.Tan. N. Sin.| ' L N.Cos. N.Cot.j N.Tan. |N.Sin.|j^|| \ririo *ya' 1 4° 1 6° 79 / N.Sin. N.Tan.iN.Cot. N.Cos.l 1 t N. Sin. N. Tan, N. Cot. N.Cos. .24192 .24933 4.0108 .97030 60 59 I .25882 .20795 3.7321 •96593 60 S9 220 964 .0058 023 910 826 .7277 585 2 249 .24995 4.0009 015 58 2 938 III .7234 578 S8 3 277 .25020 3-9959 008 57 3 966 888 .7191 570 57 4 305 056 .9910 .97001 S6 4 •25994 920 .7148 562 S6 5 .24333 .25087 3.9861 .96994 55 5 .26022 .26951 3.7105 -96555 SS 6 362 118 .9812 987 54 6 050 .26982 .7062 547 54 7 390 149 .9763 980 53 7 079 .27013 .7019 540 S3 8 418 180 .9714 973 S2 8 107 044 .6976 532 S2 9 10 II 446 211 .9665 966 51 50 49 9 10 II 135 076 •6933 524 5' 50 49 .24474 .25242 3.9617 .96959 .26163 .27107 3.6891 .96517 503 273 .9568 952 191 138 .6848 509 12 531 304 .9520 945 48 12 219 169 .6806 502 48 13 559 33^ .9471 937 47 13 247 201 .6764 494 47 '4 S87 366 •9423 930 46 H 275 232 .6722 486 46 IS .24615 .25397 3-9375 •96923 4S 15 .26303 .27263 3.6680 .96479 4S i6 644 . 428 •9327 916 44 16 33^ 294 .6638 471 44 17 672 459 •9279 909 43 17 359 326 •6596 463 43 18 700 490 .9232 902 42 18 387 357 .6554 456 42 19 20 21 728 521 .9184 894 41 40 39 19 20 21 415 388 .6512 448 41 40 39 .24756 •25552 3.9136 .96887 •26443 .27419 3.6470 .96440 784 583 .9089 880 471 451 .6429 433 22. 813 614 .9042 873 38 22 500 482 .0387 425 38 23 841 645 .8995 866 37 23 528 513 •6346 417 37 24 869 676 .8947 858 36 24 556 545 •6305 410 36 2S •24897 .25707 3.8900 .96851 3S 2S .26584 .27570 3.6264 .96402 3S 26 925 738 .8854 844 34 26 612 607 .6222 394 34 27 954 769 .8807 837 33 27 640 638 .6181 386 33 28 .24982 800 .8760 829 32 28 668 670 .6140 379 32 29 30 31 .25010 831 .8714 822 31 30 29 29 30 31 . 696 701 .6100 371 31 30 29 •25038 .25862 3-8667 •96815 .26724 .27732 36059 .96363 066 893 .8621 807 752 764 .6018 355 S2 094 924 .8575 800 28 32 780 795 •5978 347 28 33 122 955 .8528 793 27 33 808 826 •5937 340 27 34 151 .25986 .8482 786 26 34 836 858 .5897 332 26 3S ■25179 .26017 3.8436 •96778 2S 3S .26864 .27889 3.585^ .96324 2S 36 207 048 .8391 771 24 36 892 921 .5816 316 24 37 235 079 .834? 764 23 37 920 952 •5776 308 23 38 263 no .8299 7S6 22 38 948 -27983 .5736 301 22 39 40 41 291 141 .8254 749 21 20 19 39 40 41 .26976 .28015 .5696 293 21 20 19 .25320 .26172 3.8208 .96742 .27004 .28046 35656 .96285 348 203 .8163 734 032 077 .5616 277 42 376 235 .8118 727 18 42 060 109 •5576 269 18 43 404 266 .8073 719 17 43 088 140 •5536 261 '7 44 432 297 .8028 712 16 44 116 172 .5497 253 16 4S .25460 .26328 3.7983 .96705 15 4S .27144 .28203 3-5457 .96246 IS 46 488 359 .7938 697 14 46 172 234 .5418 238 14 47 S16 390 .7893 690 13 47 200 266 .5379 230 13 48 54^ 421 .7848 682 12 48 228 297 .5339 222 12 49 50 SI 573 452 .7804 675 II 10 9 49 50 SI 256 329 .5300 214 10 .25601 .26483 3.7760 .96667 .27284 .28360 3.5261 .96206 629 515 •7715 660 312 391 .5222 198 S2 6S7 S4^ .7671 6S3 8 S2 340 423 -5183 190 8 53 685 577 .7627 645 7 53 368 454 •5 '44 182 7 S4 713 608 •7583 638 6 S4 396 486 •5105 174 6 ss .25741 .26639 3-7539 .96630 5 SS .27424 .28517 35067 .96166 S i6 769 670 -7493 623 4 56 452 549 .5028 158 4 S7 798 701 .745' 615 3 S7 480 580 .4989 150 3 sB 826 733 .7408 608 2 S8 508 612 .4951 142 2 59 60 854 764 .26795 .7364 600 I 59 60 536 643 .4912 134 .25882 3-7321 •96593 .27564 .28675 3-4874 .96126 1 N.Cos. N.Cot.lN.Tan. N.Sin. t N.Cos. N. Cot. N. Tan. N.Sin. / KJ^KO •7/1° 8o 16' ir N. Sin, O I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 27564 592 620 648 676 27704 731 759 787 815 •27843 871 899 927 95? •27983 .28011 039 067 095 28123 150 178 206 234 28262 290 318 346 374 ,28680 708 736 764 792 .28820 847 875 903 931 .28959 ,28987 ,29015 042 070 ,29098 126 209 .29237 N.Tan. N.Cot. .28675 706 738 769 801 ,28832 864 895 927 958 ,28990 29021 053 084 116 29147 179 210 242 274 ,29305 Z?>7 368 400 432 ,29463 495 526 558 590 .29621 "6^3" 685 716 748 .29780 811 843 875 906 >38 .2997Q .30001 065 .30097 128 160 192 224 •30255 287 319 351 382 .30414 446 478 509 541 34874 .4836 •4798 .4760 .4722 3.4684 .4646 .4608 •4570 •4533 3^4495 •4458 .4420 .4383 .4346 3-4308 .4271 •4234 .4197 .4160 34124 .4087 •4050 .4014 •3977 3-3941 •3904 .3868 •3832 .3796 3-3759 •3723 .3687 .3652 .3616 3-3580 .3544 •3509 •3473 •3438 33402 •3367 •3332 •3297 .3261 3.3226 •3191 •3156 .3122 •3087 3-305: ■30573 N.Cos.lN.Cot •3017 .2983 .2948 .2914 32879 •2845 .2811 •2777 •2743 3.2709 N.Cos.l .96126 118 no 102 094 ,96086 078 070 062 054 ,96046 037 029 021 013 ,96005 •95997 989 981 972 .95964 956 948 940 931 •95923 9'5 907 898 890 .95882 874 865 857 849 ,95841 •832 824 816 807 •95799 791 782 774 766 •95757 749 740 732 724 •95715 707 698 690 681 •95673 664 656 647 639 .95630 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 ^9 18 t7 16 15 14 13 12 II 10 9 8 7 6 5 4 3 N.Tan. I N. Sin.| / N.Sin. N.Tan. |N. Cot. N.Cos". .29237 •30573 3.2709 -95630 60 S9 265 605 .2675 622 2 293 637 .2641 613 58 3 321 669 .2607 60? 57 4 348 700 •2573 596 56 5 .29376 •30732 32539 .95588 5S 6 404 764 .2506 579 54 7 432 796 .2472 ■571 S3 8 460 828 .2438 562 S2 9 10 II 487 860 .2405 554 51 50 49 •29515 .30891 3237 « •95545 543 923 .2338 536 12 571 ^55 ■2305 528 48 13 599 •30987 .2272 519 47 14 ^26 .31019 •2238 511 46 15 .29654 .31051 3.2205 •95502 4S 16 682 083 .2172 493 44 17 710 115 .2139 485 43 • 18 737 147 .2106 476 42 19 20 21 765 178 •2073 467 41 40 39 •29793 .31210 3.2041 •95459 821 242 .2008 450 22 849 274 •i'975 441 38 23 876 306 •1943 433 37 24 904 338 .1910 424 36 25 .29932 •31370 3^1878 •95415 35 26 960 402 .1845 407 34 27 .29987 434 .1813 398 33 28 •30015 466 .1780 389 32 29 30 31 043 498 .1748 380 31 30 29 .30071 •31530 3.1716 •95372 098 562 .1684 363 32 126 594 .1652 354 28 2,2, 154 626 .1620 ' 345 27 ^A 182 658 .1588 337 26 35 .30209 .31690 3-1556 •95328 25 36 237 722 .1524 319 24 37 265 754 .1492 310 23 38 292 786 .1460 301 22 39 40 41 320 818 .1429 293 21 20 19 .30348 .31850 3.1397 .95284 376 882 .1366 275 42 403 914 •1334 266 18 43 431 946 .1303 257 17 44 459 .31978 .1271 248 16 4S .30486 .32010 3.1240 .95240 15 46 5H 042 .1209 231 14 47 542 074 .1178 222 13 48 570 106 .1146 213 12 49 50 SI 597 139 .1115 204 II 10 9 .30625 .32171 3.1084 -95,195 653 203 •1053 186 52 680 23s .1022 177 8 53 708 267 .0991 168 7 S4 736 299 .0961 159 6 S5 •30763 .32331 3.0930 -95150 5 56 791 363 .0899 142 4 S7 819 396 .0868 133 3 S8 846 428 .0838 124 2 59 60 874 460 .0807 115 I .30902 .32492 3^0777 .95106 iN.Cos. N.Cot.jN.Tan.l N.Sin. / •ys" 79.° 18' / N. Sin. N.Tan.iN.Cot. N.Cos. .30902 .32492 3.0777 .95106 eo I 929 524 .074 '> 097 59 2 95/ 5.S6 .0716 088 58 3 .309«5 588 .0086 079 57 4 .31012 621 •0655 070 56 S .31040 •32653 3.0625 .95061 55 6 068 685 •0595 052 54 7 095 717 .0565 043 53 8 123 749 •0535 033 52 9 10 II 151 782 .0505 024 51 50 49 .31178 .32814 3.0475 •95015 206 846 .0445 ,95006 12 233 878 .0415 •94997 48 13 261 911 .0385 988 47 '4 289 943 .0356 979 46 i.S ■3^31^ •32975 3.0326 .94970 45 i6 344 •33007 .0296 961 44 17 372 040 .0267 952 43 i8 399 072 .0237 943 42 19 20 21 427 104 .0208 933 41 40 ^2 .31454 •331.S6 3.0178 .94924 915 482 169 .0149 22 510 201 .0120 906 3^ 23 537 233 .0090 897 37 24 56^ 266 .0061 888 36 2S .31593 •33298 3.0032 •94878 35 26 620 330 30003 869 34 27 648 363 2.9974 860 33 28 675 395 .9945 851 32 29 30 703 427 .9916 842 31 30 29 .31730 .33460 2.9887 .9858 .94832 7S8 492 823 32 786 524 .9829 814 28 33 813 557 .9800 805 27 34 841 589 .9772 795 26 3S .31868 •33621 2.9743 •94786 25 3" 896 654 .9714 777 24 37 923 686 .9686 768 23 3H 951 718 .9657 758 22 39 40 41* •31979 751 .9629 749 21 20 19 .32006 ■337^3 2.9600 .94740 034 816 •9572 730 42 061 848 .9544 721 18 43 089 881 •9515 712 n 44 116 913 .9487 702 16 4"; .32144 •33945 2.9459 •94693 15 46 171 •33978 .9431 684 14 47 199 .34010 .9403 674 13 48 227 043 .9375 605- 12 49 50 SI 254 075 .9347 656 II 10 9 .32282 .34108 2.9319 .94646 309 f^o .9291 637 S2 337 173 •9263 627 8 53 3H 205 •9235 618 7 54 392 238 .9208 609 6 55 .32419 .34270 2.9180 •94599 5 5^ 447 303 .9152 590 4 57 474 335 .9125 580 3 5« 502 368 .9097 571 2 59 60 529 400 .9070 561 f •32557 .34433 2.9042 .94552 N. Cos.! N. Cot. N. Tan. N.Sin. f 71° 82 20° / N. Sin. N.Tan. N. Cot. N.Cos. ^ O I .34202 •36397 2-7475 .93969 60 59 229 430 •7450 959 2 257 463 •7425 949 58 3 284 496 .7400 939 57 4 3" 529 .7376 929 56 5 .34339 .36562 2.7351 •93919 SS 6 366 595 .7326 909 54 7 393 628 .7302 899 S3 8 421 6bi .7277 889 52 9 10 II 448 694 .7253 879 51 50 49 •34475 .36727 2.7228 .93869 503 7bo .7204 859 12 530 793 .7179 849 48 13 557 82b .7155 839 47 14 584 859 .7130 829 46 15 .34612 .36892 2.710b .93819 45 lb 639 925 .7082 809 44 17 666 958 •7058 799 43 i8 694 •36991 .7034 789 42 19 20 21 721 .37024 .7009 779 41 40 39 .34748 .37057 2.b985 •93769 775 090 .b9bi 759 22 803 123 .6937 748 38 23 830 157 .6913 738 37 24 8S7 190 .b889 728 36 25 .34884 •37223 2.b8b5 •9.3718 3S 2b 912 25b .b84i 708 34 27 939 289 .b8i8 b98 33 28 9bb 322 •6794 b88 32 29 30 31 .34993 355 .b77o 677 31 30 29 .35021 •37388 2.b74b .93667 048 422 .6723 657 32 075 455 .bb99 647 28 33 102 488 .6675 637 27 34 130 521 .6652 b2b 2b 35 •35157 •37554 2.bb28 •93616 25 3b 184 588 .bbo5 bob 24 37 211 b2I •6581 596 23 3« 239 654 •6558 585 22 39 40 41 26b b87 •6534 575 21 20 19 .35293 .37720 2.651 1 .93565 320 754 .b488 5.S5 42 347 787 .b4b4 544 18 43 375 820 .b44i 534 17 44 402 853 .b4i8 524 lb 45 .35429 •37887 2.6395 •93514 15 4b 456 920 .6371 503 14 47 484 953 .6348 493 13 48 5" .37986 •6325 483 12 49 50 51 538 .38020 .b302 472 10 9 •35565 •38053 2.b279 .93462 592 08b .b25b 452 52 bi9 120 •6233 441 8 53 647 153 .b2IO 431 7 54 ' 674 1 8b .bi87 420 b 55 •35701 .38220 2.blb5 .93410 5 5b 728 253 .bi42 400 4 57 755 28b .bii9 389 3 5^ 320 .bo9b 379 2 59 60 810 353 .bo74 368 I •35837 •3838b 2.6051 •93358 N.Cos. N.Cot.lN.Tan. iN.Sin.j ' || 2r / N.Sin. N.Tan. N. Cot.|N.Cos.! 1 I •35837 .38386 2.6051 •93358 60 59 8b4 420 .b028 348 2 891 453 .boob 337 58 3 918 487 .5983 327 57 4 945 520 •5961 316 56 5 •35973 •38553 2.5938 .93306 55 b .3booo 587 .5916 295 54 7 027 b20 .5893 28s 53 8 054 654 .5871 274 52 9 10 II 08 1 b87 .•5848 2b4 51 50 49 .3bio8 •38721 2.582b .93253 135 754 .5804 243 12 Ib2 787 .5782 232 48 13 190 821 •5759 222 47 14 217 854 •5737 211 46 15 .36244 .38888 2.5715 .93201 45 lb 271 921 •5693 190 44 17 298 955 •5671 180 43 18 325 .38988 .5649 ib9 42 19 20 21 352 .39022 .5627 159 41 40 39 .36379 •39055 2.5605 .93148 40b 089 .5583 ^37 22 434 122 .5561 127 38 23 4bi 15b .5539 lib 37 24 488 190 .5517 lob 36 25 •36515 .39223 2.5495 .93095 35 2b 542 257 .5473 084 34 27 569 290 .5452 074 33 28 596 324 •5430 063 32 29 30 31 623 357 .5408 052 31 30 29 .36650 •39391 2.5386 .93042 677 425 .5365 031 32 704 458 .5343 020 28 33 731 -492 •5322 .93010 27 34 758 S2b .5300 .92999 2b 35 •36785 .39559 2.5279 .92988 25 36 812 593 .5257 978 24 37 839 b2b .5236 967 23 38 8b7 bbo .5214 956 22 39 40 41 894 694 •5193 945 21 20 19 .36921 •39727 2.5172 .92935 948 7bi •5150 924 42 •36975 79-5 •5129 913 18 43 .37002 829 .5108 902 17 44 029 8b2 .508b 892 lb 45 •37056 •39896 2.5065 .92881 15 46 083 930 .5044 870 H 47 no 963 .5023 859 13 48 137 •39997 .5002 849 12 49 50 51 ib4 .40031 .4981 838 II 9 •37191 .400b5 2.4960 .92827 218 098 .4939 816 52 245 132 .4918 805 8 53 272 ibb .4897 794 7 54 299 200 .4876 784 b 55 •37326 .40234 2.4855 •92773 5 56 353 2b7 .4834 762 4 57 380 301 .4813 751 3 58 407 335" .4792 740 2 59 60 434 369 i .4772 729 I •37461 .404031 2.4751 .92718 N.Cos. N. Cot. N.Tan. N.Sin. t fiS' 68' 22° 23° 83 / N.Sin. N.fan^jlN^Cot. N.Cos. / N. Sin. N. Tan. !N. Cot. N.Cos. .37461 .4040312.4751 .92718 60 59 I .39073 .42447 482 2.3559 .92050 60 S9 488 436 •4730 707 100 .3539 039 2 5'5 470 .4709 697 58 2 127 516 .3520 028 58 3 542 504 .4689 686 57 3 153 551 .3501 016 57 4 569 538 .4668 675 56 4 180 585 .3483 •92005 S6 S •37595 .40572 2.4648 .92664 55 5 .39207 .42619 2.3464 .91994 55 6 622 606 .4627 653 54 6 234 654 .3445 982 54 7 649 640 .4606 642 53 7 260 688 .3426 971 53 8 676 674 .4586 631 52 8 287 722 .3407 959 52 9 10 II 703 707 .4566 620 51 50 49 9 10 II 3H 757 .3388 948 51 50 49 •37730 .40741 2.4545 .92609 .39341 .4279^ 2.3369 .91936 757 775 .4525 598 367 826 •3351 925 12 784 809 .4504 587 48 12 394 860 •3332 914 48 13 811 843 .4484 576 47 13 421 894 •.3313 902 47 14 838 877 .4464 S6S 46 14 448 929 •3294 2.3276 891 46 15 •37«65 .40911 2.4443 .92554 45 15 •39474 •42963 .91879 45 i6 892 945 .4423 543 44 16 501 .42998 .3257 868 44 17 919 .40979 ■4403 532 43 17 528 •43032 •3238 856 43 18 946 .41013 .4383 521 42 18 555 067 .3220 845 42 19 20 21 973 047 .4362 510 41 40 ^2 19 20 21 581 lOI .3201 833 41 40 39 .37999 .41081 2.4342 .92499 ■39608 .43136 2.3183 .91822 .38026 115 .4322 488 635 170 .3164 810 22 053 149 .4302 477 38 22 661 205 .3146 799 38 23 080 183 .4282 466 37 23 688 239 •3127 787 37 24 o'°7 217 .4262 455 36 24 715 ^74 .3109 775 36 25 ■38134 .41251 2.4242 ■92444 35 25 •39741 ■43308 2.3090 .91764 35 26 161 285 .4222 432 34 26 768 343 .3072 752 34 27 188 319 .4202 421 33 27 795 378 •3053 741 33 28 215 353 .4182 410 32 28 822 412 .3035 729 32 29 30 31 241 387 .4162 399 31 30 29 29 30 31 848 447 •3017 718 31 30 29 .38268 .41421 2.4142 .92388 •39875 ■43481 2.2998 .91706 295 455 .4122 377 902 516 .2980 694 32 322 490 .4102 366 28 32 928 550 .2962 683 28 33 349 524 •4083 355 27 33 955 585 .2944 671 27 34 376 558 .4063 343 26 34 .39982 620 .2925 660 26 35 •38403 .41592 2.4043 ■92332 25 35 .40008 •43654 2.2907 .91648 25 36 430 626 .4023 321 24 36 035 689 .2889 636 24 37 4S6 660 .4004 310 23 37 062 724 .2871 625 23 ^« 483 694 .3984 299 22 38 088 758 .2853 613 22 39 40 41 510 728 .3964 287 21 20 19 39 40 41 "5 793 .2835 601 21 20 19 .38537 ■41763 2.394? .92276 .40141 .43828 2.2817 .91590 564 797 .3925 26s 168 862 .2799 578 42 591 831 .3906 254 18 4^ 195 897 .2781 566 18 43 617 865 .3886 243 17 43 221 932 .2763 555 17 44 644 899 .3867 231 16 44 248 .43966 .2745 543 16 45 .38671 •41933 2.3847 .92220 15 45 •40275 .44001 2.2727 .91531 15 46 698 .41968 .3828 209 14 46 301 036 .2709 519 14 47 725 .42002 .3808 198 13 47 328 071 .2691 508 13 48 752 036 .3789 186 12 48 355 105 .2673 496 12 49 50 51 778 070 •3770 175 II 10 9 49 50 51 381 140 .2655 484 II 10 9 .38805 .42105 2-3750 .92164 .40408 .44175 2.2637 .2620 .91472 461 832 139 •3731 152 434 210 52 8S9 173 •3712 141 8 S2 461 244 .2602 T^ 8 53 886 207 •3693 130 7 53 488 279 .2584 7 54 912 242 •3673 119 6 54 514 3H .2566 425 6 55 ■38939 .42276 2.3654 .92107 5 55 .40541 .44349 2.2549 .91414 5 S^ 966 310 .3635 096 4 56 567 384 .2531 402 4 57 ■38993 345 .3616 085 3 57 594 418 .2513 390 3 5« .39020 379 .3597 073 2 58 621 453 .2496 378 2 59 60 046 413 .3578 062 L 59 60 647 488 .2478 366 •39073 •42447 ! 2.3559 .92050 .40674 ■44523 2.2460 .91355 N.Cos. N. Cot. N. Tan. N.Sin.l ' \ __ N.Cos.iN.Cot. N.Tan. N.Sin. / fir fifi° 84 24° 1 N. Sin.|N.Tan.iN.Cot.|N.Cos.| || I .40674 •44523 2.2460 •91355 60 59 700 558 •2443 343 2 727 593 •2425 2)2>^ 58 3 753 627 .2408 319 57 4 780 662 .2390 307 56 5 .40806 .44697 2.2373 .91295 55 b ^Z?> 732 •2355 283 54 ,7 860 767 .2338 272 S3 8 886 802 .2320 260 52 9 10 913 837 •2303 248 51 50 49 .40939 .44872 2.228b .91236 966 907 .2268 224 12 .40992 942 .2251 212 48 13 .41019 •44977 .2234 200 47 14 045 .45012 .2216 188 46 15 .41072 •45047 2.2199 .91176 45 lb 098 082 .2182 164 44 'Z 125 117 .2165 152 43 18 151 152 .2148 140 42 19 20 21 178 187 .2130 128 41 40 39 .41204 .45222 2.2113 .91116 231 257 .2096 104 22 257 292 .2079 092 38 23 284 327 .2062 080 37 24 310 362 .2045 068 36 25 •41337 •45397 2.2028 .91056 35 2b Z^Z 432 .2011 044 34 ^Z 390 467 .1994 032 ZZ 28 416 502 .1977 020* 32 29 30 31 443 53« .i960 .91008 31 30 29 .41469 •45573 2.1943 .90996 496 608 .1926 984 32 522 643 .1909 972 28 33 549 678 .1892 960 27 34 575 l^Z .1876 948 26 35 .41602 •45748 2.1859 .90936 25 3^ 628 784 .1842 924 24 37. 65s 819 .1825 911 23 3« 68i 854 .1808 899 22 39 40 41 707 889 .1792 887 21 20 19 •41734 .45924 2-1775 .90875 760 960 . -1758 863 42 787 •45995 .1742 851 18 43 813 .46030 .1725 839 17 44 840 065 .1708 826 16 45 .41866 .46101 2.-1692 .90814 IS 46 892 136 •1675 802 H 47 919 171 .1659 790 13 48 945 206 .1642 778 12 49 50 51 972 242 .1625 766 11 10 9 .41998 .46277 2.1609 •90753 .42024 312 •1592 741 52 051 348 •1576 729 8 53 077 383 .1560 7'7 7 54 104 418 •1543 704 6 55 .42130 .46454 2.1527 .90692 5 5<^ 156 489 .1510 680 4 57 183 525 .1494 668 3 5^ 209 560 .1478 6SS 2 59 60 235 595 ! -^61 i 643 I .42262 1 .46631 1 2.1445 j .90631 1 N.Cos.|n. Cot.|N.Tan.! N. Sin.| ^J 26' f N. Sin. N.Tan. N. Cot.|N.Cos. .42262 .46631 2.1445 •90631 60 S9 288 666 .1429 618 2 315 702 •1413 606 58 3 341 737 .1396 594 57 4 367 772 .1380 S82 S6 5 .42394 .46808 2.1364 .90569 SS b 420 843 .1348 557 54 7 446 879 •1332 545 53 8 473 914 •i3'5 532 52 9 10 II 499 950 .1299 520 51 50 49 •42525 .46985 2.1283 .90507 552 .47021 .1267 495 12 578 056 •1251 483 48 13 604 092 •1235 470 47 14 631 128 .1219 458 46 15 .42657 •47163 2.1203 .90446 4S lb 683 199 .1187 433 44 17 709 234 .1171 421 43 ,18 73t> 270 .1155 408 42 19 20 21 762 305 •1139 396 41 40 39 .42788 •47341 2.1123 .90383 8'5 377 .1107 371 22 841 412 .1092 3S8 38 23 867 448 .1076 346 37 24 894 483 .1060 334 36 2S .42920 •47519 2.1044 .90321 3S 26 946 555 .1028 309 34 27 972 590 .1013 296 33 28 .42999 626 .0997 284 32 29 30 31 .43025 662 .0981 271 31 30 29 •43051 .47698 2.0965 .90259 077 IZZ .0950 246 32 104 769 .0934 233 28 zz 130 805 .0918 221 27 34 IS6 840 .0903 208 26 35 .43182 .47876 2.0887 .90196 25 3<^ 209 912 .0872 183 24 37 2.SS 948 .0856 171 23 38 261 •47984 .0840 IS8 39 40 41 287 .48019 .0825 146 21 20 19 .43313 •48055 2.0809 •90133 340 091 .0794 120 42 366 127 .0778 108 18 43 392 163 .0763 095 17 44 418 198 .0748 082 16 45 •43445 .48234 2.0732 .90070 15 46 471 270 .0717 057 14 47 ■ 497 306 .0701 045 13 48 523 342 .0086 032 12 49 50 SI 549 378 .0671 i , 019 1 II 10 9 •43575 .48414 2.0655 .90007 602 450 .0640 .89994 S2 628 486 .0625 981 8 53 654 521 .0609 968 7 S4 680 557 •0594 956 6 ss .43706 •48593 2.0579 •89943 5 56 733 629 .0564 930 4 57 759 665 •0549 918 3 58 785 701 •0533 905 2 59 60 811 737 .0518 892 I •43837 •48773 2.0503 .89879 N.Cos.l N. Cot. N.Tan. !N.Sin.| / fi/i° 64" 26° ' iN.Sin.JN.Tan.N.Cot. N.Cos 1 I •43837 .48773 2.0503 .89879 60 59 863 809 .0488 867 2 . 889 845 •0473 854 S8 3 916 881 .0458 841 57 4 942 917 •0443 828 56 S .43968 •489^3 2.0428 .89816 55 6 •43994 .48989 .0413 803 54 7 .44020 .49026 .0398 790 5^ 8 046 062 •0383 777 52 9 10 II 072 098 .0368 764 51 50 49 .44098 .49134 2.0353 .89752 124 170 •0338 739 12 151 206 •0323 726 48 13 177 242 .0308 713 47 H 203 278 .0293 700 46 IS .44229 •49315 2.0278 .89687 45 It) 255 351 .0263 674 44 17 28^ 387 .0248 662 43 18 307 423 •0233 649 42 19 20 21 333 459 .0219 636 41 40 39 •44359 •49495 2.0204 .89623 385 532 .0189 610 22 411 568 .0174 597 38 23 437 604 .0160 584 37 24 464 640 .0145 571 36 25 .44490 .49677 2.0130 •89558 35 26 51b 713 .0115 545 34 27 542 749 .0101 532 33 28 568 786 .0086 519 32 29 30 31 594 822 .0072 506 3J 30 29 .44620 .49858 2.0057 .89493 646 894 .0042 480 32 672 931 .0028 467 28 33 698 .49967 2.0013 454 27 34 724 .50004 1.9999 441 26 35 •44750 .50040 1.9984 .89428 25 3(v 77b 076 .9970 415 24 37 802 "3 •9955 402 23 38 828 149 •9941 389 22 39 40 41 854 .44880 185 .9926 376 21 20 19 .50222 1.9912 .89363 906 258 •9897 350 42 932 295 .9883 337 18 43 958 33i .9868 324 17 44 .44984 368 •9854 3" 16 45 .45010 .50404 1.9840 .89298 15 46 036 441 .9825 285 14 47 062 477 .9811 27^ 13 48 088 5H .9797 259 12 49 59 51 114 550 .9782 245 II 10 9 .45140 •50587 1.9768 .89232 166 623 •9754 219 52 192 660 .9740 206 8 53 218 696 •9725 193 7 54 243 733 .9711 180 6 55 .45269 .50769 1.9697 .89167 S 5^ 295 806 .9683 153 4 57 321 843 .9669 140 3 5^ 347 879 .9654 127 2 59 373 916 .9640 114 I 60 •45399 •50s 53 1.9626 .89101 |N.Cos.|N.Cot. N.Tan.!N.Sin.| / 27° 85 ' (n- Sin. N.Tan. N. Cot. N^ Cos. I .45399 425 .50953 .50989 1.9626 .9612 .89101 60 59 087 2 45' .51026 .9598 074 58 3 477 063 ■9584 061 57 4 503 099 •9570 048 56 5 .45529 .51136 1.9556 .89035 55 6 554 173 •9542 021 54 7 580 209 .9528 .89008 53 8 606 246 .95 '4 •88995 52 9 10 II 632 283 .9500 981 51 50 49 •45658 •51319 1.9486 .88968 684 356 •9472 955 12 710 393 •9458 942 48 13 736 430 ■9444 928 47 14 762 467 .9430 915 46 15. •45787 •51503 1.9416 .88902 45 16 813 540 .9402 888 44 17 839 577 .9388 875 43 18 865 614 .9375 862 42 19 20 21 891 651 .9361 848 41 40 39 •45917 .51688 1.9347 .88835 942 724 .9333 822 22 968 761 •9319 808 38 23 •45994 798 .9306 795 37 24 .46020 835 .9292 782 36 25 .46046 .51872 1.9278 .88768 35 26 072 909 .9265 755 34 27 097 946 .9251 741 33 28 123 •51983 •9237 728 32 29 30 31 149 .52020 .9223 715 31 30 29 •46175 •52057 1.9210 .88701 201 094 .9196 688 32 226 131 .9183 674 28 33 252 168 .9169 661 27 34 278 205 •9155 647 26 35 .46304 .52242 1.9142 .88634 25 36 330 279 .9128 620 24 37 355 316 •9115 607 23 38 381 • 353 .9101 593 22 39 40 41 407 390 .9088 580 21 20 19 •46433 •52427 1.9074 .88566 458 464. .9061 553 42 484 5°i .9047 539 18 43 510 538 •9034 526 17 44 536 575 .9020 512 16 45 .46561 .52613 1.9007 .88499 15 46 587 650 •8993 485 14 47 613 687 .8980 472 13 48 639 724 .8967 458 12 49 50 51 664 761 .8953 445 II 10 9 .46690 •52798 1.8940 .88431. 716 836 .8927 417 52 742 873 •8913 404 8 53 767 910 .8900 390 7 54 793 947 .8887 377 6 55 .46819 •52985 1.8873 .88363 5 56 844 .53022 .8860 349 4 57 870 059 .8847 336 3 58 896 09l6 .8834 322 2 59 60 921 134 .8820 308 I .46947 •53171 1.8807 .88295 N.Cos.lN.Cot.l N.Tan. 1 N. Sin.| r fiOo on° 86 28° ' N. Sin. N.Tan. N.Cot.iN.Cos.l || O I .46947 •53171 1.8807 .88295 60 59 973 208 •8794 281 2 .46999 246 .8781 267 58 3 .47024 283 .8768 254 57 . 4 050 320 •8755 240 S6 5 .47076 •53358 1.8741 .88226 ss to lOI 395 .8728 213 54 7 127 432 .8715 199 53 8 153 470 .8702 i8s S2 9 10 II 178 507 .8689 172 51 50 49 .47204 •53545 1.8676 .88158 229 582 .8663 144 12 255 620 .8650 130 48 13 281 657 .8637 117 47 H 306 694 .8624 103 46 15 .4733^ •53732 1.8611 .88089 45 lb 35« 769 .8598 075 44 17 383 807 .8585 062 43 18 409 844 .8572 048 42 19 20 21 434 882 •8559 034 41 40 39 .47460 .53920 1.8546 .88020 486 957 •8533 .88006 22 511 •53995 .8520 •87993 38 ^3 537 .54032 .8507 979 37 24 562 070 .8495 965 36 25 ■4758» •54107 1.8482 •87951 35 26 614 145 .8469 937 34 27 639 183 .8456 923 33 28 66^ 220 •8443 909 32 29 30 31 690 258 .8430 896 31 30 29 .47716 •54296 1.8418 .87882 741 333 .8405 868 32 767 371 .8392 8S4 28 33 793 409 •8379 840 27 34 818 446 .8367 826 26 35 .47844 •54484 1^8354 .87812 25 3^ 869 522 .8341 798 24 37 895 560 .8329 784 23 3° 920 597 .8316 770 22 39 40 41 946 635 .8303 756 21 20 19 .47971 .54673! 1-8291 •87743 •47997 711 .8278 729 42 .48022 748 .8265 71S 18 43 048 786 .8253 701 17 44 073 824 .8240 687 16 45 .48099 .54862 1.8228 •87673 15 4b 124 900 .8215 659 14 47 150 938 .8202 645 13 48 175 .54975 .8190 631 12 49 50 51 201 •55013 ^8177 617 II 10 9 .48226 .55051 I 1.8165 .87603 252 089 .8152 589 52 277 127 .8140 575 8 53 303 165 .8127 561 7 54 328 203 •8115 546 6 55 •48354 .55241 1.8103 •87532 5 5t> 379 279 .8090 518 4 57 405 3^7 .8078 504 3 5^^ 430 355 ^8065 490 2 59 60 456 393 1 .8053 476 .48481 .55431 ! 1.8040 .87462 N.Cos.JN.Cot.lN.Tan. N.Sin.| ' 1 29' / N.Sin.lN.Tan. N. Cot. N.Cos.l I .48481 .55431 1.8040 .87462 60 59 506 469 .8028 448 2 532 507 .8016 434 58 3 557 545 .8003 420 57 4 583 583 .7991 406 56 5 .48608 •55621 1.7979 -87391 55 b 634 659 .7966 377 54 7 659 697 .7954 363 53 8 684 736 .7942 349 52 9 fO II 710 774 •7930 335 51 50 49 .48735 i .55812 1.7917 -87321 761 850 •.79<55 306 12 786 888 •7893 292 48 13 811 926 .7881 278 47 14 837 •55964 .7868 264 46 15 .48862 •56003 1.7856 .87250 45 lb 888 041 •7844 235 44 17 913 079 .7832 221 43 18 938 117 .7820 207 42 19 20 21 964 15b .7808 193 41 40 39 .48989 •56194 1.7796 .87178 .49014 232 .7783 164 22 040 270 .7771 150 38 23 065 309 •7759 136 37 24 090 347 •7747 121 36 25 .49116 •56385 1-7735 .87107 35 2b 141 424 •7723 093 34 27 166 462 .7711 079 33 28 192 501 •7699 064 32 29 30 217 539 •7687 050 31 30 29 .49242 •56577 1.7675 .87036 268 616 .7663 021 32 293 654 •7651 .87007 28 33 318 693 •7639 .86993 27 34 344 731 •7627 978 26 3S •49369 .56769 17615 .86964 25 36 394 808 •7603 949 24 37 419 846 •7591 935 23 3ii 445 885 •7579 921 22 39 40 41 470 923 .7567 90b 21 20 19 •49495 .56962 1-7556 .86892 521 .57000 •7544 878 42 546 039 •7532 863 18 43 571 078 .7520 849 17 44 596 116 .7508 834 16 45 .49622 •57155 1.7496 .86820 15 46 647 193 •7485 805 14 47 672 232 •7473 791 13 48 697 271 .7461 777 12 49 50 51 723 309 .7449 7b2 II 10 Q •49748 .57348 1-7437 .7426 .86748 773 386 733 52 798 425 .7414 719 8 53 824 464 .7402 704 7 S4 849 503 •7391 690 6 55 .49874 •57541 1-7379 .86675 5 56 899 580 •7367 661 4 57 924 619 •7355 646 3 58 950 657 •7344 632 2 59 60 •49971 696 -7332 617 .50000 •57735 1.7321 .86603 N.Cos. N.Cot. N.Tan.! N. Sin. / 30° 31° 87 1 N.Sin. N.Tan^ N. Cot. N.Cos. ' N.Sin. N.Tan.|N. Cot. N.Cos. .5CXXX3 •57735 1. 7321 .86603 60 59 I .51504 .60086 1.6643 .85717 60 59 025 774 .7309 ' 588 529 126 .6632 702 2 050 813 .7297 573 58 2 554 165 .6621 687 58 3 076 851 .7286 559 57 3 579 205 .6610 672 57 4 lOI 890 •7274 544 56 4 604 24^ .6599 657 56 S .50126 •57929 1.7262 .86530 55 5 .51628 .60284 1.6588 •85642 55 6 151 .57968 •7251 515 54 6 653 324 .6577 627 54 7 176 .58007 •7239 501 53 7 678 364 .6566 612 53 8 201 046 .7228 486 52 8 703 403 •6555 597 52 9 10 227 085 .7216 471 51 50 49 9 10 II 728 443 .6545 582 51 50 49 .50252 .58124 1.7205 .86457 •51753 .60483 1.6534 .85567 277 162 .7193 442 778 522 .6523 551 12 302 201 .7182 427 48 12 803 562 .6512 536 48 13 327 240 .7170 413 47 13 828 602 .6501 521 47 H 352 279 •7159 398 46 , 14 852 642 .6490 506 46 15 •50377 .58318 1. 7147 .86384 45 15 .51877 .60681 1.6479 -85491 45 16 403 357 .713b 369 44 16 902 721 .6469 476 44 17 428 396 .7124 354 43 17 927 761 .6458 461 43 18 453 435 'I^^Z 340 42 18 952 801 .6447 446 42 19 20 21 478 474 .7102 325 41 40 39 19 20 21 •51977 841 .6436 431 41 40 39 .50503 ■58513 1.7090 .86310 .52002 .60881 1.6426 •85416 528 552 .7079 295 026 921 .6415 401 22 553 591 .7067 281 38 22 051 .60960 .6404 .385 38 23 578 631 .7056 266 37 23 076 .61000 •6393 370 Z1 24 603 670 .7045 251 36 24 lOI 040 ■6383 355 36 2S .50628 .58709 1.7033 .86237 35 25 .52126 .61080 1.6372 .85340 35 26 654 748 .7022 222 34 .26 151 120 .6361 325 34 27 679 787 .7011 207 33 27 '75 160 .6351 310 ZZ 28 704 826 .6999 192 32 28 200 200 .6340 294 32 29 30 31 729 865 .6988 178 31 30 29 29 30 SI 225 240 .6329 279 31 30 29 •50754 .58905 1.6977 .86163 •52250 .61280 1-6319 .85264 '779 944 •6965 148 275 320 .6308 249 32 804 .58983 •6954 nz 28 32 299 360 .6297 234 28 zz 829 .59022 •6943 119 27 zz 324 400 .6287 218 27 34 854 061 .6932 104 26 S4 349 440 .6276 203 26 35 •50879 .59101 1.6920 .86089 25 35 .52374 .61480 1.6265 .85188 25 3^ 904 140 .6909 074 24 36 399 520 .6255 173 24 37 929 179 .6898 059 23 37 423 561 .6244 157 23 3« 954 218 .6887 045 22 38 448 601 .6234 142 22 39 40 41 •50979 258 .6875 030 21 20 19 39 40 41 473 641 .6223 127 21 20 19 .51004 •59297 1.6864 .86015 .52498 .61681 1. 6212 .85112 029 336 .6853 .86000 522 721 .6202 096 42 054 376 .6842 .85985 18 42 547 761 .6191 081 18 43 079 415 .6831 970 17 43 572 801 .6181 066 17 44 104 454 .6820 956 16 44 597 842 .6170 051 16 4S .51129 .59494 1.6808 •85941 15 45 .52621 .61882 1. 6160 •85035 15 46 154 533 .6797 '926 14 46 646 922 .6149 020 14 47 179 573 .6786 911 13 47 671 .61962 .6139 •85005 13 48 204 612 .6775 896 12 48 696 .62003 .6128 .84989 12 49 50 SI 229 651 .6764 881 II 10 9 49 50 51 720 043 .6118 974 II 10 9 •51254 .59691 1-6753 .85866 •52745 .62083 1. 6107 ■84959 279 730 .6742 851 770 124 .6097 943 S2 304 770 •6731 836 8 52 794 164 .6087 928 8 53 329 809 .6720 821 7 53 819 204 .6076 913 7 54 354 849 .6709 806 6 54 844 245 .6066 897 6 55 •51379 •59888 1.6698 •85792 5 55 .52869 .62285 1-6055 .84882 5 5^ 404 928 .6687 777 4 56 893 325 ,6045 866 4 57 429 •59967 .6676 762 3 57 918 366 .6034 851 3 58 454 .60007 .6665 747 2 58 943 406 .6024 836 2 59 60 479 046 .6654 732 59 60 967 446 .6014 820 ■51504 .60086 1.6643 .85717 .52992 .62487 1.6003 .84805 N.Cos.!N.Cot.iN.Tan.| N.Sin. /■ N.Cos.|N.Cot. N.Tan. N.Sin. t KQ^ Kft°' 88 32° t N. Sin.|N.Tan.|N.Cot.|N.Cos.| || O I .52992 .62487 1.6003 .84805 60 59 •53017 527 •5993 789 2 041 568 •5983 774 58 3 066 608 •5972 759 57 4 091 • 649 •5962 743 56 5 •53115 .62689 1.5952 .84728 55 6 140 730 •594' 712 54 7 164 770 .5931 697 53 8 189 811 .5921 681 52 9 10 II 214 852 •59" 666 51 50 49 .53238 .62892 1.5900 .84650 263 933 .5890 635 12 28S .62973 .5880 619 48 13 312 .63014 .5869 604 47 14 337 055 • 5859 S88 46 15 •533^1 .63095 1.5849 •84573 45 16 386 13b •5839 557 44 17 411 177 ■5829 542 43 18 _,AJ6- 217 .5818 526 42 19 20 21 460 258 .5808 511 41 40 39 .53484 .63299 1.5798 •84495 509 340 •5788 480 22 534 380 •5778 464 38 23 558 421 .5768 448 Zl 24 583 462 .5757 433 36 25 •53607 •63503 1.5747 •84417 35 26 632 544 •5737 402 34 27 656 584 .5727 386 ZZ 28 681 625 •5717 370 32 29 30 31 705 666 •5707 355 31 30 29 •53730 •63707 1.5697 •84339 754 748 •S687 324 32 779 789 •5677 308 28 ZZ 804 830 •5667 292 27 34 828 871 •5657 277 26 35 •53853 .63912 '•5647 .84261 25 3t> 877 953 •5637 245 24 ^Z 902 •63994 .5627 230 23 3^ 926 .64035 •5617 214 22 39 40 41 951 07b .5607 198 21 20 19 •53975 .64117 1-5597 .84182 .54000 158 .5587 167 42 024 199 .5577 151 18 43 049 240 •5567 135 17 44 073 281 •5557 120 16 45 •54097 .64322 1-5547 .84104 15 4b 122 363 .5537 088 14 47 146 404 .5527 072 13 48 171 446 •5517 057 12 49 50 51 195 487 •5507 041 II 10 9 .54220 .64528 1-5497 .84025 244 569 .5487 .84009 52 269 610 •5477 .83994 8 53 293 652 •5468 978 7 S4 317 693 •5458 962 6 55 •54342 •64734 1.5448 .83946 5 5^ 36b 775 ■5438 930 4 57 391 817 .5428 915 3 S8 415 858 .5418 899 2 59 60 440 899 .5408 883 I .54464 .64941 1-5399 .83867 N.Cos. N.Cot.|N.Tan.| N.Sin.| ' || 33' t |N.Sin.|N.Tan.;N.Cot.|N.Cos.| 1 I •54464 488 .64941 1.5399 .83867 60 59 .64982 .5389 851 2 513 .65024 .5379 835 58 3 537 065 .5369 819 57 4 561 106 .5359 804 S6 5 .5458b .65148 1-5350 .83788 55 6 610 189 •5340 772 54 7 635 231 .5330 7S6 53 8 659 272 •5320 740 52 9 10 II 683 314 •53" 1.5301 724 51 50 49 .54708 •65355 .83708 732 397 .5291 692 12 756 438 •5282 676 48 13 781 480 .5272 660 47 H 805 521 .5262 64s 46 15 .54829 •65563 1-5253 .83629 45 16 854 604 .5243 613 44 7 878 646 •5233 597 43 18 902 688 •5224 581 42 19 20 21 927 729 .5214 565 41 40 39 .54951 .65771 1.5204 .83549 975 813 •5195 533 22 .54999 854 .5185 517 38 23 •55024 896 •5175 501 37 24 048 938 .5166 48s S6 25 •55072 .65980 1.5156 .83469 35 2b 097 .66021 •5147 453 34 27 121 063 .5137 437 33 28 145 105 •5127 421 32 29 30 31 ib9 H7 .5118 405 31 30 29 •55194 .66189 1.5108 .83389 218 230 •5099 373 32 242 272 .5089 356 28 ZZ 266 314 .508a 340 27 34 291 356 .5070 324 26 35 -55315 .66398 I.qo6l .83308 25 36 339 440 .5051 292 24 37 363 482 •5042 276 23 38 388 524 •5032 260 22 39 40 41 412 566 •5023 244 21 20 19 •55436 .66608 1^5013 .83228 460 650 .5004, 212 42 484 692 •4994 195 18 43 509 734 .4985 179 17 44 533 776 •4975 163 16 45 •55557 .66818 1.4966 .83147 15 46 581 860 •4957 131 14 47 605 902 •4947 115 13 48 630 944 .4938 098 12 49 50 51 654 .66986 .4928 082 II (0 9 .55678 .67028 1.4919 .83066 702 071 .4910 050 52 726 "3 .4900 034 8 53 750 155 .4891 017 7 54 775 197 .4882 .83001 6 55 .55799 .67239 1.4872 .82985 5 56 823 282 .4863 969 4 57 847 324 •4854 953 3 58 871 366 .4844 936 2 59 60 895 409 •4835 920 I .55919 .67451 1 1 .4826 .82904 1 |n.Cos.| N.Cot. N.Tan. N. Sin.| t ^•7^ KOP 34' t NTsin. N.Tan. N.Cot.|N.Cos. .55919 943 ^6745 1 493 1.4826 .82904 60 59 .4816 887 2 968 536 .4807 871 58 3 .55992 57« .4798 855 57 4 .560161 620 .4788 839 56 .S .56040 .67663 1.4779 .82822 55 6 064 705 .4770 806 54 7 088 748 .4761 790 53 8 112 790 .4751 773 52 9 10 II ^Z^ 832 .4742 757 51 50 49 .56160 .67875 917 1.4733 .82741 • 724 184 .4724 12 208 .67960 .4715 708 48 U 232 .68002 .4705 692 47 H 256 045 .4696 675 46 i.S .56280 .68088 1.4687 •82659 45 lb 305 130 .4678 643 44 17 329 173 .4669 626 43 i8 353 215 .4659 610 42 19 20 21 377 258 .4650 593 41 40 39 .56401 .68301 1. 464 1 •82577 425 343 .4632 561 22 449 3«6 .4623 544 38 23 473 429 .4614 528 37 24 497 471 .4605 5" 36 25 .56521 .68514 1.459b .82495 35 26 545 557 .4586 478 34 27 569 600 .4577 462 33 28 593 642 .4568 446 32 29 30 31 617 685 •4559 429 31 30 29 .56641 .68728 1.4550 .82413 665 771 •4541 396 32 689 814 •4532 380 28 33 713 857 •4523 363 27 34 736 900 .4514 347 26 35 .56760 .68942 1.4505 .82330 25 3^ 784 .68985, .4496 314 24 ^Z 808 .69028 .4487 297 23 3« «32 071 .4478 281 22 39 40 41 856 114 .4469 264 21 20 ^9 .56880 .69157 1.4460 .82248 904 200 .4451 231 42 928 243 .4442 214 18 43 952 286 •4433 198 17 44 .56976 329 .4424 181 16 45 .57000 .69372 1.4415 .82165 15 4b 024 416 .440b 148 14 47 047 459 .4397 132 13 48 .071 502 .4388 "5 12 49 50 51 095 545 .69588 •4379 1-4370 098 .82082 11 10 9 •57^19 143 631 .4361 065 52 167 675 .4352 048 8 53 191 718 .4344 032 7 54 215 761 •4335 .82015 6 55 •5723^ .69804 1.4326 .81999 5 5^) 262 847 .4317 982 4 57 286 891 .4308 965 3 5^ 310 934 .4299 949 2 59 60 334 .69977 .4290 932 .81915 I .57358 .70021 1.4281- N.Cos. N. Cot. N.Tan.|N.Sin. / 36° 89 / N.Sin. N.Tan. N.Cot.|N. Cos. ~ .57358 .70021 : 1.4281 .81915 899 60 59 38i 064 •4273 2 405 107 .4264 882 58 3 429 /151 •4255 865 57 4 453 1^4 .4246 848 56 5 .57477 .70238 1-4237 .81832 55 6 501 281 .4229 8iS 54 7 524 325 .4220 798 S3 8 548 368 .4211 782 52 9 10 572 412 .4202 765 51 50 49 •57596 .70455 i.4»93 .81748 619 499 .4185 73» 12 643 542 .4176 7H 48 13 667 586 .4167 698 47 14 691 629 .4158 681 46 15 .57715 .70673 1.4150 .81664 45 16 738 717 .4141 647 44 17 762 760 .4132 631 43 18 786 804 .4124 614 42 19 20 21 810 848 .4115 597 .81580 41 40 39 •57833- 857 .70891 1.4106 935 .4097 563 22 881 .70979 .4089 546 38 23 904 .71023 .4080 530 37 24 928 066 .4071 5^3 36 25 •57952 .71110 1.4063 .81496 35 26 976 154 •4054 479 34 27 .57999 198 .4045 462 33 28 .58023 242 .4037 445 32 29 30 31 047 285 .4028 428 31 30 29 .58070 •71329 1.4019 .81412 094 373 .4011 395 32 n8 417 .4002 378 28 33 141 461 •3994 361 27 34 165 505 .3985 344 26 35 .58189 .71549 1.3976 .81327 25 36 212 593 .3968 310 24 37 236 637 •3959 293 23 38 260 681 . .3951 276 22 39 40 41 283 725 .3942 259 21 20 19 .58307 .71769 1.3934 .81242 330 813 •3925 225 42 354 857 .3916 2C8 18 43 378 901 .3908 191 17 44 401 946 •3899 174 16 45 .58425 .71990 1.3891 .81157 15 46 449 .72034 .3882 140 H 47 472 078 -3874 123 13 48 496 122 .386S 106 12 49 50 51 519 167 .3857 089 10 9 •58543 .72211 1.3848 .81072 567 255 .3840, 055 52 590 299 •383I' 038 8 53 614 344 .3823 021 7 54 637 388 .3814 .81004 6 55 .58661 .72432 1.3806 .80987 5 56 684 477 • -3798 970 4 57 708 521 •3789 953 3 58 731 565 •3781 936 2 59 60 .58779 610 .3772 919 •72654 1.3764 .80902 1 N.Cos. N. Cot. N.Tan. N.Sin. / KK° .^A° 90 36° / N.Sin N.Tan.|N.Cot.|N.Cos.| || I .58779 .72654 1-3764 .80902 60 S9 802 699 •3755 885 2 826 743 •3747 867 S8 3 849 788 .3739 850 57 4 873 832 •3730 833 S6 5 .58896 .72877 1.3722 .80816 55 5 920 921 •3713 799 54 7 943 .72966 •3705 782 S3 8 967 .73010 •3697 765 S2 9 lO II .58990 055 .3688 748 51 50 49 .59014 1 .73100 1.3680 •80730 037 144 .3672 713 12 061 189 •.3663 696 48 13 084 234 .3655 679 47 14 108 278 •3647 662 46 IS •59131 •73323 1^3638 .80644 4S 16 154 368 •3630 627 44 17 178 413 .3622 610 43 18 201 457 .3613 593 42 19 20 21 225 502 .3605 576 41 40 39 .59248 •73547 1.3597 :8o558 272 592 .3588 541 22 295 637 .3580 524 3ii 23 318 681 •3572 507 37 24 342 726 •3564 489 36 2S ■59365 •73771 1.3555 .80472 3S 26 389 816 •3547 455 34 27 412 861 •3539 438 33 28 436 906 .3531 420 32 29 30 SI 459 951 .3522 403 31 30 1 29 .59482 •73996 1.3514 .80386 S06 .74041 .3506 36S 32 529 086 .3498 351 28 33 552 131 .3490 334 27 34 576 176 .3481 ^'^ 26 3S •59599 .74221 1.3473 .80299 25 36 622 267 •3465 282 24 37 646 312 .3457 264 23 3« 669 357 .3449 247 22 39 40 41 693 402 .3440 230 21 20 19 .59716 •74447 1.3432 .80212 739 492 •3424 195 42 763 S38 .3416 178 18 43 786 583 .3408 160 17 44 809 628 .3400 143 16 4S .59832 .74674 1.3392 .80125 15 46 856 719 .3384 108 14 47 879 764 •3375 091 13 48 902 810 •3367 073 12 49 50 SI 926 855 •3359 056 II 10 9 •59949 .74900 1-3351 .80038 972 946 •3343 021 S2 .59995 .74991 •3335 .Soqps 8 53 .60019 •75037 -3327 .79986 7 S4 042 082 •3319 968 6 55 .60065 .75128 i^33ii •79951 5 5^ 089 173 •3303 934 4 S7 112 219 .3295 916 3 S8 135 264 •3287 899 2 59 60 158 310 •3278 881 .60182 •75355 1.3270 .79864 N.Cos. N.Cot.N.Tan.j N. Sin.| / 37' ' |N.Sin. N.Tan. N. Cot! N. Cos I .60182 .75355 1.3270 .79864 60 S9 205 401 .3262 846 2 228 447 •3254 829 S8 J 251 492 .3246 ^ 811 57 4 274 538 .3238 793 S6 5 .60298 •75584 1.3230 .79776 ss 6 321 b29 .3222 758 54 7 344 675 .3214 741 S3 8 367 721 .3206 723 S2 9 10 II 390 767 .3198 706 51 50 49 .60414 .75812 1.3190 .79688 437 858 .3182 671 12 460 904 •3175 6S3 48 13 483 950 .3167 635 47 14 S06 •75996 .3159 618 46 15 .60529 .76042 1.3151 .79600 4S 16 553 088 .3143 583 44 17 576 '§^ •3135 565 43 18 599 180 .3127 547 42 19 20 21 622 226 .3119 530 41 40 39 .60645 .76272 1.3111 .79512 668 318 .3103 494 22 691 364 •3095 477 38 23 714 410 •3087 459 37 24 738 456 •3079 441 36 25 .60761 .76502 1.3072 .79424 3S 26 784 548 .3064 406 34 27 807 594 •3056 388 33 ■ 28 830 640 •3048 371 32 29 30 31 853 686 •3040 353 31 30 29 1 .60876 •76733 1.3032 •79335 899 779 .3024 318 32 922 825 •3017 300 28 1 33 945 871 • .3009 282 27 34 968 918 .3001 264 26 35 .60991 •76964 1.2993 .79247 25 36 .61015 .77010 •2985 229 24 ^l 038 057 •2977 211 23 38 061 103 .2970 193 22 39 40 41 084 149 .2962 176 21 20 19 .61107 .77196 1.2954 •79158 130 242 .2946 140 42 153 289 .2938 122 18 43 176 335 .2931 105 17 44 199 382 •2923 0S7 16 45 .61222 •77428 1-2915 .79069 15 46 245 475 .2907 051 14 47 268 521 .2900 033 13 48 291 568 .2892 .79016 12 49 50 51 314 615 1 .2884 .78998 II 10 9 •61337 .77661 1 1.2876 .78980 360 708 1 • .2869 962 S2 383 754! .2861 944 8 53 406 80.1 -2853 926 7 S4 429 848 1 .2846 908 6 55 .61451 •77895^ 1.2838 .78891 5 56 474 941 ■■ .2830 873 4 S7 497 .77988 ! .2822 855 3 58 520 •78035 .2815 837 2 59 60 543 082 1 .2807 819 .61566 •78129 1 1.2799 .78801 N.Cos.N.Cot.l N.Tan.| N. Sin.| / K.q^ .^9/ 3 8° 39° 91 / N.Sin. N.Tan. N.Cot.|N.Cos. ' N. Sin. N.Tan. N. Cot. N. Cos. I 2 3 .61566 .78129 1.2799 .78801 60 59 58 57 2 3 .62932 .80978 1.2349 •77715 696 678 660 60 59 58 57 589 612 635 175 222 269 .2792 .2784 .2776 783 765 747 955 .62977 .63000 .81027 075 123 •2342 •2334 •2327 4 I 658 .61681 704 316 .78363 410 .2769 1. 2761 .2753 729 .78711 694 56 55 54 4 5 6 022 .63045 068 171 .81220 268 .2320 1.2312 .2305 641 •77623 605 56 55 54 7 8 9 iO II 12 13 749 772 -- 457 504 551 .2746 .2738 •2731 676 658 640 53 52 51 50 49 48 47 I 9 10 II 12 13 090 113 135 316 364 413 .2298 .2290 .2283 586 568 550 53 52 51 50 48 47 .61795 .78598 1.2723 .78622 .63158 .81461 1.2276 .77531 818 841 864 645 692 739 .2715 .2708 .2700 604 586 568 180 203 225 5'° 558 606 .2268 .2261 .2254 513 494 476 14 16 887 .61909 932 786 •78834 881 .2693 1.2685 .2677 550 •78532 514 46 45 44 14 15 16 248 .63271 293 655 .81703 752 .2247 1.2239 .2232 458 •77439 421 46 45 44 17 18 J9 20 21 22 23 955 .61978 .62001 928 .78975 .79022 .2670 .2662 .2655 496 478 460 43 42 41 40 39 2,1 :^ 19 20 21 22 23 361 800 849 898 .2225 .2218 .2210 402 384 366 43 42 41 40 39 38 37 .62024 .79070 1.2647 .78442 •63383 .81946 1.2203 .77347 046 069 092 117 164 212 .2640 .2632 .2624 424 405 387 406 428 451 .81995 .82044 092 .2196 .2189 .2181 329 310 292 24 .62138 160 259 .79306 354 .2617 1.2609 .2602 369 •78351 36 35 34 24 25 26 473 .63496 518 141 .82190 238 .2174 1. 2167 .2160 273 .77255 236 36 35 34 ^1 28 29 30 31 32 33 183 206 229 401 449 496 .2594 .2587 •2579 315 297 279 3Z 32 31 30 29 28 27 27 28 29 30 31 32 540 563 585 287 336 38? •2153 .2145 .2138 218 199 181 33 32 31 30 29 28 27 .62251 .79544 1.2572 .78261 .63608 .82434 1.2131 .77162 274 297 320 639 686 .2564 •2557 .2549 243 225 630 675 483 .2124 .2117 .2109 144 125 107 34 342 734 .79781 829 .2542 1.2534 .2527 188 .78170 152 26 25 24 34 35 36 698 .63720 742 629 .82678 727 .2102 1.2095 .2088 088 .77070 051 26 25 24 37 38 39 40 41 42 43 411 433 456 877 924 .79972 .2519 .2512 .2504 134 116 098 23 22 21 20 19 18 17 37 38 39 40 41 42 43 765 787 810 776 825 874 .2081 .2074 .2066 033 .77014 .76996 23 ' 22 21 20 19 18 17 .62479 .80020 1.2497 .78079 •63832 .82923 1.2059 .76977 502 524 547 067 ^\^ 163 .2489 .2482 .2475 061 043 025 854 899 .82972 .83022 071 .2052 .2045 .2038 959 940 921 44 45 46 .62592 615 211 .80258 306 .2467 1.2460 .2452 .78007 •77988 970 16 15 14 44 45 46 922 •63944 966 120 .83169 218 .2031 1.2024 .2017 903 .76884 866 16 15 14 % 49 50 51 52 53 638 660 683 354 402 450 •2445 •2437 .2430 952 934 916 13 12 II 10 9 8 7 47 48 49 50 51 52 53 .63989 .64011 033 268 317 366 .2009 .2002 .1995 847 828 810 13 12 10 9 8 7 .62706 .80498 1.2423 .77897 .64056 .83415 1. 1988 .76791 728 751 774 546 594 642 •2415 .2408 .2401 879 861 843 078 100 123 465 564 .1981 .1974 .1967 772 754 735 54 796 .62819 842 690 .80758 786 •2393 1.2386 .2378 824 .77806 788 6 5 4 54 145 .64167 190 613 .83662 712 .i960 1. 1953 .1946 717 .76698 679 6 5 4 59 60 864 887 909 834 882 930 .2371 .2364 .2356 769 751 733 3 2 I 57 58 59 60 212 234 256 860 .1939 .1932 .1925 661 642 623 3 2 .62932 .80978 1.2349 •77715 .64279 .83910 1.1918 .76604 N.Cos.|N.Cot. ;N.Tan.| N.Sin. / N.Cos. N. Cot.lN.Tan.i N. Sin. r Kt ^(\' 92 40° 1 N. Sin.|N.Tan iN.Cot.|N.Cos. .64279 .83910 1 1.1918 .76604 60 59 301 .83900 .1910 586 2 323 .84009 .1903 S67 S8 3 346 059 .1896 548 57 4 368 108 .1889 530 56 S .64390 .84158 1. 1882 .76511 55 6 412 208 .1875 492 54 7 435 258 ..1868 473 53 8 457 307 .1861 455 52 9 10 II 479 357 .1854 436 51 50 49 .64501 .84407 1.1847 .76417 524 457 .1840 398 12 546 507 .1833 380 48 13 568 556 .1826 361 47 14 590 606 .1819 342 46 15 .64612 .84656 1.1812 •76323 45 lb ^35 706 .1806 304 44 17 657 7S6 .1799 286 43 18 679 806 .1792 267 42 19 20 21 701 856 .1785 248 41 40 39 .64723 .84906 1.1778 .76229 746 .84956 .1771 210 22 768 .85006 .1764 192 38 23 790 057 •1757 173 37 24 812 107 .i7?o 154 36 25 .64834 •85157 1-1743 •76135 35 2b 856 207 •1736 lib 34 ^Z 878 257 .1729 097 33 28 901 308 .1722 078 32 29 30 3^ 923 358 .1715 059 31 30 29 .64945 .85408 1.1708 .76041 967 458 .1702 022 32 .64989 509 .1695 .76003 28 33 .65011 559 .1688 .75984 27 34 033 609 .1681 965 26 35 •65055 .S5660 1. 1674 .75946 25 3^ 077 710 .1667 927 24 37 100 761 .1660 908 23 3« 122 811 •1653 889 22 39 40 41 144 862 .1647 870 21 20 19 .65166 .85912 1. 1640 .75S51 188 .85963 .1633 832 42 210 .86014 .1626 813 18 43 232 064 .1619 794 »7 44 254 "^ .1612 775 16 45 •65276 .86166 1. 1606 •W756 15 4b 298 216 .1599 738 14 47 320 267 .1592 719 13 48 342 318 .1585 700 12 49 50 51 3<34 368 .1578 680 II 10 9 .65386 .86419 ^•i57i .75661 408 470 .1565 642 52 430 521 .15.58 623 8 53 452 572 •1551 604 7 54 474 623 •1544 58s 6 55 .65496 .86674 1-1538 .75566 5 5^ 518 725 .1531 547 4 57 540 776 .1524 528 3 5^ 562 827 •1517 509 2 59 60 584 878 .1510 490 I .65606 .86929 1. 1504 •75471 N.Cos. N. Cot. N.Tan.j N. Sin. ' 41° ' iN.Sin.jN.Tan. N. Cot.] N.Cos. 1 .65606 .86929 1. 1504 .75471 60 S9 628 .86980 .1497 452 2 650 .87031 .1490 433 58 3 672 082 .1483 414 57 4 694 '^3 .1477 395 S6 . 5 .65716 .87184 1. 1470 •75375 55 b 738 236 > .1463 356 54 7 759 287 .1456 337 53 8 781 338 .1450 318 52 9 10 II 803 389 •H43 299 51 50 49 .65825 .87441 1.1436 .75280 847 492 .1430 261 12 869 543 .1423 241 48 13 891 595 .1416 222 47 14 913 646 .1410 203 46 15 •65935 .87698 1. 1403 .75184 45 lb 956 749 •1396 165 44 17 .65978 801 .1389 146 43 18 .66000 852 .1383 126 42 19 20 022 904 .1376 107 41 40 .66044 •87955 1. 1369 .75088 21 066 .88007 .1.363 069 39 22 088 059 .1356 050 38 23 109 no .1349 030 37 24 131 162 .1343 •75011 36 25 .66153 .88214 1.1336 .74992 35 2b 175 265 .1329 973 34 27 197 317 .1323 953 33 28 218 369 .1316 934 32 29 30 31 240 421 .1310 915 31 30 29 .66262 .88473 1.1303 .74896 284 524 .1296 876 32 30b 576 .1290 8S7 28 33 327 628 .1283 838 27 34 349 680 .1276 818 26 35 •66371 .88732 1. 1 270 •74799 25 36 393 784 .1263 780 24 37 414 836 .1257 760 23 38 436 888 .1250 741 22 39 40 41 458 940 .1243 722 21 20 19 .66480 .88992 1. 1 237 1.74703 501 .89045 .1230 683 42 523 097 .1224 664 18 43 545 149 .1217 644 17 44 566 201 .1211 625 16 45 .66588 •89253 1.1204 .74606 15 46 610 306 .1197 586 14 47 632 358 .1191 567 13 48 653 410 .1184 548 12 49 50 SI 675 463 .1178 528 II 10 9 .66697 718 .89515 567 1.1171 .74509 .1165 489 52 740 620 .1158 470 8 53 762 672 .1152 451 7 54 783 725 .1145 431 6 55 .66805 •89777 I. "39 •74412 5 56 827 830 .1132 392 4 57 848 883 .1126 373 3 58 870 935 .1119 353 2 59 60 891 .89988 .1113 334 .66913 .90040 I.I 106 .74314 N.Cos. N.Cot. N.Tan.j N. Sin. ^ AQ' A»° 42' ' iN.Sin. N.Tan^ NTcot. N^ Cos 1 I .66913 .90040 I.I 106 .743 « 4 60 59 935 093 .1100 295 2 95^ 146 .1093 276 S8 3 978 199 .1087 256 57 i -* .66999 251 .1080 237 56 1 S .67021 .90304 1.1074 .74217 55 043 357 .1067 198 54 7 064 410 .1061 178 53 8 086 463 .1054 159 52 9 10 IL 107 5.0 .1048 139 51 50 49 .67129 .90569 1.104 1 .7412a 151 621 •1035 100 12 172 674 .1028 080 48 13 194 727 .1022 061 47 H 215 781 .1016 041 46 IS .67237 .90834 1. 1009 .74022 45 i6 258 887 .1003 .74002 44 17 280 940 .0996 •73983 43 18 301 .90993 .0990 963 42 19 20 21 323 .91046 .0983 944 41 40 39 .67344 .91099 1.0977 .0971 •73924 300 153 904 22 3«7 206 .0964 885 3>i 23 409 259 .0958 865 37 24 430 3^3 .0951 846 36 2S .67452 .91366 1.0945 .73826 35 26 473 419 •0939 806 34 27 495 473 .0932 787 33 28 516 526 .0926 767 32 29 30 31 53« 580 .0919 747 31 30 29 •67559 •91633 1.0913 •73728 580 687 .0907 708 32 602 740 .0900 688 28 33 623 794 .0894 669 27 34 645 847 .0888 ,649 26 3S .67660 .91901 1. 088 1 .73629 25 36 688 •91955 .0875 610 24 37 709 .92008 .0869 590 23 3S 730 062 .0862 570 22 39 40 41 752 116 .0856 551 21 20 19 •67773 .92170 1.0850 •73531 511 795 224 .0843 42 . 816 277 .0837 491 18 43 ^37 331 .0831 472 17 44 859 385 .0824 452 16 45 .67880 .92439 1.0818 •73432 15 46 901 493 .0812 413 14 47 923 547 .0805 393 13 48 944 601 .0799 373 12 49 50 SI 965 655 .0793 353 II 10 9 .67987 .92709 1.0786 ■73333 .68008 763 .0780 314 S2 029 817 •0774 294 8 53 051 872 .0768 274 7 S4 072 926 .0761 . 254 6 55 .68093 .92980 ^•0755 •73234 5 5^ "5 •93034 •0749 215 4 S7 1.36 088 .0742 195 3 5« 157 H3 .0736 175 2 59 80 179 197 .0730 155 .68200 •93252 1.0724 •73135 N.Cos. N.Cot. N.Tan. N.Sin.l ' || 43° 93 I' |N. Sin. N.Tan. N.Cot. N.Cos.| l| I .68200 221 .93252 1.0724 •73'35 60 59 306 .0717 116 2 242 360 .0711 096 S8 3 264 4'5 •0705 076 57 4 285 469 .0699 056 S6 5 .68306 •93524 1.0692 •73036 ss 6 327 578 .0686 .73016 54 7 349 633 .0680 .72996 S3 8 370 688 .0674 976 52 9 10 II 391 742 .0668 957 51 50 49 .68412 •93797 1. 066 1 •72937 434 852 •0.655 917 12 455 906 .0649 897 48 13 476 .93961 .0643 877 47 H 497 .94016 •0637 8S7 46 15 .68518 .94071 1.0630 •72837 4S 16 539 125 .0624 817 44 17 ^t' 180 .0618 797 43 18 582 235 .0612 777 42 19 20 21 603 290 .0606 757 41 40 39 .68624 .94345 1.0599 •72737 645 400 •0593 717 22 666 455 .0587 697 38 23 688 510 •0581 677 37 24 709 565 .0575 657 36 25 .68730 .94620 1.0569 .72637 35 26 751 676 .0562 617 34 27 772 731 .0556 597 33 28 793 786 •0550 577 32 29 30 31 814 841 .0544 557 31 30 29 •68835 .94896 1^0538 .72537 857 •94952 •0532 517 32 878 •95007 .0526 497 28 33 899 062 .0519 477 27 34 920 118 .0513 457 26 35 .68941 •95173 1.0507 •72437 25 36 962 229 •0501 417 24 37 .68983 284 •0495 397 23 3^ .69004 340 .0489 377 22 39 40 41 025 395 •0483 357 21 20 19 .69046 .95451 1.0477 •72337 067 506 .0470 317 42 088 562 .0464 297 18 43 109 618 •0458 277 17 44 130 673 .0452 257 16 45 .69151 •95729 1.0446 .72236 15 46 172 785 .0440 216 14 47 193 841 .0434 196 13 48 214 897 .0428 176 12 49 50 51 235 .95952 .0422 156 II 10 9 .69256 .96008 1. 041 6 .72136 277 064 .0410 n6 52 298 120 .0404 095 8 53 319 176 .0398 075 7 54 340 232 •0392 055 6 55 .69361 .96288 1.0385 •72035 5 56 382 344 .0379 .72015 4 57 403 400 .0373 •71995 3 58 424 457 •0367 974 2 59 60 445 513 •0361 954 I .69466 .96569 1 1.0355 •71934 _.„ N.Cos. N. Cot. N.Tan.; N. Sin.| r 1 AT Aa^ 94 44" / N.Sin.l N.Tan.iN.Cot.JN.Cos.l | I .69466 487 .96569 1-0355 -719341 60 59 625 •0349 914 2 508 681 •0343 894 58 3 529 738 .0337 873 57 4 549 794 .0331 853 56 5 .69570 .96850 i^o325 .71833 55 6 591 907 .0319 813 54 7 612 .96963 •0313 792 53 8 633 .97020 . -0307 772 52 ' 9 lO II 654 076 .0301 752 51 50 49 .69675 .97133 1.0295 .71732 696 189 .0289 711 12 717 246 .0283 691 48 U 737 302 .0277 671 47 14 758 359 .0271 650 46 15 .69779 .97416 1.0265 .7x630 45 i6 800 472 .0259 610 44 17 821 529 •0253 590 43 18 842 586 .0247 569 42 19 20 21 862 643 .0241 549 41 40 39 .69883 .97700 1.0235 .71529 904 756 .0230 5°? 22 925 813 .0224 488 38 23 946 870 .0218 ,468 37 24 966 927 .0212 447 36 25 .69987 .97984 1.0206 .71427 35 26 .70008 .98041 .0200 407 34 27 029 098 .0194 386 33 28 049 155 .0188 366 32 29 30 31 070 213 .0182 345 31 30 29 .70091 .98270 1. 01 76 •71325 112 327 .0170 305 32 132 384 .0164 284 28 33 153 441 .0158 264 27 34 174 499 .0152 243 26 ^1 •70195 .98556 1.0147 .71223 25 36 215 613 .0141 203 24 37 236 671 •0133 182 23 38 257 728 .0129 162 22 39 40 41 277 786 .0123 141 21 20 19 .70298 .98843 1.0117 .71121 319 901 .0111 100 42 339 .98958 .0105 080 18 43 360 .99016 .0099 059 17 44 381 073 .0094 039 16 45 .70401 •991 31 1.0088 .71019 15 46 422 189 .0082 .70998 14 47 443 247 .0076 978 13 48 463 304 .0070 957 12 49 50 51 484 362 .0064 937 II 10 9 •70505 .99420 1.0058 .70916 525 478 .0052 896 52 546 536 .0047 875 8 53 567 594 .0041 855 7 54 587 652 •0035 83.4 6 55 .70608 .99710 1.0029 .70813 5 56 628 768 .0023 793 4 57 649 826 .0017 772 3 58 670 884 .0012 752 2 59 60 690 .99942 .0006 731 I .70711 1. 0000 j 1. 0000 .70711 |n Cos.|N.Cot.|N.Tan.! N. Sin.| ' A.*i' IV. CIRCULAR ARCS WITH R/ \I)IUS UNITY. 95 DEGREES. 1 MINUTES. SECONDS. O^ 0.0000000 60 61 1. 047 1 9 76 120^ 121 2.09439 5 » 2.1118484 0' , o.ooocx) 00 O", 0.0000000 0.0174533 1 .06465 08 I 0.0002909 I 0.0000048 2 0.03490 66 62 1.0821041 122 2.12930 17 2 0.00058 18 2 0.0000097 3 0.05235 99 63 1-09955 74 123 2.1467550 3 0.00087 27 3 0.0000 1 45 4 0.06981 32 64 1.11701 07 124 2.1642083 4 0.001 16 36 4 0.00001 94 5 0.08726 65 65 1. 1 3446 40 125 2.18166 16 5 0.0014544 5 0.00002 42 6 0.1047 1 9^ 66 1.15191 73 126 2.1991149 b 0.0017453 6 0.00002 91 7 0.1 22 1 7 30 67 1. 1 6937 06 127 2.2165682 7 0.00203 62 7 0.00003 39 8 0.13962 63 68 1. 1 8682 39 128 2.23402 14 8 0.00232 71 8 0.00003 88 9 10 II 0.1570796 69 70 71 1.20427 72 1.2217305 129 130 131 2.2514747 9 10 11 0.00261 80 10 11 0.00004 36 0.1745329 2.26892 80 0.00290 89 0.00004 85 0.00005 33 0.1919862 1.23918 38 2.2863813 0.0031998 12 0.20943 95 72 1.25663 71 132 2.30383 46 12 0.00349 07 12 0.00005 82 13 0.22689 28 73 1.2740904 ^33 2.3212879 13 0.00378 15 13 0.00006 30 H 0.2443461 74 1.2915436 1.34 2.3387412 14 0.00407 24 M 0.00006 79 15 0.2617994 75 1.3089969 135 2.356194? 15 0.00436 3i 15 0.00007 27 16 0.27925 27 76 1.3264502 136 2.37364 78 16 0.00465 42 16 0.00007 76 17 0.2967060 77 1-3439035 1.37 2.391 10 11 17 0.00494 5 1 17 0.00008 24 18 0.3141593 78 1.36135 68 138 2.40855 44 18 0.00523 60 18 0.00008 73 19 20 21 0.33161 26 79 80 81 1.37881 01 139 140 141 2.42600 77 19 20 21 0.00552 69 19 20 21 0,00009 21 0.34906 59 1.3962634 2.44346 10 0.00581 78 0.00009 70 0.36651 91 1.41371 67 2.46091 42 0.00610 87 o.oooio 18 22 0.38397 24 82 1.4311700 142 2.47836 75 22 1 0.00639 95 22 o.oooio 67 23 0.40142 57 83 1.4486233 143 2.49582 08 23 0.00669 04 23 0.0001 1 15 24 0.4188790 84 1.46607 66 144 2.5132741 24 0.00698 13 24 0.000 11 64 25 0.43633 23 85 1.4835299 HS 2.53072 74 25 0.00727 22 25 0.00012 12 26 0.45378 56 86 1.50098 32 146 2.5481807 26 0.0075631 26 0.00012 61 27 0.4712389 87 1. 5 1 843 64 H7 2.56563 40 27 0.00785 40 27 0.0001309 28 0.48869 22 88 1-5358897 148 2-58308 73 28 0.0081449 28 0.0001357 29 30 31 0.50614 51 89 90 91 1.5533430 149 150 151 2.6005406 29 30 31 0.00843 58 29 30 31 0.0001406 0.52359 88 1-5707963 2-6179939 0.00872 66 0.00014 54 0.54105 21 1.5882496 2.63544 72 0.00901 75 0.00015 03 32 0.5585054 92 1.60570 29 152 2.6529005 32 0.00930 84 32 0.00015 5^ 33 0.57595 87 93 1.6231562 153 2.67035 38 33 0.00959 93 33 0.0001600 34 0.59341 19 94 1.6406095 154 2.68780 70 34 0.00989 02 34 0.0001648 35 0.61086 52 95 1.6580628 iSS 2.7052603 35 0.01018 u 35 0.0001697 36 0.62831 85 96 1.67551 61 156 2.72271 36 36 0.01047 20 36 0.00017 45 37 0.64577 18 97 1.6929694 157 2.74016 69 37 0.01076 29 37 0.00017 94 3« 0.6632251 98 1.7104227 158 2.7576202 38 0.01105 38 38 0.0001842 39 40 41 0.68067 84 99 100 lOI 1.72787 60 159 160 161 2.7750735 39 40 41 0.0113446 39 40 0.0001891 0.69813 17 i.''4532 93 2.7925268 0.0116355 0.00019 39 0-7155850 1.7627825 2.8099801 0.01 192 64 41 0.00019 88 42 0.73303 ^3 102 1.78023 58 162 2.82743 34 42 0.01221 73 42 1 0.00020 36 43 0.75049 16 103 1.79768 91 163 2.8448867 43 0.0125082 43 0.00020 85 44 0.7679449 104 1.81514 24 164 2.86234 00 44 0.0127991 44 0.00021 33 45 0.78539 82 105 1-8325957 i6s 2.87979 33 45 0.0130900 45 0.00021 82 45 0.80285 15 106 1.8500490 166 2.89724 66 46 0.0133809 46 i 0.00022 30 47 0.82030 47 107 1.8675023 167 2.91469 99 47 0.01367 17 47 1 0.00022 79 48 0.83775 80 108 1.88495 56 168 2.93215 31 48 0.0139626 48 1 0.00023 27 49 50 51 0.85521 13 109 110 m 1.9024089 169 170 171 2.94960 64 49 50 51 0.01425 35 49 50 0.00023 76 0.8726646 1.91986 22 2.96705 97 0.0145444 0.00024 24 0.89011 79 I-9373I 55 2.98451 30 0.01483 53 51 0.00024 73 52 0.90757 12 112 1.9547688 172 3.0019663 52 0.01512 62 52 0.00025 21 53 0.92502 45 "3 1.97222 21 ^73 3.01941 96 53 0.01541 71 53 0.00025 70 54 0.94247 78 114 1-9896753 174 3.03687 29 54 0.0157080 54 0.00026 18 55 0.95993" 115 2.00712 86 17s 3.05432 62 55 0.0159989 55 0.00026 66 II 5^ 0.9773844 116 2.02458 19 176 30717795 56 0.0162897 56 0.00027 15 57 0.99483 77 117 2.0420352 177 3.08923 28 57 0.0165806 57 0.00027 63 5« 1.01229 10 118 2.05948 85 178 3.10668 61 58 0.01687 '5 58 0.00028 12 59 60 1.0297443 119 120 2.07694 18 179 180 3-1241394 59 60 0.01716 24 59 60 0.00028 60 1. 047 1 9 76 2.0943951 3.1415927 0.0174533 0.00029 09 DE IGREES. MINUTES. SECONDS. i| 96 V. CONVERSION OF LOGARITHMS. Base of common logarithms = 10. 1 Base of Naperian logarithms (sj3 = 2cos J (a4-/3)cos^ (a — /3). cosa — ct)S/3=— 2sin^(a + /3)sini(a — /3). Fig. sin a sin /3 = ^ cos (a — /3) — ^ cos (a + ^) . cos a cos /3 =r J cos (a — j3) + ^ cos (a + /3) . sin a cos ^ = ^ sin (a + P) + ^ sin (a — /3). sin2 a - sin2 /3 = cos^jS - cos^a = sin (a + /3) sin (a - jS). cos^ a — sin2 j3 = cos'^ /3 — sin^ a = cos (a + /3) cos (a — /3). sin 2 a = 2 sin o cos a. cos 2 o = cos- o — sin^ a. 2 sin2 ^ a = I — cos o tan tan 2 a = 2 tan g I - tan2a i«=±Vr cos a 2 C0S2 ^ o = I -f cos O. sin a I — COS a + COS a I + cos a sin a sin a + sin (o + ;r) + sin (a -f 2 jc) -f ••• + sin (a + nx) _ sin ^ (« 4- I ) jT sin (o + ^ nx) sin i^x cos a f cos {a + x)+ cos (a + 2^)+ ••• + cos (a -f «jir) _sin^ (« + .rcos(a -f ^ «^) ~ sin ^ jr ■\/ — I. ^* = cos X + «■ sin jr. cos X = - (^» -f ^-**) . 2 ^ •" = cosjT — t smx. sin jr = — (JC=-il-. 'J ^ s—a ^ s — b ^ s—c . 1 JL • ^ f2 sin /4 sin i5 Area — \ab%yi\ C = — ' 2 sin C Radius of inscribed circle = r. Diameter of circumscribed circle = \/ s{s- a)(^s-b) {s- c). sin A DIFFERENTIAL FORMULAS FOR PLANE TRIANGLES. i/A + dB + dC = o. — - cot A dA ='^ - cotBc/B = - - cot C dC. a b c da - cos Cdb + cos B dc -{- b sin CdA. a dB = sin Cdb — s'm B dc — b cos CdA. RIGHT SPHERICAL TRIANGLES (C=90°). sin « = sin ^ sin c. sin a = cotB ta.nb. a cos ^ = sin -5 cos a. cos A = tan b cot c. sin ^ = sin ^ sin c. sin <^ = cot .4 tana, cos i9 = sin ^ cos b. cos B = tan a cot r. cos c = cos a cos b = cot -r4 cot B. Fig. 3. 99 OBLIQUE SPHERICAL TRIANGLES. sin a _ sin^ _ sing sin /^ sin .5 sin C cosrt = cos(^ cosr + sin <^sinr cos/^. cos A =— cos ^ cos C 4- sin i9 sin C cos a. sin a cos i9 = cos b sin f — sin b cos r cos /4. sin A cos ?cos(5-^). sin \c s\n^{A — B)— cos ^ C sin i (« — b). sin I i r = tangent of the angular radius of the inscribed small circle. R = tangent of the angular radius of the circumscribed small circle. SPHERICAL EXCESS. E = A + B-\-C- i8o°. sm\a sin^b . s\nlt E = — ; — sin C. cos ^ c tan^^ _ tan ^rt tan^^sin C ~ I + tan ^ rt tan ^ b cos C tan2 J ^ = tan ^ J tan ^ {s - a) tan ^ (j - {z); BINOMIAL THEOREM. I I • 2 I • 2 • 3 EXPONENTIAL THEOREM.* M \ M I 2\ \ M I 3\ \ A/ I 4\ ^ = I + ;c + -^2 + 7^ + — ^ H ^ + ^ + •-. 2 6 24 120 720 * n ! denotes " factorial «," or the product i • 2 • 3 • 4 ••• «. LOGARITHMIC SERIES.* ,o.(...=,o....[.(i)--Q>;(^7-'(,7....|. log(i + x)= A/ {x - ^ x-i + -X^ --Ji-* + -x^ ). 23 4 5 l0g(l - X)=: - M {X -^ ^~X^ -\- ^ X^ + ^X* + - X^ -^ •••)• iVj// 2!^^// 3!V^'^/ 4!\^/ + -. TRIGONOMETRIC SERIES.* t I 3! 5! 7! ;r'^ , x^ x^ , cos ;r=l -\ — h""- 2! 4! 6! 3 15 315 2835 155925 cot X = X x^ x^ •^ 3 45 945 4725 93555 sec^ = I + i;c2 + -^x^ + ^x^ + ^x^ + .... 2 24 720 8064 cosec ;r = - + -;i:+ -^ x^ -\ ^ x^-{- ^x'' + •••. X 6 360 15120 604800 sin-ij = v + -y + -^ vS ^ _5_ y + _35_ y + .... 6 40 112 1152 -^2-^6-^ 40-^ 112-^ 1152-^ tan-ijj/ = J j3 4- _y jj,7 _|_ _y _ .... 3 5 7 9 >' 3r 5r 7r 9r log sin .r= log X - M ( ^-x^ + -^x^ -\- -^x^ + —^ x^ -i- "\ V6 180 2835 37800 / \ogcos X = - A/ ( ~x^ -j- — x^ + —x^ + -^x» + "•]. \2 12 45 2520 / log tan or = log.r + Afl-x^ + ^x^ + -^x^ + -^^ + -V V3 90 2835 18900 I logsin-Jj = log;/ + AI f iy^ + -^-y + Jli j^ + ...). \.6 180 5070 / logtan-i/ = log7-il/f-jj/2-i3y + ^^/ V \3 90 2835 J logsin^ = logtanjc- Afl-tan'^x - - tan* ;ir + i tan^ j; --^tan^jr + ... ). V2 4 6 8 / log tan X = log sin x + A/( - sin^ ;r + - sin* x + - sin^ x + - sin^ j: + •• • ) . \2 4 6 8 / * n ! denotes " factorial «," or the product I • 2 • 3 • 4 ... «. t The angles are expressed in circular measure. DIFFERENTIATION. d {ax + /J) = where « is even. J^ ^ ^ 3 180 1512 ' where P = Lx [^'0 + ;'„ + 2 {yo + _j/4 + . . . + j„_2) ] , ^=: 2Ax[>'i + V3 + ••• +J>'«-l], Fn'" = f{x') when x = abscissa of jn* 2. Simpson's rule : ^ = ^[JO + Jn + 2 (^2 +^4 +••• +JJ'„_2)+ 4(>'l + J3 + - +/«-!)]. 3 3. Weddle's rule (for seven ordinates) : A - ^— ^ [jj'o + 72 + ^'4 + jJ'6 + jJ^3 + 5 (/I 4-/3+ J5)]. 10 4. Prismoidal formula : F = — [^ + /4' + 4 /^m] = - [^ + ^' + 4 ^«]. 3 6 I03 Constants. Base of Naperian logs : ^ = . . Modulus of common logs : log e •=. M ■■ Degrees in arc = radius: 180'^ h- tt = Minutes in arc = radius : Seconds in arc = radius : 360^^ expressed in minutes of arc : . . . 360° expressed in seconds of arc : . . . 24 hours expressed in minutes of time : 24 hours expressed in seconds of time: TT = 314159 26535 89793 23846 logTT =0.49714 98726 94133 85435 sin i" =0.00000 48481 3681 1 07637 arc i" =0.00000 48481 3681 1 09536 . . . . 2.71828 183 . . . . . 0.43429448 . • • . 57°-29577 95i • • .3 437'-74677- • • • 2o6 264".8o6 . . . . 21 600' . I 296000" . I 440"* . 86400* . Eng. inch 0.02540 Eng. foot 0.30480 Eng. yard 0.91440 Eng. statute mile 1.60935 meter 39-3700 meter 3.28083 meter i. 09361 kilometer O.62137 sq. foot 9.29034 sq. inch 6.45163 sq. meter 10.7639 sq. centimeter 0.15500 cubic foot 0.02831 cubic inch 16.3872 cubic meter 35-3145 cubic decimeter (liter) . . . 61.0234 avoirdupois pound 453.59242 avoirdupois ounce 28.34953 Troy ounce 31.10348 grain 64.79892 kilogram 2.20462 kilogram 35.2740 kilogram 32.1507 gram 15-43235 foot-pound 0.13825 kilogram-meter 7.23300 pound per sq. in 70.3067 gram per sq. cm 0.01422 pound per cu. ft 0.0 1 601 grain per cu. in 0.00395 gram per cu. cm 62.4283 gram per cu. cm 252.8925 70 77 639 5 meters meters meters kilometers .... Eng. inches . . . Eng. feet Eng. yards .... Eng. statute miles sq. decimeters . . sq. centimeters . sq. feet sq. inches .... cubic meters . . . cubic centimeters cubic feet .... cubic inches . . . grams . . . . grams . . . . grams . . . . milligrams . . avdp. pounds avdp. ounces Troy ounces . grains . . . . kilogram-meters foot-pounds . . grams per sq. cm. 34 lbs. per sq. in. . . 84 grams per cu. cm. 425 grams per cu. cm. lbs. per cu. ft. . . grains per cu. in. Logarithms. 0.43429 448 9.6377^ 431 1. 75812 263 3.53627 38« 5.31442 513 4-33445 375 6. 1 1 260 500 3.15836249 4-93651 374 0.49714987 .68557487 — 10 -68557 487 - 10 Logarithms. 8.40483 5 — 10 9.48401 6 — 10 9.961137- 10 0.20665 O I-595165 0.515984 0.03886 3 9-79335 o - 10 0.96803 2 0.80966 9 1.031968 9.19033 I - 10 8.45204 7—10 1.214504 1-54795 3 1.785496 2.65666 6 1.452546 1 .49280 9 1.811568 0-34333 4 1.54745 4 1.50719 I 1. 18843 2 9.14068 2—10 0.85931 8 1.84699 7 8.153003 - 10 8.20461 8 — 10 7.597064- lO- 1.795382 2.40293 6 Logarithms. dynes (^jf in meters). dynes (^ in meters) 0.811568 ergs {g'\v\ centimeters) .... 4.140682 ergs (^ in centimeters), ergs per sec. watts (^ in meters) 1.881044 watts (approximately) .... 2.87273 9 32.086 528 + 0.171 293 sin2 — 0.000003 h. in feet (Harkness). = 9-779 886 + 0.052 210 sin^t^ — 0.000003 f^- i" meters (Harkness). / = 39.012 540 -f 0.208 268 sin2 — 0.000000 3 //. in inches (Harkness) = 0.990 910 -f 0.005 290 sin2 — 0.000 000 3 h. in meters (Harkness). Wt. of mass of 1 gram . . loo^ Wt. of mass of I grain . . 6.47989 2g 1 foot-pound 1 3825.5 i" 1 kilogram-meter looooo^ I watt lo'' I horse-power 76.0404^ I horse-power 746 g EXPLANATION OF THE TABLES. INTRODUCTORY. 1. When we have a number with six or more decimal places, and we wish to use only five : {a) If the sixth and following figures of the decimal are less than 5 in the sixth place, they are dropped ; thus, 0.46437 4999 is called 0.46437. (^) If the sixth and following figures of the decimal are greater than 5 in the sixth place, the fifth place is increased by unity and the sixth and following places are dropped ; thus, 0.46437 5001 is called 0.46438. (c) If the sixth figure of the decimal is 5, and if it is followed only by zeros, make the fifth figure the nearest even figure ; thus, 0.46437 500 is called 0.46438, while 0.46438 500 is also called 0.46438. The number is thus increased when the fifth figure is odd, and decreased when it is even, the two operations tending to neutralize each other in a series of computations, and hence to diminish the resultant error. 2. Hence any number obtained according to Art. i may be in error by half a unit in the fifth decimal place. 3. When the last figure of a number in these tables is 5, the number printed is too large, the 5 having been obtained according to Art. i (^) ; if the 5 is without the minus sign, the number printed is too small, the figures following the 5 having been dropped according to Art. i (a). 4. The marginal tables contain the products of the numbers at the top of the columns by i, 2, 3, •••9 tenths, and may be used in multiply- ing and dividing in interpolation. {a) To multiply 38 by .746 : 38 X .7 = = 26.6 38 X .4= 15-2; ••• 38 X .04 = 1.52 38 X .6 = 22.8 ; .-. 38 X .006 = .228 .-. 38 X .746 = 28.348 In multiplying by the second figure (hundredths), the decimal point in the table is moved one place to the left ; in multiplying by the third (thousandths), two to the left ; and so on. 88 I 3-8 2 7.6 3 1 1.4 4 15.2 s 19.0 6 22.8 7 26.6 8 30.4 9 34.2 (105) EXPLANATION OF THE TABLES. {b) To divide 28 by t,^ : Dividend, 28 38 Next less, 26.6 corresponding to •7 I 2 3.8 76 Remainder, I 4 3 11.4 Next less, I 1.4 corresponding to •03 4 5 15.2 19.0 Remainder, 26 6 7 22.8 26.6 Nearest, . Quotient, 26.6 corresponding to .007 .737 8 9 304 34-2 to the nearest third decimal place. The decimal point is moved one place to the right in each remainder, since the next figure in the quotient will be one place farther to the right. To divide 23 by 38 : Dividend, 23 22.8 corresponding to .6 0.0 corresponding to .00 2 o. Nearest, .-. Quotient, I 9.0 corresponding to .005 .605 The computer should use the marginal tables mentally. LOGARITHMS. 5. The logarithm of a number is the exponent of the power to which a given number called the dase must be raised to produce the first number. U A = i 1 tan a ^ =log — T' a' where «' is the number of minutes in the angle. Their values from 0° to 1° 40' (=100') are given at the bottom of pages 3-21 ; from i°4o'to 3° 20' at the left margin of pages 4 and 5, the first three figures being found at the top ; and from 3° to 5° on page 24. Thus, for i'= i' (pages), ^' = 6.46 373, ^' = 6.46 373 ; for 15'=: 15' (pages), ^' = 6.46 372, r' = 6.46 373 ; for 2° 40'= 160' (pages), ^'=6.46357, r'= 6.46404; for 4° 20' =260' (page 24), 6*' = 6.46 331, 7"' = 6.46 456. Each of these numbers should have —10 written after it (Art. 11). X EXPLANATION OF THE TABLES. Note. — The logarithmic cosine of a small angle is found by the ordinary method. The cotangent of an angle is the reciprocal of the tangent, and hence the logarithmic cotangent is the arithmetical complement of the logarithmic tangent. The formulas for finding the logarithmic cosine, tangent, and cotangent of angles near 90° are given on page 25. 37. To find the logarithmic sine or tangent of a small angle. — From Art. 36, we have log sin « = 6"' + log «', log tan « = 7"' + log «'. Hence, to find the logarithmic sine or tangent of an angle less than 5°, find the value of the 6"' or T^ corresponding to the angle, interpolat- ing if necessary, and add it to the logarithm of the number of minutes in the angle. Find log sin o°42'.6. Since the angle is nearer 43' than 42', we take 6"' =6.46 371 log 42.6 = 1.62 941 .*. log sin 0° 42 '.6 = 8.09 312 Find log tan i°53'.2. Since the angle is nearer i°53' (= 113') than 114', we take r' = 6.46 388 log 113.2 = 2.05 385 /. log tani°53'.2 = 8.51 773 Note. — When the angle is given in seconds, either reduce the seconds to deci- mals of a minute, or use the values of 6"" and 7^" given at the bottom of pages 3-23 and on page 24. They are defined by the equations 5"=log^-i^, and 7^" = log^, a" a" where a" is the number of seconds in the angle. Hence log sin a = .S" + log a", and log tan a— T" -\- log a". 38. To find the S7?iall angle corresponding to a given logarithmic sine or tangent. — From Art. 36, log «' = log sin « — .S', ' log a' = log tan a — T\ . or log a' = log sin a + cpl S ', log «' = log tan a + cpl T\ When the angle is less than 3°, find on pages 26-28 the value of cpl 6"' (or cpl 7") corresponding to the function, interpolating if neces- sary, and add it to log sin « (or log tan «) ; the sum will be the loga- rithm of the number of minutes in the angle. In finding the angle whose logarithmic sine is 8.09006, we see from EXPLANATION OF THE TABLES. XI theZ. Sin column (page 26) that the angle is between 0° 42' and o°43', and that the value of cpl S* must be either 3.53628 or 3.53629. The given logarithmic sine is nearer that of 42' than that of 43'; hence we take cpl ^' = 3-53628 log sin a = 8.09006 log a' = 1.62634 .*. «' = 42'.300. When the angle is between 3° and 5°, we may find S' and T' from page 24 after finding the angle approximately from pages 29 and 30. Thus in finding the angle whose logarithmic tangent is 8.77237 we find from page 29 that the angle is between 3° 23' and 3° 24', being nearer 3° 23'. Then on page 24 we have r'=: 6.46423 log tan ot = 8.77237 / .*. log tan a — 7"' = log «'= 2.30814 /. a'= 203'.30 = 3° 23'.3o. ^ -CJL-v/yv -4 \^ Angles Greater than 90°.^* M^ O*"^ ' 39. To find the logarithmic sine, cosine, tangent, or cotangent of an angle greater than 90°, subtract from the given angle the largest multi- ple of 90° contained therein. If this multiple is even, find from the tables the logarithmic sine, cosine, tangent, or cotangent of the remain- ing acute angle. If the multiple is odd, the logarithmic cosine, sine, cotangent, or tangent, respectively, of the remaining acute angle will be the function required ; thus, sin 120° = sin (90° + 30°) = cos 30°. x = I. Quadrant. a II. Quadrant. 90°+a III. Quadrant. iSo'+a IV. Quadrant. 270'+ a %\X\X — + sin a + cosa — sin a — cosa cos.r = + cosa — sin a — cosa + sin a tan X = + tana — cot a + tana — cot a cot X = -f cot a — tana + cot a - tana Or we could find the difference between the angle and 180° or 360°, and find from the tables the same function of the remaining acute angle ; thus, cos 300° = cos (360° — 60°) = cos 60°, etc. x = I. Quadrant. a II. Quadrant. 180° -a III. Quadrant. i8o'+tt IV. Quadrant. 360*— a or -a sin X = COSJf = tan X = cot X = + sin a + cosa + tana + cot a + sin a — cosa — tan a — cot a — sin a — cosa 4- tana + cot a — sin a + cosa — tan a — coto To indicate that the trigonometric function is negative, n is written after its logarithm. xii EXPLANATION OF THE TABLES. 40. To find the angle corresponding to a given function, find the acute angle a corresponding thereto, and the required angle will be «, 1 80° ± «, or 360° — a, according to the quadrant in which the angle should be placed. 41. There are always two angles less than 360° corresponding to any given function. Hence there will be ambiguity in the result unless some condition is known that will fix the angle ; thus, if the sine is positive, the angle may be in either of the first two quadrants, but if we also know that the cosine is negative, the angle must be in the second quadrant. Given One Function of an Angle, to fijid Another without Jindifig the Angle. 42. Suppose log tan a = 9.79361, and log cos a is sought. On page 57 the tabular difference for log tan « is 28, and that for log cos « is 8, the given logarithmic tangent exceeding 9.79354 by 7, Hence 28 : 7 = 8 : a: ; .*• jc = 2^ X 7 = 2 = correction to 9.92905, giving log cos a = 9.92903. In the margin are tables to facilitate the process. In the column headed ^V, the numerator is the tabular difference for the column headed Z. Cos, and the denominator that for the logarithmic tangents. The numbers in the marginal column are the values of A, — the excess of log tan a over the next smaller logarithmic tangent, found in the tables, — such that ^^^A shall be 0.5, 1.5, 2.5, etc. ; and the numbers on the left are the corrections x to be appHed to the numbers in the column headed L. Cos. Thus, if A is between 1.8 and 5.2, x is between 0.5 and 1.5, and is called i. Note that i is opposite the space between 1.8 and 5.2. In the example above, the excess A is 7, which lies between 5.2 and 8.8 ; hence x is 2, the number on the left opposite the space between 5.2 and 8.S. For example, if we have given the logarithms of the sides of a right-angled triangle, \oga = 2.98227 and log /^ = 2.90255, to find the hypotenuse, we use the formulas a /T /> tan a = -, and e = b sin a cos « The value of log tan a being found in \oga = 2.98227 (t) the column marked Z. Tan at the bot- logsin« =9.88571 (4) torn, the right column will contain the log/!' = 2.90255 (2) logarithmic sine of the corresponding log tan « = 0.07972 (3) angle. Also, the correction to 9.88563 .-. logm'^-// s^. EXPLANATION OF THE TABLES. — ^j; J^-\] TABLE V. 45. Conversion of common logarithms into Naperian^ and vice versd (page 95) . — We have \Q%y^N=M\o%,N, and \og, N = ^\og,^ N . Table V. contains the multiples of M and — by numbers from I to 100. Find the common logarithm of 2, its Naperian logarithm being 0.69314 718056. J/ X .69 =0.2996631925 J/ X .0031 = .001346312894 J/ X .000047 = .000020411841 J/ X .00000018 = .000000078173 M X .0000000005 = .00000 00002 1 7 M X .00000000006 = .oooQo 00000 26 /. Iogio2 =0.301029995651 (True value = 0.30102 99957) Find the Naperian logarithm of 0.2, its common logarithm being 9.30102 99957 — 10. Hence the true logarithm is logio 0.2 = — I H- .30102 9995 -] = — 0.69897 00043. = 1.58878 37142 = .02049 30073 28 X .000070 = .00016 1 1809 57 X .0000000043 = .00000 00099 01 /. log^ 0.2 = — 1.60943 79123 86 (True value = — 1.60943 79124) — X .0080 M M I 'm. Typography by J, S. Gushing & Co., Norwood, Mass. ? C»S^ \ Vo -r e^i^rw "^ (^.Je 3 '^^ \ y K •>. (^ (^' (L u- /■ O- *»^ A i-JTS UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine a' o OECJ 19521.1' 3Har'55JP fiiho{,Jil^^lM h\ 10ct'55KC SfP 17 1955 LU REC'D LD MAY 3 195a REC'D LD JAN 16 1959 2Feb'59MJ RECD LD JUL 1 6 1959 5sl6)4120 YC 76228