IN MEMORIAM FLORIAN CAJOR1 MxL/L^J PLANE AND SPHERICAL TRIGONOMETRY BY LEVI L. CONANT, Pii.D. ? PROFESSOR OF MATHEMATICS IN THE WORCESTER POLYTECHNIC INSTITUTE NEW YORK :. CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1909, BY LEVI L. CONANT. ENTERED AT STATIONERS' HALL, LONDON. CAJOR1 PREFACE IN this work the author has attempted to produce a text- book which should present in a concise and yet thorough manner an adequate treatment of both the theoretical and the practical sides of elementary trigonometry. The material here presented has been gathered and tested during the course of many years of experience in the class room, and the arrange- ment and method of presentation are the result of numerous experiments made for the purpose of ascertaining what could be done most effectively in the limited time usually devoted to this subject. The problems given in connection with the different cases under the solution of triangles are nearly all new, and are well graded and sufficiently numerous to give the student ample preparation for the various problems that arise in plane sur- veying and in elementary astronomical and geodetic work. That portion of the book which treats of theoretical trigo- nometry has been written in the attempt to present this aspect of the subject in the simplest and clearest manner, and at the same time with the design of equipping the student for the more advanced work in pure and applied mathematics which is pursued in the later years of his college course. The best English, French, and Italian text-books have been consulted, as well as those published in this country. For assistance in the preparation of the work thanks are due to my colleague, Professor Arthur D. Butterfield, to Professor W. B. Fite of Cornell University, to Professor O. S. Stetson of Syra- cuse University, to Mr. C. G. Brown, head of the department of mathematics in the Englewood, New Jersey, High School, and to Mr. J. A. Bollard, instructor in mathematics in the Worcester Polytechnic Institute. LEVI L. CONANT. WORCESTER POLYTECHNIC INSTITUTE, WORCESTER, MASS. 918256 ENGINEER'S TRANSIT, WITH GRADIENTEK 4 CONTENTS PLANE TRIGONOMETRY CHAPTER PAGES I. THE MEASUREMENT OF ANGULAR MAGNITUDE . . 7-19 II. TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE . 20-30 III. VALUES OF THE FUNCTIONS OF CERTAIN USEFUL ANGLES 31-35 IV. THE RIGHT TRIANGLE 36-50 V. THE APPLICATION OF ALGEBRAIC SIGNS TO TRIGO- NOMETRY . . . . . . . . . 51-73 VI. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE . . 74-84 VII. GENERAL EXPRESSION FOR ALL ANGLES HAVING A GIVEN TRIGONOMETRIC FUNCTION .... 85-91 VIII. RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS OF Two OR MORE ANGLES ...... 92-105 IX. FUNCTIONS OF MULTIPLE AND SUBMULTIPLE ANGLES . 106-113 X. INVERSE TRIGONOMETRIC FUNCTIONS .... 114-121 XI. THE GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 122-130 XII. THE OBLIQUE TRIANGLE 131-155 XIII. MISCELLANEOUS PROBLEMS IN HEIGHTS AND DISTANCES 156-165 XIV. FUNCTIONS OF VERY SMALL ANGLES HYPERBOLIC FUNCTIONS TRIGONOMETRIC ELIMINATION . . 166-175 SPHERICAL TRIGONOMETRY XV. GENERAL THEOREMS AND FORMULAS .... 177-193 XVI. SOLUTION OF SPHERICAL TRIANGLES 194-213 GREEK ALPHABET Greek is written with the following twenty-four letters : FORM NAME LATIN EQUIVALENT A a alpha a B 13 beta b r 7 gamma g A 8 delta d E epsilon e Z r zeta z H g eta e @ > <#> phi P h X % chi ch i/r psi ps n CO omega o PLANE TRIGONOMETRY CHAPTER I THE MEASUREMENT OF ANGULAR MAGNITUDE 1. The size and shape of a plane triangle can be completely determined when any three of its six parts are known, provided at least one of the known parts is a side. By means of certain ratios called trigonometric functions, which will be defined later, trigonometry enables us to investi- gate and to determine the unknown parts and the area of a tri- angle when any three of the parts are known, provided at least one of the known parts is a side. Hence, in its most elemen- tary sense, Trigonometry is that branch of mathematics which treats of the solution of triangles. During the past two centuries the sense in which the word " trigonometry " is used has been greatly extended, and it is now understood to include the general sub- ject of mathematical investigation by means of trigonometric functions. Plane trigonometry treats of plane triangles, and of plane angles and their functions. 2. Angles. In its geometric sense the word " angle " is defined as the difference in direction of two intersecting lines. In trigonometry, however, this word receives an extension of meaning, which must be fully understood at the outset. Suppose two straight lines, OA and OB, are drawn from the point in such a manner that they very nearly coincide. Let one of the lines, OA, remain fixed in position, while the other, OB, revolves on the point as a pivot. We are now free to revolve OB, either back into actual coincidence with OA, or 7 8 PLANE TRIGONOMETRY forward, so as to enlarge the opening between the lines. At any point of the revolution the angle AOB may be said to have been formed, or generated, by the revolution of the line OB. In plane geometry angles greater than 180 are seldom em- ployed, but in trigonometry the freest possible use is made of such angles. Trigonometry even considers angles greater than 360, meaning by an angle of that magnitude merely the amount of revolution that has been performed by the moving or generating line. As an illustration of the meaning of the word " angle " used in this sense, consider the movement of one of the hands of the clock. Let the minute hand start from the position it occupies at noon. In fifteen minutes it will move over or generate an angle of 90 ; in thirty minutes an angle of 180 ; in forty-five minutes an angle of 270 ; and in one hour an angle of 360. Continuing, we may say that in two hours the minute hand will move over an angle of 720, in three hours an angle of 1080, in four hours an angle of 1440, in n hours an angle of rax 360, etc. Again, suppose a runner to be competing in a two-mile race on a circular track a quarter of a mile in length. If we sup- pose a line to be drawn connecting the position of the runner with the center of the circle formed by the track, the position of the runner both on the track and in the race can be described at any instant with perfect accuracy by giving the magnitude of the angle through which this line has revolved since the beginning of the race. Thus, when the line has revolved through an angle, and hence the runner has traversed an arc, of 180, he has completed one eighth of a mile ; when he has traversed an arc of 360, he has THE MEASUREMENT OF ANGULAR MAGNITUDE !> completed one fourth of a mile ; and when he has finished the race, he has run around the track eight times. In other words, when he has finished the race the line that connects him with the center of the track has revolved through an angle of 8 x 360, or 2880. During this time the runner has traversed an arc of the same magnitude, i.e. of 2880. It is at once seen that an idea is here introduced which is an extension of the idea of the angle as it is ordinarily used in geometry. This idea, which is fundamental in all work in trigonometry involving angles, is the idea of formation or generation in connection with the angle. Evidently a defini- tion of this word is required which differs from that to which the student has become accustomed in geometry ; and in the extended sense here used, the term "angle " may be defined as follows : An angle is that relation of two lines which is measured by the amount of revolution necessary to make one coincide with the other. 3. The point about which the generating line revolves is called the origin. The generating line is called the radius vector. The line with which the radius vector coincides when in its original position is called the initial line ; and the line with which it coincides when in its final position is called the terminal line. 4. Positive and negative angles. It is convenient, and often necessary, to know not only the size of an angle, but also the direction in which the radius vector has moved while generating the angle. For this reason it is customary to speak of angles as being either positive or negative. If the radius vector moves in a direction opposite to that of the hands of a watch when the face of the watch is toward the observer, the angle it generates is said to be positive. The motion of the radius vector as it generates the angle is then said to be counter- clockivise. If the radius vector moves in the same direction as the hands of a watch when the face of the watch is toward the observer, the angle it generates is said to be negative. The motion of the radius vector is then called clockwise. 10 PLANE TRIGONOMETRY The angles AOB 1 and AOB 2 are positive angles, and the angles AOB 3 and AOB 4 are negative angles. The initial line in each case is OA, and the terminal lines are OB V OB^ OB S , OB^ respectively. The direction of rotation for each angle is indicated by the arrowhead. . 5. Angles are often described by referring them to some position with reference to two intersecting lines, at right angles to each other, of which one is horizontal and the other vertical. It is customary to regard the horizontal line extending toward the right as the initial line for all angles, when nothing is said to the contrary. If the radius vector, as shown in the figure, occupies any po- sition between OX and OY, then the angle XOB l is said to be in the first quadrant. If the radius vector is between OY and OX', the angle XOB 2 is said to be in the second quad- rant. Similarly, XOB 3 is said to be in the third quadrant, and XOB in the fourth quadrant. These expressions only mean, of course, that the terminal lines lie in the first, second, third, and fourth quadrants respectively. 6. In practical work the unit of measure that is always em- ployed in dealing with angular magnitudes is the right angle or some fraction of the right angle. This unit is chosen because: (i) The right angle is a constant angle. (ii) It is easy to draw or to construct in a practical manner, (iii) It is the most familiar of all angles, entering as it does most frequently into the practical uses of life. THE MEASUREMENT OF ANGULAR MAGNITUDE 11 In geometry the right angle is the unit universally used. In trigonometry two systems of measurement, involving the use of two different units, are in common use. 7. The sexagesimal system. In this system the unit of measure is the right angle. The right angle is divided into 90 equal parts, called degrees; each degree is divided into 60 equal parts, called minutes; and each minute is divided into 60 equal parts, called seconds. The symbols 1, 1', 1", are employed to denote one degree, one minute, and one second respectively. 60 seconds (60") = one minute. 60 minutes (60') = one degree. 90 degrees (90) = one right angle. This system is almost universally employed where numerical measurements are to be made. It is, however, inconvenient because of the multipliers, 60 and 90, which it introduces into computations. Another system, called the centesimal system, was proposed in France a little over a century ago. In this system the right angle is divided into 100 equal parts called grades, the grade is divided into 100 equal parts called minutes, and the minute is divided into 100 equal parts called seconds. The centesimal system has been used to some extent in France, but its use has never been looked upon with favor in other countries. If its use were to become general, an enormous amount of labor would have to be expended in the re-computa- tion of existing tables. For this reason the centesimal system, in spite of its intrinsic advantage over the sexagesimal system, will probably never come into general use. EXERCISE I Express the following angles in terms of a right angle : 1. 30. 3. 68 14'. 5. 228 46'. 2. 120. 4. 114 38' 12". 6. 321 14' 22". 7. The angles of a right triangle are in arithmetical progres- sion, and the greatest angle is three times the least ; what is the number of degrees in each angle ? 12 PLANE TRIGONOMETRY Show by a figure the position of the revolving line when it has generated each of the following angles : 8. | rt. angle. 11. 2^ rt. angles. 14. -150. 9. - 1 rt. angle. 12. 4| rt. angles. 15. 275. 10. -1| rt. angles. 13. 17| rt. angles. 16. 1225. 17. The angles of a triangle are such that the first contains a certain number of degrees, the second 10 times as many min- utes, and the third 120 times as many seconds ; find each angle. 18. How many degrees are passed over by each of the hands of a watch in one hour ? Represent by a figure each of the following angular magni- tudes : 19. l|-+2^ rt. angles. 23. 4 rt. angles. 20. 2| 1^ rt. angles. 24. 4n rt. angles (n integral). 21. - 4 rt. angles. 25. (4 n -f 1) rt. angles. 22. 6^ rt. angles. 26. (4 n 2) rt. angles. 8. Circular measure. Another system for the measurement of angles has, in modern times, come into vogue. It is exten- sively used in work connected with higher branches of mathematics, and is the almost universal unit employed in theoretical investigations. The unit of circular measure is the radian, which is obtained as follows : On the circumference of a circle lay off an arc, AB, equal in length to the radius of the circle, OA. The angle AOB is called a radian. Accordingly: A radian is an angle at the center of a Circle, subtended by an arc equal in length to the radius of the circle. In order to use the radian as a unit of measure, it is necessary to prove that it is a constant angle ; or, in other words, it is necessary to prove that the magnitude of the radian is the same for all circles. THE MEASUREMENT OF ANGULAR MAGNITUDE 13 9. THEOREM. The radian is a constant angle. By definition the radian is measured by an arc equal in length to the radius. Also, An angle of two right angles is measured by an arc equal to one half the circumference. Therefore, since angles at the center of a circle are to each other as the arcs by which they are subtended (Geom.), a radian radius R 1 = = 2 rt. angles semi-circumference irR TT .-. a radian = ~ of 2 right angles = 1 x 180 = 57.2958 = 57 17' 44. 8" nearly. Therefore the radian is a constant angle. 10. The reason for the use of this unit may now be readily understood. Since 1 radian = ?_L^, 7T .. TT radians = 2 rt, A = 180. Similarly, - radians = 1 rt. Z = 90. - radians = | rt. Z = 30. 6 radians = 60. o 1 TT radians = 120. f TT radians = 270. 2 TT radians = 4 rt. A = 360. 5 TT radians = 10 rt. A = 900. 18 TT radians = 36 rt. A = 3240. This gives a method for the expression of the value of an angle that is often far more convenient than that furnished by the sexagesimal system. It is especially useful in dealing with angles of great magnitude, and it greatly simplifies many of the investigations and formulas of trigonometry. 14 FLAKE TRIGONOMETRY 11. The symbol r is often used as the symbol to denote radians. Thus, 6 r would stand for 6 radians, O r for 6 radians, 7r r for TT radians, etc. When the value of the angle is expressed in terms of TT, and when the unit is the radian, it is customary to omit the r and to give the value of the angle in terms of TT alone, the r being understood. Thus, when referring to angular magnitude, TT means TT radians, ~ means radians, 6 TT means 6 TT radians, 2 etc. When the word "radians" is omitted, the student should mentally supply it, or he may readily fall into the error of sup- posing that TT alone means 180. The value of TT is the same here as in geometry, i.e. 3.14159. Neither TT nor any multiple of TT can by itself ever denote an angle. It simply tells how many radians the angle contains. Too great care cannot be exercised in keeping this distinction clear. 12. To find the number of degrees in an angle containing a given number of radians, and vice versa. Since 180 = IT radians, 1 = of a radian, 180 180 , , and l r = ol a decree. Hence, To convert radians into degrees, multiply the number of radians i 180 % ' To convert degrees into radians, multiply the number of degrees by -E-. ' y 180 EXERCISE II 1. How many degrees are there in 3 radians ? = 3 x = = m.89 nearly 7T 7T = 171 53' 24" nearly. THE MEASUREMENT OF ANGULAR MAGNITUDE 15 2. How many radians are there in 113 15' ? 113 15' = 113.25. Since 1 = ^, 180 113.25 = 11 3.25 x -^ 180 _ 113.25 x 3.14159 180 = 1.976 + radians. Express in degrees, minutes, and seconds the following angles: 5. ^- 7. ^- 9. 3?r r . Express in radians the following angles : 11. 45. 14. 225. 17. 286 38'. 20. A. Q0 12. 120. 15. 60 30'. 18. 684 26'. 21. . 7T 13. 135. 16. 115 45'. 19. n. 22. 78.126. 23. The difference between two acute angles of a right tri- angle is ^ radians; find the value of each of the angles in degrees. 5 24. If one of the angles of a triangle is 56 and a second angle is ^-^ radians, find the value of the third angle. 5 , 25. The angles of a triangle are in A. P., and the smallest is an angle of 36 ; find the value of each in radians. 26. The value of the angles of a triangle are in A. P., and the number of degrees in the least is to the number of radians in the greatest as 60 : TT ; find each angle in degrees. 27. The value of one of the interior angles of one regular polygon is to the value of one of the interior angles of another regular polygon as 3 : 4, and the number of sides in the first is to the number of sides in the second as 2 . 3 ; find the number of sides in each. 16 PLANE TRIGONOMETRY 28. Find the number of radians in one of the interior angles of a regular pentagon ; a regular heptagon ; a regular nonagon. 29. The angles of a triangle are in A. P., and the number of radians in the least angle is to the number of degrees in the mean angle as 1:120; find the value of each angle in radians. 30. The angles of a quadrilateral are in A. P., and the greatest is double the least; find the value of each angle in radians. 31. Express in degrees and in radians the angle between the hour hand and the minute hand of a clock at (1) five o'clock ; (2) quarter-past nine ; (3) half -past ten. 32. At what time between four and five o'clock are the hour and the minute hands of a clock 90 apart ? At what time are they 180 apart ? 13. THEOREM. The circular measure of an angle whose vertex is at the center of a circle is the ratio of its intercepted arc to the radius of the circle. By geometry, arc AC Z.AOB arc A B a radius' arc AC ' X/ _ AOB radius arc A : x a radian. radius Hence, the number of radians in any angle is found by dividing the arc which subtends that angle by the radius of the circle. The formula just obtained is often expressed in the following convenient, though somewhat incorrect, form : arc = angle x radius. (1) The meaning of this formula is, that the length of any arc of a circle is equal to the length of the radius of the circle multiplied by the number of radians in the angle subtended by the arc. THE MEASUREMENT OF ANGULAR MAGNITUDE 17 EXERCISE III 1. Find in degrees the angle subtended at the center of a circle whose radius is 30 ft. by an arc whose length is 46 ft. 6 in. In this circle the arc which subtends an angle equal to a radian is 30 ft. in length, and the required angle is subtended by an arc whose length is 46.5 ft. ,. ^radians = *?? x i* = 88.8'. Ans. 30 300 TT 2. In a circle whose radius is 8 ft., what is the length of the arc subtended by an angle at the center, of 26 38' ? Let x = the length of the required arc. ber of radians in 26 . (See Art. 13.) Then, - = the number of radians in 26 38' 8 x = 3.72 ft. nearly. 3. In running at a uniform speed on a circular track, a man traverses in one minute an arc which subtends at the center of the track an angle of 3| radians. If each lap is 880 yd., how long does it take him to run a mile ? Let x the number of yards traversed during each minute. Then, x = 3^ x R. (See Art. 13.) 99 = x 140 = 440 yards. Since Y& = 4, therefore he can run a mile in 4 min. 4. The radius of a circle is 15 ft. ; find the number of radians in an angle at the center subtended by an arc of 26^ ft. 5. The radius of a circle is 32 ft.; find the number of degrees in a central angle subtended by an arc of 5 TT ft. 6. The fly wheel of an engine makes 3 revolutions per second ; how long will it take it to turn through 5 radians ? 7. The minute hand of a tower clock is 2 ft. 4 in. long ; through how many inches does its extremity move in half an hour ? CONANT'S TRIG. 2 18 PLANE TKIGONOMETRY 8. A horse is picketed to a stake ; how long must the rope be to enable the horse to graze over an arc of 104.72 yd., the angular measurement of this distance being 150 ? 9. What is the difference between the latitude of two places, one of which is 150 mi. north of the other, the radius of the earth being reckoned as 4000 mi. ? 10. The angle subtended by the sun's diameter at the eye of an observer is 32' ; find approximately the diameter of the sun, if its distance from the observer is 92,500,000 mi. NOTE. In this example the diameter of the sun, which is really the chord of an arc of which the observer's eye is the center, may be regarded as coinciding with the arc which it subtends. 11. Calling the earth a sphere, and the arc of a great circle on its surface subtended by an angle of 1 at the center 69 J mi., what is the radius of the earth ? 12. A railway train is traveling at the rate of 60 mi. an hour on a circular arc of two thirds of a mile radius ; through what angle does it turn in 10 sec. ? 13. The radius of a circle is 3 m.; find approximately, in radians, the arc subtended by a chord whose length 'is also 3 m. 14. How many seconds are there in an angle at the center of a circle subtended by an arc one mile in length, the radius of the circle being 4000 mi. ? 15. In the circle of Ex. 14, what is the length of an arc that subtends an angle of 3' at the center? 16. What is the ratio of the radii of two circles, if the semi- circumference of the greater is equal in length to an arc of the smaller which subtends an angle of 225 at the center? 17. If an arc 1.309 m. long subtends at the center of a circle whose radius is 10 m. an angle of 7 30', what is the ratio of the circumference of a circle to its diameter ? 18. The circumference of a circle is divided into four parts which are in A. P., and the greatest part is twice the least ; find the number of radians in the central angle subtended by each of the respective arcs into which the circumference is divided. THE MEASUREMENT OF ANGULAR MAGNITUDE 19 19. The diameter of a circle is 80 m., and an arc whose length is 15.75 m. subtends a central angle of 22 30'; find the value of TT to four decimal places. 20. How many radians are there in a central angle subtended by an arc of 20" ? 21. The semicircumference of a certain circle is equal to its diameter plus a given arc ; find the central angle subtended by that arc. 22. Find the radius of a globe such that the distance of 3 in. on its surface, measured on an arc of a great circle, may subtend at the center an angle of 1 45'. 23. At what distance does a telegraph pole, 24 ft. high, sub- tend an angle of 10', the eye of the observer being on the same level as the foot of the pole ? NOTE. The suggestion made in connection with Ex. 10 applies to this problem also. When a chord and its arc differ but little from each other it is often convenient to use the arc in place of the chord. 24. At what distance will a church spire 100 ft. high subtend an angle of 9', the angle being measured from the level on which the church stands? 25. The difference between two angles is - - radians, and y their sum is 76 ; what is the value of each of the angles ? 26. If an incline rises 5 ft. in 300 ft., find the angle it makes with its projection on the horizontal plane. 27. How many radians are there in an angle of a? 28. How many radians are there in an angle of 10" ? CHAPTER II TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 14. In the present chapter only acute angles will be con- sidered. In Chapter V the definitions here given will be extended to angles of any magnitude. Let any line having a given initial position OA begin to revolve on as a pivot, in a direction opposite to the direction A ' in which the hands of a clock move. Let the angle which it generates be the acute angle A OA' . A From any point in either side of the angle, asP in the side OA', let fall a perpendicular PM to the other side of the angle. The trigonometric functions, or ratios, of the angle AOA are then denned as follows : The sine of the angle A OA is the ratio ^ = side opposite OP hypotenuse The cosine of the angle AOA' is the ratio M = side adjacent OP hypotenuse The tangent of the angle A OA is the ratio = e opposite OM side adjacent The cotangent of the angled OA is the ratio ^ = side adjacent MP side opposite The secant of the angle AOA f is the ratio = hypotenuse > OM. side adjacent The cosecant of the angle A OA is the ratio QL = h .Ypotenuse MP side opposite 20 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 21 In addition to these there are two other functions, less frequently used, versed sine of A OA' = 1 cosine of AOA', coversed sine of A OA 1 = 1 sine of A OA'. In writing, it is customary to abbreviate the words " sine," "cosine," "tangent," etc., and to express the functions of any given angle, a?, as follows : sin x, cos x, tan a?, cot a?, sec #, esc #, vers a?, covers x. It should be noted at the very beginning that these functions are mere numbers, and their values can be expressed numerically whenever the angle to which they belong is known. Thus, sin x may equal J, J, or any other proper fraction ; tan x may equal 2, 5, 18, or any other real number whatever. The expression sin a;, for example, is a single symbol, and is to be regarded as the name of the number which expresses the value of the particular ratio in question. The expressions sin, cos, etc., have no meaning unless some angle is asso- ciated with them. 15. The trigonometric functions are always constant for the same angle. From any points in either side of the angle x, as A, A f , A", drop perpendiculars AB, A'B', A"B" to the other side. Then, by geometry, the triangles A OB, A' OB 1 , A" OB" are similar, and their homologous sides are proportional. Therefore, A BA_B'A' _B"A" _ OA~ OA' = '' OA" = OB OB' OB" OA OA' OA" = cos a:, U B" A B and similarly for the other functions. Hence, the value of any function of x remains unchanged as long as the value of the angle itself remains unchanged. Any increase or decrease in the size of the angle pro- duces a change in the value of the function, or ratio. From this it is readily seen why these ratios are called functions of the angle. 22 PLANE TRIGONOMETRY From the -last paragraph the following important results may now be stated : (1) To every acute angle there corresponds one and only one value of each trigonometric function. (2) Two unequal acute angles have different trigonometric functions. (3) To each value of any trigonometric function there is but one corresponding acute angle. 16. Fundamental relations between the trigonometric func- tions of an acute angle. From the definitions given in Art. 14 it follows immediately that the sine of the angle x is the recip- rocal of the cosecant x ; also that cosine x is the reciprocal of secant #, and that tangent x is the reciprocal of cotangent x. That is, . cscrr' cos# = , or cos # sec # = 1, sec # tan x = , or tan ^ cot a: = 1. cot x Also, it follows from the definitions that sin x T cos a: tan#=- , and cot x cosx sin a; In the right triangle ABC, a 2 + b 2 = c 2 . Therefore, c , and _ = ! + a 2 a 2 From these equations it follows that ^ sin 2 a? + cos 2 x - 1 , (3) sec 2 x 1 + tan 2 x, (4) esc-' JT, = 1 + cot 2 x. (5) TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 23 17. From the definitions of the trigonometric functions, p. 20, it follows that in any right triangle any function of either of the acute angles is equal to the corresponding co- function of the other -acute angle. For example, sinJ. = -, and cos=-. .-. sin A cos B= cos(90 A}. G c Similarly, cos A = sin B = sin (90 J.), tan A = cot B = cot (90 - J.), cot A = tan B = tan (90 - A), sec A = esc B = esc (90 - A), esc A = sec B = sec (90 - A), vers A = covers B = covers (90 A), covers A = vers B = vers (90 A). Hence, Any function of an acute angle is equal to the corresponding co- function of its complement. The meaning of the prefix co, in cosine, cotangent, cosecant, and coversed sine appears from the above. The cosine of an angle is the complement-sine, i.e. the sine of the complement of that angle: the tangent of an angle is the cotangent of its com- plementary angle; and a similar statement may be made for the secant and for the versed sine of an angle. ORAL EXERCISES Prove the following relations : 1. sin A cot A = cos A. SOLUTION. Using only the left number of the equation, we proceed as follows : ___ A sin A cot A = sin A -. (Art. 16, (2).) sin A. = cos A . .*. sin A cot A = cos A . 2. cos A tan A = sin A. 3. (sec A tan A) (sec A + tan A) = 1. 4. (esc A cot ^4)(csc A -f- cot A) = 1. 5. (tan A + cot A) sin A cos ^4 = 1. 24 PLANE TRIGONOMETRY 6. (tan A cot A) sin A cos A = sin 2 A cos 2 A. 7. sin 2 -*- esc 2 6 = sin 4 0. 8. sin 4 0- cos 4 0=sin 2 0-cos 2 0. 9. (sin - cos 0) 2 = 1 - 2 sin cos 0. 10. (sin - cos 0) 2 + (sin + cos 0) 2 = 2. 11. sec cot = esc 0. 12. (tan + cot 0) 2 = sec 2 + esc 2 (9. 13. cot 2 9 cos 2 = cot 2 - co., 2 0. 14. sin 2 + esc 2 6 + 2 = (sin (9 + esc 0) 2 . 15. vers 6 (1 + cos 0) = sin 2 0. 16. sin 2 (9 + vers 2 (9 = 2(1- cos (9). 17. sec sin tan = cos 0. 18. esc 6 cos cot 6 = sin 0. 19. sec 2 (9 - tan 2 = sin 2 + cos 2 0. 20. esc 2 (9 -cot 2 = sin 2 (9 + cos 2 0. EXERCISE IV Prove the following identities : 1. cos 4 d - sin 4 = 2 cos 2 6-1. SOLUTION. Using only the left member of the equation, we proceed as = (1) (cos 2 B - sin 2 0) = cos 2 - (1 - cos 2 0) (Art. 16, (3).) = 2cos 2 0- 1. 2. sin 3 + cos 3 = (sin + cos 0)(1 - sin cos 0). 3 smA +1+008^2080^. 1 + cos A sin A 4. (1 + sin a + cos a) 2 = 2(1 + sin a)(l + cos a). 5. ( COS 3 - sin 3 0) = (cos - sin 0)(1 + sin cos 0). 6. cos 2 /3 (sec 2 /3- 2 sin 2 /3) = cos 4 /3 + sin 4 /3. .< t TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 25 sin (S 1 4- sin /3 9 ,., 7. :: :~ 5 + H = sec 2 ff (csc/3 + 1). 1 sin /8 sin 8. tan a + tan /3 = tan a tan /:? (cot a + cot /3). 9. cot a 4- tan /3 = cot a tan /3 (tan a + cot /3). 10. cos 6 a 4- sin 6 a = 1 3 sin 2 a cos 2 a. /I sin A 11. \/ 7 = sec A tan A. *1 4- sin A 12. sin 2 <9 tan 2 04-cob 2 cot 2 = tan 2 4- cot 2 <9- 1. 13 . CSC / + CSC / =2 sec 2 A. esc JL 1 esc J. -h 1 esc A 14. = cos ^L. cot A + tan A 15. (1 sin a cos ) 2 (1 + sin a + cos a) 2 = 4 sin 2 a cos 2 a. 16. (sec A H- cos A) (sec A cos J.) = tan 2 A + sin 2 ^. 17. = sin A cos A. cot ^4. H- tan A 1 tan A _ cot J. 1 1 + tan A ccff-A+1' 19. sjn 3 J. cos J. + cos 3 A sin ^4 = sin A cos A. * 20. sin 2 J. cos 2 ^L + cos 4 A = 1 sin 2 A. esc a sec a cot a tan a cot a -f- tail a esc a + sec a 1 + tan 2 A sin 2 A 21. 22. 23. sec A-tauA l _ 2sQGA tan A + 2 tan2 ^ sec J. 4- tan ^4. 24. tan 2 a sec 2 a 4- cot 2 a esc 2 a = sec 4 a esc 4 a 3 sec 2 a esc 2 a. tan .A cot A 25. T + ^ = sec ^4. esc y 4- 1. 1 cot A 1 tan ^1 cos A sin ^4. ,, , ,, 26. - = sin A + cos A. 1 tan A 1 cot A 26 PLANE TRIGONOMETRY 27. COt 4 A + COt 2 A = CSC 4 A CSC 2 A. 28. Vcsc 2 A I = cos A esc ^4. 29. tan 2 ^4 - sin 2 A = sin 4 .A sec 2 A. 30. (1 + cot^4. csc A) (I + tan J. + sec A) =2. 1 11 1 31. 32. csc A cot A sin .A sin A csc .A + cot A cot .A cos A. cot A cos vl cot A + cos A cot A cos ^4. 33. 2 - vers 2 = sin 2 0+2 cos 0. 34. sin 8 A cos 8 .A = (sin 2 A cos 2 A)(l 2 sin 2 A cos 2 A). cos ^1 esc A - sin A sec ^ 3g cos ^ + sin ^. tan A + sec ^4. 1 _ 1 + sin A tan A sec A + 1 c< >s A 37. (tan a + csc ) 2 (cot fi sec a) 2 = 2 tan a cot /3(csc a.+ sec /3) 38. 2 sec 2 a sec 4 a 2 csc 2 a + csc 4 a = cot 4 a tan 4 Tx. 39. (sin a + csc a) 2 + (cos a + sec a) 2 = tan 2 a + cot 2 a -f 7. v x . sec A csc, A 40. (1 + cot A + tan A) (sin J. cos A)= - csc 2 J. sec 2 A 2 41. 2 vers^l -f cos 2 J. = 1 + vers 42. S6C x ~ tan * = 1 + 2 tan x (tan g - sec x). sec a; + tan x 2 sin 6 cos cos 6 43. - - - - r = COL (7. 44. (sin a cos /3 + cosa sin /3) 2 4- (cos a cos /S sin a sin/^) 2 =l 45. (ta,^ + sec^ = l^. 1 sin 6 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 27 18. Limits of the values of the trigonometric functions of an acute angle. Since sin 2 A + cos 2 A = 1, and since each term, being a square, is positive, neither sin 2 A nor cos 2 A can be greater than unity. Hence, neither sin A nor cos^l can be numerically greater than unity. Since esc A is the reciprocal of sin A, and sec A is the recip- rocal of cos A, both sec A and esc A can have any values numeri- cally greater than unity, but neither can ever be numerically less than unity. Since sec 2 A=\ + tan 2 J., tan A = Vsec 2 ^. 1. Hence, tan A can have any value between and oo. And since cot A is the reciprocal of tan A, therefore cot A can have any value between oo and 0. These results are summarized as follows : When A is an acute angle, sin^L can take any value between and + 1, cos A can take any value between -f- 1 and 0, tan A can take any value between and +00, cot A can take any value between + GO and 0, sec A can take any value between -f 1 and -foo, esc A can take any value between +00 and + 1. These results also follow directly from the definitions of the functions of an acute angle, p. 20. 19. To express all the trigonometric functions in terms of any one of them. From any point in either side of the angle A let fall a perpendicular upon the otlu r side. Let the hypotenuse of the right triangle thus formed be taken as unity, 28 PLANE TRIGONOMETRY and call the perpendicular a. Then the remaining side of the right triangle is Vl a 2 . Then, sin A = - a = sin A, cos A = Vl a 2 = Vl sin 2 A, sin A tan A = Vl - a 2 Vl - sin 2 A cot A = Vl - sin 2 A sin A sec . = Vl - a 2 Vl - sin 2 csc . = -= , sin .A This gives the value of each of the functions, except the vers A and the covers A, in terms of sin A. To express the values of the functions in terms of cos J., tan A, or of any other given function of A, proceed in a similar manner. It is not best, however, to assume the hypotenuse equal to unity for all cases. Sometimes the side opposite the given angle should be taken as unity, and sometimes the side c adjacent. For example, to find the other functions of A in terms of tan A, assume the side adjacent A equal to unity, and let the side oppo- A B site the same angle equal a ; then the hypotenuse of the right triangle equals VI + a 2 , and the* required values are found as follows : tan A = - = a = tan A, Vl + a 2 Vl + taii 2 ^. -V TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 29 cos .4 = - = , tan a tan A In this work it will be noticed that the side adjacent to A is assumed equal to unity, while in the preceding the hypotenuse was assumed to be unity. Any other supp6sition might be made with equal correctness, but no other would be equally convenient. EXERCISE V 1. Express all the other functions of 6 in terms of cos 6. This problem can be solved, and the required values found, in a manner similar to that employed in finding the values of each of the other functions in terms of sin 0, or tan 0, which has just been illustrated. Or the values can be found by means of the relations deduced in Art. 16. Thus : = vi- cos 2 6, si ii# Vl cos tan = cos cos A ~~ : ,etc. Sill 2. Express all the other functions of in terms of cot 6. 3. Express all the other functions of in terms of sec 0. 4. Express all the other functions of 6 in terms of esc 6. 5. Given sin 6 = f , find cos 6 and tan 0. cos B = Vl - sin 2 = Vl - ^ = i V21. cos0 5 5 5 V21 V21 21 30 PLANE TRIGONOMETRY 6. Construct the angle 6 if tan 6 = f . The angle 6 may be considered one of the acute angles of a right triangle. Hence, to construct 6 we have only to construct a right triangle whose legs are respectively 2 and 7. Since tan = f , is the acute angle opposite the Bide 2 ' 11. 7. If sin = -j3_, find sec 0. 8. If sin A = J-J, find vers A. 9. If cos = f, find esc 6 and tan 0. 10. If cos = |, find cot and sec 0. 11. If tan A = J-J, find sec J. and cos A. 12. If tan. A = |, find esc A and cos .4. 13. If cot A = | , find sin J. and cos A. 14. If sec B = 5, find siri J5 and tan .#. 15. If sec B = |^, find tan .5 and vers B. 16. If esc A = 8, find cos A and tan A. 17. If esc JL = f , find sin A and sec A. 18. Find all the functions of each of the acute angles, ^4, B, of the right triangle whose sides are 8, 15, 17. 19. Find all the functions of each of the acute angles, A, B, of the right triangle whose sides are x + y, 2 xu x y x-y 20. . If sin 2 + cos = 2|, find tan 0. 21. If tan 2 - sec 6 = f>, find cos 0. 22. If 10 sin 2 6 - 5 cos = - |, find esc 6. 23. If sin + cot = 4y, find 0080. 24. If sin 0, = a and tan 6 = ft, prove (1 - a 2 )(l - ft 2 ) = 1 . ' \ \ CHAPTER III VALUES OF THE FUNCTIONS OF CERTAIN USEFUL ANGLES 20. Functions of an angle of 0. If the angle A is very small, then in considering the value of sin A, that is, the ratio CB - , it is at once seen chat the numerator, CB, is very small in comparison with the denominator, AB. Hence, the numerical value of sin A is very small when the angle A is very small. Also, if A decreases, the numerator of the fraction will also decrease, while the denominator will remain constant ; and as the angle approaches as a limit, the sine of the angle will also approach as a limit. When the angle becomes 0, that is, when A B coincides with A C, we shall have CB = 0, and AB = AC. Hence, sin 0= -=0, When we say that sin A when A = 0, we simply mean that, if A is made small enough, we can make the value of CB, and hence the value of sin A as small as we please ; or, to ex- press the same statement in different words, we can make sin A smaller than any assignable quantity. Hence, as stated above, sin A approaches as a limit when A approaches as a limit. In a similar manner, we interpret the statements cosO = l, tan 90 = oo , etc,, as meaning that cos A approaches 1 as a limit, tan A approaches oo as a limit, etc., when A approaches as a limit, when A approaches 90 as a limit, etc. 31 32 PLANE TRIGONOMETRY 21. Functions of an angle of 30. Let OAC be an equilateral triangle ; then is it also equiangular. From the vertex draw OB perpendicular to AC. Then in the right triangle GAB the angle A = 60, and the angle A OB = 30. Also, theleg AB=IAC= Let AB = a. Then OA = 2 a, and = V 6U 2 - AB* = V4 a 2 - a 2 = V3 a 2 = a V3. The trigonometric functions of 30 can now be found as follows : OA tan 30= B A = -A_ = JL = 1 V3, av /3 V3 3 BA V3 esc 30 = = = 2. BA a 22. Functions of an angle of 45. Let OAB be an isosceles right triangle. Each of the acute angles is 45, and the leg OB equals the leg AB. ^4 Let AB = a. Then OB = a and OA = a V2, and we have : sin 45' cos45=^ = - 2 FUNCTIONS OF CERTAIN USEFUL ANGLES - BA - a --\ ~ ~' 33 CSC 45=^= = V2. BA a 23. Functions of an angle of 60. Let OAC be an equilateral triangle. Then is it also equiangular. From the vertex A draw AB perpendicular to 00. Then in the right triangle OAB, angle 0= 60, and angle OAB= 30. Also, OB = OC = \ OA. Let OB = a. Then OA = 2 a, and AB = V OA* - OB* = V42_ a 2 = V3a2 = 6? V3. The trigonometric functions of 60 can now be found as follows : *.!. V 5 ,i - OB a cot60 = ^^ = -^=- J L = i BA aV3 V3 3 fO 4 9 /, 9 9 csc 60 - ^ = ^^ =-^- = O a 24. Functions of an angle of 90. Let the angle AOB (p. 34) be very nearly a right angle. Then the angle A is very small, and B the foot of the perpendicular from A to OB is very near the vertex 0. When the angle approaches a right angle, AB CON A NT'S TKIO. - 3 34 PLANE TRIGONOMETRY will approach coincidence with AO, and B will approach coinci dence with 0. Hence, . o BA OA - YA one cos 90 = = OA OA BA OA 0, _ BA OA * =-^=ir OA OA The real meaning of these equations is that, as the angle ap- proaches 90 as a limit, the sine of the angle approaches 1 as a limit, the cosine approaches as a limit, the tangent approaches oo as a limit, etc. It is, however, customary to say sin 90 = 1, cos 90 = 0, tan 90 = oo , etc. A more complete discussion of the values of the trigonometric functions for limiting cases such as the above is given later. See Art. 41, p. 59. 25. In the following table are collected the results obtained in the last five sections. These results are exceedingly im- portant, and the student should become thoroughly familiar with them before proceeding further. 30 45 60 90 sine 1 JvS |V3 1 cosine 1 JV3 *V2 i tangent iV3 1 V:] CO cotangent CO V3 1 *V3 secant 1 I jVS V2 o -a cosecant cc j 2 V2 |v3 1 FUNCTIONS OF CERTAIN USEFUL ANGLES 35 It is necessary to commit to memory only one half of this table. The remainder can be obtained at any time by means of the relations which were found in Art. 17, of which the -following is a condensed statement : Any trigonometric function of an acute angle is equal to the corresponding co-function of its complement. EXERCISE VI Verify the following : , 1. cos + sin 30 + sin 90 = 2. 2. cos cos 60 -f sin sin 60 -f sin 30 = 1. 3. tan 2 30+ sec 2 30 = If. 4. cos 2 60 + cos 2 45 4- cos 2 30 = f . 5. sin 60 cos 30 -f- cos 60 sin 30 = 1. 6. sin 2 30 tan 2 45 + sec 2 60 sin 2 90 = 4 J. 7. (sin 30 + cos 60) (sec 45 + c.sc 45; - 2 V2. 8. sin 30 sin 45 sin 60 tan 60 = f V2. 9. cot 30 tan 60 sin 45 cos 45 = f . 10. tan 2 45 + sin 2 30 - cos 2 30 - f tan 2 30 = J. Prove the following identities : 11. sin A cos (90 - A) sec (90 -A) = sin A. 12. cos A cos (90 - A) sin (90 - A) esc A == cos 2 A. 13. tan (90 - A) cot (90 - A) tan A = cos (90- A) esc (90- A). cos (90 -A) cot(90-J.) 14. = sin A. esc (90 - A) sin A 15. cos 2 A sec 2 (90 - A) tan 2 A cot 2 (90 -A) = tan 2 A. 16 tan 2 (90-^) cos 2 .4 csc 2 (90-^) ^ 4 A sin 2 A ' cot 2 (yO - A) ' sec 2 (yo - A) " 17. cos (90 - A) 1 - cos (90 - A) _ t . m A covers A sin ( ( JO A) 18. secMEEl +cos 2 (90-^)csc 2 (90-7l). 19. csc 2 ^ = 1 + sin 2 (90 - A) sec 2 (90 - A). 20 cot 2 (90 - A) tan 2 (90 -A')^ 1 esc 2 (90 - A) ' sin 2 (90 -A) CHAPTER IV THE RIGHT TRIANGLE 26. In order to solve a right triangle, two parts besides the right angle must be given, of which at least one must be a side. The known parts may be : 1. An acute angle and the hypotenuse. 2. An acute angle and the opposite leg. 3. An acute angle and the adjacent leg. 4. The hypotenuse and either leg. 5. The two legs. 27. In the preceding sections we have seen that the trigono- metric functions are pure numbers ; and in the case of the angles 0, 30, 45, 60, and 90, the values of these functions have been ascertained. From a trigonometric table the values of the functions of any angle can be found ; and by the aid of these values the solution of any triangle can be effected. The method for each case arising under right triangles is illustrated by the following examples : CASE 1 Given A = 61 22', c = 46.2; find B, a, b. fit 22' B = 90 - 61 22' = 28 38'. b (1) (2) sin A--- .-.a=csiuA c = 46.2 x 0.8777. .-. a = 40.54. (3) cos^l =-. .-. b = coos ,4 c = 46.2 x 0.4792. .-. b = 22.14. 30 THE RIGHT TRIANGLE CASE 2 Given A = 31 17', a = 321 ; find B, c, b. (1) B = 90 - 31 17' = 58 43'. (2) sin.4 = ^ .' = -;? 321 = 618.14. 0.5193 .-. c = 618.14. (3) tan J. = - /. b = b tan A 321 0.6076 .-. b = 528.31. CASE 3 Given A = 43 42', b = 38.6 ; find B, a, , 43'42' (1) = 90 -43 42' =46 18'. (2) tan /I = - .. a b tan A (3) = 38.6 x 0.9556. .-. a = 36.89. * cos A _ 38.6 0.7230* .-. c = 53.39. CASE 4 Given a = 36. 4, NOTE. In solving problem 70 natural functions have been employed. On p. 156 a method will be given by means of which problems of this kind can be solved by the use of logarithms. In the following problems it is recommended that natural functions be employed. v> 71. At a point on a level plain the angle of elevation of the top of a church spire is 45, and at a point 50 ft. nearer, and in the same straight line with the first point and the church, the corresponding angle of elevation is 60. What is the height of the spire? ^ 72. From the top of a cliff 150 ft. high the angles of depres- sion of the top and bottom of a tower are 30 and GO , respec- tively. What is the height of the tower? 73. The angles of elevation of the top of a tower, taken at two points 268 ft. apart and in the same straight line with the tower, are 21 14' and 53 4G', respectively. What is the height of the tower? 74. At the foot of a mountain the angle of elevation of the summit is 45 ; one mile up the slope of the mountain, which rises at an inclination of 30, the angle of elevation of the sum- mit is 60. What is the height of the mountain ? 75. At a certain point south of a tower the angle of eleva- tion of the top of the tower is 60, and at a point 300 ft. east of the point the corresponding angle of elevation is 30. What is the height of the tower ? 30. The isosceles triangle. The perpendicular from the vertex, (7, of an isosceles triangle to the base divides the triangle into two equal right triangles. Any two parts of either of these right triangles being given, one or both of which are sides, the right triangle can be completely determined. Therefore the isosceles triangle also can be completely determined. Denoting the base of the isosceles triangle by *" 20. Compute the area of a regular heptagon if the length of one of its sides is 13.88. v 21. The radius of the circumscribed circle of a regular dodecagon is 27. What is the area? CHAPTER V THE APPLICATION OF ALGEBRAIC SIGNS TO TRIGO- NOMETRY 32. In the preceding work no attempt has been made to apply the definitions of any of the trigonometric functions to any except positive acute angles. These definitions will now be extended so as to apply to negative as well as to positive angles, and to angles of any magnitude whatever. 33. The coordinate axes. The location of a point or a line lying in a given plane is often described by referring it to two intersecting straight lines in that plane, called coordinate axes. These lines are usually drawn perpendicular to each other. Let the two lines XX' and YY' intersect at right angles. Then the plane of these lines is divided into four quadrants, designated as the first, second, third, and fourth quadrants, respectively. These quadrants are numbered as indicated in the figure. 34. Coordinates of a point in a plane. The location of any point in the plane determined by the axes XX' and YY' is described by means of its perpendicular distances from these axes. Y The distance of a point from YY' measured along a line parallel to XX' is called the abscissa of the point ; and the distance of a point from XX', measured on a X- line parallel to YY' is called the ordinate of the point. The abscissa of a point is usu- ally designated by the letter x, y' 51 52 PLANE TRIGONOMETRY and the ordinate by the letter y. These two distances, taken together, are called the coordinates of the point. The line XX' is called the axis of abscissas, and the line YY' is called the axis of ordinates. These axes are, for the sake of brevity, often called the #-axis and the #-axis, respec- tively. Their point of intersection, 0, is called the origin. Any abscissa measured to the right of YY' is considered positive, and any abscissa measured to the left of YY' is con- sidered negative. Any ordinate measured above XX' is considered positive, and any ordinate measured below XX' is considered negative. The coordinates of a point determine its position completely. For example, if the point A is 4 units from YY 1 and 6 units from XX', its position can be located as follows : measure off on XX' a distance equal to 4 units, and through the point thus found draw a line parallel to YY'., Also, measure off on YY 1 a distance equal to 6 units, and through the point thus deter- mined, draw a line parallel to XX'. The intersection, A, of these two lines is the required Y 12 point. The abscissa of A is 4, and its ordinate is 6, and this point, whose location is given by means of its coordinates, is called the point (4, 6). The point B, located in a similar manner, has for its coordi- nates x = 3 and y = 4 ; and this point B is called the point (3, 4). The point is called the point (4, 5); and the point D is called the point (6, 3). In a similar manner we can locate any other point (#, 5), where a and b are any real quantities whatever, either positive or negative. 35. Trigonometric functions of any angle. Let the line OA (p. 53) start from OX and revolve in a positive direction until it occupies a position in any one of the four quadrants. From any point P in the revolving line draw a perpendicular PM to the axis of abscissas, XX' . In each of the four figures we have THE APPLICATION OF ALGEBRAIC SIGNS 53 CM= x and MP = y. Let the distance OP = r. The trigono- metric functions of the angle XOA, which may be represented by 0, are then, for all positions of OA, defined as follows : M x X X x x sill = ordinate revolving line r /j abscissa x cos u = - : = -> revolving line r = = abscissa x , a abscissa x cot u = - = -> ordmate y a revolving line r sec 6 = - = -> x abscissa esc a __ revolving line _ r ordinate y 54 PLANE TRIGONOMETRY The functions vers and covers are denned in a manner similar to that employed in the case of the right triangle, as follows: T ers 0=1 -cos*, covers 6 = 1 sin 6. NOTE. In the case of cot 0, esc 0, tan 90, sec 90, cot 180, esc 180, tan 270, sec 270, cot 360 and esc 360, these definitions fail. For, taking as an illustration the tangent of 90, we have in that case a fraction whose numera- tor is r and whose denominator is 0. The value of tan 90 is, then, if we attempt to use the above definition, given by this fraction whose numerator is r, and whose denominator is 0. But there is no such thing as division by 0, hence, according to the definition given, the symbol tan 90 has no mean- ing. This and other similar cases will be discussed later. (See pp. 57-63.) 36. In a manner precisely similar to that employed in Art. 16 it can be proved that, for any value whatever of the fol- lowing relations are true : sin 2 6 + cos 2 6 = 1$ (1) sec 2 6 = 1 + tan 2 6 ; (2) esc 2 = 1 + cot 2 6. (3) Also, from the definitions of the functions, the following rela- tions are immediately derived : sin 6 = -^-r, .-. sin esc 6 = 1, (4) CSC U -^ cos0 = - -, .-. cos 6 sec = 1, (5) sec# - - cot u , .-. tan6cot0 = l. (6) Also, since, cos# = -, . . a? = reos6, (7) (8) (9) THE APPLICATION OF ALGEBRAIC SIGNS 55 37. Signs of the trigonometric functions. In dealing with the functions of an acute angle of a right triangle (Art. 14, p. 20), no attention was paid to the question of positive or negative signs. All lines employed in that connection were considered positive; hence the value of each of the functions was considered positive. But in dealing with the general angle we have to consider both positive and nega- tive lines, and as a result the signs of the functions undergo certain changes as the revolving line passes from quadrant to quadrant. First Quadrant. Assume that the revolving line is always positive, and let it occupy any position in the first quadrant. In this position both x and y are positive; hence, since r is also positive, both numerator and denominator are positive in the case of each of the functions. Therefore all the trigono- metric functions are positive for the angle in the first quadrant. Y Second Quadrant. Let the revolving line occupy any posi- tion in the second quadrant. In this case x is negative and y is positive; and we have the following results: The sine is a fraction whose numerator and denominator are both positive ; therefore the sine of an angle in the second quadrant is positive. The cosine is a fraction whose numerator is negative and whose denominator is positive; therefore the cosine of an angle in the second quadrant is negative. The tangent is a fraction whose numerator is positive and whose denominator is negative; therefore the tangent of an angle in the second quadrant is negative. 56 PLANE TRIGONOMETRY The cotangent is a fraction whose numerator is negative and whose denominator is positive; therefore the cotangent of an angle in the second quadrant is negative. The secant is a fraction whose numerator is positive and whose denominator is negative; therefore the secant of an angle in the second quadrant is negative. The cosecant is a fraction whose numerator and denominator are both positive; therefore the cosecant of an angle in the second quadrant is positive. Third Quadrant. Let the revolving line occupy any position in the third quadrant. In this case both x and y are negative; therefore the following results can at once be obtained: The sine is negative. The cosine is negative. The tangent is positive. The cotangent is positive. The secant is negative. The cosecant is negative. Fourth Quadrant. Let the revolving line occupy any position in the fourth quadrant. In this case x is positive and y is negative; therefore the following results can at once be obtained: The gine . g negative< The cosine is positive. The tangent is negative. The cotangent is negative. The secant is positive. The cosecant is negative. THE APPLICATION OF ALGEBRAIC SIGNS 57 The above results are conveniently grouped together by means of the following table: sine + sine -f- cosine cosine + tangent cotangent secant tangent + cotangent -f secant + cosecant + cosecant + Y sine sine cosine cosine + tangent + cotangent + secant tangent cotangent secant + cosecant cosecant Y' From the definitions of the versed sine and of the coversed sine it follows that these two functions are always positive. 38. Changes in sign and magnitude of the trigonometric func- tions as the angle increases from to 360. As before, we assume for the revolving line a constant length, r. As the revolving line starts from its initial position we have x = r, and y = 0. As the angle 0, which is generated by the revolution of this line, increases from to 90, y increases and x decreases; and when OA coincides with OY, we have x = 0, and y = r. Hence, as the angle increases from to 90, x decreases from r to 0, and y increases from to r. As the angle increases from 90 to 180, x decreases in- creases numerically from to r and y decreases from r to 0. As the angle increases from 180 to 270, x increases de- creases numerically from r to and y decreases increases numerically from to r. As the angle increases from 270 to 360, x increases from to r and y increases decreases numerically from r to 0. Inasmuch as all changes in sign and magnitude among the trigonometric functions are directly dependent on the changes just noted, the following results are now obtained without difficulty. 58 PLANE TRIGONOMETRY X- X X- Y Y Y Y 39. Sine. As the angle increases from to 90 the numera- tor of the fraction that expresses the value of the sine increases from to r, and the denominator r remains constant. Hence the sine increases from to 1. As the angle increases still further, the numerator begins to decrease, the denominator still remaining constant, and at 180 the numerator becomes 0. Hence as the angle increases from 90 to 180 the sine decreases from 1 to- 0. As the revolving line enters the third quadrant, y becomes negative and continues to decrease algebraically, becoming r when the angle equals 270. Hence in the third quadrant the sine is negative, and as the angle increases from 180 to 270 the sine decreases from to - 1. In the fourth quadrant y continues negative ; but as the angle increases y increases algebraically, and when the revolving line reaches its original position, y again becomes 0. Hence as the angle increases from 270 to 360 the sine is negative, and increases from 1 to 0. Collecting the above results for the sake of convenience we have the following statement : THE APPLICATION OF ALGEBRAIC SIGNS 59 In the first quadrant the sine increases from to 1; in the second it decreases from 1 to ; in the third it decreases from to 1 ; in the fourth it increases from 1 to 0. 40. Cosine. In a manner similar to that employed in the case of the sine, the following results are obtained: As the angle increases from to 90 the cosine decreases from - to -, i.e. from 1 to 0. As the angle increases from r r 90 to 180 the cosine decreases increases numerically from - T to ^^, i.e. from to 1. As the angle increases from 180 to 270 the cosine increases decreases numerically from' to -, i.e. from 1 to 0. As the angle increases from 270 r r to 360 the cosine increases from - to -, i.e. from to 1. r r 41. Tangent. The value of the tangent is the value of the fraction ^ When the angle is very small, the numerator of x this fraction is very small, and the denominator is very nearly equal to r. Hence the tangent of the angle is very small ; or, as it is commonly expressed, when the angle equals 0, the tangent of the angle is also equal to 0. As the angle increases the numerator y increases and the denominator x decreases. Hence the tangent of the angle increases. When the angle is nearly 90, the numerator is very nearly equal to r\ and as the angle approaches 90 the value of the numerator continually increases, approaching r as its limit. At the same time the value of the denominator con- tinually decreases, approaching as its limit. Hence, as approaches 90 the value of tan can be made to exceed any finite number previously assigned, no matter how great that number may be. This is usually expressed by saying that when the angle is equal to 90, the tangent of the angle is equal to infinity. Hence, ~ In the first quadrant the tangent increases from - to -, i.e. from to QO . In the second quadrant the denominator x becomes negative while the numerator y remains positive. Hence the tangent 60 PLANE TRIGONOMETRY of an angle in the second quadrant is negative. When the angle is but little greater than 90, the numerator is very nearly equal to r and the denominator is very small, and nega- tive. Therefore, as the revolving line enters the second quadrant, the numerical value of the tangent can be taken to be greater than any negative finite limit previously assigned. That is, when the angle is in the second quadrant and differs from 90 by an amount that is less than any finite number assigned in advance, no matter how small that number may be, the tangent of the angle is negative and is numerically greater than any finite limit assigned in advance. To express this we shall 'say that tan 90 = oo. It is thus seen that tan 90 will be called equal to either +00 or oo according as the angle is approaching the limit 90 from the positive direction, or as the revolving line is leaving the position at which the angle equals 90 and is just entering the second quadrant. As the angle increases, the numerator decreases and the denominator, which is negative, increases numerically. Hence, the tangent decreases numerically increases algebraically and when the angle becomes equal to 180, the tangent becomes equal to 0. Hence, In the second quadrant the tangent increases from - to , i.e. from oo to 0. In the third quadrant both numerator and denominator are negative. Hence the tangent is positive. The numerator in- creases numerically from to r, and the denominator de- creases numerically from r to 0. Hence, In the third quadrant the tangent increases from - to -^, i.e. from to oo. In the fourth quadrant the numerator is negative and the denominator is positive. Hence the tangent is negative. The numerator decreases numerically from r to 0, and the de- nominator increases from to r. Hence, In the fourth quadrant the tangent increases from -^ to , i.e. from oo to 0. The same restriction is to be observed with respect to the value of tan 270 as was noted in connection with tan 90. That is, if the angle is in the third quadrant and is approaching 270 as its limit, the tangent of the angle can be made to exceed THE APPLICATION OF ALGEBRAIC SIGNS 61 in magnitude any finite positive limit previously assigned. If it is in the fourth quadrant, the tangent is negative and can be made to exceed in numerical magnitude any finite limit _previously assigned. For this reason it is customary to say that tan 270 = oo . 42. Cotangent. The value of the cotangent is the value of the fraction -. When the angle is very small, the numerator y is nearly equal to r and the denominator is nearly equal to 0. Hence the value of the cotangent of is infinity. Then, letting the angle increase, and reasoning in the same manner as in the case of the tangent, we obtain the following results : In the first quadrant the cotangent is positive and decreases T from to , i.e. from GO to 0. r In the second quadrant the cotangent is negative and de- r creases from to , i.e. from to -co. r In the third quadrant the cotangent is positive and decreases r from to , i.e. from oo to 0. -r In the fourth quadrant the cotangent is negative and de- T creases from - to -, i.e. from to oo. -r Remarks similar to those made in connection with tan 90 and tan 270 apply to cot 0, cot 180, and cot 360. 43. Secant. The value of the secant is the value of the fraction -. The numerator remains constant for all positions x of the revolving line, while the denominator varies. When the angle is very small, the numerator and the denominator are approximately equal. Hence the secant of is equal to unity. As the angle increases the denominator x decreases, thus caus- ing the value of the secant to increase. When the angle is nearly equal to 90, the denominator is nearly equal to 0, and approaches as its limit. Therefore the secant can be 62 PLANE TRIGONOMETRY made to exceed any finite limit previously assigned. We shall express this by saying that sec 90 = . Hence, As the angle increases from to 90 the secant increases from - to , i.e. from +1 to +00 . r When the revolving line enters the second quadrant the denominator x becomes negative and begins to increase numeri- cally decrease algebraically becoming equal to r when the angle becomes 180. Hence, beginning with a negative value numerically greater than any finite limit assigned in advance, the secant increases decreases numerically until it reaches the value 1. Hence, As the angle increases from 90 to 180 the secant increases T T from -^ to , i.e. from oo to 1. r In the third quadrant the denominator continues negative, but begins to decrease increase numerically as soon as the revolving line enters the quadrant. At 270 the denominator becomes 0. Hence, As the angle increases from 180 to 270 the secant decreases from to -, i.e. from 1 to oo. r In the fourth quadrant the denominator again becomes posi- tive, and increases from to r as the angle increases from 270 to 360, returning to its original value when the revolving line completes one entire revolution. Hence, As the angle increases from 270 to 360 the secant decreases from -^ to -, i.e. from oo to 1. r The same restriction is to be observed with respect to the value of sec 270 as was noted in connection with sec 90. That is, if the angle is in the third quadrant and is approaching 270 as its limit, the secant of the angle can be made to exceed in numerical magnitude any finite negative limit assigned in advance. If the angle is in the fourth quadrant, the secant is positive, and can be made to exceed in magnitude any finite positive limit assigned in advance. We shall express this by saying that sec 270 = cc . THE APPLICATION OF ALGEBRAIC SIGNS 63 44. Cosecant. The value of the cosecant is the value of the fraction -. Remembering that the numerator remains con- y stant, and tracing out the changes in sign and magnitude of the denominator, as in the case of the secant, we obtain the fol- lowing results : As the angle increases from to 90 the cosecant decreases from - to -, i.e. from oo to 1. r As the angle increases from 90 to 180 the cosecant increases 7* 7* from - to -, i.e. from 1 to oo. r As the angle increases from 180 to 270 the cosecant increases y v* from - to , i.e. from oo to 1. r As the angle increases from 270 to 360 the cosecant decreases '/* '/* from to -, i.e. from 1 to oo. r Remarks similar to those made in connection with sec 90 and sec 270 apply to esc 0, esc 180, and esc 360. The changes that take place in the sign and magnitude of the different trigonometric functions are conveniently grouped together in the following table : r FIRST QUADRANT increases from to 1 SECOND QUADRANT sine decreases from 1 to cosine decreases from to 1 tangent increases from GO to cotangent decreases from to oo secant increases from x to 1 cosecant increases from 1 to oo X' sine cosine decreases from 1 to tangent increases from to oo cotangent decreases from oo to secant increases from 1 to oo cosecant decreases from oo to 1 THIRD QUADRANT sine decreases from to- 1 cosine increases from 1 to tangent increases from to oo cotangent decreases from oo to secant decreases from 1 to oo cosecant increases from -co to 1 FOURTH QUADRANT sine increases from 1 to cosine increases from to 1 tangent increases from oo to cotangent decreases from to oo secant decreases from oo to 1 cosecant decreases from 1 to -co T' 45. After the changes in sign and magnitude have been obtained for the first three functions, the corresponding changes for the last three can be found by remembering that the 64 PLANE TRIGONOMETRY cotangent, secant, and cosecant are the reciprocals of the tangent, the cosine, and the sine respectively. The student should verify the above results by obtaining them in this manner also. In connection with the general definitions of the trigonometric functions given on p. 53, it was noted that these definitions failed in the case of certain functions for certain values of the angle. These cases have been explained in some detail in Arts. 41-44, and we now have definitions of the tangent, the cotan- gent, the secant, and the cosecant of any angle from to 360 inclusive ; and hence, by the usual considerations, Arts. 50-57, definitions of these functions for any angle whatever. In order that the relations between tan 90 and cot 90, tan 270 and cot 270, sec 90 and cos 90, etc., may be the same as that between the same functions in the case of other angles we shall say that = 0, and - = oo . But the student is cautioned oo that "oo " is not a number in the usual sense of the word, and that these two equations are not to be taken literally. They are used merely for the sake of expressing concisely the result of a definite limiting process, a process much more complicated than that of ordinary division. 46. Geometrical representation of the trigonometric functions. The trigonometric functions are pure numbers, the value in each case being a ratio between two given magnitudes. These magnitudes are represented by lines, and if the length of the revolving line is properly chosen, it is possible to represent the values of the functions themselves by lines. Let the revolving line be the radius of a circle, and let its value be assumed to be unity. The sine of the angle AOB is CD -. But since OD=\, we may OD say CD CD Similarly, THE APPLICATION OF ALGEBRAIC SIGNS 65 vers <9 = 1 - cos = (L4 - 00= AC, covers = 1 - sin (9 = OG - OE = GE. For an angle of the second quadrant the so-called "line values" of the trigonometric functions are obtained as follows: vers = 1 - cos = 0.4 - 00= OA+00= CA, covers = 1 - sin = OG- OE=E&. The change in sign when 00 is replaced by 00 in obtain ing the value of the versed sine should be noted carefully. CONANT'S TRIG. 5 66 PLANK TRIGONOMETRY For angles of the third and fourth quadrants the line values are obtained in a manner similar to that employed in connec- tion with angles of the first two quadrants. The figures are lettered so that the following values hold for both : oc an = = 0(J OA an esc OI> OH OH = - = - OH, CD oa i vers 0=1- cos 0=OA- OC= OA, covers 0=1 - sin = GO- - OE=Ea. The signs of the trigonometric functions when used as lines are, of course, the same as when they are used as ratios. It will be noticed that when the line that represents the sine extends upward from the axis of abscissas, or horizontal di- ameter, the sine is positive ; when it extends downward, the THE APPLICATION OF ALGEBRAIC SIGNS 67 sine is negative. The cosine is positive when the line that represents its value extends toward the right from the origin, negative when it extends toward the left. The tangent is positive when its line extends upward from the axis of abscissas, or horizontal diameter, negative when it extends downward. The cotangent is positive when its line extends toward the right from the axis of ordinates, or vertical diameter, negative when it extends toward the left. The secant and the cosecant are positive when their respective lines extend in the same direction from the origin as the revolving line, negative when they extend in an opposite direction. The versed sine is con- sidered as extending toward the right from the foot of the sine, and the coversed sine upward from the foot of the perpen- dicular dropped from the extremity of the revolving line to the vertical diameter. Both are always positive. The trigonometric functions were originally used as lines ; and the numerical value was, in each case, the length of the line in terms of the revolving line, or the radius of the circle, taken as a unit. There are certain advantages connected with the use of these line values, but for general purposes the ratios are so much more convenient than the line values that they have now come into almost universal use. 47. Limiting values of the trigonometric functions. In dis- cussing the variation in the values of the different functions the following limits were found. In the case of the sine the positive limit was 1, and the negative limit was 1. For the cosine also these limits were -f 1 and -- 1 respectively. For the tangent and the cotangent the limits were +00 and -co. For the secant and the cosecant it was found that the positive values that these functions could take were comprehended between + 1 and + oo, and the negative values between 1 and co. Hence, we can make the following definite state- ment respecting the limits between which the different func- tions can vary : The sine can take any value between -f- 1 and 1. The cosine can take any value between + 1 and 1. The tangent can take any value between -f oo and oo. The cotangent can take any value between + oo and GO. 68 PLANE TRIGONOMETRY The secant can take any value between + 1 and + oo, and between 1 and oo. The cosecant can take any value between -f- 1 and + oo, and between 1 and oo. From the definitions of the versed sine and the coversed sine it follows that each of these functions can take any value be- tween and -f 2. 48. Graphs of the trigonometric functions. The graphs of the trigonometric functions can be plotted in the ordinary manner if the values of the angles are taken as ordinates and the corresponding values of the functions as abscissas. Sine. For the sine we form the following table of values from the equation y ~ s j n Xt In this table the values of the sine are, for convenience in plotting, given decimally, instead of in the ordinary common fractious. /scf 30 00 QO 120 f 50 -A' X y 30 .5 45 .71 60 .87 90 1 120 .87 135 .71 150 .5 180 225 - .71 270 - 1 315 - .71 360 415 .71 450 1 495 .71 etc. etc. Continuing this table, and plotting the points thus deter- mined, we find that the graph is a curve consisting of an infinite number of waves like those in the figure. By using negative values of the angle we obtain similar waves at the left of the origin. The curve is called the sine curve, or sinusoid. THE APPLICATION OF ALGEBRAIC SIGNS 69 Cosine. The graph of the equation y = cos x is found in a similar manner. Forming a table of values, and plotting the points determined by these values, we find that the cosine curve has the following form. 90 30 60 Y Tangent. The table of values for x and y formed from the e( l uation # = tan* is as follows. X y 30 .58 45 1 60 1.73 90 00 120 - 1.73 135 - 1 150 - .58 180 210 .58 225 1 240 1.73 270 CO 300 - 1.73 315 -1 330 -.58 360 390 .58 etc. etc. -270 -96 30 60 90 I 2O 150' I /ISO 270 70 PLANE TRIGONOMETRY Continuing the table, and plotting the points determined by the values thus found, we obtain the tangent curve, which con- sists of an infinite number of branches, each like one of those in the figure. Negative values of the angle give an infinite number of like branches at the left of the origin. Cotangent. The graph of the equation y = cot x is similar to that of y = tan #, except that the points where the different branches cross the #-axis are 90 to the right of those where the tangent curve branches cross, and the curvature is toward the right instead of toward the left. The form of the graph is shown in the following figure. X y CO 30 1.73 45 1 60 .58 90 120 -.58 135 - 1 150 - 1.73 180 CO 210 1.73 225 1 240 .58 270 300 -.58 315 -1 330 -1.73 360 CO 390 1.73 etc. etc. Secant. The table of values for the equation y sec x can readily be found if it is remembered that sec# is the recip- rocal of cosx. The graph has the form shown in the first figure on p. 71. Cosecant. The graph of the cosecant is similar in form to that of the secant, but the relative position of the various THE APPLICATION OF ALGEBRAIC SIGNS Y 71 -36O -i branches with respect to the #-axis is different. The graph is shown in the following figure. 49. Periods of the trigonometric functions. In considering the changes in value through which the functions pass as the angle increases, it is seen that the sine, for example, takes all its pos- sible values, in both increasing and decreasing order of change, while the angle is increasing from to 360. As the angle increases from 360 to 720 the values of the sine which were obtained in the first 360 are repeated, this repetition of values occurring in the original order. The same cycle of values will again occur in the next 360, and so on, for each complete revolution of the generating, or revolving line. The angle formed by the generating line while this regular recurrence of values takes place is called the period of the sine; and in accordance with this result we may say that The period of the sine is 360, or 2 IT. A similar course of reasoning shows us that 360 is also the period of the cosine, of the secant, and of the cosecant. 72 PLANE TRIGONOMETRY The values of the tangent repeat themselves completely with each increase of 180 in the angle. Hence, The period of the tangent is 180, or IT. The period of the cotangent is the same as the period of the tangent. EXERCISE X 1. Trace the changes in sign and magnitude of sin 6 as varies from - ^ to - ^ ; from - 270 to - 450. 2. Trace the changes in sign and magnitude of cos A as A varies from TT to 2 TT. 3. Trace the changes in sign and magnitude of tan A as A varies from 180 to 540. 4. Trace the changes in sign and magnitude of sec A as A varies from - 90 to -270. Find the value of each of the following: 5. sin 6 + cos 6 when 6 = 60. 6. sin 2 6 + 2 cos when = 45. 7. sin A 4- tan A when A = 135. 8. sin 60 + tan 240. 9. cos cos 30 + tan 135 cot 315. 10. cos tan 60 - sec 2 30 cot 225. 11. 2 sin 90 sec 2 30 + cos 180 tan 315. 12. 2sec 2 7rcos0 + 3sin 3 ^-csc^. 2 2t 13. COSTT tan !L - sec 2 11^ tan 2 ii 464 14. For which of the following values of 6 is sin 6 cos 6 positive and for which is it negative? (9 = 0; (9=60; = 120 ; (9=210; (9=240; (9=300; 6 = 330. THE APPLICATION OF ALGEBRAIC SIGNS 73 15. For which of the following values of is sin -f cos positive and for which is it negative? (9 = 135; (9 = 210; 0=300; 0=315; 0=330. 16. Prove that sec 6 - tan 6 0=3 sec 2 tan 2 + 1. 9 79 17. If cos = a ~ _ , find sin and tan 0. a 2 4- 6 2 18. If tan = a + a , find cos and sin 0. 2a + 1 19. Prove the equation sin0 = # + - impossible for all real values of x. 20. Prove the equation sec 2 = ^ . impossible unless CHAPTER VI TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 50. Functions of an angle 9 in terms of functions of 6. Let the revolving line OA generate an angle 0, of any mag- nitude. The final position of OA is, then, in any one of the four quadrants, as shown in the figures. Also, let the line OA' generate an angle 0, equal in magnitude to the positive angle 0, generated by OA. Take OB= OB 1 , and from B and B' draw perpendiculars BC, B'C', to XX'. Then are the triangles OBC, OB'O', equal geometrically, since they are right triangles having the hypote- nuse and an acute angle of one equal respectively to the hypote- nuse and an acute angle of the other. Hence the points 6 Y , 6 Y/ , coincide, BO=B'Q'< and 00= 0' C' . TRIGONOMETRIC FUNCTIONS OF ANY ANGLK 75 For convenience, let OB = r, OB' = r f , BC = y, B'C'=y', 00 = x, OC' = z' -, then for each of the four figures we have sin ( - 0) = ^ = 1 == - sin (9, r r cos ( - ) = - = - = cos 0, r r tan (-0)=^ = = - tan0, COt ( 0) = ^ = =r COt y -y A\ ^ r sec ( u) = = = sec # a; CSC ( 0) = = ^- = CSC 0. y ~ y EXAMPLES. 1. sin ( - 30) = - sin 30 = J, A/2 2. cos (-45)= cos 45 = -^> 40 3. tan (- 60) = - tan 60 = - V. 51. Functions of an angle 90 6 in terms of functions of 0. Let the revolving line OA (p. 76) generate an angle 0, of any magnitude, and at the same time let OA' generate an angle whose magnitude is 90 0. As before, take OB OB' , and from B, B', draw perpendic- ulars BO, B'C', to XX'. The triangles OB C, OB'C', are, in each of the four figures, equal geometrically. The proof should be supplied by the student. With the same notation as in the previous figures we have, considering only the actual lengths of the lines, and paying no attention to positive and negative signs, r = >', y = or', x y' . 76 PLANE TRIGONOMETRY The following equations then hold true for all possible cases: sin (90 - 0) = -' = - = cos 0, r r cos (90 - 0) = - = = r r tan (90 - 0) = V- = - = cot 0, x 1 y y sec (90 - (9) = - = - = esc 0, x' y esc (90 - 0) = - t = - = sec 0. ' NOTE. For the special case that occurs when 6 is an acute angle, these relations were established independently in connection with the definitions of the functions of an acute angle of a right triangle (Art. 17, p. 23). TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 77 52. Functions of an angle 90 + 6 in terms of functions of 6. Let the revolving line OA generate an angle 0, of any magni- tude, and at the same time let OA' generate an angle whose ^nagnitude is 90 + 6. As in each of the previous cases, take OB = OB', and from B, B', draw perpendiculars BC, B 1 C', to XX'. The triangles OBC, OB' C' are, in each of the four figures, equal geometri- cally. The proof should be supplied by the student. With the notation used in the previous cases we have, con- sidering only the actual lengths of the lines, and paying no attention to positive and negative signs, r = r',x = y f , y = x f . If positive and negative signs are taken into account, these equa- tions become r = r', x = y', y = x'. The following equations then hold true for all possible cases: sin (90 + 6) = '4 = - = cos 0, r' r cos (90 + 0) = ^ = ^ = - sin 0, r r 78 PLANE TRIGONOMETRY tan (90 + 0) = = = - cot 6, x' -y cot (90 + 0) = - = ^ = - tan 6, ' sec (90 + 9) = r - t = = - esc 0, x -y esc (90 + 0) = -= - = sec0. y x EXAMPLES. 1. sin (9(T 4- 30) = cos 3CP = 2. cos (90 + 45) = -sin45 = -| 3. tan (90 + 60) = - cot 60 =- 4. cot (90 + 120) = - tan 120 = - (- V3) = V3, 5. sec (90 + 135) = ~ esc 135 = - V2, 6. esc (90 + 150) = sec 150 = - f V3. 53. Functions of an angle 180 9 in terms of functions of 9. Let the revolving line OA generate an angle 0, of any magni- tude, and at the same time let OA' generate an angle whose magnitude is 180 - 0. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 79 As in the previous cases, take OB = OB', and from B, B', draw perpendiculars BO, B'C', to XX'. .The triangles OBC, OB'C', are, in each of the four figures, equal geometrically. The student should supply the proof. With the notation used in the previous cases we have, con- sidering only the actual lengths of the lines, and paying no attention to positive arid negative signs, r = r', x == x', y = y' . If positive and negative signs are taken into account, the second equation becomes x = x' . The following equations then hold true for all possible cases : sin (180 - 0) = ^ = 2 = sin 0, r' r cos (180 - 0) = t = - = - cos 6, r ' r tan (180 - 0) = tf- = -- = - tan 0, x' x cot (180 -6) = - = - = - cot 0, y y sec (180 - 0) = r , = - - - - sec 0, x' x csc (180 - (9) = - t = - = esc 0. 1 1 y y EXAMPLES. 1. sin (180 - 80) = sin 30 = \. 2. cos (180 - 60) = - cos 60 - -|> 3. tan (180 - 45) = - tan 45 - 1, 4. cot (180 - 120) = - cot 120 = - f--^2)=4 \ o / o 5. sec (180- 135) = - sec 135 = - (- V2)= V2, 6. esc (180 - 150) = esc 150 = 2. 80 PLANE TRIGONOMETRY 54. Functions of an angle 180 -f in terms of functions of 6. Let the revolving line OA generate an angle #, of any magni- tude, and at the same time let OA' generate an angle whose magnitude is 180 + 6. As in the cases already considered, take OB = OB', and from B, B', draw perpendiculars BC, B'C', to XX'. The triangles OB 0, OB' C' , are, in each of the four figures, equal geomet- rically. The student should supply the proof. X- With the notation used in the previous cases we have, con- sidering only the actual lengths of the lines, and paying no attention to positive and negative signs, r = r r , x = x\ y = y*. If positive and negative signs are taken into account, the last two equations become x = a;', y = y* respectively. The following equations then hold true for all possible cases : sin (180 + 0) = tf- = ^ = - sin <9, r r cos (180 + 0) = ~ = = - cos 0, TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 81 tan (180 + 0) = - = = tan 0, x' x cot (180 + 0) = ^ = = cot 0, y' -y sec (180 + 0) = ^ = ^~ = - sec 0, x' -y esc (180 + 0) = r - t = -- = - esc 0. y -* EXAMPLES. 1. sin (180 +30) = - sin 30 = -l 2. cos (180 + 45) = -cos 45=-iV2, 3. tan (180 +60) = tan 60= V3, 4. cot (180 + 120)= cotl20 = -iV3, 5. sec (180 + 1 35) = - sec 135 = - ( - V2) = V2, 6. esc (180 + 150)= -esc 150 = -2. 55. In a manner precisely similar to that employed in the preceding sections, we can determine the functions of an angle 270 6 in terms of functions of 6. The figures for each quadrant should be constructed by the student, and the values obtained, as in the cases which have just been considered. These relations, true for all values of 0, are as follows : sin (270 -0) = - cos 0, cos (270 -0)=- sin 0, tan (270 -(9)= cot0, cot (270 -6)= tan0, sec (270 - 0) = - esc 0, esc (270 -0)=- sec 0. 56. The corresponding values of functions of an angle 270 -f in terms of functions of can also be obtained in a manner similar to that employed in the cases already discussed (Art. 50-54). These values are as follows: sin (270 + 0) = - cos 0, cos (270 +0)= sin0, CONANT'S TRIG. 6 . vV 82 PLANE TRIGONOMETRY tan (270 + #) = -cot (9, cot (270 4- 0)=- tan 6, sec (270 + #)= csc0, esc (270 + 0)= -sec (9. EXAMPLES. sin (270 - 210) = - cos 210 = - ( - \ V3) = J V3, cos (270 - 150):= - sin 150= - |, tan(270 + 185) = -cot 135 =-(-l)=l, cot (270 -2 10 = tan240 = V3 sec (270 + 30) = esc 30 = 2, esc (270 + 60) = - sec 60 = - 2. 57. Functions of an angle 360 + 9 in terms of functions of 6. When the revolving line lias generated an angle 360 + #, its position is the same as that occupied after it has generated the angle 6. Hence, The functions of an angle 360 + 6 are the same as the cor- responding functions of 6. Also, since the revolving line returns to its initial position after any number of complete revolutions, in either a positive or negative direction, it follows that, when n is any positive or negative integer or zero, Functions of an angle n x 360 4- 6 are equal to the correspond- ing functions of 6. In a similar manner it may be shown that the functions of n x 360 9 are equal to the corresponding functions of 6. 58. By means of the equations contained in Arts. 50-57, pp. 74-82, the functions of any angle can be found in terms of functions of an angle less than 90. For example, gin 2m > = ^ ( - x m , + 3510) = sin :JJ31 = sin (270 + 81) = - cos 81. Similarly, cos ( - 2058) = cos 2008 = cos (5 x 860 + 258) = cos -_>:>s = cos (270 - 12) = - sin 12. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 80 o By reductions of this kind it is easy to find the values of func- tions of any large angle, either positive or negative. Multiples of 360 should first be subtracted, and the remainder of the reduction performed by the theorems of this chapter. 59. The following table contains the values of the functions of the angles between and 360 which are of most frequent occurrence in elementary mathematics. sine cosine tangent cotangent secant cosecant 30 45 GO 90 120 135 150 180 270 1 2 M ivs 1 H H 1 2 - 1 1 H H 1 2 i 2 -I* -H -1 () |V3 i V3 00 - >/3 -i > CO 00 V3 i H -H - 1 -Va *> 1 N V2 2 co -2 -V2 o -5 VI - 1 co 00 2 V2 h 1 fvs V2 2 CO _ 1 NOTE. In the above table the double sign, which is used wherever the value co occurs, signifies that either the positive or the negative value is obtained according as the revolving line approaches the given position from the one or from the other side. For example, tan 90 = + co if the revolving line approaches the positive portion of the y-axis from the right, i.e. through positive rotation; tan 90 = co if the revolving line approaches the same position from the left, i.e. through negative rotation. EXERCISE XI Prove that 1. sin 210 tan 225 + cos 300 cot 315 = - 1. 2. cos 240 cos 120 - sin 120 cos 150 = 1. 3. tan 120 cot 150 + sec 120 esc 150 = - 1. 4. tan 675 sec 540 + cot 495 esc 450 = 0. 84 PLANE TRIGONOMETRY 5. For what values of A between and 360 are sin A and cos .4 equal? For what values are tan A and cot A equal? 6. What sign has sin A -{-cos A for the following angles? A = 120; A=135 ; A = 150; A = 300; A=315; A = 690 ; x 7. What sign has sin A cos A for each of the following angles? A = 210; A =225; A = 240; A = 300; A =625; A = . o 8. What sign has tan A cot A for each of the following angles? A =60; A = 120; A =135; A = 150; A = 210; A = 225; A = il^. 6 9. Find all the angles less than 360 that satisfy the fol- lowing relations : =_-^?; (6) cos 6 = - ^?; rr+ ITT. Another general expression for the same angles is (2 n -f 1) TT | TT. GENERAL EXPRESSION FOR ALL ANGLES 89 Find the general value of which satisfies each of the follow- ing equations : 4. sin0=lV2. 13. tan0 = JV3. 5. sin 0=1. 14. sin20 = . 6 . sin0=-|V8. 15 COS 20 = |. 7 ' s in * = -J- 16 . 8 tan 2 = 1. 8. COS0=W3. 17. 3 sec 2 = 4. 9. cos0 = - 1 |V2. 18. cot 2 = 1. 10. cos = 0. 11. cos0=-l. 19 - tan 2 0=2 sin 2 0. 12. tan = 1. 20. 2 tan 2 = sec 2 0. 21. What is the general value of that satisfies both of the following equations ? sin = J V3, and cos = J. 22. What is the general value of that satisfies both of the following equations? sin = l, and cos = J V3. In the following five examples, show that the same angles are indicated by both the given expressions. 23. mr + -, and 24. W7r +(_ 25. M7T ^, and WTT + TT 6 6 26. WTT + ^, and mr + o o 27. \\ 90 PLANE TRIGONOMETRY 68. An equation involving trigonometric functions of an unknown angle is called a trigonometric equation. The solution of a trigonometric equation involves the de- termination of all angles that satisfy the equation. In solving a trigonometric equation, the smallest positive angle that satisfies it should first be determined, and then the general value should be found for all angles that satisfy it. This has been illustrated in the examples of Exercise XII, and will be still further shown in those of the following set. EXERCISE XIII 1. Solve the equation cos 2 6 + 2 sin 2 6 = J. This may be written 2-2cos 2 = . .-. cos' 2 = f . The smallest positive angle whose cosine is - V3 is 30, or -. Therefore, using the positive result, = 2 rnr - Alp, the smallest positive angle whose cosine is - \/3 is 150, or f TT. Therefore, using the negative result, = 2 rnr | TT, or (2 n + 1) TT -. 6 These two sets of values may be combined in the single expression 2. Solve the equation 2 cos 2 V3 sin + 1 = 0. This may be written 2-2 sin 2 - V3 sin 6 + 1 = 0. 2 sin 2 + V3 sin 0-3 = 0. Factoring, (sin + V3) (2 sin - V3) = 0. .-. sin = - V3, or sin = | V3. The sine of an angle cannot be numerically greater than 1; therefore, the first equation gives no solution. The smallest positive angle that satisfies the equation sin = \ V3, is = 60, or 5, o and (Art. 62) the general expression for the value of all angles that have the same sine as 60 is TT 8 = n7r+ (~l) n |- Therefore, the most general expression for all angles that satisfy the original equation is mr + ( l) n f GENERAL EXPRESSION EOR ALL ANGLES 91 3. Solve the equation tan 4 cot 3 6. This may be written tan 4 = tan I * - 3 0\ by Art. 51, = tan (nir + f - 3 6\ by Art. 65. .-. 4 = riTT + - - 3 0, Solve the following equations, finding the general value of in each case : 4. 2 sin 2 0- cos = 1. 16. sin 30 = sin 90. 5. tan 2 + sec = 1. 17. cos 6 = cos 2 0. 6. cot 2 0- esc = 1. 18 C os40 = cos50. 7. cos 2 - sin 0=1. 19. cos mv = cos r&0. 8. 2 sin 2 + 3 cos = 0. 20. cos 4 = sin 2 0. 9. 2 cos 2 + cos 0= 1. 21. sin 4 .0 = cos 2 0.. 10. sin 2 - 2 cos + I = 0. 11. 3sin 2 0-2sin0=l. 22. "tan 2 = tan 30. 12. 23> cot 5 6 = COt 2 13. csc20-cot0=3. 24 - tan 4 = cot 5 0. 14. tan 2 + cot 2 = 2. 25. tan ^0 4- cot nO = 0. 15. sin 50= sin 2 0. 26. tan 2 tan = 1. M CHAPTER VIII RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS OF TWO OR MORE ANGLES 69. Sine and cosine of the sum of two angles. Let x and y be acute angles, and let x -f y be either acute or obtuse. In both figures the lettering is so arranged that the following demonstrations apply to either case, o D F o F = Z From C, any point in OB, draw CD _L XX', and CE JL OA ; and from E draw EH \\ XX' and EFXX'. Since Z x - Z OEH = 90 - ..-. Z.x = Z.HCE. T\f~V Then we have sin (x + y) = =- : - ^ J) OC Also, OC = oc + oc ~ ~OE ~OC + ~CE OC = sin x cosy + cosZ HCE sin y. . . sin (a? + ?/) = sin x cos y + cos a? sin y. OP OC OF - DF (1) cos (x + y} = OC = OF HE OC OC = OF OE HE CE OE OC CE OC cos x cos y sin Z HCE sin y. cos (x + y) = cos as cos y sin a? sin ?/. 92 (2) RELATIONS BETWEEN TWO OR MORE ANGLES 93 70. The above proofs are given only for the case when both x and y are acute. To prove the formulas true for all values of x and y we _proceed as follows : Let x and y be acute angles, and let x l = 90 + x; .then we have (Art. 52), sin x } = cos x, and cos a^ = sin x* (1) Then, sin (aj + #) = sin (90 + x + y) = cos (z + ?/), (Art. 52) (2) where # and y are both acute angles. But (Art. 69, p. 92) when # and y are both acute angles, cos (x + y) cos x cos y sin # sin #. Substituting in this equation the values given in (1) and (2), we have sin (a?i + y} sin a?i cos y + cos a?i sin ?/. Q.E.D. In like manner, cos (x^ + ?/) = cos (90 + a; + ?/) = -sin(a; + y), (Art. 52) (3) where x and y are both acute angles. But (Art. 69, p. 92) when x and y are both acute angles, sin (a: + ?/) = sin # cos ?/ cos x sin y. Substituting in this equation the" values given in (1) and (3), we have cos (a?i + t/) = cos xi cos ?/ sin oc\ sin ?/. Q.E.D. Formulas (1) and (2) (Art. 69, p. 92) have now been proved for the case when x is obtuse and y is acute. Letting y^ = (90 + /), and proceeding in the same manner, we can establish these formulas for the case when both angles are obtuse. Then, letting a; 2 =90 + a: 1 , ^ = 90 + ^, x s =90-\-x^ etc., and proceeding in a precisely similar manner, we can establish the formulas for all possible values of x and y. 71. Sine and cosine of the difference of two angles. Let x and y be two acute angles, placed as represented in the figure. It is here assumed that x > y. From C, any point in the final position of the generating line OA, draw CD 1 OX and CE A. OB. Prolong DC, and from E draw EH \\ OX, inter- secting DC produced in H. Since / X 94 PLANE TRIGONOMETRY Then, sin (x - y) = - = FE- HC OC = FE OE HC EC OE OC EC OC sin x cos y cos Z ECU sin y. .-. sin (a? -y}= sin a? cosy cosa?siny. (1) In like manner, nn OF + 06' = OF EH OC OC = OFOEKH EC_ OE OC EC OC = cos x cos ij + sili Z EC/7 sin y. . : cos (a? y ) = cos . cos ?/ + sin x sin ?/. (2) These proofs have been given on the assumption that x > y. To prove that they are true when x < y, we proceed as follows : sin (x -, y} = sin [ - (y - x)] = sin (?/ #), (Art. 50, p. 75) = sin ^ cos -f cos y sin x, or, rearranging the terms and the factors in each term, sin (a? y) = sin a? cos 2/ cos a? sin?/. Q. E. D. (3) In like manner, cos (x - y} = cos \_-(y - x~}~\ - cos (y - x) (Art. 50) = cos y cos x -f- sin ?/ sin #, or, rearranging the factors in each term, cos (x y} = cos x cos y + sin a? sin y. Q. E.D. (4) 72. The formulas of Art. 71 have now been proved for all cases when x and y are both acute angles. To prove that they are true for all possible values of x and ?/, we proceed as follows : Let x and y be acute angles, and let x\ 90 + x. Then, sin X L = cos x, and cosa,^ = sin a:. (1) Then we have sin (x l y) = sin (90 + x y) = cos(x - y). (Art. 52) (2) RELATIONS BETWEEN TWO OR MORE ANGLES 95 But since .r and // are acute angles, cos (x y} = cos x cos y + sin x sin y. (3) Substituting in (3) the values given in (1) and (2), we have sin(a?i y} = sin a?i cosy - cos^isiny. Q. E. D. (4) In like manner, cos (.r t - y) = cos (90 + x - y) = - sin (x - y). (5) But since x and y are acute angles, - sin (x - y} = - (sin x cos y - cos x sin y). (Art. 71, p. 94) (6) Substituting in (6) the values given in (1) and (5), we have cos (a?i y) = cos a?i cos y + sin a?! sin t/. Q. E. D. (7) Formulas (1) and (2) (Art. 71, p. 94) have now been proved for the case when x is an obtuse angle and y is an acute angle. Letting y 1 = 90 + ?/, and proceeding as before, we can es- tablish these formulas for the case when both angles are obtuse. Then, letting x 2 = 90 + ^, # 2 = 90 + y v x 8 = 90 + x v etc., and proceeding in a precisely similar manner, we can establish the formulas for all possible values of x arid y. EXERCISE XIV 1. Find the value of sin 75. sin 75 = sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30 = _L ^1 + JL1 V2 2 \/:2 2 = V3 + 1 2\/2 2. Find the value of sin 15. si nl 5 = sin (45 -30) = sin 45 cos 30 - cos 45 sin 30 1 \/3 1 1 ~ \/2 2 V2 2 = V3-1 2V2 96 PLANE TRIGONOMETRY 3. Find the value of cos 105. cos 105 = cos (60 + 45) = cos 60 cos 45 - sin 60 sin 45 = i i Va i 2 V2 2 V2 2 "s/2 4. If sin a = | and sin /3 = ^|, find sin ( /3). 5. If sin a = | and cos ft = if, find cos (a -f /:?). ^ 6. If cos a = &, and cos ft = -|, find cos (a /3). Prove that 7. si 8. sin 105 + cos 105 = cos 45. 9. sin 75- sin 15 = cos 105 + cos 15. 10. sin (45 - 6) cos (45 -)- cos (45 - 0) sin (45 - 0) = sin ((/> 6). HINT. Let # = 45 - and y = 45 - <. Then compare with (1), Art. 66. The converse application of the x-y formulas, as illustrated by this example, is of frequent occurrence. 11. sin (45 + 6) cos (45 - () + cos (45 + 0) sin (45 - 0) = cos (6 $). 13. cos (30 + ) cos (30- a) + sin (30 + ) sin (30 - ) = cos 2 a. 14. cos a cos (/3 ) sin sin (/3 a) = cos /3. 15. sin (n + 1) sin (w 1) + cos (n + 1)<* cos (w- 1) = cos 2 . 16. sin (n + l)a sin (n + 2)a -f cos (n + 1) cos (n + 2) = cos a. 17. sin (a - jB + 15) cos (/3 - + 15) - cos (a 0+ 15) sin (/ a + 15) = sin (2 2 yS). RELATIONS BETWEEN TWO OR MORE ANGLES 97 The following examples are of especial importance, and are often used as standard formulas. 18. sin 75 = cos 15 = _ 2V2 19. sin 15 - cos 75 - 2V2 20. cos (a? + y) cos (a? y) = cos 2 a? sin2 y. 21. sin (a? + -*/) sin (ac y} = cos 2 y cos 2 sc. 73. Tangent of the sum and of the difference of two angles. For all values of x and y we have (Art. 69) sin (x + y) = sin a; cos y + cos a: sin y, and cos (a; + y) = cos # cos y sin a; sin y. tan Q + y) ^ sin *' cos ^ + cos x sin ^ . cos x cos y sin x sin ?/ Dividing both numerator and denominator by cos a: cosy, we have sin x cos y cos ar sin y ,. cos a: cos \i cos a: cos y tan (a? -f y) = 1 tan a? tan y In like manner, . sn arsn cos a: cosy (1) tan * - cos (a: - y) _ sin x cos y cos x sin y cos x cos y + sin x siti y sin a; cos y cos a: sin y _ cos a; cos y cos a: cos y ~~ cos a: cos y sin a: sin y cos a; cosy cos a: cosy sin a: _ sin y cos a: cos y . (2) 1 -f tan 35 tan i/ CONANT S TRIG. 98 PLANE TRIGONOMETRY 74. Cotangent of the sum and of the difference of two angles. For all values of x and y we have Expanding cos (x -f ?/) and sin (x -f- ^), dividing both numerator and denominator by sin x sin ?/, and reducing, we - (1) coty In a similar manner it can be proved that uin*^ \j\rii y ~t /"O\ coty cota? 75. Formulas (1) and (2), Art. 69, (1) and (2), Art. 71, (1) and (2), Art. 73, and (1) and (2), Art. 74, are often re- ferred to as the addition and subtraction formulas. The addi- tion formulas are sometimes known as the x + y formulas, and the subtraction formulas as the x y formulas. When refer- ence is made to both groups together, the general expression, "the x-y formulas," is often employed. 76. From the formulas for the functions of the sum of two angles the formulas for the functions of the sum of three angles are at once obtained, as follows : sin ( x + y + z) = sin [(x + y) + z] sin\(.r + y) cos z -f cos (x + /y) sin z\ = (sin x cos y + cos x sin y) cos z + (cos x cos// sin a: sin y} sin z. .*. sin (x + y + z) = sin a; cos// cos z -f cosx siny cosz -f cos x cosy sin z sin x siuy sinz. (1) In like manner it can be proved that cos (x + y + z) cos x cosy cosz - cosx siny sinz sin x cosy sin z sin x sin y cosz, (2) and that cos (x + y + z) _ tan x + tan y + tan z tan x tan /y tan z ,.,, 1 tan x tan y - tan x tan z tan y tanz ^ RELATIONS BETWEEN TWO OK MORE ANGLES 99 EXERCISE XV 1. If tan = | and tan ft 1, iind tan ( + ft). 2. If tan a = | and tan /3 = f f, find tan (_). 3. If tan = f and cot ft = -f%, find cot (a -f /8). 4. If tan = | and ft = 45, find tan ( + /3). 5. If tan = -|- and tan /8 = , find tan (2 a -f ft). 6. If tan a = ^ and tan /9 = - - - , find tan (a + ft). n + l 2 W -f 1 7. If tan a = | and tan ft = Jj, prove that a -f ft 45. The next four examples are of especial importance, and are tan 75" = cot 16 = 2 often used as standard formulas. 8. , , cot (9-1 12. cot - + = 10. tanl5-cot75=2-V3. 14. tan [ -f 15. Prove the identity cos ( 4- ft) cos ft -+- sin (a + ft) sin ft = cos a. HINT. Let a + /? = x and /? = y. Then compare with (2), Art. 69. Many of the remaining examples can be worked without difficulty by applying the addition or subtraction formulas directly. 16. sin 2 a cos a -f- cos 2 a sin a = sin 3 a. 17. sin 3 a cos cos 3 a sin = sin 2 a. 18. cos 3 a. cos 2 a sin 3 a sin 2 a = cos 5 a. sec esc (x. 20. sin (60 + ) cos (30+ a) - cos (60 + a) sin (30 + a) = J. 21. tin2+tan = tftn 3 g> 1 tan 2 tan a 22. - =tan2c . 1 Un( + /8)tan( - ft) 100 PLANE TRIGONOMETRY tana- tu .- 23 . 1 + tan a tan (a cot 3 a cot 2 a 4- 1 24. = cot a. cot 2 a cot 3 a 25. tan 2 - tan 6 = tan sec 2 0. 26. sec 2 6 = 1 + tan 2 tan 0. 27. csc20 = cot0-cot20. tan 3 6 tan 2 tan 4 6 tan 3 tan 3 tan 20 1 + tan 4 tan 3 4tan0 29. tan (45 4- 0) - tan (45 - 0) = JL tan u sin (# 4- y) cot # 4- cot y 3O. - ^ - = - cos (x T- y) 1 4- cot # cot y 77. The algebraic sum of two sines or of two cosines in the form of a product. For all values of x and y we have (Arts. 69 sin (x 4 ?/) = sin x cosy 4 cos x sin y, and sin (a; y) = sin a: cos y cos # sin y. Adding and subtracting, we have sin (x 4 y) 4 sin (a; y} = 2 sin # cos y, (1) and sin (x 4 #) sin (x #) = 2 cos x sin y. (2) Also (Arts. 69 and 71), cos (a: 4 y) = cos x cosy sin a; sin y, and cos (a? y) = cos a: cos y 4 sin re siny. Adding and subtracting, as before, we have cos (x 4 y) 4 cos (ar ?/) = 2 cos a: cos y, (3) and cos (a; 4 y) cos (a; y~) = 2 sin a; sin y. (4) Let a: 4- y u, and x y = v. Solving these two equations for x and y, . Substituting these values of a: and ?/ in (1), (2), (3), and (4), we have smu 4 sinv = 2sm^^cos^^$ (5) ** A^--\r sin w - sin v = 2 cos ^ + ^ sin^-^T (6) cost* + cost? - 2cos^pcos^p; (7) cost* - cos v = - 2sin^^ sin-^. (8) RELATIONS BETWEEN TWO OR MORE ANGLES ; I01 These formulas are among the most important of all' the formulas of trigonometry. The student should commit them carefully to memory, and become perfectly familiar with their -application. They will sometimes be referred to as the u-v formulas. As illustrations of the manner in which certain expressions can be simplified by the application of one or more of these processes, the following examples are given : 2. sin 75 - sin 1 5 _ 2 2 = 2 cos 40 sin 30 = cos 40. 75 + 15 . 75 -15 2 cos sin cos 75 + cos 15 n _ _75 + 15 75 - 15 cos _ 2 _ 2 cos 45 sin 30 ~ 2 cos 45 cos 30 = tan 30 = iV3= 0.57735. (sin 6 + sin 2 0)(cos 2 - cos 4 6) (sin 5 6 + sin 0)(cos 3 - cos 5 0) _ (2 sin 4 cos 2 0)(2 sin 3 sin 0) = ~ (2 sin 3 cos 2 0) (2 sin 4 9 sin 0) EXERCISE XVI Prove the following relations : l. sin 70 + sin 50 = V3 cos 10. 2 sin80-sin60 = tan ^ 3 . sin 2 + sin 6 cos 8 6 + cos 6 6 cos 2 6 + cos 6 6 sinSg-rintf 8g 4g- sin 2 ^ + sin 2 fl = fan ^ ^ ^ _ B sml A- sin 2 5 K.)L' . PLANE TRIGONOMETRY cos cos 20 2 * cosJ.+ cos cos B cos ^4. 2 9. sin ( J. + JB) + cos (A - J5) = 2 sin(45 + 5) cos (45 - cos b A cos 3 ^4 , cos 2 A cos 4 ^4 si n A 10. sin <) A sin 3 .A sin 4^4 sin ^ A cos 4 ^4 cos 3 A 11. sin (60 + A)- sin (60 - A) = .sin A 12. cos (30 - 0) + cos(30 + 0; - V^ cos 0. 13. 14 sin 6 + sin 3 + sin 5 + sin 7 = t 4 ^ cos H- cos 30 + cos 50 + cos 7 15 sin - sin 50 + sin 9 - sin 18 = cot Q cos cos 5 cos 90 + cos 13 16 _ sin x sin ^/ cos x -\- cos v ^ ^ 17. = cot ' cot cos x cos y 2 2 18. cos 3 + cos 5 + cos 7 + cos 150=4 cos 4 cos 5 cos 6 0. 19. 20. sin 50 + sin 10 -sin 70 = 0. 2 sin (3 A + i?) + sin(J. 3 ^ 22. sin 80 + sin 70 - sin 10 - sin 20 = -t sin 40 4- sin 50. 23. cos x + cos 2 # + cos 4 # + cos 5 # = 4 cos ~ cos cos 3 r. RELATIONS BETWEEN TWO OR MORE ANGLES 103 24. sin O + P + 7) + sin (a - yS - 7) + sin (a + /3 - 7) -f- sin ( @ -f 7) = 4 sin a cos /? cos 7. 25. sin 2 a + sin 2 /3 + sin 2 7 sin 2 (a -f ft -f- 7) = 4 sin (/3 + 7) sin (7 + a) sin (a = cos 3 6 cos 3 6 + 2! cos 5 6 + cos 7 ~ cos 5 sin 30 + 2 sin 50 + sin 10 . c 27. __^ =sm 5 S1H0 + 2 sin 30 + sin 00 sin J + # - 2 sin A + sin Qi - 28. cos (^4 + ^) 2 cos J. + cos ( A B} 29. cos( sin (x + y -|- 2) + sin( ^ + y + ^; sin (2: y-\-z)+ s\ = cot ^. so. cos 20 + cos 100 + cos 140 =0. 78. The product of two sines, of two cosines, or of a sine and a cosine expressed in the form of an algebraic sum. In (1), (2), (3), and (4), Art. 77, the u-v formulas are ex- pressed in a form which is quite as important as that already considered, and which is so convenient, and of such frequent application that the formulas are here reproduced in that form. Using the left for the right and the right for the left members, they are 2 sin oc cos y = sin (& + y} + sin (oc - y) ; (1 ) 2 cos ac sin y = sin (x + y} - s :n (& - y} ; (2) 2 cos w cos y = cos (oc + y} + cos (x y} ; (3) 2 sin x sin y = cos (a? + y) - cos (x y). (4) These formulas are the converse of the u-v formulas, and may be conveniently referred to by that name. The two groups taken together are useful in solving problems and in performing investigations which, without them, could be handled only with the greatest difficulty. 104 PLANE TRIGONOMETRY EXERCISE XVII 1. Express in the form of a sum or difference 2 sin 6 sin 4 6. 2 sin 60 sin 4 = - (cos(60 + 40)&cos(60 - 40)) = - (cos 100 -cos 20) = 00820- COS 100, 2. Express in the form of a sum or difference cos (A 2B) sin (04 + 2 B). cos(^ -2B)sin(A +25)= | (sin 2/1 - sin (-45)) = (sin 2^4 + sin 45). 3. Find the value of 2 sin 75 sin 15 . 2 sin 75 sin 15 = cos (75 - 15) - cos (75 + 15) = cos 60 - cos 90 = i-o = * Express as a sum or difference the following: 4. 2 sin 60 cos 26. 30 8. COS - COS . 5. 2 cos 40 sin '20. 6. cos ? sin ?*. 9- 2 sin (2 ,1 + 10 cos (A-*). 2 2 -* ** 10. 2 cos 3 J. cos 01 2*). o u 7 " 1 T C "T* 11. sin (60 + 0) cos (60 - 0). Prove the following identities : 12. cos (120 + 0) cos (120 - 0) = (2 cos 2 - 1). 13. cos (30 - 0) cos (60 - 0) = | (2 sin 20 + V 3). 14. sin (120 - 0) cos (60 + 0) = J- (sin 60 - 2 0). 15. sin (0 + 45) sin (0 - 45) = - | cos 2 0. 16. cos 3 sin 2 - cos 4 sin = cos 2 sin 0. 17. sin 3 sin 6 + sin sin 2 = sin 4 sin 5 0. 18. sin 20 cos + sin 6 cos = sin 3 cos 2 + sin 50 cos 20. 19. cos (40 - 0) cos (40 + 0) + cos (50 + 0) cos (50 - 0) = cos 2 0. RELATIONS BETWEEN TWO OR MORE ANGLES 105 20. sin A cos {A + B) cos A sin (A B) = cos 2 -4 sin B. 3 7T 4 7T . 47T, 10 7T A 21. 2 cos - cos -f cos - -f- cos = 0. 22. 4 sin A sin ^ sin Q = sin (5 + (7 J.) -f sin ((7+ .A B) + sin ( j. + B - Q) - sin (A + ^ -f 6 Y ). cos 3 ^4. sin 2 A cos 4 J. sin A _ _ t o >4 cos 5 A cos "2 A cos 4 ^4. cos 3 .A 24. 4 sin sin (60 + 0) sin (60 - (9) = sin 3 0. 25. 4 cos cos + cos Z - = cos 3 6. v 8 / \ o 26. sin 20 sin 40 sin 80 = \ V3. 27. cos 20 cos 40 cos 80 =i. CHAPTER IX FUNCTIONS OF MULTIPLE AND SUBMULTIPLE ANGLES 79. Functions of an angle in terms of functions of half the angle. If in the addition formulas, Arts. 69, 71, 73, and 74, we put x = y, we have sin (x 4- x) = sin x cos x -\- cos x sin a?, cos (x + x) = cos x cos x sin x sin #, tan (* + *)= *" * + **"*, 1 tan x tan # and c cot x -f- cot x sin 2 x = 2 sin a? cos a? 5 (1) cos 2 = cos 2 a? - sin 2 a?; (2) ; (3) (4) In these formulas 2# may have any value whatever; or, in other words, 2 # is any angle whatever. Hence, these formulas are to be regarded as formulas for expressing the values of functions of an angle in terms of functions of half the angle. They may also, of course, be re- garded as formulas for expressing the functions of twice an angle in terms of functions of the angle itself. 80. If we let 2x= 0, we have the formulas in the following useful form : ~ ^ sin6 = 2 sin - cos -; (1) 10<5 MULTIPLE AND SUBMULT 1PLE ANGLES 107 cos = cos 2 - - sin 2 - (2) 2 2 2cos 2 --l. 2tan- (3) 1 - tan 2 - 2 cot 2 - - 1 cot 6 = - 2cot- 2 81. Functions of an angle 3 & in terms of functions of oc. If in the addition formulas we put y = 2 x, we obtain expres- sions for the value of functions of 3 # in terms of functions of #, as follows : sin (a; + 2 a;) = sin x cos 2 x 4- cos x sin 2 x = sin x (cos 2 a; sin 2 a:) + cos a: 2 sin a: cos a; = sin x (1 2 sin 2 a;) + 2 sin x (1 sin 2 a;) = sin x 2 sin 3 a, 1 + 2 sin a: 2 sin 3 a:. .-. sin 3 3C = 3 sin a? 4 sin 3 cc. (1) In like manner, cos (a; + 2 a:) cos a: cos 2 a; sin x sin 2 a: = cos a: (cos 2 a; sin 2 a:) sin x 2 sin a? cos a: = cos a: (2 cos 2 a; 1) 2 (1 cos 2 a;) cos a; = 2 cos 3 x cos x 2 cos x + 2 cos 3 a:. .. cos 3 a; = 4 cos 3 x 3 cos a% (2) Also, ten 3*= - 1 - tan z tan 2 x ^ _ fcftn ^ 2 tan x 1 tan 2 a; 3 3 tan a? ^tan 3 x XON /. tan 3 a? = - - . (3) In a similar manner it is possible to obtain formulas for the functions of higher multiples of x in terms of functions of x. 108 PLANE TRIGONOMETRY 82. Functions of an angle expressed in terms of functions of twice the angle. Since cos 2 x = 1 - 2 sin 2 x, we have 2 sin 2 ;r = 1 cos 2 x. .. sin a? = Also, cos 2 x = 2 cos 2 a: 1, 2 cos 2 s = 1 4- cos 2*. 1 + cos2ag *2\ Dividing (1) by (2) we have (3) These formulas are often given in the following form, where *=- (5) In this form they are to be regarded as formulas for express- ing the values of functions of a half-angle in terms of functions of the angle itself. The magnitude of the angle determines which of the two signs preceding the radical is to be employed. EXERCISE XVIII 1. If sin 6 = 1, find sin 2 9 and sin 3 0. 2. If sin = \, find cos 2 d and cos 3 0. 3. If cos 6 = |, find sin 20 and cos 30. 4. If tan 6 == l, find tan 2 and tan 3 0. 5. If tan 6 = 1, find sin 2 and tan 3 6. MULTIPLE AND SUBMULTIPLE ANGLES Prove the following identities : 6. cos 4 sin 4 = cos 2 0. cot tan 1 a an COt + tail 7. tan 4- cot = 2 esc 2 0. )t j ^T tan -2 cot 2 0-1 H5j T ^ The next six equations are especially important, and may^e regarded as standard formulas. 1 11 f S in 6 4 D S 6 Y 11 -in 9 18 r*e 2 ~ sec2 * s\\ ^ ' ^"g h *ij - sec 20 12. tan 9^ 8in26 ;,* sec0-.l J l + cos29 1 2 - 2sec l TO * A S * n ^ " A f A\ , XI i oA* 2O tanf 77 "-!- 1 2DI 1 i T\r 1 COS 2 U ^ u - ^ /i ttl11 I i 1 "c\ I' i p 1 - sm \4 2/ ^| V <* 9 i _ eos Q } 1 4- t*in i i Z> ; *** 1 1 6 2 sin 9 \ & ta "V9 ?- -| i , 9 _ 1 4- cos 9 eos (Jby ^/\ ^\^jf 2 sin 9 2 /A e\ 2 cos 20 _, ,, ro ^ 16. [ sin cos ] =1 sin 9. ^ i i ^i o /} 4 \ " */ ^ s^^ \ \* /i &i /-> o0 1 -h sec sin 3 cos 30 /- * 2 sec sin cos 11 ^ 1 cos J. + cos B cos (A + ./?) ^4 ^^-B & 1 + cos 4 cos B cos ( A + .#) " 2 ~ 2 r ~j 25. tan (45 + 0) + tan (45 - 0) = 2 ^ ' N^ 26. tan 2 - sec sin = tan sec 2 0. ^x v i ^*. 1 \ T*, siii 2 -sm 2 /3 Hn / rt + N T * . /i /-> """ Lail V 1 r*x sin a cos a sin p cos p ** \ f i OQ cos + sin cos sin _ 9 ^ o ^ h '-O cos sin cos + sin i' V ^ - L 110 PLANE TRIGONOMETRY 2g cos(0 + 15) _ sin (9 - 15) = 4 cos 2 sin (0 + 15) cos (6 - 15) 1 + 2 sin 20 30. 32 sin 20 + sin ;- pos :; +si "^=ta, 1 + cos 20 + sin 20 - tan + 1 = 1 - sin 2 tan + 1 cos 2 33 sin 2 1 - cos = 1 - cos 2 " cos '" 2 ' 34 sin (n + 1)0 + sin (ft- 1)0 + 2 sin w0 = cot cos (n 1)0 cos (w +1)0 2 35. . COS Sill 36. sin 6 + sin 4 - sin 2 = 4 sin 2 cos cos 3 0. 37. (sec 20 + 1) Vsec 2 - 1 = tan 2 0. 38. 4 cos cos (60 - 0) cos (60 + Q = cos 3 # 39. 16 cos 20 cos 40 cos 60 cos 80 = 1. 40. tan (45 + 0) = X /f M sin 2 41 sin (n + 1)0 - sin (n - 1)0 _ t 0. cos (ft + 1) + cos (n 1) + 2 cos w0 ~~ 2 42. cos 2 O + 1)0 - cos 2 nO = - sin (2 n + 1)0 sin 0. 83. Identities that are true for angles whose sum is 180 or 90. When three angles are involved whose sum is either 90 or 180, many relations are found to exist that do not hold true for angles in general. For, if A + B + C= 180, we have (Art. 53, p. 78), sin (A + B) = sin C, cos (^4 + B) = - cos (7, tan (A+B)=- tan (7, MULTIPLE AND SUBMULT1FLE ANGLES 111 and similar relations hold between functions of the sum of any two of the given angles, and the corresponding functions of the third angle, since the sum of any two is the supplement of the third. Also, if (- +- = 90, the sum of any two of these angles is the complement of the third. Therefore, in (I + f ) = cosf , cosg H- f ) = sinf , tan (f + f)= cotf , sn and similar relations hold between functions of the sum of any two of the angles and the corresponding co-functions of the third. Ex. 1. If A + B + O= 180, prove that sin 2 A -f sin 2 B sin 2 = 4 cos A cos B sin O. Left member = 2 sin (A + 5) cos (4 - B) - 2 sin C cos C = 2 sin C cos(.4 - ) + 2 sin C cos (A + ) = 2 sin C [cos (.1 + B) + cos (.4 - 5)] = 2 sin C (2 cos A cos 7?) = 4 cos -4 cos R sin C. Ex. 2. If A -h 5 + # = 180, prove that ABO cos A + cos # + cos (7= 1 + 4 sin sin sin . Left member = 2 cos -- cos - +1-2 sin a /^ A T> SI = 1 + 2 sin cos ^- - 2 sm 2 . = 1 + 2 sin 112 PLANE TRIGONOMETRY Ex.3. If A + B+ (7=180, prove thai tan A + tan B + tan (7 = tan A tan B tan 0. Since A + B = 180 - C, tan (^4 + 5) = - tan C ; tan A + tang = _ tftn c< 1 tan /I tan B Clearing of fractions, tan A + tan B = tan C + tan A tan .-. tan .4 + tan B + tan C = tan /I tan B tan C. EXERCISE XIX If A + B + C= 180, prove that 1. sin 2 A 4 sin 2 B + sin 2 C = 4 sin A sin B sin C. 2. cos 2 A -f- cos 2 .B 4 cos 2 (7= 1 4 cos A cos .B cos 3. cos 2 A - cos 2 4 cos 2 (7= 1 -4 sin A cos B sin (7. 4. sin 2 A sin 2 B sin 2 "(7= 4 sin A cos B cos (7. A .B (7 5. cos, A + cos B cos 0= 1 + 4 cos cos sin . A B C 6. sin A 4 sin B 4 sin (7=4 cos cos cos . 7. sin A -f sin I? sin C 4 sin mt cos . i- ^ ' 8. sin 2 A + sin 2 B sin 2 (7=2 sin A sin .Z? cos (7. 9. cos 2 A + cos 2 5 - cos 2 (7= 1-2 sin A sin .B cos (7. 14 - CW - '- : 0j f a,/;.: sin A 4 sin .B sin (7 A, .Z? * A*>;*>; 4 >l 1 I T- ' 1 1 < ! CHAPTER XI THE GENERAL SOLUTION OF TRIGONOMETRIC EQUA- TIONS 89. A trigonometric equation is an equation in which the un- known quantity or quantities appear in the form of trigono- metric functions. These equations have been used with the utmost freedom in previous chapters, though no formal definition has been given until the present time. They have been used in many differ- ent ways, involving one or more functions, one or more angles, and one or more values of the given angles in any single equation. At first the only angles used were acute angles, and an equa- tion was understood to involve functions of an acute angle only. Then the idea was introduced of an angle unrestricted in magnitude ; and after this had been done, all results were freed from the restraints which had previously been imposed by the fact that we were dealing with acute angles only. A large class of the equations with which we have previously been concerned consist of trigonometric identities, that is, equa- tions in which both sides had the same value for all possible values of the angles employed, though the form might be different. Examples of these are the formulas that have been proved from time to time, as, sin 2 6 + cos 2 1 ; sin (x -f /) = sin x cosy + cos x sin y ; etc. Equations of this kind are true for all possible values of the angle or angles involved. But trigonometric equations are, of course, not ordinarily true for all values of the angles involved. For example, if we consider the equation cos Q _. \ we see at once that we can assign but two values of 6 between and 360 that satisfy this equation. In other words, 122 GENERAL SOLUTION OF TK1GONOMETIUC EQUATIONS 123 cos 6 = is true only for = 60 and 6 = 300, as long as is restricted to values between and 360. If angles of unre- stricted magnitude are allowed, cos 6 = J is satisfied by all values of 6 that are included in the general expression 0-Siirdbg, O and by no other values. In like manner, the equation is satisfied by all values of that are given by the general expression 0=W7T+^, o and by no other values ; sin 6 = J V2 by those values of 6 that are given by the expression and by no other values ; and so on for other examples that might be given. In all these illustrations it is to be under- stood that n is any positive or negative integer or zero. The solution of an equation is the determination of the value of the angle or angles that satisfy the equation. In Art. 67, p. 88, a method of solution was given by means of which some of the simpler forms of trigonometric equations could be treated. But at that time only a limited number of the formulas of transformation were at our disposal. Hence, the number of classes of equations that could be handled was necessarily quite limited. The methods of reduction and transformation that are now available make it possible to solve many classes of equations that were formerly quite out of our reach, and also to sim- plify some of the methods previously employed. The present chapter will illustrate some of the simpler cases of this kind. This work should be looked upon as an extension of that given in Art. 68, p. 90. 124 PLANE TRIGONOMETRY 90. Solution of equations of the form a cos 6 + b sin 6 = c. (1) A simple method of solving equations of this form is fur- nished by the introduction of what are termed auxiliary angles, as follows : Assume a right triangle whose legs are a, 6, and designate by (/> the angle lying opposite the leg b. The hypotenuse of this right triangle is Va 2 -}- 2 , and we now have cos < = , and sin = Dividing each member of the original equation by Va 2 -}- 5 2 , we have cos + sin 6 = (2) Substituting cos and sin

cos H- sin sin = : G or, cos (0 <) = Since a, b, and c are known, cos (6 <) is known, and 9 < can at once be found from the tables. Calling this angle , for convenience we have cos (# <) = cos a. . . 6 = 2 MTT a, Art. 64, p. 87. # = 2 ftTT + ^ a. The cosine -of an angle can never be numerically greater than unity. Hence, in dealing with the equation cos (0 ) - it is to be remembered that we must have c ^ Va 2 + 2 . If c > Va 2 -f- b' 2 , there is no real value of <$> which will satisfy the equation. GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 125 Ex. i. Solve the equation V3 cos + sin 6 = V2. Dividing both sides of the equation by V3 + 1, i.e. by 2, we have In this case we have a = V3, 6 = 1, and Va' 2 + 6 2 = 2. Hence, the auxiliary angle is equal to 30. The original equation then becomes cos 30 cos 6 + sin 30 sin = |V2. cos(0-30)=|V2. But \/2 is the cosine of 45. Hence, we write cos (0-30) = cos 45. 4' 6 4* Ex. 2. Solve the equation 5 cos 6 4- 2 sin = 4. In this problem we have a = 5, and 6 = 2. Dividing both sides of the equation by Va 2 + ft 2 , we have JL cos + -?= sin = -t= (1) A/29 V29 A/29 In the preceding example we were able to find the value of < from the familiar coefficients and -, which we already knew were the cosine and sine respectively of 30. But in this example we have unfamiliar values to consider. From the figure on the margin of the page we see that is an o i^i angle whose cotangent is ~. Turning to the tables, we find that the o value of 4-45 ). For convenience we replace cos x 4- sin x by y, and then, form- ing the equation y = V2 sin (x + 45), we form the following table of values. Plotting the graph by the method explained in Art. 48, we have the following result. X y 1 45 V2 90 1 135 180 -1 225 -V2 270 -1 315 360 1 /T\ 273 ~Kd Since the greatest value that the sine of any angle can have is 1, the maximum value of this expression occurs when sin (x 4- 45) = 1, i.e. when z = 45. This gives V2 as the maximum value of the expression sin x 4- cos x. In like manner, the minimum value of the expression is - V2, which corresponds to the angle x = 225. COKANT'S TRIG. 9 130 PLANE TRIGONOMETRY If the table of values is extended, and the graph is plotted for values of x greater than 360, the values of ?/, i. e. of cos x + sin #, will be repeated in their original order ; that is, cos x + sinx is a periodic function with a period of 360. (See Art. 49, p. 71.) 93. When a or 5, or both a and , are different from unity, the process is slightly modified, as follows : a cos x+ b sin x = Va 2 + b 2 l ^ - cos x -\ - sin x ] = Va 2 -f- b 2 (cos x cos a + sin x sin a) = Va 2 -f- b 2 cos (x a) . Here, as is readily seen from the figure on the b margin of the page, it has been assumed that a is the angle whose cosine is a and whose sine is When a and b are known, a can be found, as in Va 2 + b 2 Art. 90, p. 124. The table of values can then be obtained and the graph con- structed, as in the preceding case. Since cos (x a) has 1 for its maximum value and 1 for its minimum value, the expression a cos x + b sin x has Va 2 + b 2 for its maximum value and Va 2 b 2 for its minimum value. NOTE. In computing the table of values for the purpose of constructing the graph, the values of y can always be obtained directly from the expres- sion as it is originally given, without any reduction whatever. This is sometimes preferable; and in certain cases, as for example the functions given in Examples 7, 9, 10, and 11 in the following set, it is easier to com- pute the values directly than to compute them after transforming the expression. EXERCISE XXIII Trace the changes in sign and magnitude of the following expressions as x increases from to 360. Find the period and construct the graph in each case. 1. sin x cos x. 5. sin x + V3 cos x. 9> cos 3 9. 10 ' Sm 8 ' 2. V3sinz + cosz. 6 . 2 sin x + 3 cos x. 11. tan 20. 3. sin* + V3cos*. 7. cos 20. ^ sin 20- sin* 4. V3 sin x cos x. 8. sin 6 cos 6. cos 2 + cos 6 CHAPTER XII THE OBLIQUE TRIANGLE 94. The law of sines. Let A, B, denote the angles of a triangle, and a, b, c respectively the sides opposite. From any vertex, as (7, draw CD perpendicular to AB, meet- ing AB, or AB produced, in D. A D B A B From the first figure we have Also, = b sin A. - = sm B. a .. h = a sin B. Equating these values of h we have b sin A = a sin B. From the second figure we have - = sm A. b = sn A. Also, whence as before, - = si b sin A = a sin 131 132 PLANE TRIGONOMETRY Therefore in either case we have the same result, b sin ^4.= a sin J9; a - sin A In like manner drawing perpendiculars from the vertices A and B to the opposite sides respectively we can prove that b _ sin G sin and c sm The results obtained in (1), (2), and (3) enable us to state the law of sines as follows : The sides of a triangle are proportional to the sines of the opposite angles. Equations (1), (2), and (3) are often combined and written in the following manner : sin A sin B sin 95. The geometric meaning of each of the three ratios just stated will be understood from the following : Let ABO be any triangle, and let a circle be circumscribed about the triangle. From the center to the vertices of the triangle draw the radii OA, OB, 00, respectively, and also draw OD perpendicular to AB. By geometry From this we have = r sin C. .*. c = 2 r sin C. In like manner it can be proved that a= 2rsin A* and b = 2 r sin B. . THE OBLIQUE TRIANGLE 133 Equating the values of 2 r obtained from these three equa- tions we have a i c 2r = - - = -r = -^-~ - That is, Bin .4 sin If sin O The ratio of any side of a triangle to the sine of the opposite angle is equal to the diameter of the circumscribed circle. 96. The law of cosines. Let ABO be any triangle, and let (7Z), the perpendicular from the vertex to the opposite side, meet AB, produced if necessary, in D. D B A From the first figure we have = 52 + ^2 _ 2 c - b cos A. 2 be From the second figure we have 2 = h* + BD* = h? + (AD - c) = 2 + AD*-2c = b 2 + c 2 2 c - b cos A. 2 be Therefore, the same result is obtained for both triangles. In like manner, drawing perpendiculars from A and B to the opposite sides respectively, we can prove that and cos B = cos C' = (2) (3) 134 PLANE TRIGONOMETRY Equations (1), (2), and (3) are often useful when expressed in the following form : (4) C. The law of cosines can now be stated as follows : The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice their product into the cosine of the included angle. yv 97. The law of tangents. We have already proved that, in . . i a sin A any triangle, - = Therefore, considering this equation as a proportion, and taking the four quantities by division and composition, a b__ sin A sin B a + b sin A + sin B 2 sin *** cos ^p? ~ L cot^i^tanAzi^. a-b 2 2 In like manner it can be proved that A-C tan and THE OBLIQUE TRIANGLE 135 The law of tangents can now be stated as follows : The difference of two sides of a triangle is to their sum as the tangent of half the difference of the opposite angles is to the tan- gent of half their sum. NOTE. In using the formulas of this section it is better to let the greater side and the greater angle precede the smaller in all cases. The formulas are true, whichever order is used ; but if the smaller side and the smaller angle precede the greater side and the greater angle respectively, negative numbers are introduced, and if logarithms are to be employed, these num- bers should be avoided whenever it is possible to do so. 98. The given parts. In the solution of oblique plane tri- angles four cases occur. In each case three parts are given, as follows : 1. One side and two angles. 2. Two sides and the angle opposite one of them. 3. Two sides and the included angle. 4. Three sides. The formulas developed in Arts. 94-97 are sufficient for the solution of every possible case that can arise. These cases will now be considered separately. 99. CASE 1. Given one side and two angles. Let the given angles be A and J5, and the given side a. The formulas for solution are as follows : b _ sin B 7 _ a sin B 2. - -, . . - ; a sin A. c sin > a sin JL' sin A Ex. i. Given a = 467, A = 56 28', B = 69 14'; find the re maining- parts. The work may be conveniently arranged as follows : C = 180 - (.4 + B) = 54 18'. (1) By natural functions. b = a x sin B - sin A = 467 x 0.9350 t 0.8336 = 523.8. c = a x sin C *- sin .4 = 467 x 0.8121 -* 0.8336 = 454.95. \\ 136 PLANE TRIGONOMETRY (2) By logarithms. log b = log a + log sin B log sin A = log + log sin B + colog sin A. log c = log a + log sin C log sin A = log a + log sin C + colog sin A. log a = 2.66932 log a = 2.66932 log sin B = 9.97083 - 10 log sin C = 9.90960 - 10 colog sin A = 0.07906 colog sin A = 0.07906 2.71921 2.65798 b = 523.85 c = 454.97 NOTE. To insure accuracy the student should check all results by solving each problem by a second method, entirely independent of the first ; or by the same method, using a different combination of parts. In the case under consideration it is usually sufficient to employ the same method, i.e. the law of sines, combining the parts in a manner different from that employed in the first place. For example, after c has been found we can solve again for b by the formula b = csm / f , as follows : sin C log c = 2.65798 log sin B = 9.97083 - 10 colog sin C = 0.09040 log b = 2.71921 b = 523.85 check. EXERCISE XXIV Solve the following triangles : 1. Given a = 438.3, A = 43 50' 24", B= 69 30' An*. C= 66 39' 24", b = 592.74, c = 580.*. 1*1 2. Given b = 421, A = 31 12', B = 36 20'. Ans. (7=112 28', a = 368.08, c = 656.63. { 3. Given a = 412, 4 = 58U', B = 65 37'. Ans. 0= 56 9', 5 = 441.37, c = 402.45. 4. Given 6 = 81.5, B = 43 44' 18", 0= 75 2' 42". 4 =61 13', a =103.32, c= 113.89. 5. Given c = 77.93, B = 22 15' 20", O= 41 50' 30". Aw. A = 115 5& 10", a = 105.07, 5 = 44.23. 6. Given c = 6.98, A = 25 7' 10", (7= 36 12' 24". Ans. B = 118 40' $", 4 = 5.016, b = 10.37. THE OBLIQUE TRIANGLE 137 7. Given a = 928.4, A = 61 17' 15", 6 V = 58 18' 40". Am. B = 60 24' 5", c = 900.78, ft = 920.45. 8. Given a = 328.4, A = 29 41' 12", B = 37 50' 24". Ana. C =11 2 28' 24", 5 = 406.77, c = 612.73. 9. Given A = 64 35', 0= 73 49', a = 213.47. Ans. B = 41 36', 5=156.92, c= 226.98. 10. Given ^1 = 41 23' 47", B = 124 49', 5 = 65.536. Am. 0= 13 47' 13", a = 52.788, c = 19.023. 11. Two points, A and ^, are separated by a body of water. In order to find the distance between them a line AQ is meas- ured 612.3 ft. in length, and the angles BAG, ACB are meas- ured and are found to be 49 17' and 68 11' respectively. What is the distance from A to B ? 12. It is desired to find the distance of a lighthouse A to each of two stations B, C, situated on shore, and in the same horizontal plane with the base of the lighthouse. BC is 21 miles, Z.ABO is 39 38', and ZACB is 74 56'. Find AB and AC. 13. The angles of elevation of a balloon that has ascended vertically between two stations one mile apart on a level plain, and in the same vertical plane with the balloon, are 29 41' and 37 17' respectively. What is the distance of the balloon from each station, and what is its vertical height above the plain ? 14. Solve the preceding problem on the supposition that both the stations are on the same side of the balloon. 15. To find the width of a stream a line AB, 351 ft. long, is measured on one side, parallel to the bank. On the opposite side of the stream a stake C is set, and the angles CAB, CBA, are observed and are found to be 38 17' and 31 29' respec- tively. What is the width of the stream ? 16. From the top and bottom of a column the angles of elevation of the top of a tower 236 ft. high standing on the same horizontal plane are 44 27' and 61 31' respectively. What is the height of the column ? 138 PLANE TRIGONOMETRY 17. When the altitude of the sun is 49 52', a pole standing on the slope of a hill inclined 16 55' to the level of the plain casts a shadow directly down the hill a distance of 45 ft. 8 in. What is the height of the pole ? 18. An observer in a balloon measures the angle of depres- sion of an object on the ground and finds it to be 63 58'. After ascending vertically 582 ft. he finds the angle of depression of the same object 74 49'. What was the height of the balloon at the time of the first observation ? 19. From a ship the bearings of two objects were found to be N.N.W. and N.E. by N., respectively. After sailing due east 10 miles the two objects were in a line bearing W.N.W. How far apart were the objects ? NOTE. For an explanation of the term "bearing," and for instruction in reading angles by means of the compass, see p. 176. 20. From a ship a lighthouse bears N. 21 12' E. After the ship has sailed S. 25 12' E. 2| miles the lighthouse bears due north. Find the distance of the lighthouse from the last point of observation. 100. CASE 2. Given two sides and the angle opposite one of them. Let the given parts be the sides a and &, and the angle A. The required parts can be found in the following manner : By the law of sines (1) sin A a a From this equation the angle B can be found. Then, C= 180 - (4 + B). Also, ^ = ^4, .'.c = ^?. (2) a sin A sin A In solving for the angle opposite the second side, in this case the angle B, it is to be noted that two solutions are pos- sible, since the sines of supplementary angles are equal (Art. 53, p. 79). The following considerations will determine the number of solutions for any given set of conditions. THE OBLIQUE TRIANGLE 139 If a > b, then A > B, and B is necessarily an acute angle, since a triangle can have but one obtuse angle. Therefore there is one and only one solution. If a = b, then A = B, and both A and B are acute angles. There- fore there is one and only one solu- tion, an isosceles triangle. If a < b, then A < B, and A is an FlG - 1- acute angle. In this case B may One solution, a> 6 be either acute or obtuse, and there will be two solutions if a > CD, the perpendicular drawn from the vertex C to AB, produced if necessary. That is, either of the two triangles AB l C, AB 2 C, will satisfy the given conditions. But the perpen- dicular CD = b sin A.- Therefore, if A is acute and #<&, and c c b sin A FIG. 2. Two solutions, a > b sin A FIG. 3. One solution, a = b sin A if a > b sin A, there are two solutions. The angles AB^C, are supplementary, since /.AB 1 C=/.B 1 B^C. Both angles are given by the formula If a = b sin A, that is, if a is equal to the perpendicular CD, there is but one solution, a right triangle. This is also seen from the fact that when a= b sin A, the value of sin B reduces to unity. This gives B = 90. If a < b sin A, that is, if a is less than the perpendicular CD, there is no solution, and the triangle is impos- sible. This is also seen from the fact that when a b sin A. One solution. (0) A obtuse and a > b. (5) A acute and a = b sin A. (c?) A acute and a > b. No solution. (a) A acute and a < b sin A. (b) A obtuse and a = b or a < b. To determine the number of solutions, first note whether A is acute or obtuse. Then, on examining the different cases just studied, it is seen that there can never be more than one solu- tion unless A is acute and the Me opposite A is less than the side adjacent. In this case there may be two solutions, one solution, or no solution. The comparison between a and b sin A is often most con- veniently made by means of the natural value of sin A. In many cases the computation can be performed mentally ; for all that is now desired is to determine whether a is less than, equal to, or greater than b sin A. If logarithms are used, we compute log sin J5. The results are as follows. (a) log sin 1?>0, no solution. (b) log sin B = 0, one solution, a right triangle. (V) log sin B < 0, one solution if a > 5, and two solutions if a< b and A is acute. The student should bear in mind that the given parts are not necessarily a, b, and A ; they, may be any two sides and the angle opposite one of them. If other parts are given than those mentioned above, the proper modifications should be made in the formulas for determining the number of solutions. Ex. 1. Given a = 26, b = 72, A = 30 ; find the remaining parts. Since sin A \, we have b sin A = 36. Hence, the triangle is impossible as a < 36. THE OBLIQUE TRIANGLE 141 Ex. 2. Given a = 88, b = 103, A = 120; find the remaining parts. Here A is obtuse and a < b ; therefore the triangle is impossible. Ex. 3. Given a =738.4, 6 = 1185.7,. ^ = 79 38' 40"; find the remaining parts. Solving by logarithms we proceed as follows : a logb = 3.07397 log sin A = 9.99287 - 10 colog a ="7.13171-10 Since log sin5 >> there is no log sin 5 = 10.19855 -10 Ex. 4. Given a = 28.2, e = 45.65, A = 38 1' 7.5" ; find the remaining parts. Proceeding as in Ex. 3 we have , a logc = 1.65944 ... C = 90, and the triangle is a log sin A = 9.79081 - 10 right triangle. colog a = 8.54975 - 10 log sin C - 10.00000 - 10 Solving for B and b by the usual methods employed in the case of right triangles (Arts. 26 and 27, pp. 36-38), we find B = 51 50' 52.5", b= 35.998. Ex. 5. Given a = 67.53, b = 56.82, A = 77 14' 19" ; find the remaining parts. Here a > b and A is acute; therefore there is but one solution. The unknown parts are found in the following manner : log b = 1.75450 log sin A = 9.98914 - 10 colog a = 8.17050 -10 C = 180-(A+B) __ 4gO 00 1 KAff log sin B = 9.91414- 10 *= 55 8 ' 47 "' Check: log b = 1.75450 log = 1-82950 log sin C = 9.86843 - 10 log sin C = 9.86843 - 10 colog sin A = 0.08586 colog sin A = 0.01086 log c = 1.70879 log c = 1.70879 .-. c= 51.143. c = 51.143 142 PLANE TRIGONOMETRY Ex.6. Given = 168.32, 5=221.46, 4 = 33 39' 16"; tind the remaining parts. In this case the simplest method of finding the number of solutions is to obtain the value of b sin A by multiplying the value of b, 221.46, by the natural value of sin A, and comparing the result with 168.32, the value of a. The sine of 33 39' 16" is approximately 0.55. Hence, it is seen at a glance that b sin A is a trifle over one half of 221.46; that is, much less than a Hence, since A is acute and a < &, there are two solutions. The work of computation, exhibited in compact form, is as follows : log b = 2.34529 log a = 2.22613 log sin A = ,9.74365 - 10 log sin C = 9.99396 - 10 colog a = 7.77387 - 10 colog sin A = 0.25635 2.22613 9.35729 - 10 0.25635 log sin B = 9.86281 - 10 log c = 2.47644 1.83977 .-. B l = 46 48' 50", .-. c x = 299.53, c 2 = 69.147. B 2 = 133 11' 10". .-. C = 99 31' 54", or, 13 9' 34". NOTE. The method of checking results is the same as that used in con- nection with Case 1. In Ex. 5 above the check' work for c is given. After a little practice this work can be performed with great rapidity. Every result obtained by the student should, be subjected to a check of some kind. EXERCISE XXV 1. Determine the number of solutions in each of the follow- ing cases : (1) a = 30, 5 = 60, 4 = 30. (2) a = 20, 5 = 60, 4 = 30. (3) = 40, 5=<;o, 4=30. (4) a = 750, 5 = 638, A = 69 30'. (5) a = 38. 8, 5 = 45.5, 4 = 60. (6) a = 226, 5 = 196, 4 = 123 40'. 2. Given a=l 09.68, e = 467, A= 13 35'; find (7=90', ^ = 76 25', 5 = 453.94. 3. Given a =392, 5 = 124, A = 36 41' 42"; find .5=10 53' 45" <7=13224'33" = 484.37. 4. Given a = 168.2, 5 = 218.6, 4 = 3422 ; 50"; \f r~" find . g 1 = 4712'49", 6\ = 9824'21", e 1 = 294.67. 5o=13247'll", (7 9 =12 49' 59", .16. THE OBLIQUE TRIANGLE 143 5. Given 6 = 8472.2, c = 3211.7, (7=16 33' 45"; find ^ = 114 40' 42", ^ = 48 45' 33", 1 = 10238. ^ 2 = 32 11' 48", B 2 =UIU' 21", 2 =6003.4 6. Given a = 506, 6 = 432, ^ = 367'12"; find ,6 = 30 13', 6 7 =113 39' 48", c= 7-86.22. 7. Given a = 36.27, 6 = 23.96, 5=30 26' 14"; find ^4 1 = 50 C 4'24", ^ = 99 29' 22", ^ = 46.65, A 2 =129 55' 36", (7 2 =1938'10", c a =l ' 8. Given = 283.4, 5 = 268.5, JL = 60 40' 26"; find J5=5541 / 23", (7= 63 38' 11, c= 291.25. 9. Given a = 158, 6 = 179, J. = 2117' 22"; find ^ = 24 17' 18", 6\ = 13425' 20", ^=310.8, 5 2 = 15lf p .42'42", ^.^^4=10^46-10 B + A = 60 o 23/33" colog(6 + a)= 7.43790 - 10 Q 75 A n " A log tan 9.62557 10 **~ A = 22 53' 31" 2 2 A= 37 30' 2" ^r''~- 22 58 ' 31 " B= 83 17' 4" CONANT'S TRIG. 10 146 PLANE TRIGONOMETRY Check: loga= 2.14192 log b = 2.35447 log sin C = 9.93494 - 10 . log sin C = 9.93494 - 10 colog sin A = 10.21554 - 10 colog sin B - 0.00299 logc= 2.29240 log c = 2.29240 c= 196.06 c= 196.06 NOTE. In the solution of this problem b precedes a since b > a. (Compare Art. 97, p. 134.) In finding c we use A rather than B, because B is so near 90 that any solution obtained by means of its sine is likely to be inaccurate. NOTE. In Ex. 1 the check solution gives a result exactly equal to that obtained by the original solution. In the work near the top of p. 136 the check solution also gave a result exactly equal to that obtained in the origi- nal solution. In general, however, the check solution may be expected to differ slightly from the original. Ex. 2. Given a = 7, c = 9, B = 60 ; find the third side 6. In this problem the second method furnishes the solution with the smallest amount of labor. fe 2 = a 2 + c 2 2 etc cos B, b = V49 + 81 - 2 .7 9 = VtJ7. .-. b = 8.1854. EXERCISE XXVI 1. Given a = 426, 6 = 582, 0= 52 18'; find A = 46 21' 16", ^=81 20' 44", c = 465.8. 2. Given 6 = 123, c = 211, 4 = 115 22'; find ^ = 41 46' 45", 0= 22 51' 15", a = 286.16. 3. Given a = 121. 6, c = 192.2, B =114 .42'; find ^=24 26' 49", 0= 40 51' 11", 6 = 266.94. 4. Given a = 619, 6 = 515, 6^=39 17'; find A = 84 46' 10", B= 55 56' 50", c= 393.56. < 5. Given 6 = 35.218, c = 54.176, A = 32 48' 14"; find ^=37 49' 35", 0= 109 22' 11", a =31.112. 6. Given a = 46.792, c = 61.234, ^=45 29' 16"; find ^ = 49 34' 5", 0= 84 56' 39", 6 = 43.836. THE OBLIQUE TRIANGLE 147 7. Given b = 718.01, c = 228.88, A = 68 55' 2"; find B = 92 30' 47", (7= 18 3-1' 11", a = 670.61. 8. Given 5 = 2478.1, c = 5134.8, A = 137 8' 49"; find 5 =13 37' 43. 5", 0= 29 13' 27. 5", a =7152. 5. 9. Given a = 4.1203, 5 = 4.9538, O= 65 38' 52"; find A = -&4' 18", B = 65 16' 50", c = 4. 9683. 10. Given a = 0.59217, 5 = 0.21833, (7= 41 37' 4"; find ^1=119 42' 18", ^=18 40' 38", c = 0.4528. 11. Two objects A and B are separated by a body of water. In order to find the distance between them a third point C is chosen from which each of these points is visible, and the following measurements are made: CA = 2560 ft., (7.5=3120 ft., and Z ACB = 105 35'. Find the distance from A to B. 12. If two sides of a triangle are 68.6 ft. and 92.2 ft. respectively and the included angle is 112 42', what is the third side ? 13. Find the distance between two points A, B, which are separated by a marsh, when the distances of these points from a third point C are 4214 ft. and 6932 ft. respectively, and the angle A CB is 51 11. 14. In an isosceles triangle each of the equal sides is 9 and the included angle is 60. Find the third side. 15. In an isosceles triangle each of the equal sides is 9 and the included angle is 120. Find the third side. 16. There are two points, A, B, on the bank of a river, but owing to a curve in its course it is impossible to measure the distance between them directly. A third point C is chosen such that the distances AC=l6Q ft. and 5(7=1680 ft. can be measured, and the angle ACB is found to be 68 42' 30". What is the distance from A to B? 17. In a given triangle two of the sides are 6 and 9 respec- tively, and the included angle is 38. What is the third side? 18. The diagonals of a parallelogram are 8 and 10 respec- tively, and they intersect at an angle of 60. What are the sides of the parallelogram? 148 PLANE TRIGONOMETRY 19. If two sides of a triangle are 1468 and 2136 respectively and the included angle is 72 21' 14", what are the values of the other angles? 20. There are two points, A, B, so situated that they are not visible from each other, and there is no other point from which both can be seen. To find the distance from A to B two other points (7, .Z), are selected so that A and D are visible from (?, and B and are visible from D\ and the following measure- ments are made: CD = 826.5 ft., ZACD = 121 12',ZOZ) = 58 55', ^ADC= 49 12', ^ADB = 62 38'. What is the dis- tance from A to B? 102. Given the three sides a, b, c. When the three sides of a triangle are given, the angles can be found directly from the formulas proved in Art. 96, p. 133. * < In order to obtain a form suitable for computation by means of logarithms we proceed as follows : Let the sum of the sides of the triangle # + >-h=66.9; find all the angles. The work of solving for A and B is as follows : a =79.3 s- a = 40.9 b = 94.2 s - b = 26 c = 66.9 s-c = 53.3 2 s = 240.4 s = 120.2 s = 120.2 THE OBLIQUE TRIANGLE 151 log (*-&)= 1.41497 log - c ) = 1 .72673 colog (s- ) = 8.38828 -10 colog 6- = 7.92010-10 2)19.45008 -20 log tan ^ = 9.72504-10 .-. |- = 2757'56". A = 55 55' 52". log (s-c)= 1.72673 log (s- a) =1.61172 colog (s - b) = 8.58503 - 10 colog .s=^.92010 -10 2)19.84358-20 log tan = 9.92179-10 ... ^ = 39 52' 6.9". B = 79 44' 13.8". ^+B = 13540'5.8' / . .-. (7 = 44 19' 54.2". If the value of C is found by logarithms in the same manner as were the values of A and B, it will be found to be 44 19' 56.8", which is 2.6" larger than the value found by subtracting the sum of A and B from 180. The sum of the three angles, when all are found independently, is 180 0' 2.6". The sum of the three angles determined in this manner is rarely equal to exactly 180. This is due to the fact that logarithmic computation is almost always slightly inexact. It is customary in practical work to divide the error among the three angles according to the probable amount for each angle. Ex. 2. Solve the preceding example by the use of formulas (8), (9), arid (10). In solving by this method it is best to find all the logarithms at the outset, and then to subtract the logarithms of s a, s b, s c, respectively, from the logarithm of r. A com- pact arrangement of the work can be secured by following the model below. log (s- a) = 1.61172 log (s- b) =1.41497 log (s-c) = 1.72673 colog s = 7.92010 - 10 log r 2 = 2.67352 log r= 1.33676 s = 120.2 Check. Check. log tan ^ = 9.72504 -10 log tan | = 9.92179 - 10 log tan = 9.61003 -10 A 2 B_ 2 C _ 2 A = B = C = 27 57' 39 52' 22 9' 55 55' 79 44' 44 19' 56" 6.9" 58.4" 52" 13.8" 56.8" 152 PLANE TRIGONOMETRY EXERCISE XXVII 1. Given a = 56, ft = 58, c = 64 ; find ^=54 22' 43", = 57 20' 32", (7= 68 16' 44". If fe^ ' 1 2. Given a = 54, 5 = 52, e = 68 ; find 4 = 51 24' 3.8", B = 48 48' 52.8", O= 79 47' 7.6". 3. Given a = 35, ft = 41, c = 47 ; find .4 = 46 15' 5", =57 48' 16", C = 75 56' 41.5". 4. Given a = 73, b = 82, c = 91 ; find A = 49 34' 58", ^=58 46' 58", C=7138'4". 5. Given a = 47, ft = 51, c = 58; find .4 = 50 35' 18", .B = 56 58' 4", 6 Y = 72 26' 40". 6. Given a = 286, ft = 321, c = 463 ; find J. = 37 33' 57", = 43 10' 46", G 7 = 99 15' 23". 7. Given a = 138, ft = 246, c == 321 ; find ^=23 47' 23", ^=45 58' 41", 0=110 14' 8. Given a = 196, ft = 211, t \\^^ ({ 19. The sides of a triangle are to each other as 3:4:5. Find \(L I ! - (& all the angles. i S ^ ~~ ~~ 20. In a given triangle a =8, 6 = 8, c = S. Find all the angles. 21. Three cities are respectively 22.6, 21.4, 19.6 miles apart. If the curvature of the earth is disregarded, what angles are made by the lines joining the cities? 22. In discussing the solution of a triangle when two sides and the angle opposite one of them are given, it was noted that two solutions were possible when an angle was found by means of its sine. Why does not a similar ambiguity exist when an angle is found by means of formula (4), p. 149? 23. The sides of a triangle are a = 7, b = 8, c 5. Find the angle A. 24. The sides of a triangle are a = 7, 6 = 5, c 3. Find the angle A. 25. An object 16.2 ft. in length is so situated that one end is 17J ft. and the other is 11.9 ft. from the eye of an observer. What angle does the object subtend at the eye? ff# -' >i I a * 154 PLANE TRIGONOMETRY 103. Area of a triangle. In geometry it was proved that the area of a triangle (A) can be found by either of the following formulas: A = | base x altitude, or, A = Vs (s a)(s b)(s in 60 sin 30 x = sin 3IT 50-1 x ^ = 218.5 ft. 107. If the height of an object is to be determined, and no two points can be found that are in the same st might line, and at ilie same time conveniently situated for observation, the following method is often employed : From A measure AB in any convenient direction. Let the angle of elevation of the top of the object D, measured at A* be , and let the distance AB be a. At A and B measure the angles DAB=& and DIM = 7, respectively. Then in the triangle AD* Therefore, AD _ _ 81117 _ _ siu 7 ~o~ " sin (1W - ( + 7)) ~ sin ( + 7) 1 -ing the value of AD obtained from this equation, we have . sin ( + 7) 158 PLANE TRIGONOMETRY MISCELLANEOUS EXAMPLES THE RIGHT TK I ANGLE 1. The angle of elevation of the top of a vertical cliff 426.28 ft. high, taken from a point on the same level as the foot of the cliff, is 59 51' 14". What is the distance from the foot of the cliff to the point where the observation was taken? 2. A pole 36 ft. high casts a shadow 39 ft. long. What is the angle of elevation of the top of the pole, measured at the extremity of the shadow? 3. The height of a room is 12.62 ft. and its length is 14.44 ft. What is the angle of e! of one of the upper corners of the room taken at the lower corner on the same side? 4. What is the elevation of the sun when a tree 31.6 ft. high casts a shadow 42.9 ft. in length ? 5. What angle does a ladder 25.2 ft. long make with the ground when it just reaches the sill of a window 18.95 ft. above the level on which the foot of the ladder rests ? 6. The angle of depression of a point on the ground, meas- ured from the top of a building 49.27 ft, high, is 34 6' 36". What is the distance from the foot of the building to the given point ? 7. The length of the diagonal of a rectangular field is 247.39 ft., and the angle between the diagonal and the shorter side of the field is 29 40' 36". What is the width of the field? 8. A path measuring 256.4 ft. in length leads diagonally across a rectangular plot of ground, making with one of the sides an angle of 61 12' 52". What is the length of the side ? 9. The angle of elevation of a balloon measured at a certain point is 71 14' 12", and from this point to a point directly below the balloon the horizontal distance is 415.9 ft. What is the height of the balloon and its distance from the point of observation ? PROBLEMS IX HEIGHTS AND DISTANCES 159 10. A kite is fastened to a string 483.2 ft. long, and the string makes an angle of 63 19' 28" with the level of the ground. What is the vertical height of the kite above the ground, no allowance being made for the sagging of the string ? 11. To ascertain the width of a river a distance AB is meas- ured along one of the banks 262.38 ft. Directly across the river from B is a point (7, and the angle BAG is found upon measurement to be 41 38' 20". Required the width of the river. 12. Two forces, of 198.52 Ib. and 393.13 Ib. respectively, are acting at right angles to each other. What is the resultant of the two forces, and what is the angle which the direction of each force makes with the resultant ? 13. What is the radius of the parallel passing through a point on the earth's surface whose latitude is 43 15', the radius of the earth being reckoned as 3956 mi. ? 14. The angle of elevation of the top of aitill viewed from a certain point is 29 17', and from a point 362.4 ft. nearer, measured directly toward the hill, the angle of elevation is 48 12'. Required the height of the-MH-. - 15. From the top of a mountain the angles of depression of two milestones 2 mi. apart and in the same vertical plane with the top of the mountain are 10 14' 42" and 5 38' 46" respec- tively. What is the height of the mountain? 16. A flagstaff which is broken at a certain distance above the ground falls so that its tip touches the ground at a distance of 13.5 ft. from the foot of the portion which remains standing. The length of the part broken over is 35.1 ft. What was the total height of the staff before it was broken over ? 17. If the angle of depression of the visible horizon, observed from the top of a mountain 3 mi. in height, is 2 13' 59", what is the diameter of the earth ? 18 A ladder 30 ft. long when set in a certain position between two buildings will reach a point 20 ft. from the ground on one of the buildings, and on being turned over without having the position of its foot changed it reaches a 1HO PLANE TRIGONOMETRY point on the other building 15 ft. from the ground. What is the angle between the two positions of the ladder ? (Solve by natural functions.) 19. A lighthouse 50 ft. high stands on the top of a rock. The angle of elevation of the top of the rock and of the top of the lighthouse, measured from the deck of a vessel, are 6 5' and 6 58" respectively. What is the height of the rock, and the distance from the vessel to the foot of the rock ? (Solve by natural functions.) 20. At any point on the earth's surface a line is drawn tan- gent to the surface at that point. If the earth is considered a sphere whose diameter is 7912.4 mi., how far from the surface will the line be at the end of 1 mi.? 21. A building 50 ft. high stands at the foot of a hill, and from the top of the hill the angles of depression of the top and of the bottom of the building are 45 15' and 47 12' respectively. What is the height of the hill ? 22. The angles of a triangle are 1:2:3, and the perpendicu- lar from the greatest angle to the side opposite is 15 ft. Required the sides of the triangle. 23. A bridge of five equal spans crosses a river, each span measuring 100 ft. from center to center. From a boat moored in line with one of the middle piers the length of the bridge subtends a right angle. What is the distance from the boat to the bridge? (Solve by natural functions.) 24. An observer on a vessel at anchor sees another vessel due north of him; at the end of fifteen minutes it bears E., and half an hour later it bears S.E. How long after it is first seen is it nearest the observer, and in what direction is it sail- ing, its course being supposed to be in a straight line from the time of the first to the time of the last observation? (Solve by natural functions.) 25. A statue on a column subtends the same angle at dis- tances of 27 and of 33 ft. from the column. If the tangent of the angle equals T ^, what is the height of the statue ? (Solve by natural functions.) PROBLEMS IN HEIGHTS AND DISTANCES 161 26. A tower 145 ft. high stands on an elevation 75 ft. high. At what point in the plain on which the elevation stands must an observation be made in order that the tower and the height of the elevation may subtend equal angles? (Solve by natural functions.) 27. A flagstaff 50 ft. high stands in the center of a plot of ground in the form of an equilateral triangle. Each side of the triangle subtends at the top of the staff an angle of 60. What is the length of one of the sides of the triangle ? (Solve by natural functions.) * 28. A tower stands on the slope of a hill that has an inclination of 15 to the level of the plain. At a point 80 ft. farther up the hill it is found that the tower subtends an angle of 30. Prove that the tower is 40(VJ- V) ft. in height. 29. At a distance of 300 ft. from the foot of a tower the angle of elevation is one third as great as it is at a distance of 60 ft. What is the height of the tower? THE OBLIQUE TRIANGLE 30. The angles of elevation of a balloon measured at the same instant at two points on level ground and in the same vertical plane as the balloon are 41 56' and 28 14' respectivel} T . The two points from which the angles are measured are 3462 ft. apart and on the same side of the balloon. Required its height at the time of observation. 31. The angle of depression of an object viewed from the top of a tower is 50 12' 56", and the angle of depression of a second object 250 ft. farther away, and in a straight line with the first object and the foot of the tower is 31 19' 54". What is the height of the tower ? 32. The angles of depression of two objects on a level plain, viewed from an elevation in the same vertical plane with the objects, are 48 12' and 29 17' respectively, and the distance between the two points is 362.4 ft. Required the height of the point of observation. CON ANT'S TKIG. 11 162 PLANE TKIGONOMETKY 33. The sides of a triangular plot of ground are 138 ft., 246 ft., and 321 ft. respectively. What is the greatest angle formed by the sides? 34. Two objects are separated by a building, and it is re- quired to find the distance between them. At a third point, distant 268 ft. and 315 ft. respectively from the given ob- jects, the angle subtended by the line connecting the objects is measured and is found to be 108 17'. What is the distance between the objects ? i^. 35. What is the distance between two points 4, B, when the distances from these points to a third point C are 6282 ft. and 2344 ft. respectively, and the angle :S*UL is 119 40' 40"? Is more than one solution possible ? Why ? (See Art. 100, p. 138.) 36. The distance between two points A, B, cannot be ob- tained directly by the use of the chain or tape because of an intervening body of water. A third point C is chosen from which both A and B are visible, and the following measure- ments are then made: 4(7=3101.8 ft., Z CAB = 51 28', Z. ABQ = 70 37' 33". What is the required distance ? 37. In a system of triangulation the sides of a triangle con- necting the stations on the tops of three hills have been com- c puted and have been found to be 54,692.73 ft., 61,284.39 ft., and 42,798.64 ft. respectively. What are the values of the angles of this triangle as computed from the sides ? 38. An observation station A is set up in a field along one side of which runs a straight, level road. Two points of ob- servation on the road, J5, (7, one fourth of a mile apart, are *o chosen, on opposite sides of the first station and the angles ABO, ACB, are measured and found to be 46 20' 28" and 38 24' 48" respectively. What is the distance from the station A to the road ? 39. The distances from a point on shore to two buoys are known to be 1286 ft. and 2466 ft. respectively, and the angle subtended at that point by the line connecting the buoys is 42 14' 16". What is the distance between the buoys? PROBLEMS IN HEIGHTS AND DISTANCES 163 40. A tripod is set up on a rock, and to find the distance from the tripod to the shore a line 8500 ft. in length is meas- ured along the shore, and at each extremity of the line the angle is measured which subtends the line connecting the tripod with the other end of the line. The angles are found to be 46 28' and 43 32' respectively. Find the distance from the tripod to the line of measurement along the shore. 41. Two vessels lying at anchor 1 mi. apart are observed from a third vessel sailing east to be in a straight line due north. After sailing an hour and a half one of the vessels bears N.W. and the other W.N.W. Find the rate at which the vessel is sailing. \ \ ^ ^ 42. The distance between two points A, B, is to be deter- mined, where B is accessible and -4Ms not. A kite is sent up and made fast, and its position is determined to be 517.3 yd. vertically above D, which is on the same level with A and B. The following angles are then measured: A 6^5 = 13 15' 15", CAD= 21 9' 18", DBC=<2& 15' 34". \What is the dis- tance from A to B? J $ v % $~ I 43. Two forces, of 410 Ib. and 320 Ib. respectively, are act- ing at an angle of 51 37'. Required the direction and in- tensity of the resultant. 44. A kite A has been sent up and is fastened to the ground at a point Q. The kite has drifted a certain distance and now stands directly above a point B, which is on the same level as (7, but is separated from it by obstacles which render direct measurement impracticable; and the height of the kite is de- sired. To ascertain this a line is measured from to a point Z), 4262.4 ft. in length, and the following angles are meas- ured: ACB= 31 17' 14", ACD= 66 14' 52", CDA = 52 51' 38". Required the vertical height of the kite above the point B. (See Art. 107.) 45. Two rocks are to be charted. To ascertain the distance between them the angles of elevation of a point at the top of a cliff 527.4 ft. high are taken and are found to be 21 8' 16" and 23 14' 20" respectively, and the angle subtended by the 164 I'LAM-: TUGOXOMHTKY line connecting the rocks, measured at a point at the top of the cliff, is 16 3' 30". Required the distance between the rocks. 46. A balloon, J., is sighted at the same instant from two points, B, C, which are on the same level, and are 262.4 ft. apart. The angle of elevation of the balloon at B is 41 15' 24", ZABC = 62 48' 14", ZACB=59 14' 21". What is the height of the balloon at the instant of observation? ^ ^\ ^ t *^ \ 47. A tower stands on the slope of a hill which makes an angle of 16 with the horizon. At a distance of 95 ft. from the foot of the tower, measured directly up the side of the hill, the height of the tower subtends an angle of 38. What is the height of the tower? 48. A tree stands E.S.E. of an observer, and at noon the extremit}^ of the shadow of the tree is directly N.E. of the position in which he is standing. The length of the shadow is 60 ft., and the angle of elevation of the top of the tree viewed from the position of the observer is 45. What is the height of the tree? (Solve by natural functions.) 49. It is required to find the distance between two points, A, B, neither of which is accessible. For that purpose a base line, (7Z>, 4968 ft. long, is measured, and the following angles are observed: ACD=W8 14', 6^Z) = 41 15', 7X7=11,5 21', ADO= 39 42'. What is the distance from A to B> 50. Two points are so situated that it is not possible to measure directly from one to the other, but observations can be taken at either point. Two other points, (7, D, are chosen, 5226 ft. apart, and the following angles are measured: ACB = 15 18' 24", DAO= 21 12' 46", DBC= 23 18' 42", AT)C = BDC= 90. What is the distance from A to #? 51. To find the distance between two inaccessible points, A, B, two other points, (7, D, are chosen,' so situated that from either of them the three other points can be seen; and the fol- lowing measurements are then made: 6 Y .Z) = 826.5 ft., ZACD = 121 12', Z BOD = 58 55', Z.ADC= 49 12', Z.ADB = 2 38'. What is the distance from A to B'> PROBLEMS IN HEIGHTS AND DISTANCES 165 52. Two points, A, B, are so situated that only one point, (7, can be found which is conveniently situated for observation, from which both can be seen. A fourth point, D, is found from which A and Q can be seen, and a fifth point, J, from which B and C can be seen. The following measurements are taken, from which it is required that the distance from A to B shall be computed: CD = 6428.72 ft., OE = 5872.54 ft., = 64 21'. 53. Two points, A, B, are so situated that no point can be found from which both can be seen. Two other points, (7, 1), are found, so placed that A and D can be seen from C and B from D, and also two additional points, E, F, so placed that A and can be seen from F, and B and D from E. The following data can now be obtained for the determination of the distance from A to B: (7Z)=1254 ft,, OF =1216 ft., 7)^=1216 ft., Z.AFC = 78 14' 15", ZFCA = 53 51' 40", Z.4CZ) = 52 17' 18", Z. CDB = 155 24' 20", ZBDE=53 49' 8", ZD^ = 82 57'. What is the length of the line AB? CHAPTER XIY FUNCTIONS OF VERY SMALL ANGLES HYPERBOLIC FUNCTIONS TRIGONOMETRIC ELIMINATION 108. Trigonometric functions of very small angles. Let A OB be any angle less than 90* With as a center and any radius OA describe a circle. Draw BQ perpendicular to OA, and produce it to intersect the circle in B 1 ' . Draw tangents to the circle at B, B'. These tangents will, by geometry, inter- sect OA produced in the same point D. Then chord BB' < arc BB' < BD + B' D. Dividing by 2, CB < arc AB < DB. CB arc AB BD 'OB OB " OB ' CB . /j BD ;\rcAB , But - = sm0, = tan 0, and = the circular OB OB OB measure of the angle 0, or of the arc AB (Art. 13, p. 16). Therefore, sin < < tan 0. This important result may be expressed as follows : When 6 < 90, sin 6, 6, and tan are in the ascending order of magnitude. 109. Dividing the inequality just obtained by sin 0, we have 1< sin<9 < sec 0, or, 166 FUNCTIONS OF VERY SMALL ANGLES 167 Therefore, lies between 1 and cos 6 for all values of 6 u between and - But as approaches as its limit, cos approaches 1 as its limit; and at the same time approaches 1 as its limit. COS0 Therefore, when is very small, and is approaching as its limit, lies between 1 and a quantity that may be made to differ from 1 by a quantity e which may be made as small as we please ; and as approaches as its limit, e also approaches as its limit. t, . * ; , t - ^<( l^ e^ . &-^ In other words, when fl x approaches as its limit, S1 " ap- u proaches 1 as its limit. This fact is often expressed by the statement that when is very small, sin = 0, approximately. In like manner it can be shown that as approaches as its limit, tan will also approach the limit ; that is, when is very small, tan = approximately. From the above it follows also that when is very small, sin = tan 0, approximately. In this discussion it should be remembered that is ex- pressed in^circular measure ; i.e. is the number of radians in the angle or arc under consideration. EXERCISE XXVHI 1. Find the sine and the cosine of V. Let x be the circular measure of 1'. Therefore, since x > sin x > 0, (Art. 108) sin 1' lies between and 0.000290889. Also, cosl' = Vl-sin 2 l > Vl - (0.00029088^) 2 ^O QQQQQQQ >0.9999999. .-.cosT = 0.9999999+. (1) But (Art. 108, p. 166), sin x > x cos x. .-. sin l'> 0.000290888 x 0.9999999 .kO.000290887. (2) 168 PLANE TRIGONOMETRY Therefore, sinl' lies between (1) and (2); i.e. sin 1' = 0.00029088+, and the next decimal place is either 7 or 8. Find approximately the values of the following : 2. sin 10'. 4. sin 1'. 6. cos 15'. S.^cosTO'. .5. sin 15'. 7. sin 8". HYPERBOLIC FUNCTIONS 110. In the differential calculus it is proved that the follow- ing equations are true for all values of x : ein ~ _ ~ X X X> i . . . -fl\ bill X JU -f -T- ' , {1J cos ^_^_^__|_?: ^__^_...j (2) where e = 2.7182818 is the base of the natural system of log- arithms. In (1) and (2) x is the value of the angle or arc expressed in radians. If in (3) x is replaced by ix, where i = V 1, we have X 2 The series in the first parenthesis is the same as the right member of (2), and that in the second parenthesis is the same as the right member of (1). Hence, replacing these series by their values, we have equation (4) in the following form : e** = cos x + i sin x. (5) In a precisely similar manner it may be shown that e~ lr = cos x i sin x. (6) HYPERBOLIC FUNCTIONS Adding (5) and (6), and dividing by 2, we have . (7) Subtracting (6) from (5) and dividing by 2 i, and the cor- responding value for sin x is obtained : (8) These equations give the values of the sine and the cosine of any angle whatever in exponential form. 111. If in (5) and (6) of the preceding section we replace x by ix, the following equations are obtained : e~ x = cos ix-\- i sin ix\ (1) e x = cos ix i sin ix. (2 ) By addition and subtraction we obtain from these the results below : e x , e -x cos ix = - -~ - ; (3) It will be noticed that the exponential functions which occur in the right-hand members of (3) and (4) possess a striking similarity to those which appear in (7) and (8) of the preced- ing section. It has been found convenient to make use of this similarity, and, corresponding to the exponential values of sin x and cos x given in those equations, to give the following definitions : - is called the hyperbolic cosine of x, oX p X and is called the hyperbolic sine of x. ft These functions are written in abbreviated form cosh x and and sinh x respectively. Accordingly we have g.r I g X cosh x y " = cos ix ; (5) sinh x = '- = i si nia?. (6) 170 PLANE TRIGONOMETRY The name hyperbolic is applied to these functions because they bear to the equilateral hyperbola a relation analogous to that which sin a; and cos x bear to the circle. (Art. 46, p. 64.) The other hyperbolic functions are denned as follows : - cosh# sinh x sech x = 1 ; (9) cosh x - .. sinli x 112. Ex. l. Prove the relation sinh = 0. By (6), Art. Ill, we have (10) . 2 2 Ex. 2. Prove the relation sinh (x -f y) = sinh x cosh y + cosh x sinh y. By definition sinh (x + y) = - i (sin (ix + iy) ) = i (sin t.r cos /# + cos ix sin z/y) = i (i sinh x cosh ?/ + i cosh x sinh #) = sinh x cosh y + cosh x sinh ?/. Ex. 3. Prove the relation sinh x + sinh y = 2 sinh ^^ cosh By definition sinh x + sinh y i (sin ix + sin z HYPERBOLIC FUNCTIONS 171 EXERCISE XXIX Prove the following identities : 1. cosh = 1. 9. sin ( ix) = sin ix. . i TTI . 10. cos( iz} = uosix. 2. smn = i. 11. tan ix = i tanh x. 3. cosh = 0. 12 - sinh (-2:)= -sinh a;. 13. cosh ( x) = cosh x. 4. sinh7n'=0. , , . 14. coth ( x) = coth #. 5. cosh9r* = -l. 15 sec h(-:r)=sech*. 6. sinh2mr = 0. 16> C sch ( - ^) = - csch a;. 7. cosh2mr = l. 17. 8. tanh = 0. 18. sech 2 x + tanh 2 x = 1. 19. csch 2 x coth 2 a; = 1 . 20. cosh (x + /) = cosh x cosh y -}- sinh x sinh ^. 21. sinh 2x 2 sinh a: cosh x. 22. cosh 2 # = cosh 2 x + sinh 2 #. 23. sinh # sinh y = 2 cosh ^^ sinh H 24. cosh a; + cosh y= 2 cosh ^-^ cosh ~ 2i 2 25. cosh x cosh y = 2 sinh x ^ sinh ' y ~^ 2 2 113. The notation for inverse hyperbolic functions is the same as for inverse circular functions (Art. 84, p. 114). If y = sinh x, then, x = sinli" 1 ?/. But by (6), p. 169, y = e ll. Solving this equation for #, we have 1). 172 PLANE TRIGONOMETRY In like manner, cosh' 1 ^ = log (?/ -f V/ 2 1) ; (2) tanh- 1 *, = 1 log i^; (3) J- y coth- 1 y = tanh- 1 - = \ log 1 ; (4) y * y -* sechr 1 / = cosh a - = log *- ; (5) csch- 1 y = sinh- 1 = log A ^. (6) EXERCISE XXX Prove the following relations : 1. tanh- 1 -^_ = 2 tanh- 1 2:. 2. sinh" 1 2 # = 2 sinh" 1 a; cosh" 1 a;. 3. sinh" 1 z = cosh" 1 Vl + x z . 4. sinh" 1 a; = tanh" 1 VI T -4- 5. tanh" 1 x+ tann" 1 y = tanh" 1 - ELIMINATION 114. It often happens that two or more equations are given that contain trigonometric functions of some angle, or perhaps of more than one angle. From these equations a single equa- tion is to be obtained from which all trigonometric functions have been eliminated. In theory the required elimination can always be performed, but in practice this often involves processes that are some- what complicated ; and the desired results are obtained with a greater or less degree of difficulty. No general rule for work of this kind can be given ; and the process is best illustrated by a few examples. TRIGONOMETRIC ELIMINATION 173 115. Ex. i. Find the values of r and 6 from the equations r sin 6 = a ; (1) r cos d = b. (2) Squaring and adding, r 2 (sin 2 + cos 2 0) = a 2 + 6 a , r 2 = 2 + b 2 , r = \ / a 2 + b*. Also, dividing (1) by (2), tan0 = , 6 = tan- 1 ? . o Ex. 2. Find the equation of relation between a and b if sin 3 = a, and cos 3 = 6. From the values here given we have sin = , and cos = 6*. But for all values of 0, sin 2 + cos 2 0=1. Therefore, substituting, a ^ + 6 7 = 1, which is the equation desired. Ex. 3. Eliminate 6 from the equations, a cos + b sin # = c, d cos + e sin =/. Solving by any of the ordinary methods of elimination, d c<7 a/* sm0 = - J-, od ae bd ae Substituting these values of sin and cos in sin 2 + cos 2 0= 1, and reducing, we have - * (j/_ ce y + (c,i ._ a f) 2 = (bd - ae) Ex. 4. Eliminate from the equations cot + tan = a ; (1) sec cos = b. (2) From (1) a = - 1 - + tan = 1 + ta f . tan tan S6C \/ xo\ a= te^- (3) 174 PLANE TRIGONOMETRY From (2) 6 = sec - = sec sec . sec0 From (3) and (4) 2 6 = sec 3 0, and a& 2 = tan 3 0. But sec 2 0- tan 2 0=1. (4) = 1, or, aV - aV = 1. Ex. 5. Eliminate from the equations - cos 6 - & sin 6 = cos 20; (1) # -sin<9 + ^cos<9:=2sin20. (2) a 6 Multiplying (1) by cos and (2) by sin and adding the resulting equa- tions, we obtain x - = cos cos 20 + 2 sin 2 sin a = cos cos 2 + sin sin 20+ sin sin 2 = cos + 2 sin 2 cos 0. (-1) In like manner, multiplying (1) by sin and (2) by cos0 and subtracting, = 2 sin 2 cos - cos 2 sin = sin0+2sin0cos 2 0. (4) we obtain 1t = 2 sin 2 cos - cos 2 sin b Adding (3) and (4), - + 1 - cos + sin + 2 sin cos (cos + sin 0) = (cos + sin 0) ( 1 + 2 sin cos 0) = (cos + sin 0)(cos 2 + sin 2 + 2 sin cos 0) = (cos0 + sin0) 3 . i (5) By subtracting (4) from (3) and reducing the result, we find that () Squaring (5) and (6) and adding the results, we obtain the following, which is the desired equation : x y __ ^ a b TRIGONOMETRIC ELIMINATION 175 Ex. 6. From the following simultaneous equations, find the values of r, , 6 : r sin 0cos< = a; (1) r cos cos < = & ; (2) r sin < = c. (3) Dividing (1) by (2), ten0 = |. .-. = tan-*S. (4) Squaring (1) and (2) and adding, r 2 cos 2 < = a 2 + 6 2 . (5) Taking the square root of (5), and then dividing (3) by this result, c (6) Va 2 + b 2 Va 2 + 6 2 Squaring (3) 'and adding the result to (5), r 2 = a 2 + b' 2 + c 2 , r = Va 2 + 2 + c 2 . EXERCISE XXXI 1. Find r and if r sin = 1.25 and r cos = 2.165. Eliminate from the equations following : 2. cos + 6 sin = c, and 6 cos 6 a sin = c?. 3. - cos + f sin = 1, and - sin ^ cos = 1. a b a b 4. a sec 06 tan = = 6, and + c = a. 13. sin + sin <$> = a, cos + cos $=b, and $ = <*. 14. a cos 2 + 6 sin 2 = ccos 2 (, #sin 2 + 6cos 2 = c?sin 2 <, and c tan 2 - d tan 2 = 0. 176 PLANK TRIGONOMETRY SPHERICAL TRIGONOMETRY CHAPTER XV GENERAL THEOREMS AND FORMULAS 116. Spherical trigonometry is that branch of trigonometry which treats of the solution of spherical triangles. 117. The following definitions and theorems are to be found in works on solid geometry. For a discussion of the defini- tions and for proofs of the theorems the student is referred to any text-book on that subject. DEFINITIONS AND THEOREMS 1. The curve of intersection of a plane and a sphere is a circle. 2. A great circle is a circle formed by a plane that passes through the center of the sphere. 3. A small circle is a circle formed by a plane that inter- sects the sphere without passing through its center. 4. Through any two points on the surface of a sphere one and only one great circle can be passed, unless these points are at opposite extremities of a diameter of the sphere. 5. A spherical angle is the angle between two arcs of great circles. It is equal to the angle between the tangents to the two circles drawn at their point of intersection ; it is also equal in angular magnitude to the dihedral angle formed by the planes of the two great circles. 6. A spherical polygon is a portion of the surface of the sphere bounded by three or more arcs of great circles. 7. A spherical trian'gle is a spherical polygon of three sides. CONANT'S TRIG, 12 177 178 SPHERICAL TRIGONOMETRY 118. Let ABC be any spherical triangle, and the center of the sphere on whose surface the triangle is drawn. The vertices are represented geometri- cally by the letters A, B, C, and the same letters are used to designate the angles lying at these vertices respectively. The sides opposite these angles are designated by the corresponding letters a, b, c. Since is the center of the sphere, OA = OBOC, each being a radius of the same sphere. Also, the arcs a, b, c, are the meas- ures of the central angles BOC, AGO, A OB, respectively. THEOREMS. The following theorems on spherical triangles were proved in solid geometry. I. The sum of any two sides of a spherical triangle is greater than the third side.* II. In any spherical triangle the greatest side is opposite the greatest angle, and conversely. Also, equal sides are opposite equal angles. III. Any angle of a spherical triangle is less than 180. IV. The sum of the angles of a spherical triangle is greater than 180 and less than 540 ; i.e. 180 < A -f B + 0< 540. V. Any side of a spherical triangle is less than 180. VI. The sum of the sides of a spherical traingle is less than 360 ; i.e. a + b + c< 360. VII. The difference of any two angles of a spherical triangle has the same sign as the difference of the corresponding opposite sides ; e.g. A B and a b are of the same si In order to establish the truth of the theorem for all values of a and b we pro- ceed as follows : Let b be greater than 90. Produce the arcs CA and OB until they intersect again in C'. Since AC > 90, we have AC' < 90. Therefore in the tri- angle ABC', A( j, 90 o and, by hypothesis, AB < 90, while BC' is unrestricted. GENERAL THEOREMS AND FORMULAS 181 Applying (1), Art. 119, to the triangle ABC', we have cos a' = cos b' cos c -f- sin b r sin c cos Z C'AB. (1) But (Art. 53, p. 78), cos a' = cos a, cos b r = cos b, and cos Z C'AB = cos A. Substituting these values in (1), we have cos a = cos b cos c -f- sin b sin c cos ^4. In like manner it can be shown that the theorem remains true if a and b are both greater than 90. Hence, it is true for all spherical triangles which come within the scope of our work. Also, by drawing the perpendiculars DE< DF, from some point in the radius OB in the planes BOC, BOA, respectively, in the figure of Art. 119, we can obtain a corresponding formula for expressing the value of cos b ; and by drawing these per- pendiculars from some point in the radius 00, in the planes OOA, COB, respectively, a similar formula for the value of cos c. Therefore, cos a = cos b cos c 4- sin b sin c cos A, cos b = cos c cos a-\- sine sin a cos B, (2) cos c = cos a cos b + sin a sin b cos C. The above are relations involving the sides and one of the angles of a spherical triangle. From these equations the following are at once derived : cos a cos b cos c cos A = T> cos b cos c cos a cos -o = COS sin c sin a cos c cos a cos b sin a sin b sin b sin c (3) These relations express the values of the cosines of the angles of a spherical triangle in terms of the sides of the triangles. 182 SPHERICAL TRIGONOMETRY 121. After the first of the three formulas in (2) or in (3) in the preceding article has been obtained the others can be derived from it by a cyclic interchange of the letters a, >, c, replacing at the same time A by B, and B by 0. 122. The law of sines. From plane trigonometry we have the relation . . sin 2 A = 1 cos 2 A. Replacing cos 2 A by its value from (3) in the preceding section, sin 2 A = 1 ( cos a ~ cos fr cos g ) 2 sin 2 b sin 2 c _ sin 2 b sin 2 c (cos a cos b cos . sin a sin a sin o sin c In a precisely similar manner it can be proved that and . * sin b also that : have the same value. Therefore, since each of sin c these ratios has the same value, they are equal to each other. sin A _ sin B __ sin ^^\ sin a sin b sin c ' which is the law of sines. It may be stated in words as follows : The sines of the sides of a spherical triangle are to each other as the sines of the opposite angles. GENERAL THEOREMS AND FORMULAS 183 An inspection of (1) shows that a cyclic interchange of the letters a, 5, c, and A, J5, (7, leaves the right member of the equa- tion unchanged, while the left member is changed into and . Q sin b successively. Hence, after (1) has been proved, (2) can sin c be established by cyclic interchange of letters. 123. To derive a relation involving the angles and one of the sides of a spherical triangle. Let A'JS 1 '0' and ABC be two spherical triangles polar to each other. Then (Art, 118, p. 1T9), a' = 180 - A, b f = 180 - B, c' = 180 - 0. By (1), Art. 119, p. 180, cos a' = cos b' cos c' + sin b f sin c'cos A! . But, by (1), cos a' = cos A, cos b' = cos B, cos c 1 = cos C. sin b 1 = sin _Z?, sin c' = sin (7, cos A' = cos a. Substituting these values in (2), we have cos A = cos B cos C sin .Z? sin (7 cos a. In like manner we can obtain corresponding values for cos B and for cos 0. Therefore, cos A = cos B cos (7+ sin B sin C cos cos B = cos (7 cos ^4 + sin (7 sin A cos 6, cos = cos J. cos B + sin .A sin B cos cos b cos (7 + sin b sin + vsin B sin (7 cos a. GENERAL THEOREMS AND FORMULAS 185 Replacing 1 cos 2 B by its value, sin 2 B, and dividing both sides of the equation by sin B, we obtain the desired relation, cos A sin B cos a sin cos c sin A cos B. (1) In like manner we can obtain corresponding expressions for the value of cos A sin, (7, cos B sin J., etc. Therefore, cos A sin 1? = cos a sin (7 cos c cos ^ sin A, cos A sin (7 = cos a sin .5 cos b cos (7 sin A, cos (7 sin 5 = cos c sin A. cos a cos jB sin (7, cos (7 sin A = cos T . sin c , , sin C .-, i Replacing - by its equal -, tins becomes sin a sin -4 cot a sin b = cos b cos (7+ sin C sin A . *. cot a sill b = cos 5 cos C -f- sin (7 cot A. In like manner we can obtain corresponding expressions for the value of cot a sin c) 2 sin 6 sin c . .. sn = 2 sin 6 sin c In like manner, 1 . j _ sin b sin c cos b cos and corresponding values for cos-- and sin-' (Art. 127, p. 186), we have cos fA+.lP\ m /sin g sin Q - a} ^ /sin * sin ( \2 2y ^ sin b sin c ^ sin a si 8-5) Sill /sin (8 b) sin (s sin b arid sin B - sin c GENERAL THEOREMS AND FORMULAS 191 From (3), Art. 123, p. 183, formulas for finding either of the oblique angles when the opposite leg and the other oblique angle are given. cos A = cos a sin B, 1 cos B = cos b sin A. J From (2) and (3) are derived the following formulas for finding an oblique angle when the hypotenuse and the adjacent leg are given. cos A = tan b cot 90; there- fore b > 90 (Ex. 7, p. 193). After c has been computed b and B may be found, if other formulas than those given above are desired, by the following (Art. 131, p. 191): cos b = COS Jt5 = cos a tan a tan c These formulas give unique solutions for b and B, but for obtaining sin a, there is no solution ; for in that case sin B > 1, which is impossible. If sin A sin b < sin #, (1) is satisfied by two supplementary values of B. But - - and - - are necessarily of the same 2 2 species. Therefore, if both these values of B satisfy this con- dition, there are two solutions ; if not, there is but one. NOTE. To make use of the test just given it is necessary that we first solve for B. There are several methods of testing for the number of solu- tions without first finding B, but it is not thought best to include any of them in this work. For a full explanation of them the student is referred to more extended treatises on the subject of Spherical Trigonometry. Ex. i. Given a = 56 30', b = 31 20', A = 105 11' 10"; find B, C, c. Since in this case sin A sin b < sin a, there may be either one or two solutions. To test for the number of solutions we find the possible values of B. log sin A = 9.98456 - 10 log sin b = 9.71602 -10 colog sin a = 0.07889 log sin B = 9.77947 - 10 B = 37 0' 3", or, B = 142 59' 57". We have from data given, ! < 90. This shows that only the smaller of the two values of B is admissible. Therefore there is but one solution. SOLUTION OF SPHERICAL TRIANGLES 205 The work of solution may be compactly and conveniently arranged as follows : 2 a - b = 25 10' ^=12 35 A+B = U2U f 13" A + B , A-B = 68 II' 7" 2 log ain = 9.1*501 -10 lo S sin = 9-34112- 10 colog sin ^ ^ = 0.25140 log tan ^Lzi = 9.34874 - 10 col S sin ^T = ' 66182 log tan ^9.57605 -10 log tan ^ = 9.83053 - 10 log cot ^ = 0.33347 - = 20 38' 38" r = 24 53' 31" c = 41 17' 16" C = 49 47' 2" EXERCISE XXXVI 1. In a spherical triangle given a = 71 14', b = 122 27' 18", '4 = 77 23' 24"; find B, (7, c. 2. In a spherical triangle given a = 80 5' 16", b = 82 4', .4 = 83 34' 12"; find B, <7, c. 3. In a spherical triangle given a = 151 22' 30", b = 133 31' 25", A= 143 32' 28"; find B, 0, c. 4. In a spherical triangle given a =30 38', b = 31 29' 45", A = 87 53' 20" ; find the remaining parts. 147 CASE 4. Given two angles A, B, and the side oppo site one of them a; to find C, 6, c. As in the preceding case one of the parts, in this case 6, can be found by means of the law of sines, from which we have (Art. 122, p. 182) ~~ sin A &1D a ' 206 SPHERICAL TRIGONOMETRY The values of c and O can then be found by means of the fourth and the second of Napier's Analogies : . A + B sin - c tan 2 = sn C A-B -*" ' (3) The solution is ambiguous, the value of b being determined by means of its sine. If sin B sin a > sin A, there is no solution; for in that case sin b > 1, which is impossible. If sin B sin a < sin A, (1) is satisfied by two supplementary values of b. To ascertain whether or not both these values are admissible we proceed in a manner similar to that employed in the last case. If both values of b satisfy the condition im- posed by the fact that - -^ and a are of the same species, there are two solutions ; otherwise there is but one. NOTE. The number of solutions can always be determined by forming the polar of the given triangle and then determining by the tests under Case 3 the number of solutions of that triangle. The number of solutions of the given triangle is always the same as the number of solutions of its polar. Ex. i. Given A = 29 43' 12", B = 45 4' 18", a = 36 19' 32"; find 6, c, C. In this case sin B sin a < sin A ; therefore there may be either one or two solutions. Solving for 5, we proceed as follows : log sin B = 9.85003 - 10 log sin a = 9.77260 - 10 colog sin A = 0.30173 log sin b = 9.92736 - 10 b = 57 48' 38", or, b = 122 13' 22". We have from data given, A + SOLUTION OF SPHERICAL TRIANGLES 207 % Both of the values of b just found satisfy this condition. Hence, there are two solutions. The values of and c can now be found in the ordinary manner, both values of b being employed. EXERCISE XXXVII 1. In a spherical triangle given A = 109 20' 10", .#=134 16' 24", a= 148 48' 40"; find 6, c, 0. 2. In a spherical triangle given J. = 113 30', .8=125 31' 34", a = 66 44' 40"; find 5, c, 0. 3. In a spherical triangle given A = 28 35' 5", J5 = 47 51' 15", a = 38 41 '32"; find b, c, 0. 4. Iii a spherical triangle given A = 24 30', 5=38 15', a = 65 22'; find 5, c, 0. 148. CASE 5. Given a side c and the two adjacent angles A, B ; to find a, 6, C. The third and fourth of Napier's Analogies may be used for determining the values of a and b (Art. 130, p. 190) : A-B cos - 2 A-B From these formulas the values of a and b can be obtained. The value of C can then be found by means of the first of Napier's Analogies : a-b cos a 2 .A + B tan - = -- cot 2 a + b 2 COS ~2~ The solution is unique. 208 SPHERICAL TRIGONOMETRY Ex. l. Given A =108 28' 55", B = 38 11' 27", c = 52 29'; find a, ft, 0. = 35 8' 44 = 73 20' 11' r = 26 14' 30" log cos - - = 9.91259 - 10 log tan = 9. 69282- 10 colog cos = 0.54250 log tan - = 10.14791 - 10 ^-^ = 54 34' 24.4" = 16 30' 1.3" 2 a-b a = 71 4' 26" & = 38 4' 23" log sin = 9.76016 - 10 log tan | = 9.69282 - 10 colog sin A + B . = 0.01863 log tan ^~ = 9.47161 - 10 ^-=^=16 30' 1.3" log cos = 9.98174 - 10 lo cot colog cos = 9.47599 - 10 = 0.23682 log tan ^ = 9.69455 - 10 | = 26 19' 56" C = 52 39' 52" EXERCISE XXXVIII 1. In a spherical triangle given ^. = 126 40' 50", j5=81 45' 42", c = 51 56' 12"; find a, b, 0. 2. In a spherical triangle given B= 27 27' 36", C = 40 44' 20", a =155 16'; find 6, c, A 3. In a spherical triangle given J. = 127 19' 38", (7=108 41' 30", b = 125 22' 32"; find a, c, .5. 4. In a spherical triangle given A = 154 20' 42", B == 79 16' 22", c = 85 24' 28"; find a, b, C. 149. CASE 6. Given the three angles A, B, C; to find the three sides a, b, c. Any of the three groups of formulas in Art. 128, p. 187, can be used. The formulas for the tangents are recommended in preference to those for the sines or for the cosines. SOLUTION OF SPHERICAL TRIANGLES 209 - cos - o If all three of the sides are to be found, it is convenient to proceed in a manner similar to that employed in Art. 144, p. 200, where three sides were given and three angles were to be found. Multiplying both numerator and denominator of the fraction under the radical sign in (1) by cos ($ A) we have Putting tan R = " ^, A)t!OS(CO8( ^, g) we may write tan - = tan R cos (S A) . Making the corresponding changes in (2) and (3), we have the three equations tan - = tan R cos ($ J5), 2 tan | = tan R cos (# - (7). The solution is unique. COXANT'S TRIG. 14 210 SPHERICAL TRIGONOMETRY Ex. l. Given A = 221, B = 128, 0= 153 ; to find a. The formula for tan -, with the algebraic sign of each factor written above it for convenience, is as follows : tan fl = / cos 5 cos (S A ) 2 A _ \c os(S-B)cos(S- C) A = 221 log- cos S = 9.51264 - 10 B = 128 log cos(S -.-0 = 9.93753- 10 C = 153 colog cos (S - B) = 0.26389 2 S = 502' colog cos (5 - C) = 0.85644 2)20.57050- 20 S = 251 s -A = 30 log tan ^=10.28525 - -10 s -B = 123 - = 62 35' 35 o s -C = 98 a = 125 11' 10" The result is real (Art. 128, p. 187), the four negative signs under the radical producing a positive quantity. Ex. 2. Given A = 21 26' 20", B = 56 46' 28", (7=115 23' 4"; find a, 5, c. Proceeding by the second method, we first find the value of log tan R. The following is suggested as a convenient arrangement of the work: tan R= \ cos (S- A) cos (S - B) cos (S - C) A _ 910 9f>' ()f\r> log tan 2 = 9.30865 - 10 B = 56 46' 28" 2 C = 115 23' 4" log tan* =9.70007 -10 25 = 193 35' 52" 2 5 = 96 47' 50" log tan c - = 9.87055 - 10 S - A = 75 21' 36" iS'- = 40 T28" 5 = 11 30' 17.5" S-C = -19 24' 52" log cos S = 9.07330 - 10 o = :}l 39/ 43 " colog cos (5 - /I ) = 0.59732 ^_ Qf>o o-/ 4 f>// colog cos (S -B) = 0.11590 2 " colog cos (S - C) = 0.02542 a = 23 0' 35" log tan 2 R = 9.81 194 - 10 b = 63 19' 26" log tan R = 9.90597 - 10 c = 73 10' 9" SOLUTION OF SPHERICAL TRIANGLES 211 EXERCISE XXXIX 1. In a spherical triangle given A = 121 40' 24", B= 60 12' 22", O= 105 40'; find a, b, c. 2. Iii a spherical triangle given ,4 = 58 20' 27", =8430'30", (7=61 35' 10"; find a, 6, c. 3. In a spherical triangle given A = 105 14' 4", B=55 31' 24", =88 51 '6"; find a, 5, flf ' \ sm cos ( 2 sin- sin - ) c cos- (Art. 77, p. 100) . sin - sin - m . _____ ^ V sin s sin ( s a) sin (g 6) sin ( s c} cos c_ sin a sin 5 2 (Art. 127, p. 186) Replacing sin a and sin b by their values (Art. 80, p. 106) and canceling, we have E _ Vsin 8 sin (g a) sin (g 6) sin (s c) Sill . ' 2 n a b c L COS - COS - COS - (2> L'HUILIER'S THEOREM. This theorem, which expresses the value of E by means of its tangent, is derived as follows : ~Ei A tan = A + , TT- (7 cos - - + cos - (Art, 77, p. 100) SOLUTION OF SPHERICAL TRIANGLES 213 a-b c cos cos - cos ^_ 2 2 (Art> 129 , p . 189) a -\-b c C cos-^- + cos- sm- sin sin __JL_ -JLcotS. ( Art - 77 > P- 10 ) (Art. 127, p. 186) (8) All other cases may be solved by first finding the three sides or the three angles, and then applying the proper formula. ANSWERS 1. 1. 2. If 3. 0.7581+. PLANE TRIGONOMETRY Exercise I. Pages 11, 12 4. 1.2737+. 5. 2. 54 19-. 6. 3.5693+. 7. 40, 60, 80. 17. 5, 25, 150. 18. 30, 360, 21600. 4. 5. 6. 7. 8. 9. 10. 30. 120. 36. 54. 270. 150. 540. 2700. 11. = 3' T Exercise II. Pages 14-16 12 2?r 17 8599 TT 23. 27, 63. 3 5400 24. 52, 66, 72. 3?r . lg 20533 TT 7T 7T 7 7T 13. 5400 25. f, 3' Is" STT 19 W7r . 26. 30, 60, 90. 14. 180 4 20 ^- 27. 4, 6. 121 7T 180 oa 3?r 57T 77T 360 21. J. 28. --, 7 9 16. 463 TT oo 13021 TT OQ 1 T 2?r 1 720 30000 ' 2' 3 ' 3 / 2 5?r 2ir 31. 150, ; 82 30', 11^; 135. 37T 9 ' 3 ' 6 ' 24 ' 4 32. minutes past four ; 54 T 6 r minutes past four. 5. 1.77. 6. 28 7' 30". 7. 0.265 sec. 8. 40yd. 9. 2 8' 52.8". 10. 861,031 mi. (approximately) . 11. 3962.95. 12. 14 19' 26.2". 13. 1.047 radians, Exercise III. Pages 17-19 14. 51.56. 21. 65 24' 30.4". 15. 102 ft. (approximately) . 16. 5:4. 17. 3.1416. 18. -, -T, - 399 19. 3.1416. 20. 0.000097+. 215 22. 98 ft. 23. 1 mi. 908 ft. nearly. 24. 7 mi. 1237.2 ft. 25. 18 and 58. 26. 19.099'. 27 60 ^ 10800 28. 0.00004848. 216 ANSWERS Exercise V. Pages 29, 30 7. T 4 r V7. 11. Ii, $?. 15. U> e 8. |f. 12. |, |. 16. f A/7, 9. f, f. 13. ft A/61, ft A/61. 17. f, ft A/14 10. & A/16, f 14. |Vfl,2VO. 18. sin A = T 8 7 , cos A = tf , etc. ; sin 5 = jf , cos J5 = T 8 7 , etc. 1. Sill ^1 x* + y' - , ws ^i , CIA;. , iii jj , V>V_/0 J-f ~ 20. f. 21. i. 22. f. ^23T}f Exercise VIII. Pages 42-48 32. 60. 43. $ a 2 cot A. 53. 23 11' 55". 63. 355.34. 33. 45188. 44. \a?\,*\\B. 54. 38 9' 25". 64. 74.335. 34. 6. 45. c 2 sin A cos A. 55. 80.49,105.64. 65. 42.838. 35. 124.71. 46. 29 22'. 56. 74 43' 54". 66. 313.1. 36. 182.8. 47. 60 38'. 57. 124.27. 67. 38.13. 37. 1143.4. 48. 20.48. 58. 560.88. 68. 43.03. 38. 1916.64. 49. 33.64. 59. 25.165, 36.458. 69. 39 11'. 39. 36157.5. 50. 41 36'. 60. 89.44. 71. 118.3. 40. 498.51. 51. 24 54' 16". 61. 46.71. 72. 100. 41. 52444.44. 52. 42 42' 34". 62. 122.53. 73. 145.58. 42. iaVc 2 -a 2 . Exercise IX. Pages 49, 50 1. 64 20' 26". 2. 75 32' 50". 9. M 2 sin-cos . 16. A = 309.01. A = 29.82. 3. 243.57. 10. wrt 2 sin A cos A. 17. A = 104.71. 4. 175.068. 11. nh' 2 cotA. 18. A = 12.312. 5. 148.91'. 6. 80 17'. 12. A = 69.24. 19. 13. A = 1325.46. 20. A = 115.92. A = 700.616. 7. 91.204. 14. A = 3741.18. 21. A = 2186.95. 8. 3 34' 8". Exercise X. Pages 72, 73 K V3 + 1 V2-2 Q V3 + 2 11. L^. 12. -2. 2 2 2 c 1 + 2 V2 3 A/3 1Q 3 A/3 -4. 13. -|. 6 2 " 8> 2 3 , etc. ANSWERS 217 14. Positive for 60, 120, 210, 330 ; negative for 0, 240, 300. 15. Positive for 330; negative for 210, 300; zero for 135. 2ab 2ab 2a + l 2 a 2 + 2 a ' a 2 + 6 2 ' a 2 - 6 2 2 a 2 + 2 a + 1 ' 2 a- + 2 a + 1 Exercise XI. Pages 83, 84 5. 45 and 226 ; 45, 135, 22o, 315. 6. Positive for 120 and 690 ; negative for 150, 300, and ; zero for 135 and 315. 7. Positive for 210 and 780; negative for 240, 300, 625 and ; zero for 225. 8. Positive for 60, 150, and ^^ ; negative for 120 and 210 ; zero for 135 and 225. 9. (a) 240 and 300; (6) 210 and 330; (c) 135 and 315; (d) 30 and 210. 14. 3. 15. cot 2 A esc A. Exercise XII. Pages 88, 89 I. 14. * = i. 4 6 15. = n7r-- 2 ^4 s\ /j ( I \ n ^T 6' 7. e =mr-(-\y\. 6 17. e = nr -. 8. 6 = 2mr^- 18 ^ = W7r 7r > 7T 9. ^ =(2 w + !)TT - v 19. ^ WTT or mr - 2 20. 6= mr - 4 11. 0=(2W + 1)7T. 21. <9^2w7r+-. 12. f = iw + 7- 3 4 22. *<1.-H), + S. Exercise XIII. Pages 90, 91 4. 2 nir , or (2 n + I)TT. 5. 2 mr, or 2 nTr |. 3 g W7r _)_ (_ 1) ^, or MTT + ( 1) M ^-^- 6 2 218 ANSWERS 7. W T+(-l)?. 2 W7T 8. 2 mr ~ 18. 2 HIT, or 9. 2 WTT *, or (2 w + I)TT. 19. -1^-, O r 2 r?r 8\ y " - m n m + n 10. 2 nir -. 20. nir -, or + . 3 4 3 12 11. 2 nir + -, or sin = - |. 21. WTT + ^, or 7 -^ + -JL. ^j 12. W7T -. 22. 7i7T. 13. WTT + -, or cot 6 = 2. 23. . 4 3 14. HT^. 24. 5? + -. 15. (2n + l)or. 25. r 1 3 2(m 16. + ,orH. 26 . (2)1 + Exercise XIV. Pages 95-97 4 - -if- 5- ;;:; 6. ^ Exercise XV. Pages 99, 100 1. 1. 2. &\. 3. - II 4. - 4. 5. 3. Exercise XVIII. Pages 108-110 - >, u. , 4V2 23 2 7 3\/15 3. it, - Hi 3VI6 9 ' 27* 8' 16 4. |, 6. 5 Exercise XXI. Pages 120, 121 1. ^V2, 6 V ^ 9. I. 13. iv 2. \ V2. 2V?' 10. 1 or - i. 14. If- 3. 1. 7. V3. 11. or .]. 15. 1- 4. x imaginary. -3Vl7 12. 1 or \. 16. *V5. 5. 13. 4 17. aft 19 ab 20. V3. vV 1 -f Vft Va 2 - 1 + V6 2 -1 21. 2. 18. WTT or n r+J, 4 ANSWERS 219 Exercise XXII. Page 127, 128 1. 2r, or2*w- 11. mr +(- 1) W 36 52', or 2 mr -. 3 2 . 12. 2n7r-3652'. 2. 2W7T + -, or (2n 13. , or . 439 14. f,or,|. -' 7. 2W7T + , or2w7r- . 16. O 17. ^ + |, 18. 19. S;qr-2!:+(--I)-i. 20. 2n7r, or 10. 2/i7r, or 2 WIT + 112 38'. 21. 2 mr, (2 n + 1) - , or (2 n + 1) - 22. (2w+l)^,(2rc + l)^,or(2N + l). '2 4 o 23. 2 7Z7T, 01' ?I7T y o _ 35. 2 mr, or 2 7i?r + 25. ttr,or(2 + l). ^ B)r _ ^ or .yr +( _ 1)n jr . 26. - 27. , ,r 2 , , , 4 2 29. n7r^, or(2n+l)- 41. 5E f or-| 30. (2* + l)f,or^.f(-l)*|. 42. WTT, or nir- 31. 7i7T. 3 32. (2w+ 1)|, or WIT |- 43. nr, or 33. ?ITT, or WTT 44. WTT, or ~ + 220 ANSWERS Exercise XXIV. Pages 136-138 11. 640.65ft. 12. AC = 8332.2 ft., AB = 12163.53 ft. 13. Distances 2841.2 ft., 3475.46 ft. Height 1721.08 ft. 14. Distances 11975.68 ft., 24182.77 ft. Height 19769.54 ft. 15. 121.04ft. 16. 171.15ft. 17. 110.39ft. 19. 4.588 mi. 20. 4.506 mi. Exercise XXVI. Pages 146-148 11. 4536.4 ft. 13. 5402.6 ft. 15. 15.6. 12. 134.49 ft. 14. 9. 16. 1781.2 ft. 19. A= 39 46' 0.4", B = 68 2' 45.6". 20. 4494.3 ft. 17. 5.65. 18. 4.58: 9.81 Exercise XXVII. Pages 152, 153 17. 43 55' 13". 20. 60, 60, 60. 18. 49 8' 46". 21. 66 44' 2", 60 26' 53", 52 49' 9". 19. 30, 60, 90. 23. 60. 24. 120. 25. 73 44'. Miscellaneous Examples. Pages 158-165 1. 247.56 ft. 3. 41 9' 7". 5. 48 45' 44". 7. 122.48 ft. 2. 42 42' 34". 4. 36 22' 21". 6. 72.75ft. 8. 123.47ft. 9. Height = 1224.3 ft.; distance = 1292.9 ft. 10. 431.78 ft. 11. 233.27 ft. 12. 440.36 Ib. ; 63 12' 26", 26 47' 34". 13. 2881.46 mi. 15. 2304.52ft. 17. 7912.8 mi. 14. 407.61ft. 16. 67.5ft. 18. 108 11'. 19. Height = 350.67 ft., distance =3205.15 ft. 20. 8.0076 in. 21. 746 ft. 22. 17.32, 30, 34.64. 23. 244.95. 24. tan- 1 f ; 3 \ of an hour. 25. 6ft. 26. 136.13 ft. from the foot of the tower. , 27. 61.24ft. 29. 109.9ft. 31. 308.66ft. 33. 110^'.^" 35. 4782.2ft. 30. 4621.1ft. 32. 407.61ft, 34. 473.3ft. 36. 2785.6ft. 37. 60 20' 8", 76 49' 18", 42 50' 29". 38. 595.84 ft. 39. 1743.36 ft. 40. 4244.4 ft. 41. 9.1 mi. arc-hour. 42. 383.37 yd. 43. Resultant = 658.36 Ib. ; angle bet. resultant and greater force 22 23' 43". 44. 2019.62 ft. 47. 63.08. 50. 3883 ft. 52. 13451.52 ft. 45. 410.35ft. 48. 45.92ft. 51. 4494.3ft. 53. 1949.77ft. 46. 178.88 ft. 49. 10520.49 ft. Exercise XXVIII. Pages 167, 168 2. 0.0029089. 4. 0.002036. 6. 0.99999. 3. 0.9999958. 5. 0.004363. 7. 0.00003878. ANSWERS 221 Exercise XXXI. Page 175 1. r = 2.5, = 2.165. 5. 2 c 2 - ad 2 = bed. 2. 3. 4. 9. 10. 11. a 2 + b' 2 = c 2 + cf 2 . 6. a 4 2 a 2 + & 2 = 0. z 2 , y 2 _ 2 7. 2 & - b = 2. 2 6 2 8. (a 2 - & 2 ) 2 - 16 ab = 0. a 2 + ^2 _ 6 - 2 + C 2. (a 2 + I) 2 + 2 &(a 2 + l)(a + 6) - 4(a + ft) 2 = 0. (a + 6)* + f a _ 5)1 - 2 12 - ( ~ & ) tan + & = - | i _| 13. a 2 - & 2 - 2 cos a - 2 = 0. SPHERICAL TRIGONOMETRY Exercise XXXIII. Pages 199, 200 1. 6 = 14 25' 20", A 47 30' 46", B = 44 25' 26" , 2. c = 77 56' 37", A = 81 50' 9", B = 34 28' 58" . 3. Impossible. 4. c = 69 55' 18", b = 130 15' 58", A = 115 33' 51". 5. a = 49 30' 54", b = 131 41' 29", B = 124 6' 53" , 6. a = 34 20' 53", b = 42 23' 40", c = 52 25' 39" 7. a = 80 28' 44", b = 78 38' 54", B = 83 47 '23". 8. b = 145 13' 27" , A = 35 2' 7", B = 118 8' 2". 9. b = 155 23' 47" , c = 71 18' 48", A = 98 54' 34". 10. &i = 153 59' 53" , Ci = 69 36', BI = 152 6' 47" .62 = 26 0' 7", Co = 110 24', B-2 = 27 53' 13" 11. c = 62 33' 19", A = 68 51' 35", B = 39 59' 48". 12. a = 49 53' 28", b = 58 26', c = 70 17' 27". Exercise XXXIV. Page 202 1. A = 113 51' 22" ,B = 66 17' 20", C = 960'18". 2. A = 65 10', B = 98 50' 37", C = 125 17' 48". 3. A = 129 22' 58", , B = 109 41 '38", c = 97 21' 36". 4. A = 23 16' 48", B = 62 13' 34", G = 107 54' 18' Exercise XXXV. Page 203 1. A = 117 33' 50", B = 46 37' 46", c 62 36' 45". 2. A = 116 0' 7", B = 51 34' 15", c 57 51' 26". 3. B = 101 4' 47", C = 40 8' 22", a = 57 31' 43". 4. A = 35 18' 32", c = 126 39' 6", b 77 10' 36". 5. B = 69 28' 26", c = 42 13' 34", a = 76 17' 36". Exercise XXXVI. Page 205 1. B = 119 34' 43", C = 96 55' 26", c = 105 36' 14". 2. 7? = 631'40", C = 84 50' 28", c = 80 51' 28". 3. BI = 63 55' 10". Ci = 3351'5", ci = = 26 41' 4". B = 1 16 4' 50", Impossible. 222 ANSWERS Exercise XXXVII. Page 207 1. 6 = 156 51' 40", c = 30 57' 43", 2. b = 125 22' 40", c = 155 48' 12", 3. bi = 75 38' 40", ci = 102 0' 42", 4. b. 2 = 104 21' 20", c 2 = 134 30' 27", Exercise XXXVIII. Page 208 1. a = 129 29' 29", b = 107 45' 45", 2. 6 = 36 23' 38", c = 122 53' 23", 3. = 123 21' 30", c = 84 15' 24", 4. a = 153 51' 21", 6 = 89 26', Exercise XXXIX. Page 211 1. a = 142 5' 25", b = 38 47' 39", 2. a = 4920'39", 6 = 62 31' 13", 3. a = 107 45' 46", b = 54 27' 19", 4. a = 118 52' 50", b = 34 20' 45", (7=69 37 '20". C= 155 50' 58". Ci = 48 27' 53". C 2 = 146 55' 13". C = 54 54' 16". A = 161 1' 28". 5 = 129 4' 47". (7= 78 21' 23". c = 135 57' 44". c = 5137'5". c = 99 18' 46". c = 84 53' 32". FIVE-PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES BASED OX THE TABLES OF F. G. GAUSS ARRANGED BY LEVI L. CONANT, PH.D. PROFESSOR OF MATHEMATICS IN THE WORCESTER POLYTECHNIC INSTITUTE NEW YORK :. CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1909, BY AMERICAN BOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON. CONANT TRIG. TAliLES. W. P. I INTRODUCTION 1. A logarithm is the exponent by which a number a must be affected in order that the result shall be a given number m. That is, if a x = m, then x is called the logarithm of m to the base a. The above equation written in logarithmic form is log a m = x. Any positive number except 1 may be used as the base of a system of logarithms. In practical work involving numerical computation 10 is the base that is universally employed. All computations by means of logarithms are based on the following theorems : 2. The logarithm of a product is equal to the sum of the logarithms of the factors. PROOF. Let m and n be any two positive numbers, and let x and y be their logarithms respectively. Then m-n = 10*-1 O^IO^. /. log(w^i) = x -f- y = log m + log n. 3. The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. PROOF. = ! =10*-". n I0 y .'. log = x y = log m log n. n 4. The logarithm of any power of a number is equal to the loga- rithm of the number multiplied by the index of the power. PROOF. m y = (10*)* = 10**. .'. log m y = xy y log m. 5. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. PROOF. Vm = ^10^ = 10*. ]ow m 6. The logarithm of any integral power or root of 10 is an integral number. The logarithms of all other positive numbers are fractions. Negative numbers have no logarithms. If any logarithmic computation is to be performed which involves negative numbers, the problem should be solved as though the numbers were all posi- tive ; and the algebraic sign of the result should then be deter- mined by the usual methods of algebra. 7. The logarithm of a number consists of two parts, an integral part and a decimal. The integral part is called the characteristic, and the decimal part the mantissa. As logarithms are usually printed the mantissa is always positive. The characteristic may be positive, negative, or zero. The characteristic of the logarithm of any number may be found by one of the following rules : I. The characteristic of the logarithm of a number greater than one is positive, and is one less than the number of digits in the integral part of the number. II. The characteristic of the logarithm of a decimal fraction is negative, and is numerically one greater than the number of ciphers immediately after the decimal point. For example, the characteristic of the logarithm of 3286 is 3 : of 294645 is 5 ; of 0.0241 is -2 ; of 0.000649 is -4. For the sake of convenience a negative characteristic is often changed in form by adding to it and subtracting from it the number 10. For example, if the characteristic of a logarithm is 2, and the mantissa is .38416, the logarithm may be written 8.38416 10. If the characteristic is 1 and the mantissa is .74925, the logarithm may be written 9.74925 10. If the negative forms of the charac- teristics are retained, the above logarithms are written 2.38416 and 1.74925 respectively. When it is remembered that the mantissas are positive, the reason for writing the negative sign of a charac- teristic above instead of before it will be readily understood. In all work connected with the logarithms in the following tables the characteristics, when negative, are to be understood as being increased and diminished by 10. TABLE I Directions for finding the logarithm of a number. 8. When the number is between i and 100. The entire logarithm, including both characteristic and man- tissa, is given on p. 9. 9. Numbers containing one or two significant figures. The mantissa is found on p. 9. It is the same for all numbers containing the same significant figures arranged in the same order, no matter where the decimal point is placed. The characteristic is found by means of the rules given above. For example, log 53 = 1.72428, log .53 = 9.72428 - 10, log 5.3 = 0.72428, log .053 = 8.72428 - 10. 10. Numbers containing three significant figures. The number, no attention being paid to the decimal point, is found at the left of the page in the column headed No. The mantissa is found on a line with the number, and in the column headed 0. The characteristic is found as before, by one or the other of the rules on p. 4. For example, log 763 = 2.88252, log .0763 = 8.88252 - 10, log 76.3 = 1.88252, log .00763 = 7.88252 - 10. 11. Numbers containing four significant figures. The first three significant figures are found in the column headed No., and the fourth is at the top of the page. On a line with the first three figures, and in the column headed by the fourth figure, the mantissa is found. The characteristic is deter- mined as in the previous cases. For example, log 296300 = 5.47173, log .2963 = 9.47173 - 10, log 29,63=1.47173, log .0002963 = 6.47173 - 10. 12. Numbers containing more than four significant figures. Let the number whose logarithm is required be 61487. Since the number lies between 61480 and 61490, the logarithm of the re- quired number lies between the logarithms of those numbers, i.e. between 4.78873 and 4.78880. Now log 61490 = 4.78880 and log 61480 = 4.78873 giving a difference of .00007 Hence, we see that an increase of 10 in the number produces an increase of .00007 in the logarithm. But the actual increase we have to consider in the number is 7. Now if an increase of 10 in the number produces an increase of .00007 in the logarithm, an increase of 7 in the number will produce an increase of -^ of .00007, or .000049. Calling this correction .00005, we have log 61480 = 4. 78873 correction = .00005 .-. log 61487 = 4.78878 It is here assumed that an increase in the number is accom- panied by a proportional increase in the logarithm of the number. This is not true ; but in obtaining logarithms from a table, that assumption is always made. If greater accuracy is desired, it will be necessary to use tables containing a greater number of figures. Directions for finding the number corresponding to a given logarithm. 13. Logarithms whose mantissas are found in the table. When the exact mantissa of a logarithm is found in the table, the first three significant figures of the number corresponding to the logarithm are found in the column headed No., and on a line with the given mantissa. The fourth significant figure is at the top of the column in which the given mantissa is found. For example, 2.68529 is the logarithm of 484.5. See p. 17. 9.68529-10 is the logarithm of 0.4845. 7.68529 - 10 is the logarithm of 0.004845. 5.68529 is the logarithm of 484500. 14. Logarithms whose mantissas are not found in the table. When the exact mantissa of the given logarithm is not found in the table, the first four significant figures of the number corre- sponding to the logarithm are the same as the first four significant figures of the number corresponding to the next smaller logarithm. The remaining figures are found by interpolation, as illustrated in the following. To find the number corresponding to the logarithm 3.44127. Number corresponding to 3.44138 is 2763 See p. 13. Number corresponding to 3.44122 is 2762 .00016 ~T Thus we see that an increase of .00016 in the logarithm corresponds to an increase of 1 in the number. But the given logarithm, 3.44127, is .00005 greater than the logarithm of the number 2762. Therefore, the increase in the required number is ;$$$}f, or, more simply, -^g- of 1. This gives .31 as the required increase. Hence 2762.31 is the number whose logarithm is 3.44127. Similarly, 78.565 is the number whose logarithm is 1.89523. 58317.5 is the number whose logarithm is 4.76580. .17532 is the number whose logarithm is 9.24383 - 10. 15. Cologarithms. The cologarithm of a number is the logarithm of the recipro- cal of that number. Since the reciprocal of a number is unity divided by that num- ber, and since the logarithm of unity is 0, it follows that the cologarithm of a number is found by subtracting the logarithm of the number from 0, or from 10 10. For example, colog 256 = log 2 iff = log 1 - log 256 = - 2.40824 = - 2.40824. To avoid the use of negative logarithms the above work is performed, and the value of the above result is expressed as follows: log 1 = 10. 00000 -10 log 256= 2.40824 .-. colog 256= 7.59176-10. From the definition of a cologarithm it follows that the effect of subtracting the logarithm of a number is the same as that of adding its cologarithm. For example, finding the logarithm of HI by each of the two possible methods, we have : BY LOGARITHMS BY COLOGAKITHMS log 293 = 12.46687 - 10 log 293 = 2.46687 log 478= 2.67943 colog 478= 7.32057 - 10 Subtracting, 9.78744 - 10 Adding, 9.78744 - 10 The result is the same in both cases. TABLE III This table contains the logarithmic sine and tangent for every second from 0' to 3', and the logarithmic cosine and cotangent for every second from 89 57' to 90 ; the logarithmic sine, cosine, and tangent for every ten seconds from to 2, and the logarithmic sine, cosine, and cotangent for every ten seconds from 88 to 90 ; and the logarithmic sine, cosine, tangent, and cotangent for every minute from 1 to 89. I 16. The logarithmic sine, cosine, tangent, or cotangent of an angle less than 90. If the angle is less than 45, use the column having the name of the proper function at the top, and the column of minutes at the left of the page; if the angle is between 45 and 90, use the column having the name of the proper function at the bottom, and the column of minutes at the right of the page. To illustrate the use of this table, let us find the logarithm of sin 26 28' 12". % P- 48 > log sin 26 28' = 9.64902 - 10. The difference between log sin 26 28' and log sin 26 29' is .00025. Increasing the former by ^| of this difference, or .00005, we have log sin 26 28' 12" = 9.64907 - 10. As a further illustration, find log tan 71 3/ 10". % P- 44 > log tan 71 38' = 10.47885 - 10. Increasing this by J J of .00042, i.e. by .00013, we have log tan 71 38' 19" = 10.47898 - 10. If the logarithm of a cosine or of a cotangent is to be found, the correction for seconds must be subtracted, since these functions decrease as the angle increases from to 90. 17. The angle corresponding to a logarithmic sine, cosine, tan- gent, or cotangent. Find the angle whose log tan = 9.65647 10. The next smaller logarithmic tangent is (p. 47) 9.65636 10, which corresponds to an angle of 24 23'. The difference between this logarithm and the log tan 24 2i' is .00033, and the difference between log tan 24 23' and the given logarithm is .00011. There- fore, the angle corresponding to the next smaller logarithm, i.e. 24 23', must be increased by 1J of 60", i.e. by 20". Hence, 9.65647 - 10 = log tan 24 23' 20". In the case of the logarithm of the cosine or of the cotangent we work from the next larger logarithm to the next smaller, in- stead of from the smaller to the larger as in the case of the sine and the tangent. TABLE IV This table contains the numerical or natural values of the sine, cosine, tangent, and cotangent for every minute from to 90. - TABLE I THE COMMON OR BRIGGS LOGARITHMS OF THE NATURAL NUMBERS FEOM 1 TO 10000 MOO No. Log. No. Log. No. Log. No. Log. No. Log. 2O 1.30103 21 1.32222 22 1.34242 23 1.36173 24 1.38021 4O 1.60206 41 1.61278 42 1.62325 43 1. 63 347 44 1.64345 6O 1.77815 61 1.78533 62 1 . 79 239 63 1. 79 934 64 1. 80 618 8O 1.90309 81 1.90849 82 1.91381 S3 1. 91 908 84 1.92428 1 0. 00 000 2 0. 30 103 3 0.47712 4 0.60206 5 0. 69 897 6 0.77815 7 0.84510 8 0.90309 9 0. 95 424 25 1. 39 794 26 1.41497 27 1.43136 28 1.44716 29 1.46240 45 1. 65 321 46 1.66276 47 1.67210 48 1.68124 49 1.69020 65 1. 81 291 66 1. 81 954 67 1.82607 68 1.83251 69 1. 83 885 85 1. 92 942 86 1.93450 87" 1.93952 88 1.94448 89 1.94939 MBh i.ooooo 11 1. 04 139 12 1.07918 13 1. 11 394 14 1. 14 613 3O 1.47712 31 1. 49 136 32 1.50515 33 1,51851 34 1.53148 5O 1.69897 51 1.70757 52 1.71600 53 1. 72 428 54 1. 73 239 * 7O 1.84510 71 1.85126 72 1.85733 73 1.86332 74 1.86923 9O 1.95424 91 1.95904 92 1. 96 379 93 1.968-JS 94 1.97313 15 1. 17 609 16 1.20412 17 1. 23 045 18 1.2-5527 19 1.27875 35 1.54407 36 1. 55 630 37 1. 56 820 38 1.57978 39 1. 59 106 55 1. 74 036 56 1.74819 57 1.75587 58 1.76343 59 1.77085 75 1. 87 506 76 1. 88 081 77 1.88649 78 1.89209 79 1. 89 763 95 1. 97 772 96 1. 98 227 97 1.98677 98 1 . 99 123 99 1. 99 564 2O 1. 30 103 4O 1.60206 6O 1.77815 8O 1.90309 1OO 2.00000 MOO It) 100-149 No. 01234 56789 1OO 00000 00043 00087 00130 00173 00217 00260 00303 00346 00389 101 00432 00475 00518 00561 00604 00647 00689 00732 00775 00817 102 00860 00903 00945 00988 01030 01072 01115 01157 01199 01242 103 01 284 01 326 01 368 01 410 01 452 01494 01536 01578 01620 01662 104 01 703 01 745 01 787 01 828 01 870 01912 01953 01995 02036 02078 105 02 119 02 160 02 202 02 243 02 284 02325 02366 02407 02449 02490 106 02531 02572 02612 02 653 02 694 02735 02776 02816 02857 02898 107 02938 02979 03019 03060 03100 03141 03181 03222 03262 03302 108 03342 03383 03423 03463 03503 03 543 03 583 03 623 03 663 03 703 109 03743 03782 03*822 03862 03902 03941 03981-04021 04060 04100 110 04139 04179 0-1218 04258 04297 04336 04376 04415 04454 04493 111 04532 04571 04610 04650 04689 04727 04766 04805 04844 04883 112 04922 04961 04999 05038 05077 05 115 05 154 05 192 05 231 05 269 113 05 308 05 346 05 385 05 423 05 461 05500 05538 05576 05614 05652 114 05 690 05 729 05 767 05 805 05 843 05881 05918 05956 05994 06032' \ 115 06070 06108 06145 06183 06221 06258 06296 06333 06371 06408 116 06446 06483 06521 06558 06595 06633 06670 06707 06744 06781 117 06819 06856 06 893 06930 06967 07004 07-041 07078 07115 07151 118 07 188 07 225 07 262 07 298 07 335 07372 07408 07445 07482 07518 119 07555 07591 07628 07664 07700 07737 07773 07809 07846 07882 12O 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 121 08279 08314 OS 3"50 08386 08422 08458 08493 08529 08565 08600 122 08636 08672 08707 08743 08778 08814 08849 08884 08920 08955 123 08991 09026 09061 09096 09132 09167.09202 09237 09272 09307 124 09342 09377 09412 09447 09482 09517 09552 09587 09621 09656 125 09691 09726 09760 09795 09830 09864 09899 09934 09968 10003 126 10037 10072 10106 10140 10175 10209 10243 10278 10312 10346 127 10380 ]0415 10449 10483 10517 10551 10585 10619 10653 10687 128 10721 10755 10789 10823 10857 10890 10924 10958 10992 11025 129 11 059 11 093 11 126 11 160 11 193 11227 11261 11294 11327 11361 13O 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 131 11727 11760 11793 11826 11860 11893 11926 11959 11992 12024 132 12057 12090 12123 12156 12189 12222 12254 12287 12320 12352 133 12385 12418 12450 12483 12516 12548 12581 12613 12646 12678 134 12710 12743 12775 12808 12840 12872 12905 12937 12969 13001 135 13033 13066 13098 13130 13]62 13194 13226 13258 13290 13322 136 13354 13386 13418 13450 13481 13513 13545 13577 13609 13640 137 13672 13704 13735 13767 13799 13830 13862 13893 13925 13956 138 13988 14019 14051 14082 14114 14145 14176 14208 14239 14270 139 14301 14333 14364 14395 14426 14457 14489 14520 14551 I45^H 140 14613 14644 14675 14706 14737 14768 14799 14829 14860 1489^ 141 14922 14953 14983 15014 15045 15 076 15 106 15 137 15 168 15 198 142 15229 15259 15290 15320 15351 15381 15412 15442 15473 15503 143 15534 15564 15594 15625 15655 15685 15715 15746 15 776 15806 144 15836 15866 15897 159^7 15957 15987 16017 16047 16077 16107 145 16137 16167 16197 16227 16256 16286 16316 16346 16376 16406 - 146 16435 16465 16495 16524 16554 16584 16613 16643 16673 16702 147 16732 16761 16*791 16820 16850 16879 16909 16938 16967 16997 148 17026 17056 17085 17114 17143 17173 17202 17231 17260 17289 149 17319 17348 17377 17406 17435 17464 17493 17522 17551 17580 No. O 1*2 3 4 56789 100-149 150-199 ii No. 1 2 3 4 5 6 7 8 9 150 17609 17638 17667 17696 17725 17754 17 782 17811 17840 17869 151 17898 17926 17 955 17984 18 013 18041 18070 18099 18127 18156 152 18184 18213 18241. 18270 18298 18327 18355 18384 18412 18441 153 18469 18498 18526 18554 18 583 18611 18639 18667 18 696 18724 ^154 18752 18780 18808 18837 18865 18893 18921 18949 18977 19005 155 19033 19061 19089 19117 19145 19173 19201 19229 19257 19285 156 19312 19340 19368 19 396 19424 19451 19479 19507 19535 19562 157 19590 19618 19645 19673 19700 19728 19756 19783 19811 19838 158 19866 19893 19921 19 948 19976 20003 20030 20 058 20085 20112 159 20140 20167 20194 20222 20249 20276 20303 20330 20358 20385 16O 20412 20439 20466 20493 20520 20548 20575 20602 20629 20 656 161 20683 20710 20737 20763 20790 20817 20844 20871 20898 20925 162 20 952 20978 21005 21 032 21059 21085 21 112 21139 21 165 21 192 163 21219 21 245 21272 21299 21325 21 352 21378 21405 21 431 21458 164 21484 21511 21537 21 564 21590 21617 21643 21669 21696 21722 165 21 748 21775 21801 21 827 21 854 21880 21906 21932 21 958 21985 166 22011 22 037 22063 22 089 22115 22 HI 22167 22194 22220 22246 167 22272 22298 22324 22 350 22376 22 401 22427 22453 22479 2T2505 168 22531 22557 22 583 22608 22634 22660 22686 22712 22737 22763 169 22789 22814 22840 22866 22891 22917 22943 22968 22994 23019 17O 23045 23070 23096 23121 23 If 7 23172 23198 23223 23249 23274 171 23300 23325 23 350 23376 23401 23426 23 452 23477 23,502 23528 172 23553 23378 23603 23629 23 654 23679 23 704 23729 23754 23779 173 23805 23830 23855 23 880 23905 23930 23955 23980 24005 24030 174 24055 24080 24105 24130 24155 24180 24204 24229 24254 24279 175 24304 24 329 24353 24378 24403 24428 24452 24477 24 502 24527 176 24551 24576 24601 24625 24650 24674 24699 24724 24748 24773 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 178 25 042 25066 25091 25115 25 139 25 164 25188 25 212 25 237 25261 179 25285 25310 25 334 25358 25 382 25406 25431 25455 25479 25503 180 25527 25 551 25575 25600 25624 25 648 25672 25696 25720 25744 181 25768 25 792 25816 25 840 25864 25888 25 912 25 935 25 959 25983 182 26 007 26031 26055 26079 26102 26126 26150 26174 26198 26221 183 26245 26269 .26293 26316 26340 26364 26387 26411 26435 26458 184 26482 26 505 26529 26 553 26576 26600 26623 26647 26670 2 694 185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 1B6 26951 26975 26998 27021 27045 27068 27091 27114 27138 27161 ^gjl ^27184 27207 27231 27254 27277 27300 27323 27346 27370 27393 M: 416 27439 27462 27485 27508 27531 27554 27577 27600 27623 i V; 646 27669 27 692 27715 27738 27761 27784 27807 27830 27852 W^ 27 875 27898 27921 27944 27967 27989 28012 28035 28058 28081 191 1 28 103 28126 28149 28171 28194 28217 28240 28262 28285 28307 192 28330 28 353 28375 28398 28421 28443 28466 28488 28511 28533 193 28556 28 578 28601 28623 28646 28668 28691 28713 28735 28758 194 28780 28803 28825 28847 28870 28892 28914 28937 28959 28981 195 29003 29026 29048 29070 29092 29115 29137 29 159 29181 29203 196 29226 29248 29270 29292 29314 29 336 29 358 29380 29403 29425 197 29 447 29469 29491 29 S13 29535 29 557 29579 29601 29623 29645 198 29667 29688 29710 29732 29754 29776 29798 29820 29842 29 863 199 29885 29907 29929 29951 29973 29994 _3a?i?^ 30016 30038 30060 30081 No. O 1 2 3 4 5 6 7 8 9 150-199 12 200-249 No. O 1 2 3 4 5 6 7 8 9 2OO 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 201 30320 30341 30363 30384 30406 30428 30449 30471 30492 30514 202 30 535 30557 30578 30 600 30621 30643 30664 30685 30707 30728 203 30750 30771 30792 30814 30835 30 856 30878 30899 30920 30942 204 30963 30984 31006 31027 31048 31069 31091 31112 31 133 31154 205 31 175 31 197 31218 31239 31260 31281 31302 31323 31345 31366 206 31387 31408 31 429 31450 31471 31492 31 513 31534 31 555 31576 207 31 597 31618 31 639 31660 31 681 31 702 31 723 31 744 31765 31 785 208 31806 31 827 318-18 31869 31 890 31911 31931 31 952 31973 31994 209 32015 32035 32056 32077 32098 32118 32139 32160 32181 32201 21O 32222 32 243 32263 32284 32305 32 325 32346 32 366 32387 32408 211 32428 32449 32469 32490 32510 32 531 32 552 32572 32593 32613 212 32634 32654 32 675 32 695 32715 32736 32 756 32777 32797 32818 213 32838 32 858 32879 32 899 32 919 32940 32 960 32980 33001 33021 214 33041 33062 33082 33102 33122 33143 33 163 33183 33203 33224 215 33244 33264 33284 33304 33325 33 345 33 365 33385 33405 33425 216 33445 33465 33 486 33 506 33 526 33 546 33 566 33 5S6 V 33 606 33 626 217 33646 33666 33 686 33 706 33 726 33 746 33766 33786 33 806 33826 218 33 846 33866 33885 33 905 33 925 33 945 33 965 33985 34005 34025 219 34044 34064 34 084 34104 34124 34143 34163 34183 34 203 34223 22O 34242 34 262 34282 34301 34 321 34 341 34361 34380 34400 34 420 221 344^9 34 459 34479 34498 345ft 34 537 34 557 34 577 34 596 34 616 222 34635 34 655 34674 34 694 34713 34 733 34753 347^2 34 792 34811 223 34830 34850 34869 34889 34 908 34 928 34 947 34 967 34 986 35 005 224 35025 35044 35 064 35083 35102 35 122 35 141 35160 35 ISO 35 199 225 35 218 35 238 35 257 35 276 35295 35315 35334 35353 35 372 35 392 226 35411 35430 35 449 35 468 35 488 35 507 35 526 35 545 35564 35583 227 35603 35 622 35 641 35 660 35 679 35 698 35717 35736 35 755 35 774 228 35 793 35813 35 832 35851 35 870 35 889 35908 35 927 35 946 35965 229 35 984 36003 36021 36040 36059 36078 36 097 36116 36135 36 154 23O 36 173 36192 36211 36229 36248 36 267 36 86 36305 36324 36342 231 36361 36380 36399 36418 36 436 36 455 36474 36 493 36511 36 530 232 36 549 36 568 36 586 36605 36 624 36 642 36 661 36 680 36698 36 717 233 36736 36 754 36 773 36 791 36810 36 829 36847 36 866 36 884 36 903 234 36922 36940 36959 36^977 36 996 37014 37033 37051 37070 37088 235 37107 37 125 37 144 37162 37 181 37 199 37218 37236 37254 37 273 236 37 291 37310 37328 37346 37365 37383 37401 37420 37 438 37457 237 37475 37493 37511 37 530 37 548 37 566 37 585 37 603 37621 238 37 658 37 676 37 694 37712 37 731 37 749 37 767 37 785 378031 239 37840 37858 37876 37 894 37912 37931 37949 37967 37 98| 24O 38021 38 039 38057 38 075 38093 38112 38130 38148 38166 241 38202 38220 38 238 38 256 38274 38 292 38310 38328 38 346 38 364 242 38382 38399 38417 38435 38 453 38471 38 489 38 507 38 525 38 543 243 38 561 38578 38 596 38614 38 632 38 650 38668 38686 38703 38721 244 38739 38 757 38775 38792 38810 38828 38846 38863 38881 38899 245 38917 33 934 38952 38970 38987 39005 39023 39041 39 058 39076 246 39 094 39111 39129 39 146 39164 39182 39199 39217 39 235 39 252 247 39^70 39287 39305 39 3-22 39 340 39 358 39375 39 393 39410 39428 248 39 445 39 463 39480 39498 39515 39 533 39 550 39 568 39 585 39602 249 39620 39637 39655 39672 39690 39707 39724 39742 39 759 39 777 No. O 1 2 3 4 5 6 7 8 9 200-249 250-299 13 No. O 1 2 3 4 56789 25O 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 251 39967 39985 40002 40019 40037 40054 40071 40088 40106 40123 252 40140 40157 40175 40192 40209 40226 40243 40261 40278 40295 253 40312 40329 40346 40364 40381 40398 40415 40432 40449 40466 N 254 40483 40500 40518 40535 40552 40 569 40 586, 40 603 40 620 40 637 255 40654 40671 40688 40705 40722 40739 40756 40773 40790 40807 256 40824 408-11 40858 40875 40892 40909 40926 40943 .40960 40976 257 40993 41010 41027 41044 41061 41078 41095 41111 47128 41145 258 41162 41179 41196 41212 41229 41 246 41 263 41 280 41 296 41 313 259 41330 41347 41 363 41 380 41397 41414 41430 41447 41464 41481 26O 41 497 41 514 41 531 41 547 41 564 41581 41597 41614 41631 41647 261 41664 41681 41697 41714 41731 41747 41764 41780 41797 41814 262 41 830 41 847 41 863 41 880 41 896 41 913 41 929 41 946 41 963 41 979 263 41996 42012 42029 42045 42062 42078 42095 42111 42127 42144 264 42 160 42 177 42 193 42 210 42 226 42 243 42 259 42 275 42 292 42 308 265 42 7 25 42341 42357 42374 42390 42406 42423 42439 42455 42472 266 42488 42504 42521 42537 42553 42570 42586 42602 42619 42635 267 42651 42667 42684 42700 42716 42732 42749 42765 42781 42797 268 42813 42830 42846 42862 42878 42894 42911 42927 42943 42959 269 42975 42991 43008 43024 43040 43 056 43 072 43 088 43 104 43 120 27O 43 136 43 152 43 169 43 185 43 201 43217 43233 43249 43265 43281 271 43297 43313 43329 43345 43361 43377 43393 43409 43425 43441 272 43457 43473 43489 43505 43521 43537 43553 43569 43584 43600 273 43 616 43 632 43 648 43 664 43 680 43 696 43 712 43 727 43 743' 43 759 274 43775 43791 43807 43823 43838 43854 43870 43886 43902 43917 275 43933 43949 43965 43981 43996 44012 44028 44044 44059 44075 276 44 091 44 107 44 122 44 138 44 154 44170 44185 44201 44217 44232 277 44248 44264 44279 44295 44311 44326 44342 44358 44'373 44389 278 44404 44420 44436 44451 44467 44483 44498 44514 44529 44545 279 44560 44576 44592 44607 44623 44638 44654 44669 44685 44700 28O 44 716 44 731 44 747 44 762 44 778 44793 44809 44824 44840 44855 281 44871 44886 44902 44917 44932 44948 44963 44979 44994 45010 282 45025 45040 45056 45071 45086 45102 45117 45133 45148 45163 283 ! 45 179 45 194 45 209 45 225 45 240 45255 45271 45286 45301 45317 284! 45332 45347 45362 45378 45393 45408 45423 45439 45454 45469 285 1 45 484 45500 45515 45530 45545 45561 45576 45591 45606 45621 286 45637 45652 45667 45682 45697 45 712 45*28 45 743 45 758 45 773 287 45 788 45 803 45 818 45 834 45 849 45864 45879 45894 45909 45924 288 45 939 45 954 45 969 45 984 46 000 46015 46030 46045 46060 46075 289 46090 46105 46120 46135 46150 46165 46180 46195 46210 46225 29O 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 291 46389 46404 46419 46434 46449 46464 46479 46494 46509 46523 292 46538 46553 46568 46583 46598 46613 46627 46642 46657 46672 293 46687 46702 46716 46731 46746 46761 46776 46790 46805 46820 294 46 835 46 850 46 864 46 879 46 894 46909 46923 46938 46953 46967 295 296 46982 46997 47012 47026 47041 47129 47144 47159 47173 47188 47056 47070 47lfc 47100 47 !M 47202 47217 47232 47246 47 2B1 297 47276 47290 47305 47319 47334 47349 47363 47378 47392 47407 298 47422 47436 47451 47465 47480 47494 47509 47524 47538 47553 299 47567 47582 47596 47611 47625 47640 47654 47669 47683 47698 No. 01234 56789 250-299 14 300-349 No. O 1 2 3 4 5 6 7 8 9 300 47712 47727 47741 47756 47770 47784 47799 47813 47828 47 842 301 47 857 47871 47885 47900 47914 47929 47 94.3 47958 47972 47986 302 48001 48 015 48029 48 044 48058 48073 48087 48101 48116 48130 303 48144 48159 48173 48187 48202 48216 48230 48244 48 259 48273 304 48287 48302 48316 48330 48344 48359 48373 48387 48401 48416 305 48430 48444 48458 48473 48487 48501 48515 48530 48544 48 558 306 48572 48586 48601 48615 48629 48643 48657 48671 48686 48700 307 48714 48728 48742 48756 48770 48785 48799 48813 48 827 48 841 308 48 855 48869 48 883 48897 48911 48926 48 940 48 954 48968 48982 309 48996 49010 49024 49038 49052 49066 49080 49094 49108 49122 31O 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 311 49 276 49290 49304 49318 49332 49346 49360 49374 49388 49402 312 49 415 49429 49443 49457 49471 49485 49499 49513 49527 49 541 313 49 554 49 568 49582 49 596 49610 49624 49638 49651 49665 49679 314 49693 49707 49721 49734 49748 49762 49776 49790 49803 49817 315 49831 49845 49859 49872 49886 49900 49914 49927 49941 49955 316 49969 49982 49996 50010 50024 50037 50051 50 065 50079 50092 317 50 106 50 120 50133 50 147 50161 50174 50188 50 202 50 215 50 229 318 50 243 50 256 50270 50284 50297 50311 50325 50 338 50352 50365 319 50379 50393 50406 50420 50433 50447 50461 50474 50488 50501 320 50515 50529 50542 50 556 50 569 50583 50 596 50610 50623 50637 321 50651 50664 50678 50691 50705 50718 50 732 50745 50759 50772 322 50 786 50799 50813 50 826 50 840 50853 50 866 50880 50893 50907 323 50 920 50934 50947 50961 50974 50987 51001 51014 51028 51041 324 51 055 51068 51081 51095 51108 51121 51135 51148 51162 51175 325 51188 51202 51 215 51228 51242 51 255 51268 51282 51295 51308 326 51322 -51335 51348 51 362 51 375 51388 51402 51415 51428 51441 327 ^1415/51468 51481 51495 51508 51521 51534 51548 51561 51574 328 sStffefc 51601 51614 51627 51 640 51654 51667 51680 51693 51706 329 l '5l'20 51733 51746 51759 51772 51786 51799 51812 51825 51838 33O 51 851 51865 51 878 51891 51904 51917 51930 51943 51957 51970 331 51983 51996 52009 52022 52 035 52 048 52061 52075 52088 52101 332 52114 52127 52140 52 153 52 166 52179 52192 52205 52218 52231 333 52244 52257 52 270 52284 52 297 52310 52323 52336 52 349 52362 334 52375 52388 52401 52414 52427 52 440 52453 52466 52479 52492 335 52504 52 517 52530 52543 52556 52569 52582 52595 52608 52621 336 52634 52647 52 $60 52673 52686 52699 52711 52724 52737 52 750 337 52763 52776 52789 52802 52815 52827 52840 52853 52866 52879 338 52892 52905 52917 52 930 52943 52 956 52 969 52982 52994 53007 339 53020 53033 53046 53058 53071 53084 53097 53] 10 53122 53135 340 53148 53161 53 173 53186 53199 53 212 53224 53237 53250 53263 341 53275 53 288 53 301 53314 53326 53 339 53352 53364 53377 53390 342 53403 53415 53428 53441 53 453 53466 53479 53491 53504 53 517 343 53529 53542 53555 53567 53580 53 593 53 605 53 618 53631 53 643 344 53656 53668 53681 53694 53706 53719 53732 53744 53757 53769 345 53782 53794 53807 53820 53832 53845 53 857 53870 53882 53895 346 53908 53920 53 933 53945 53958 53 970 53 983 53 995 54008 54020 347 54033 54045 54058 54070 54083 54095 54108 54120 54133 54 145 348 54 158 54170 54183 54 195 54208 54220 54 233 54 245 54258 54270 349 54283 54295 54307 54320 54332 54345 54357 54370 54382 54394 No. 1 2 3 4 g 6 7 8 9 300-349 350-399 15 No. O 1 2 3 4 56789 350 351 352 353 354 54407 54419 54 432 \5 4 444 54456 54531 54543 54555 54568 54580 54654 54667 54679 54691 54704 54 777 54 790 54 802 54 814 54 827 54900 54913 54925 54937 54949 54469 54481 54494 54506 54518 54593 54605 54617 54630 54642 54716 54728 54741 54753 54765 54839 54851 54864 54876 54888 54962 54974 54986 54998 55011 355 356 357 358 359 55023 55035 55047 55060 55072 55 145 55 157 55 169 55 182 55 194 55267 55279 55291 55303 55315 55388 55400 55413 55425 55437 55509 55522 55534 55546 55558 55084 55096 55108 55121 55133 55206 55218 55230 55242 55255 55328 55340 55352 55364 55376 55449 55461 55473 55485 55497 55570 55582 55594 55606 55618 36O 361 362 363 364 55630 55642 55654 55666 55678 55751 55763 55 775 55 787 55799 55871 55883 55895 55907 55919 55991 56003 56015 56027 56038 56110 56122 56134 56146 56158 55 691 55 703 55 715 55 727 55 739 55811 55823 55835 55*847 55859 55931 55943 55955 55967 55979 56050 56062 56074 56086 56098 56170 56182 56194 56205 56217 365 366. 367 368 369 56229 56241 56253 56265 56277 56348 56360 56372 56384 56396 56467 56478 56490 56502 56514 56585 56597 56608 56620 56632 56703 56714 56726 56738 56750 56289 56301 56312 56324 56336 56407 56419 56431 56443 56455 56526 56538 56549 56561 56573 56644 56656 56667 56679 56691 56761 56773 56785 56797 56808 37O 371 372 373 374 56820 56832 56844 56855 56867 56937 56949 56961 56972 56984 57054 57066 57078 57089 57101 57171 57183 57194 57206 57217 57287 57299 57310 57322 57334 56879 56891 56902 56914 56926 56996 57008 57019 57031 57043 57113 57124 57136 57148 57159 57229 57241 57252 57264 57276 57345 57357 57368 57380 57392 375 376 377 378 379 57403 57415 57426 57438 57449 57519 57530 57542 57553 57565 57634 57646 57657 57669 57680 57749 57761 57772 57784 57795 57864 57875 57887 -57898 57910 57461 57473 57484 57496 57507 57576 57588 57600 57611 57623 57692 57703 57715 57726 57738 57807 57818 57830 57841 57852 57921 57933 57944 57955 57967 38O 381 '382 383 384 57978 57990 58001 58013 58024 58092 58104 58115 58127 58138 58206 58218 58229 58240 58252 58320 58331 58343 58354 58365 58433 58444 58456 58467 58478 58035 58047 58058 58070 58081 58149 58161 58172 58184 58195 58263 58274 58286 58297 58309 58377 58388 58399 58410 58422 58490 58501 58512 58524 58535' 385 386 387 388 389 58546 58557 58569 58580 58591 58659 58670 58681 58692 58704 58771 58782 58794 58805 58816 58883 58894 58906 58917 58928 58995 59006 59017 59028 59040 58602 58614 58625 58636 58647 58715 5^726 58737 58749 58760 58827 58838 58850 58861 58872 58939 58950 58961 58973 58984 59051 59062 59073 59084 59095 39O 391 392 393 394 59106 59118 59129 59140 59151 59218 59229 59240 59251 59262 59329 59340 59351 59362 59373 59439 59450 59461 59472 59483 59550 59561 59572 59583 59594 59162 59173 59184^59195 59207 59273 59284 59295 59306 59318 59384 59395 59406 59417 59428 59494 59506 59517 59528 59539 59605 59616 59627 59638 59649 395 396 397 398 399 59660 59671 59682 59693 59704 59770 59780 59791 59802 59813 59879 59890 59901 59912 59923 59988 59999 60010 60021 60032 60097 60108 60119 60130 60141 59715 59726 59737 59 74S-.597Sa 59824 59835 59846 59857 5956$ 59934 59945 59956 59966 59977 60043 60054 60065 60076 60016 60152 60163 60173 60184 60195 No. O 1 2 3 4 56789 350-399 lli 400-449 ]$0. 1 2 3 4 5 6 7 8 9 4OO 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 401 60314 60325 60 336 60347 60358 60369 60379 60390 60401 60412 402 60423 60 433 60 444 60455 60466 60477 60487 60498 60509 60520 403 60531 60541 60552 60 563 60574 60584 60595 60606 60617 60627 404 60638 60649 60660 60670 60681 60692 60703 60713 60724 60735 405 60746 60 756 60767 60778 60788 60799 60810 60821 60831 60842 406 60 853 60863 60874 60 885 60895 60906 60917 60927 60938 60949 407 60959 60970 60981 6Q991 61002 61013 61023 61034 61045 61 055 408 4 61 066 61077 61087 61 098 61 109 61 119 61 130 61140 61 151 61 162 409 61 172 61183 61194 61204 61215 61225 61236 61247 61257 61268 41O 61 278 61289 61300 61310 61321 61331 61342 61 352 61363 61374 411 61 384 61 395 61 405 61416 61426 61437 61448 61458 61469 61479 412 61490 61500 61511 61521 61532 61542 61553 61563 61 574 61584 413 61 595 61606 61616 61627 61637 61648 61 658 61 669 61679 61690 414 61700 61 711 61 721 61731 61742 61752 61763 61773 61784 61 794 415 61805 61815 61826 61836 61 847 61 857 61 868 61878 61888 61899 416 61909 61920 61930 61941 61951 61962 61972 61982 61 993 62003 ^417 62014 62024 62034 62045 62 055 62066 62 076 62086 62097 62107 418 62118 62128 62138 62149 62 159 62170 62180 62190 62201 62211 419 62221 62232 62242 62252 62263 62273 62284 62294 62304 62315 420 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 421 62428 62439 62 449 62 459 62469 62 480 62490 62500 62511 62521 422 62 531 62 542 62552 62562 62 572 62 583 62593 62603 62613 62 624 423 62 634 62644 62655 62665 62675 62 685 62696 62 706 62716 62726 424 62 737 62747 62757 62 767 62778 62 788 62 798 62808 62818 62829 425 62 839 62849 62859 62870 62880 62890 62900 62910 62921 62931 -126 62941 62951 62961 62972 62 982 62992 63 002 63012 63022 63 033 427 63043 63053 63 063 63 073 63 083 63094 63 104 63114 63 124 63134 428 63 141 63 155 63165 63175 63185 63 195 63 205 63215 63 225 63236 429 63 246 63256 63266 63276 63286 63296 63306 63317 63327 63337 430 63347 63357 63367 63377 63 387 63397 63407 63417 63428 63438 431 63 448 63458 63468 63478 63 488 63 498 63 508 63 518 63 528 63538 432 63548 63 658 63568 63579 63 589 63 599 63 609 63619 63629 63639 433 63649 63 659 63669 63679 63 689 63 699 63709 63719 63729 63739 434 63749 63759 63769 63779 63789 63 799 63809 63819 63829 63 839 435 63 849 63 859 63869 63 879 63889 63899 63909 63919 63929 63939 436 63949 63 959 63 969 63979 63 988 63 998 64008 64018 64028 64 038 437 64048 64058 64068 64 078 64088 64098 64108 64118 64128 64137 438 64 147 64 157 64167 64 177- 64 187 64 197 64207 64217 64227 64237 439 64 246 64256 64266 64276 64286 64296 64306 64316 64326 64335 44O 64 345 64 355 64365 64375 64385 64395 64 404 64414 64424 64434 441 64 444 64 454 64464 64 473 64483 64493 64 503 64513 64 523 64532 442 64542 64 552 64562 64572 64582 64 591 64601 64611 64621 64631 443 64640 64650 64660 64670 64680 64689 64699 64709 64719 64729 444 64738 64748 64758 64768 64777 64787 64797 64807 64816 64826 445 64836 64846 64856 64 865 64 875 64885 64895 64904 64914 64 924 446 64933 64943 64953 64963 64972 64 982 64992 65 002 65011 65021 447 65031 65 040 65050 65 060 65 070 65 079 65 089 65 099 65108 65 118 448 65128 65 137 65147 65 157 65 167 65 176 65 186 65 196 65 205 65 215 449 65 225 65 234 65244 65254 65 263 65273 65283 65292 65302 65312 1 No. O 1 2 3 4 5 6 7 8 9 I 400-449 450-499 17 No. 1 2 3 4 5 6 7 8 9 450 65 321 65 331 65 341 65350 65360 65369 65 379 65 389 65 398 65 408 451 65418 65 427 65 437 65 447 65 456 65466 65 475 65485 65495 65 504 452 65 514 65 523 65 533 65 543 65 552 65 562 65571 65581 65 591 65600 453 65 6 10 65 619 65 629 65 639 65648 65 658 65667 65677 65686 65696 454 65 706 65715 65725 65734 65744 65753 65763 65772 65782 65792 \ 455 65 801 65811 65 820 65 830 65839 65 849 65 858 65868 65877 65887 456 65 896 65 906 65916 65 925 65 935 65944 65 954 65963 65 973 65982 457 65 992 66001 66011 66020 66 030 66039 66049 66 058 66068 66 077 458 66087 66096 66 106 66115 66124 66134 66143 66153 66162 66172 459 66181 66191 66200 66 210 66219 66229 66238 66247 66257 66266 46O 66 276 66285 66295 66304 66314 66323 66332 66342 66351 66361 461 66 370 66380 66389 66398 66408 66417 66427 66436 66445 66 455 462 66 464 66 474 66483 66492 66502 66511 66521 66 530 66539 66549 463 66 558 66 567 " 66577 66 586 66 596 66605 66614 66624 66633 66642 464 66652 66 661 66671 66680 66689 66 699 66 708 66717 66727 66736 465 66 745 66 755 66764 66773 66783 66792 66801 66811 66820 66829 466 66839 66848 66857 66867 66876 66 885 66894 66904 66913 66922 467 66932 66 941 66950 66960 66969 66978 66987 66997 67006 67015 468 67025 67034 67043 67 052 67062 67071 67080 67089 67099 67108 469 67 1.17 67127 67136 67145 67154 67164 67173 67182 67191 67201 470 67210 67219 67228 67237 67247 67 256 67265 67274 67284 67293 471 67 302 67311 67321 67330 67339 67348 67357 67367 67376 67385 472 67394 67403 67413 67422 67431 67 440 67449 67 459 67468 67477 473 67486 67495 67 504 67514 67523 67 532 67541 67 550 67 560 67569 474 67578 67587 67596 67605 67614 67624 67633 67642 67651 67660 475 67669 67679 67688 67697 67706 67715 67724 67733 67742 67 752 476 67761 67770 67 779 67788 67797 67806 67815 67825 67834 67843 477 67852 67861 67870 67879 67888 67897 679J&6 67 916 67925 67934 478 67 943 67 952 67961 67 970 67979 67988 67997 68006 68015 68024 479 68 034 68043 68052 68061 68070 68079 68088 68097 68106 68115 480 68124 68133 68 142 68 151 68160 68169 68178 68187 68196 68 205 481 68215 68224 68 233 68242 68 251 68260 68269 68278 68287 68296 482 68 305 68314 68323 68332 68341 68350 68 359 68368 68377 68386 483 68395 68404 68 413 68422 68431 68440 68449 68 458 68467 68476 484 68485 68494 68502 68511 68520 68529 68538 68547 68556 68565 485 68574 68 583 68 592 68601 68610 68619 68628 68637 68646 68655 486 68664 68673 68681 68690 68 699 68708 68717 68726 68735 68744 487 68753 68762 687T1 68780 68789 68797 68806 68815 68824 68833 488 68 842 68851 68860 68869 68878 68.886 68 895 68904 68913 68922 489 68931 68940 68949 68 958 68 966 68 975 68984 68993 69002 69011 49O 69020 69028 69037 69046 69055 69064 69073 69082 69090 69 099 491 69 108 69117 69126 69135 69 144 69 152 69161 69170 69179 69188 492 69197 69 205 69214 69223 69232 69 241 69249 69 258 69267 69276 493 69 285 69294 69 302 69311 69320 69329 69338 69346 69 355 69364 494 69373 69381 69390 69 399 69408 69417 69425 69434 69443 69452 495 69461 69469 69478 69487 69496 69 504 69513 69522 69531 69 539 496 69 548 69 557 69 566 69574 69 583 69 592 69601 69 609 69618 69627 497 69636 69644 69 653 69662 69671 69679 69688 69697 69 705 69 714 498 69723 69732 69740 69749 69758 69 767 69775 69 784 69793 69801 499 69810 69819 69 827 69836 69845 69 854 69862 69871 69880 69888 No. O 1 2 3 4 5 O 7 8 9 450-499 18 500-549 No. O 1 2 3 4 5 6 7 8 9 5OO 69 897 69906 69914 69923 69932 69940 69949 69958 69966 69 975 501 69984 69992 70001 70-010 70018 70027 70036 70044 70053 70062 502 70070 70079 70088 70096 70105 70114 70122 70131 70140 70148 503 70 157 70165 70174 70183 70191 70200 70209 70217 70 226 70234 504 70243 70252 70260 70269 70278 70286 70295 70303 70312 70321 505 70329 70338 70346 70355 70364 70372 70381 70389 70398 70406 506 70415 70424 70432 70441 70449 70 458 70467 70475 70484 70492 507 70 501 70509 70518 70526 70535 70544 70 552 70561 70569 70578 508 70586 70 595 70603 70612 70621 70629 70 638 70 646 70655 70663 509 70672 70680 70689 70697 70706 70714 70723 70731 70740 70749 510 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 511 70842 70851 70859 70868 70 876 70885 70893 70 902 70910 70919 512 70927 70935 70944 70 952 70961 70969 70 978 70986 70995 71003 513 71012 71020 71029 71037 71046 71054 71 063 71071 71079 71088 514 71096 71105 71113 71122 71130 71 139 71 147 71155 71164 71172 515 71181 71189 71198 71206 71214 71223 71231 71240 71248 71 257 516 71265 71273 71282 71290 71 299 71 307 71315 71324 71332 71341 517 71 349 71357 71366 71374 71383 71391 71399 71408 71416 71 425 518 71433 71 441' 71450 71458 71466 71475 71483 71492 71 500 71508 519 71517 71525 71533 71 542 71 550 71559 71567 71575 71584 71592 520 71600 71609 71617 71625 71634 71642 71650 71659 71667 71675 521 71684 71692 71700 71709 71 717 71725 71734 71742 71750 71759 522 71767 71775 71784 71792 71800 71809 71817 71825 71834 71842 523 71850 71858 71867 71875 71 883 71892 71900 71908 71917 71925 524 71933 71941 71950 71958 71966 7197,5 71983 71991 71999 72008 525 72016 72024 72032 72041 72049 72057 72066 72074 72082 72090 526 72099 72107 72115 72123 72132 72140 72148 72156 72165 72173 527 72181 72189 72198 72206 72214 72222 72 230 72239 72247 72255 528 72263 72272 72280 72288 72296 72304 72313 72321 72329 72337 529 72346 72354 72362 72370 72378 72357 72 395 72403 72411 72419 53O 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 531 72 509 72518 72 526 72534 72 542 72 550 72 558 72567 72575 72583 532 72591 72599 72607 72616 72624 72 632 72 640 72648 72656 72665 533 72673 72681 72689 72697 72705 72713 72722 72 730 72738 72746 534 72754 72762 72770 72779 72787 72795 72803 72811 72819 72827 535 72 835 72843 72852 72860 72868 72876 72884 72892 72900 72908 1 536 72916 72925 72933 72941 72949 72957 72965 72973 72981 72989 537 72997 73006 73014 73022 73030 73038 73046 73054 73062 73070 538 73078 73086 73094 73 102 73 111 73119 73127 73135 73143 73151 539 73159 73167 73175 73183 73191 73199 73207 73215 73223 73231 54O 73239 73247 73 255 73263 73272 73280 73288 73296 73304 73312 541 73320 73328 73336 73344 73 352 73360 73368 73376 73384 73392 542 73 400 73408 73416 73424 73432 73440 73448 73456 73464 73472 543 73480 73488 73496 73 504 73512 73 520 73528 73536 73 544 73552 544 73560 73568 73576 73 584 73592 73600 73608 73616 73624 73632 545 73 640 73648 73 656 73664 73 672 73679 73687 73695 73703 73711 546 73719 73727 73 735 73743 73751 73759 73767 73775 73783 73791 547 73 799 73807 73815 73823 73830 73838 73846 73 854 73862 73870 548 73878 73 886 73894 73902 7391$ 73918 73926 73933 73941 73 949 549 73957 73 965 73973 73981 73989 73997 74005 74013 74020 74028 No. 1 2 3 4 5 6 7 8 9 500-549 550-599 19 No. O 1 2 3 4 5 6 7 8 9 550 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 551 74115 74123 74131 74139 74147 74155 74162 74170 74178 74186 552 74194 74202 74210 74218 74225 74233 74241 74249 74257 74265 553 74273 74280 74 288 74296 74304 74312 74320 74327 74335 74 343 554 74351 74359 74367 74374 74382 74390 74398 74406 74 414' 74421 \ 555 74429 74437 74445 74453 74461 74468 74476 74484 74492 74500 556 74507 74515 74523 74531 74539 74547 74 554 74562 74570 74578 557 74 586 74593 74601 74609 74617 74624 74632 74640 74648 74656 558 74663 74671 74679 74 687 74695 74702 74710 74718 74726 74733 559 74741 74749 74757 74764 74772 74780 74788 74796 74803 74811 560 74819 74827 74834 74842 74850 74 858 74865 74873 74881 74889 561 74896 74904 74912 74920 74927 74935 74943 74950 74 958 74966 562 74974 74981 74 989 74997 75005 75 012 75020 75028 75035 75 043 563 75 051 75 059 75 066 75074 75082 75089 75097 75105 75 113 75120 564 75128 75 136 75 143 75 151 75159 75166 75174 75 182 75 189 75197 565 75205 75213 75220 75 228 75236 75243 75251 75259 75266 75274 566 75282 75289 75297 75305 75312 75320 75328 75335 75343 75351 567 75358 75366 75374 75 381 75389 75397 75404 75412 75420 75427 568 75435 75 442 75 450 75 458 75465 75473 75481 75488 75496 75504 569 75511 75519 75526 75534 75542 75549 75 557 75565 75572 75580 570 75 587 75595 75603 75610 75618 75626 75633 75641 75648 75656 571 75 664 75 671 75679 75686 75694 75702 75709 75717 75 724 75732 572 75740 75747 75 755 75762 75 770 75 778 75 785 75793 75800 75808 573 75815 75823 75831 75838 75846 75853 75861 75868 75876 75884 574 75891 75899 75906 75914 75921 75929 75937 75944 75952 75959 575 75967 75974 75982 75989 75997 76005 76012 76020 76027 76035 576 76042 76050 76057 76065 76072 76080 76087 76095 76103 76110 577 76118 76125 76133 76140 76148 76155 76163 76170 76178 76185 578 76193 76200 76208 76215 76223 76230 76238 76245 76 253 76260 579 76268 76275 76283 76290 76298 76305 76313 76320 76328 76 335 580 76343 76 350 76358 76365 76373 76380 76388 76395 76403 76410 581 76418 76 425 76433 76440 76448 76455 76462 76470 76477' 76485 582 76492 76500 76 507 76515 76522 76530 76537 76 545 76552 76559 583 76567 76574 76 582 76 589 76597 76604 76612 76619 76626 76634 584 76641 76649 76 656 76664 76671 76678 76686 76693 76701 76708 585 76716 76723 76730 76738 76 745 76753 76760 76768 76775 76782 586 76790 76797 76805 76812 76819 76827 76834 76842 76849 76856 587 76864 76871 76 879 76886 76 893 76901 76908 76916 76923 76930 588 76938 76945 76 953 76 960 76967 ' 76975 76982 76989 76997 77004 589 77012 77019 77026 77034 77041 77048 77056 77063 77070 77078 59O 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 591 77 159 77166 77173 77181 77188 77195 77203 77210 77217 77225 592 77232 77240 77247 77 254 77262 77269 77276 77283 77291 77298 593 77 305 77313 77320 77327 77335 77342 77349 77357 77 364 77371 594 77379 77386 77393 77401 77408 77415 77422 77430 77437 77444 595 77 452 77 459 77 466 77 474 77481 77488 77495 77503 77510 77517 596 77525 77532 77539 77 546 77 554 77 561 77 568 77576 77583 77590 597 77597 77605 77612 77619 77627 77634 77641 77648 77 656 77663 598 77670 77677 77685 77692 77699 77706 77714 77721 77728 77 735 599 77743 77750 77757 77764 77772 77779 77786 77793 77801 77808 No. 1 2 3 4 5 6 7 8 9 550-599 20 600-649 No. O 1 2 3 4 5 G 7 8 9 GOO 77815 77822 77830 77837 77 844 77 851 77 859 77866 77873 77 880 601 77887 77895 77902 77909 77916 77924 77931 77938 77 945 77 952 602 77960 77 967 77974 77981 77 988 77996 78003 78010 78017 78 025 603 78032 78 039 78046 78053 78 061 78068 78075 78082 78089 78 097 604 78104 78111 78118 78125 78132 78140 78147 78154 78161 78168 605 78176 78183 78190 78197 78204 78211 78219 78226 78233 78240 606 78247 78254 78262 78269 78276 78283 78290 78297 78305 78312 607 78319 78326 78333 78340 78347 78355 78362 78369 78376 78 383 608 78390 78398 78405 78412 78419 78426 78433 78440 78447 78455 609 78462 78469 78476 78483 78 490 78 497 78504 78512 78519 78 526 61O 78 533 78540 78 547 78554 78 561 78569 78576 78583 78590 78 597 611 78604 78611 78618 78 625 78 633 7S6HO 78647 78 654 78661 78668 612 78 675 78682 78 689 78696 78704 78711 78718 78725 78732 78739 613 78746 78 753 78760 78 767 78774 78 7S1 78789 78796 78 803 78810 614 78817 78824 78'S31 78838 78845 78852 78859 78866 78873 78880 615 78888 78895 78902 78909 78916 78923 78930 78937 78944 78951 616 78 958 78 965 78972 78979 78 986 78 993 79000 79007 79014 79021 617 79029 79036 79 043 79 050 79 057 79064 79071 79078 79085 79092 618 79099 79106 79 113 79120 79127 79134 79 141 79148 79 155 79162 619 79169 79176 79183 79190 79197 79204 79211 79218 79 225 79232 62O 79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 621 79309 79316 79323 79330 79 337 79344 79351 79358 79365 79372 622 79379 79386 79393 79400 79 407 79414 79421 79428 79435 79442 623 79449 79 456 79463 79470 79 477 79 484 79 491 79498 79 505 79511 624 79518 79525 79532 79539 79546 79553 79 560 79567 79574 79581 625 79 588 79595 79602 79609 79616 79623 79630 79637 79644 79650 626 79657 79664 79671 79678 79 685 79692 79699 79706 79 713 79720 627 79727 79734 79741 79748 79 754 79761 79768 79 775 79782 79 789 628 79796 79803 79810 79817 79824 79831 79837 79 844 79851 79 858 629 79865 79872 79879 79886 79 893 79900 79 906 79913 79920 79927 63O 79934 79941 79948 79955 79962 79969 79975 79982 799S9 79996 631 80003 80010 80017 80024 80030 80037 80044 80051 80058 SO 065 632 80072 80079 80085 80092 80099 80106 80113 80120 80127 80134 633 80140 80147 80154 80161 80168 80175 80182 80188 80195 SO 202 634 80209 80216 80223 80229 80236 80243 80250 80257 SO 264 SO 271 635 80277 80284 80291 80298 80305 80312 80 318 80 325 80 332 80 339 636 80346 80 353 80359 80 366 80373 80380 80387 80393 80400 80407 637 80414 80421 80428 80 434 80441 80448 80 455 80462 SO 468 80 475 638 80482 80489 80496 80502 80509 80516 80 523 80 530 SO 536 80 543 639 80550 80557 80564 80570 80577 80 584 80591 80598 80604 SO 611 640 80618 80625 80632 80638 SO 645 80652 80 659 80665 SO 672 80679 641 80686 80693 80699 80706 80713 80720 80786 SO 733 80740 SO 747 642 80754 80760 80767 80774 80781 80787 80794 80801 80 SOS. SO 814 643 80821 80828 80835 80841 80848 80855 80862 SOS6S SO 875 80882 644 80889 80895 80902 80909 80916 80922 80929 80936 80943 80949 645 80956 80963 80969 80976 80983 80990 80996 SI 003 81010 81017 646 81023 81030 81037 81 043 81 050 81057 81064 81 070 81077 SI 084 647 81090 81097 81 104 81111 81 117 81 124 81131 81137 81144 SI 151 648 81 158 81 164 81171 81178 81 184 81191 81 198 81204 81211 81218 649 81224 81 231 81238 81245 81251 81258 81265 81271 8127S 81285 No. O 1 2 3 4 5 G 7 8 9 600-649 650-699 21 No. O 1 2 3 4 50789 65O 651 652 653 654 81291 81298 81305 81311 81318 81358 81365 81371 81378 81385 81425 81431 81438 81445 81451 81491 81498 81505 81511 81518 81 558 81 564 81 571 81 578 81 584 81 325 81 331 81 338 81 345 81 351 81391 81398 81405 81411 81418 81458 81465 81471 81478 81485 81525 81531 81538 81544 81551 81591 81598 81604 81611 81617 655 656 657 658 659 81624 81631 81637 81644 81651 81 690 81 697 81 704 81 710 81 717 81 757 81 763 81 770 81 776 81 783 81823 81829 81836 81842 81849 81889 81895 81902 81908 81915 81 657 81 664 81 671 81 677 81 684 81 723 81 730 81 737 81 743 81 750 81 790 81 796 81 803 81 809 81 816 81 856 81 862 81 869 81 875 81 882 81921 81928 81935 81941 81948 66O 661 662 663 664 81954 81961 81968 81974 81981 82020 82027 82033 82040 82046 82 086 82092 82099 82105 82112 82151 82-158 82164 82171 82178 82217 82223 82230 82236 82243 81987 81994 82000 82007 82014 82053 82060 82066 82073 82079 82119 82125 82132 82138 82145 82184 82191 82197 82204 82210 82249 82256 82263 82269 82276 665 666 667 668 669 82282 82289 82295 82302 82308 82347 82354 82360 82367 82373 82413 82419 82426 82432 82439 82478 82484 82491 82497 82504 82543 82549 82556 82562 82569 82315 82321 82328 82334 82341 82380 82387 82393 82400 82406 82445 82452 82458 82465 82471 82510 82517 82523 82530 82536 82575 82582 82588 82595 82601 67O 671 672 673 674 82607 82614 82620 82627 82633 82672 82679 82685 82692 82698 82737 82743 82750 82756 82763 82802 82808 82814 82821 82827 82866 82872 82879 82885 82892 82640 82646 82653 82659 82666 82705 82711 82718 82724 82730 82769 82776 82782 82789 82795 82834 82840 82847 82853 82860 82898 82905 82911 82918 82924 675 676 677 678 679 82930 82937 82943 82950 82956 82995 83001 83008 83014 83020 83059 83065 83072 83078 83085 83 123 83 129 83 136 83 142 83 149 83 187 83 193 83 200 83 206 83 213 82963 82969 82975 82982 82988 83027 83033 83040 83046 83052 83 091 83 097 83 104 83 110 83 117 83 155 83 161 83 168 83 174 83 181 83219 83225 83232 83238 83245 68O 681 682 683 684 83251 83257 83264 83270 83276 83315 83321 83327 83334 83340 83378 83385 83391 83398 83404 83442 83448 83455 83461 83467 83506 83512 83518 83525 83531 83283 83289 83296 83302 83308 83347 83353 83359 83366 83372 83410 83417 83423 83429 83436 83474 83480 83487 83493 83499 83537 83544 83550 83556 83563 685 686 687 688 689 83569 83575 83582 83588 83594 83632 83639 83645 83651 83658 S3 696 83 702 83 708 83 715 83 721 83 759 83 765 83 771 83 778 83 784 83822 83828 83835 83841 83847 83601 83607 83613 83620 83626 83664 83670 83677 83683 83689 83727 83734 83740 83746 83753 83790 83797 83803 83809 83816 83853 83860 83866 83872 83879 690 691 692 693 694 83885 83891 83897 83904 83910 83948 83954 83960 83967 83973 84011 84017 84023 84029 84036 84073 84080 84086 84092 84098 84 136 84 142 84 148 84 155 84 161 83916 83923 83929 83935 83942 83979 83985 83992 83998 84004 84042 84048 84055 84061 84067 84105 84111 84117 84123 84130 84167 84173 84180 84186 84 If 2 695 696 697 698 699 84198 84205 84211 84217 84223 84261 84267 842^ 84280 84286 84323 84330 84336 84342 84348 843S6 84392 84398 84404 84410 84448 84454 84460 84466 84473 84230 84236 84242 84248 A0d? 84292 84298 84305 84311 84317 84354 84361 84367 84373 84379 84417 84423 84429 84435 84442 84 479 84 485 84 491 84 497 84 504 No. O 1 2 3 4 56789 650-699 9,9, 700-749 No. O 1 2 3 4 5 6 7 89 700 701 702 703 704 84510 84516 84522 84528 84535 84572 84578 84584 84590 84597 84634 84640 84646 84652 84658 84696 84702 84708 84714 84720 84757 84763 84770 84776 84782 84541 84547 84553 84559 84566 84603 84609 84615 84621 84628 84665 84671 84677 84683 84689 84726 84733 84739 84745 84751 84788 84794 84800 84807 84813 705 706 707 708 709 84819 84825 84831 84837 84844 84880 84887 84893 84899 84905 84942 84948 84954 84960 84967 85003 85009 85016 85022 85028 85065 85071 85077 85083 85089 84850 84856 84862 84868 84874 84911 84917 84924 84930 84936 84973 84979 84985 84991 84997 85034 85040 85046 85052 85058 85 095 85 101 85 107 85 114 85 120 710 711 712 713 714 85 126 85 132 85 138 85 144 85 150 85 187 85 193 85 199 85 205 85 211 85 248 85 254 85 260 85 266 85 272 85309 85315 85321 85327 85333 85370 85376 85382 85388 85394 85156 85163 85169 85175 85181 85 217 85 224 85 230 85 236 85 242 85278 85285 85291 85297 85303 85339 85345 85352 85358 85364 85400 85406 85412 85418 85425 715 716 717 718 719 85431 85437 85443 85449 85455 85 491 85 497 85 503 85 509 85 516 85552 85558 85564 85570 85576 85612 85618 85625 85631 85637 85673 85679 85685 85691 85697 85461 85467 85473 85479 85485 85522 85528 85534 85540 85546 85582 85588 85594 85600 85606 85643 85649 85655 85661 85667 85703 85709 85715 85721 85727 72O 721 722 723 724 85 733 85 739 85 745 85 751 85 757 85794 85800 85806 85812 85818 85854 85860 85866 85872 85878 85914 85920 85926 85932 85938 85974 85980 85986 85992 85998 85 763 85 769 85 775 85 781 85 788 85824 85830 85836 85842 85848 85884 85890 85896 85902 85908 85944 85950 85956 85962 85968 86004 86010 86016 86022 86028 725 726 727 728 729 86034 86040 86046 86052 86058 86094 86100 86106 86112 86118 86153 86159 86165 86171 86177 86213 86219 86225 86231 86237 86273 86279 86285 86291 86297 86064 86070 86076 86082 86088 86124 86130 86136 86141 86147 86183 86189 86195 86201 86207 86243 86249 86255 86261 86267 86303 86308 86314 86320 86326 73O 731 732 733 734 86332 86338 86344 86350 86356 86392 86398 86404 86410 86415 864S1 86457 86463 86469 86475 86510 86516 86522 86528 86534 86570 86576 86581 86587 86593 86362 86368 86374 86380 86386 86421 86427 86433 86439 86445 86481 86487 86493 86499 86504 86540 86546 86552 86558 86564 86599 86605 86611 86617 86623 735 736 737 738 739 86629 86635 86641 86646 86652 86688 86694 86700 86705 86711 86747 86753 86759 86764 86770 86806 86812 86817 86823 86829 86864 86870 86876 86882 86888 86658 86664 86670 86676 86682 86717 86723 86729 86735 867-41 86 776 86 782 86 788 86 794 86 800 86835 86841 86847 86 853 86859 86894 86900 86906 86911 86917 74O 741 742 743 744 86923 86929 86935 86941 86947 86982 86988 86994 86999 87005 87040 87046 87052 87058 87064 87099 87105 87111 87116 87122 87157 87163 87169 87175 87181 86953 86958 86964 86970 86976 87011 87017 87023 87029 87035 87070 87075 87081 87087 87093 87128 87134 87140 87146 87151 87186 87192 87198 87204 87210 745 746 747 748 749 87216 87221 87227 87233 87239 87274 87280 87286 87291 87297 87332 87338 87344 87349 87355 87390 87396 87402 87408 87413 87448 87454 87460 87466 87471 87245 87251 87256 87262 87268 87303 87309 87315 87320 87326 87361 87367 87373 87379 87384 87419- 87425 87431 87437 87442 87477 87483 87489 87495 87500 No. O 1 2 3 4 56789 700-749 750-799 23 No. O 1 2 3 4 56789 750 751 752 753 754 87506 87512 87518 87523 87529 87564 87570 S7~576 87581 87587 87622 87628 87633 87639 87645 87679 87685 87691 87697 87703 87737 87743 87749 87754 87760 87535 87541 87547 87552 87558 87593 87599 87604 87610 87616 87651 87656 87662 87668 87674 87708 87714 87720 87726 87731 87766 87772 87777 87783 87789 755 756 757 758 759 87795 87800 87806 87812 87818 87 852 87858 87864 87869 87875 87910 87915 87921 87927 87933 87967 87973 87978 87984 87990 88024 88030 88036 88041 88047 87823 87829 87835 87841 87846 87881 87887 87892 87898 87904 87938 87944 87950 87955 87961 87996 88001 88007 88013 88018 88053 88058 88064 88070 88076 76O 761 762 763 764 SSOSl 88087 88093 88098 88104 88138 88144 88150 88156 88161 88195 88201 88207 88213 88218 88252 88258 88264 88270 88275 88 309 88315 88321 88326 88332 88110 88116 88121 88127 88133 88167 88173 88178 88184 88190 88224 88230 88235 88241 88247 88281 88287 88292 88298 88304 88338 88343 88349 88355 88360 765 766 767 768 769 88366 88372 88377 88383 88389 88423 88429 88434 88440 88446 88480 88485 88491 88497 88502 88536 88542 88547 88553 88559 88593 88598 88604 88610 88615 88395 88400 88406 88412 88417 88451 88457 88463 88468 88474 88508 88513 88519 88525 88530 88564 88570 88576 88581 88587 88621 88627 88632 88638 88643 770 771 772 773 774 88649 88655 88660 88666 88672 88705 88711 88717 88722 88728 88762 88767 88773 88779 88784 88818 88824 88829 88835 88840 88874 88880 88885 88891 88897 88677 88683 88689 88694 88700 88734 88739 88745 88750 88756 88790 88795 88801 88807 88812 88846 88852 88857 88863 88868 88902 88908 88913 88919 88925 775 776 777 778 779 88930 88936 88941 88947 88953 88986 88992 88997 89003 89009 -89042 89048 89053 89059 89064 89098 89104 89109 89115 89120 89154 89159 89165 89170 89176 88958 88964 88969 88975 88981 89014 89020 89025 89031 89037 89070 89076 89081 89087 89092 89126 89131 89137 89143 89148 89182 89187 89193 89198 89204 780 781 782 783 784 89209 89215 89221 89226 89232 89265 89271 89276 89282 89287 89321' 89326 89332 89337 89343 89376 89382 89387 89393 89398 89432 89437 89443 89448 89454 89237 89243 89248 89254 89260 89293 89298 89304 89310 89315 89348 89354 89360 89365 89371 89404 89409 89415 89421 89426 89459 89465 89470 89476 89481 785 786 787 788 789 89487 89492 89498 89504 89509 89542 89548 89553 89559 89564 89597 89603 89609 89614 89620 89653 89658 89664 89669 89675 89708 89713 89719 89724 89730 89515 89520 89526 89531 89537 89570 89575 89581 89586 89592 89625 89631 89636 89642 8^647 89680 89686 89691 89697 89702 89735 89741 89746 89752 89757 790 791 792 793 794 89763 89768 89774 89779 89785 89 818 89823 89829 89834 89840 89873 89878 89883 89889 89894 89927 89933 89938 89944 89949 89982 89988 89993 89998 90004 89790 89796 89801 89807 89812 89845 89851 89856 89862 89867 89900 89905 89911 89916 89922 89955 89960 89966 89971 89977 90009 90015 90020 90026 90031 795 796 797 798 799 90037 90042 90048 90053 90059 90091 90097 90102 90108 90113 90146 90151 90157 90162 90168 90200 90206 90211 90217 90222 90255 90260 90266 90271 90276 90064 90069 90075 90080 90086 90119 90124 90129 90135 90140 90173 90179 90184 90189 90195 90227 90233 90238 90244 90249 90282 90287 90293 90298 90304 No. O 1 2 3 4 5 G 7 8 9 750-799 800-849 No. O 1 2 3 4 5 6 7 8 9 80O 90309 90314 90320 90 325 90331 90 336 90342 90347 90 352 90 35S 801 90363 90369 90374 90380 90385 90390 90 396 90401 90 407 90412 802 90417 90423 90428 90 434 90 439 90445 90 450 90 455 90 461 90466 803 90 472 90 477 90 4 82 904SS 90493 90 499 90504 90 509 90515 90520 804 90526 90531 90536 90542 90547 90553 90558 90563 90569 90574 805 90 580 90585 90590 90596 90601 90607 90 612 90617 90623 90628 806 90 634 90 639 90 644 90650 90 655 90660 90 666 90671 90 677 90 682 807 90687 90693 90698 90703 90709 90714 90720 90 725 90 730 90 736 808 90741 90747 90752 90757 90763 90768 90773 90779 90 7S4 90 7 89 809 90795 90800 90806 90811 90 816 90822 90827 90 832 90 838 90843 810 90849 90854 90 859 90865 90S70 90875 90 SSI 90 886 90891 90 S97 811 90902 90 907 90 913 90918 90924 90 929 90934 90 940 90 945 90 950 812 90956 90961 90966 90 972 90 977 90 982 90 9SS 90 993 90 998 91 004 813 91009 91014 91020 91025 91030 91036 91041 91 046 91 052 91 057 814 91062 91068 91073 91 078 91084 91 OS9 91094 91100 91 105 91 110 815 91 116 91 121 91 126 91 132 91 137 91 142 91 148 91 153 91 158 91 164 816 91 169 . 91 174 91 ISO 91 185 91 190 91 196 91 201 91 206 91212 91217 817 91222 91228 91233 91238 91243 91 249 91 254 91 259 91 265 91270 818 91275 9128L 91 286 91 291 91297 91302 91307 91 312 91318 91323 819 91328 91334 91339 91344 91350 91355 91360 91365 91371 91376 82O 91381 91387 91 392 91397 91403 91 408 91413 91418 91424 91429 821 91434 91440 91445 91450 91 455 91461 91 466 91 471 91477 91 482 822 91487 91492 91 498 91 503 91 508 91514 91519 91 524 91 529 91535 823 91540 91 545 91 551 91556 91 561 91566 91 572 91 577 91 582 91 5S7 824 91 593 91 598 91603 91609 91614 91 619 91624 91 630 91 635 91 640 825 91645 91651 91656 91661 91666 91672 91677 91 682 91687 91693 826 91698 91 703 91 709 91714 91 719 91 724 91 730 91735 91 740 91 745 827 91751 91 756 91 761 91766 91772 91 777 91 782 91 787 91 793 91798 828 91803 91808 91814 91 819 91S24 91S29 91 834 91 840 91 845 91 850 829 91855 91 861 91866 91871 91876 91 882 91SS7 91892 91 S97 91 903 830 91 90S 91913 91918 91924 91929 91934 91939 91944 91 950 91 955 831 91960 91965 91971 91976 91 981 91 986 91 991 91 997 92002 92 007 832 92012 92018 92023 92028 92 033 92 038 92044 92049 92 054 92 059 833 92065 92070 92075 920SO 92 085 92 091 92 096 92101 92 106 92111 834 92117' 92122 92 127 92 132 92137 92143 92 148 92153 92158 92163 835 92169 92174 92179 92 184 921S9 92195 92200 92205 92210 92215 836 92221 92226 92231 92236 92 2-11 92 247 92 252 92 257 92 262 92267 837 92 273 92278 92 283 922SS 92293 92298 92 304 92 309 92314 92319 838 92324 92330 92335 92340 92345 92 350 92355 92361 92366 92371 839 92376 92381 92387 92392 92397 92402 92407 92412 92418 92423 840 92428 92433 92 438 92443 92449 92 454 92 459 92464 92469 92474 841 92480 92485 92 490 92495 92 500 92 505 92 511 92 516 92 521 92526 842 92531 92536 92542 92 547 92 552 92 557 92 562 92567 92572 92 5 78 843 92583 92588 92593 92598 92 603 92 609 92614 92619 92624 92629 844 92634 92639 92645 92650 92655 92660 92665 92670 92675 92681 845 92686 92691 92696 92701 92706 92711 92716 92722 92727 92 732 846 92 737 92742 92747 92 752, 92 758 92763 92 768 92773 92 778 92 783 847 92788 92 793 92799 92804 92809 92 814 92 819 92824 92 829 92 834 848 92840 92 845 92850 92855 92 860 92 865 92 870 92 875 92 SSI 92SS6 849 92891 92896 92901 92906 92911 92916 92921 92927 92 932 92 937 No. O 1 2 3 4 5 6 7 8 9 800-849 850-899 25 No. O 1 2 3 4 5 6 7 8 9 850 92942 92947 92952 92 957 92962 92967 92973 92978 92983 92988 851 92 993 92 Si98 93003 93008 93013 93018 93024 93029 93034 93039 852 93044 93049 93 054 93 059 93064 93069 93075 93080 93085 93090 853 93095 93 100 93105 93110 93115 93120 93125 93131 93136 93 141 854 93146 93151 93156 93161 93166 93171 93176 93181 93186 93192 855 93197 93202 93207 93212 93217 93222 93227 93232 93237 93242 856 93247 93252 93 258 93263 93268 93 273 93278 93283 93 288 93293 857 93 298 93303 93308 93313 93 318 93323 93328 93334 93339 93344 858 93 349 93 354 93 359 93364 93369 93374 93379 93384 93389 93394 859 93 399 93404 93409 93414 93420 93425 93430 93435 93440 93445 860 93450 93 455 93460 93 465 93470 93475 93480 93485 93490 93495 861 93 500 93 505 93510 93 515 93 520 93 526 93531 93536 93 541 93546 862 93551 93 556 93 561 93566 93571 93576 93 581 93 586 93591 93596 863 93601 93606 93611 93616 93621 93626 936.31 93636 93641 93646 864 93651 93656 93661 93666 93671 93676 93682 93687 93692 93697 865 93702 93707 93712 93717 93722 93727 93732 93737 93742 93747 866 93 752 93 757 93762 93767 93772 93777 93782 93787 93792 93797 867 93802 93807 93812 93817 93822 93827 93832 93837 93842 93847 868 93852 93857 93862 93867 93872 93877 93882 93887 93892 93897 869 93902 93907 93912 93917 93922 93927 93932 93937 93942 93947 870 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 871 94002 94007 94012 94017 94022 94027 94032 94037 94042 94047 872 94 052 94057 94 062 94067 94072 94077 94082 94086 94091 94096 873 94101 94106 94111 94116 94121 94126 94131 94136 94141 94146 874 94151 94156 94161 94166 94171 94176 94181 94186 94191 94196 875 94201 94206 94211 94216 94221 94226 94231 94236 94240 94245 876 94 250 94255 94260 94265 94270 94275 94280 94285 94290 94295 877 94 300 94305 94310 94315 94320 94325 94330 94335 94340 94345 878 94 349 94354 94359 94364 94369 94374 94379 94384 94389 94394 879 94399 94404 94409 94414 94419 94424 94429 94433 94438 94443 88O 94448 94453 94 458 94463 94468 94473 94478 94483 94488 94493 881 94498 94 503 94507 94512 94517 94522 94527 94532 94537 94542 882 94 547 94 552 94557 94 562 94 567 94571 94576 94581 94586 94591 883 94596 94601 94606 94611 94616 94621 94626 94630 94 635 94640 884 94645 94650 94655 94660 94665 94670 94675 94680 94685 94689 885 94694 94699 94 704 94709 94714 94719 94724 94729 94734 94738 886 94743 94748 94753 94 758 94763 94768 94773 94778 94783 94787 887 94792 94797 94802 94 807 94812 94817 94822 94827 94832 94836 888 94841 94846 94851 94856 94861 94866 94871 94876 94880 94 885 889 94890 94895 94900 94905 94910 94915 94919 94924 94929 94934 89O 94939 94944 94949 94954 94 959 94963 94968 94973 94978 94983 891 94988 94993 94998 95002 95007 95 012 95 017 95022 95 027 95 032 892 95036 95041 95046 95051 95 056 95061 95 066 95071 95075 95080 893 95085 95090 95095 95 100 95 105 95109 95 114 95119 95 124 95 129 894 95134 95139 95143 95148 95153 95 158 95163 95168 95173 95177 895 95 182 95187 95192 95197 95202 95 207 95211 95216 95221 95 226 896 95 231 95 236 95 240 95 245 95 250 95 255 95260 95265 95 270 95 274 897 95279 95284 95289 95294 95299 95 303 95308 95313 95318 95323 898 95328 95332 95337 95342 95 347 95 352 95357 95361 95 366 95 371 899 95376 95381 95386 95390 95395 95400 95405 95410 95415 95419 No. O 1 2 3 4 5 6 7 8 9 850-899 26 900-949 No. O 1 2 3 4 5 6 7 8 9 9OO 95424 95429 95434 95439 95444 95448 95 453 95458 95463 95468 901 95472 95477 95 482 95487 95492 95497 95501 95 406 95511 95516 902 95 521 95 525 95530 95535 95 540 95545 95 550 95 554 95559 95564 903 95569 95574 95578 95 583 95588 95 593 95 598 95602 95607 95612 904 95617 95622 95626 95631 95636 95641 95646 95650 95655 95660 905 95 665 95670 95674 95679 95684 95689 95694 95698 95703 95708 906 95713 95718 95 722 95727 95732 95737 95 742 95746 95751 95 756 907 95761 95 766 95770 95775 95 780 95785 95789 95794 95799 95804 908 95809 95813 95818 95823 95828 95832 95837 95 842 95 847 95 852 909 95856 95861 95866 95871 95875 95880 95885 95890 95895 95899 910 95904 95909 95 914 95918 95923 95928 95933 95 938 95942 95947 911 95952 95957 95961 95966 95 971 95976 95980 95985 95990 95995 912 95999 96004 96009 96014 96019 96023 96028 96033 96 038 96042 913 96 047 96052 96057 96061 96 066 96071 96076 96080 96085 96090 914 96095 96099 96104 96109 96114 96118 96123 96128 96133 96137 915 96142 96147 96152 96 156 96161 96166 96171 96175 96180 96185 916 96 190 96194 96 199 96204 96209 96213 96218 96223 96227 96232 917 96237 96242 96246 96251 96 256 96 261 96265 96270 96275 96280 918 96284 96289 96294 96298 96303 96308 96313 96317 96322 96327 919 96332 96336 96341 96346 96350 96355 96360 96365 96369 96374 92O 96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 921 96426 96431 96435 96440 96445 96450 96454 96459 96464 96468 922 96473 96478 96483 96487 96492 96497 96501 96506 96511 96515 923 96520 96 525 96530 96 534 96539 96 544 96 548 96 553 96558 96562 924 96567 96572 96577 96581 96586 96591 96595 96600 96605 96 609 925 96614 96619 96624 96628 96633 96638 96642 96647 96652 96656 926 96661 96666 96670 96675 96680 96685 96689 96694 96699 96703 927 96708 96713 96717 96722 96727 96731 '96736 96741 96745 96750 928 96755 96759 96764 96 769 96774 96778 96783 96788 96792 96797 929 96802 96806 96811 96 816 96820 96825 96830 96834 96839 96844 93O 96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 931 96895 96900 96904 96909 96914 96918 96923 96928 96 932 96937 932 96942 96946 96951 96956 96 960 96965 96 970 96974 96979 96984 933 96988 96993 96997 97002 97007 97011 97016 97021 97025 97030 934 97035 97039 97044 97049 97053 97058 97063 97067 97072 97077 935 97081 97086 97090 97 095 97100 97104 97109 97114 97118 97123 936 97128 97132 97137 97142 97146 97151 97 155 97160 97165 97169 937 97174 97179 97183 97188 97192 97197 97202 97206 97211 97216 938 97220 97225 97230 97234 97239 97243 97248 97253 97257 97262 939 97267 97271 97276 97280 97285 97290 97294 97299 97304 97308 940 97313 97317 97322 97327 97331 97336 97340 97345 97350 97354 941 97359 97364 97368 97373 97377 97382 97387 97391 97396 97400 942 97405 97410 97414 97419 97424 97428 97 433 97437 97442 97 447 943 97451 97 456 97 460 97465 97470 97474 97479 97483 97488 97493 944 97497 97 502 97506 97511 97516 97 520 97 525 97529 97534 97539 945 97543 97548 97 552 97557 97562 97566 9757,1 97575 97580 97585 946 97 589 97594 97598 97603 97607 97612 97617 97621 97626 97630 947 97 635 97640 97 644 97649 97653 97658 97663 97667 97672 97676 948 97681 97685 97690 97695 . 97 699 97704 97708 97713 97717 97722 949 97727 97731 97736 97740 97745 97749 97 754 97759 97763 97768 No. O 1 2 3 4 5 6 7 8 9 900-949 950-1000 27 No. o 1 2 3 4 5 6 7 8 9 950 97772 97777 97782 97 786 97791 97795 97800 97804 97 809 97813 951 97818 97823 97827 97 832 97836 97841 97 845 97850 97855 97859 952 97864 97868 97873 97877 97882 97886 97891 97896 97900 97905 953 97909 97 914 97918 97923 97928 97932 97937 97941 97946 97950 954 97955. 97959 97964 97968 97973 97978 97982 97987 97991 97996 955 98000 98005 98009 98014 98019 98023 98028 98032 98037 98041 956 98046 98050 98055 98059 98064 98068 98073 98078 98082 98087 957 98091 98096 98100 98105 98109 98114 98118 98123 98127 98132 958 98137 98141 98146 98150 98155 98 159 98164 98 168 98173 98177 959 98182 98186 98191 98195 98200 98204 98209 98214 98218 98223 96O 98227 98 232 98 236 98241 98 245 98250 98254 98259 98263 98268 961 98272 98277 98281 98286 98290 98295 98299 98304 98308 98313 962 98318 98322 98327 98331 98336 98340 98345 98349 98354 98358 963 98363 98367 98372 98376 98 381 98385 98390 98394 98399 98403 964 98408 98412 98417 98421 98426 98430 98435 98439 98444 98448 965 98453 98457 98462 98466 98471 98475 98480 98484 98489 98493 966 98498 98502 98507 98511 98516 98520 98525 98529 98534 98538 967 98 543 98547 98552 98556 98561 98565 98570 98574 98579 98 583 968 98588 98592 98597 98601 98605 98610 98614 98 619 98623 98628 969 98632 98637 98641 98646 98650 98 655 98659 98664 98668 98673 97O 98677 98682 98686 98691 98695 98700 98704 98709 98713 98717 971 98722 98726 98731 98735 98740 98744 98749 98753 98758 98762 972 98767 98771 98776 98780 98 784 . 98789 98793 98798 98802 98807 973 98811 98816 98820 98825 98829 98834 98838 98843 98847 98851 974 98856 98860 98865 98869 98874 98878 98883 98887 98892 98896 975 98900 98905 98909 98914 98918 98923 98927 98932 98936 98941 976 98945 98949 98954 98958 98963 98967 98972 98976 98981 98985 977 98989 98 994 98998 99003 99007 99012 99016 99021 99025 99029 978 99034 99038 99043 99047 99052 99056 99061 99065 99 069 99074 979 99078 99083 99087 99092 99096 99100 99105 99 109 99114 99118 980 99123 99127 99131 99136 99 140 99 145 99149 99154 99158 99162 981 99167 99171 99176 99 180 99185 99189 99193 99198 99202 99207 982 99211 99216 99220 99224 99 229 99233 99238 99242 99247 99251 983 99255 99260 99264 99269 99273 99277 99282 99286 99291 99 295 984 99300 99304 99308 99313 99317 99322 99326 99330 99335 99339 985 99 344 99348 99352 99357 99361 99366 99370 99374 99379 99383 986 99 388 99392 99396 99401 99405 99410 99414 99419 99423 99427 987 99432 99436 99441 99445 99449 99454 99458 99463 99467 99471 988 99476 99480 99 484 99489 99493 99498 99 502 99 506 99511 99515 989 99520 99524 99528 99533 99537 99542 99546 99550 99 555 99559 990 99 564 99568 99 572 99 577 99 581 99 585 99590 99594 99599 99603 991 99607 99612 99616 99621 99625 99629 99 634 99638 99642 99647 992 99651 99656 99660 99664 99 669 99673 99677 99682 99686 99691 993 99695 99699 99704 99708 99712 99717 99721 99726 99730 99734 994 99739 99743 99747 99752 99756 99760 99765 99769 99774 99778 995 99782 99787 99791 99795 99800 99804 99808 99813 99817 99822 996 99826 99 830 99835 99839 99843 99848 99 852 99 856 99861 99865 997 99870 99874 99878 99883 99887 99891 99896 99900 99904 99909 998 99913 99917 99922 99926 99930 99935 99939 99944 99948 99952 999 99957 99961 99965 99970 99974 99978 99983 99987 99991 99996 1OOO 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 No. 1 2 3 4 5 6 7 8 9 950-1000 TABLE II -USEFUL CONSTANTS AND THEIR LOGARITHMS LOG Circumference of the Circle in Degrees 360 2. 55 630 250 Circumference of the Circle in Minutes = 21 600 4.3344537.S Circumference of the Circle in Seconds 1296000 6.11 260500 If the radius = 1, the semi-circumference is TT 3. 14 159 265 358 979 323 846 764 338 378 0.49714987 ALSO LOG 7r 2 = 9.86960440 0.99429975 27r= 6.28318531 0.79817987 = 0.10132118 9.00570025 - 10 47r = 12.56637061 1.09920986 7T- = 1.57079633 0.19611988 VV = 1. 77 245 385 0.24857494 2 = 1.04719755 0.02002862 JL =0.56418958 9. 75 142 506 - 10 3 VTT ~= 4.18879020 0. 62 208 861 /I 3 \/- = 0.97 720 502 9.98998569- 10 = 0.78539816 9.89508988- 10 4 /4 \/-= 1.12837917 0. 05 245 506 = 0.52359878 9.71899862- 10 \7T 6 i -v/TT = 1.46459189 0.16571662 -i.= 0.31830989 9. 50 285 013 - 10 7T 1 -^ = 0.68278406 9.83428338-10 = 0. 15 915 494 9.20182013-10 VTT 2-jr = 0.95492966 9.97997138- 10 vV = 2. 14 502 940 0.33143325 7T 8/~5~" ^-= 1.27323954 0.10491012 V /A. = 0.62 035 049 \4w 9.79263713 - 10 7T = 0. 23 873 241 9. 37 791 139 - 10 \ 8 / = 0.80 599 598 9.90633287- 10 4?r \ 6 Angle 0, whose arc is equal to the radius r, is 180 in degrees, = =57.29577951 7T 1.75812263 10800 in minutes, 0' 3437 74677' 3.53627388 7T 648000 in seconds 0" 206264806" 5.31 442 513 Angle 2 0, whose arc is equal to twice the radius, 2 r, is in degrees, 20= = 114. 59 155 903 2. 05 915 263 7T in minutes, 2 0' - 21 60 -687549354' 3.837303*88 7T in seconds 20" 12 96000 412529612" 5.61545513 7T If the radius r = 1, the length of the arc is : for 1 degree = -~- = = 0. 01 745 329 . 8.24187737-10 J 180 for 1 minute = -L = ?_ = 0. 00 029 089 , 6.46372612- 10 0' 10 800 for 1 second = -L- = - = 0. 00 000 485 4. 68 557 487 ~ 10 0" 648000 for & degree = -^ = ^ =0.00872665 .... 7.94084737- 10 for 2 minute - 1 TT c 1.00014544 .... 6.16269612 10 2 0' 21 600 for .> second - 1 r - .00000242 .... 4.38454487 10 20" 1296000 Sin 1", when the radius r - 1, is . . .=0.00000485 . . . . 4.68557487- 10 TABLE III LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS From 0' to 3', and from 89 57' to 90, for every second From to 2, and from 88 to 90, for every ten seconds From 1 to 89, for every minute NOTE. The characteristic of every logarithm in the following table is too large by 10. Therefore, 10 should be written after every logarithm. L sin and L tan U I< sin and L tan //. 1' 6.46 373 6.47 090 6.47 797 6.48 492 6.49 175 2' 6.76476 6.76 836 6.77 193 6.77 548 6.77 900 // // O' 6.16270 6.17694 6.19072 6.20 409 6.21 705 1' 6.63 982 6.64 462 6.64 936 6.65 406 6.65 870 2' 6.86 167 6.86455 6.86 742 6.87 027 6.87 310 // O 1 2 3 4 6O 59 58 57 56 30 31 32 33 34 30 29 28 27 26 4.68 557 4.98 660 5.16270 5.28 763 5 6 7 8 9 5.38 454 5.46373 5.53 067 5.58 866 5.63 982 6.49 849 6.50512 6.51 165 6.51 808 6.52 442 6.78 248 6.78 595 6.78 938 6.79278 6.79616 55 54 53 52 51 35 36 37 38 39 6.22 964 6.24 188 6.25 378 6.26 536 6.27 664 6.66 330 6.66 785 6.67 235 6.67 680 6.68 121 6.87 591 6.87 870 6.88 147 6.88 423 6.88 697 25 24 23 22 21 10 11 12 13 14 5.68557 5.72 697 5.76476 5.79952 5.83 170 6.53 067 6 53 683 6.54 291 6.54 890 6.55 481 6.79952 6.80 285 6.80615 6.80 943 6.81 268 50 49 48 47 46 40 41 42 43 44 6.28 763 6.29 836 6.30 882 6.31 904 6.32 903 6.68 557 6.68 990 6.69 418 6.69 841 6.70 261 6.88 969 6.89 240 6.89 509 6.89 776 6.90 042 2O 19 18 17 16 15 16 17 18 19 5.86 167 5.88969 5.91 602 5.94085 5.96433 6.56064 6.56 639 6.57 207 6.57 767 6.58 320 6.81 591 6.81 911 6.82 230 6.82 545 6.82 859 45 44 43 42 41 45 46 47 48 49 6.33 879 6.34 833 6.35 767 6.36 682 6.37 577 6.70676 6.71 088 6.71 496 6.71 900 6.72 300 6.90306 6.90 568 6.90 829 6.91 088 6.91346 15 14 13 12 11 20 21 22 23 24 5.98660 6.00 779 6.02 800 6.04 730 6.06579 6.58 866 6.59 406 6.59 939 6.60 465 6.60 985 6.83 170 6.83479 6.83 786 6.84 091 6.84394 40 39 38 37 66 50 51 52 53 54 6.38 454 6.39315 6.40 158 6.40 985 6.41 797 6.72 697 6.73 090 6.73 479 6.73 865 6.74 248 6.91 602 6.91 857 6.92 110 6.92 362 6.92 612 1O 9 8 7 6 25 26 27 28 29 6.08351 6.10055 6.11694 6.13 273 6.14 797 6.61 499 6.62 007 6.62 509 6.63 006 6.63 496 6.84 694 6.84 993 6.85 289 6.85 584 6.85 876 35 34 33 32 31 55 56 57 58 59 6.42 594 6.43 376 6.44 145 6.44 900 6.45 643 6.74 627 6.75 003 6.75 376 6.75 746 6.76 112 6.92 861 6.93 109 6.93 355 6.93 599 6.93 843 5 4 3 2 1 30 6.16270 59' 6.63 982 58' 6.86 167 57' 30 6O 6.46373 59' 6.76 476 58' 6.94 085 57' O " " " " L cos and L cot 89' L cos and L cot 29 30 / // L sin L tan L cos // / / // L sin L tan I cos // / On 10 00000 fiO 1O 746^7^ 7 46 37"? 1000000 fft V 10 5.68557 5.68557 10.00000 V/ \J VJ 50 J_ Vr \J 10 / . I O O t O / . i O O / O \J.\J\J \J\J\J 7.47090 7.47091 10.00000 U Ovr 50 20 5.98660 5.98660 10.00000 40 20 7.47797 7.47797 10.00000 40 30 6.16270 6.16270 10.00000 30 30 7.48491 7.48492 10.00000 30 40 6.28763 6.28763 10.00000 20 40 7.49175 7.49175 10.00000 20 50 6.38454 6.38454 10.00000 10 50 7.49 849 7.49849 10.00000 10 1 6.46373 6.46373 10.00000 59 11 7.50512 7.50512 10.00000 049 10 6.53067 6.53067 10.00000 50 10 7.51165 7.51165 10.00000 50 20 6.58866 6.58866 10.00000 40 20 7.51808 7.51809 10.00000 40 30 6.63982 6.63982 10.00000 30 30 7.52442 7.52443 10.00000 30 40 6.68557 6.68557 10.00000 20 40 7.53067 7.53067 10.00000 20 50 6.72697 6.72697 10.00000 10 50 7.53683 7.53683 10.00000 10 2 6.76476 6.76476 10.00000 58 12 7.54291 7.54291 10.00000 048 10 6.79952 6.79952 10.00000 50 10 7.54890 7.54890 10.00000 50 20 6.83170 6.83170 10.00000 40 20 7.55481 7.55481 10.00000 40 30 6.86167 6.86167 10.00000 30 30 7.56064 7.56064 10.00000 30 40 6.88969 6.88969 10.00000 20 40 7.56639 7.56639 10.00000 20 50 6.91602 6.91602 10.00000 10 50 7.57206 7.57206 10.00000 10 3 6.94085 6.94085 10.00000 57 13 7.57767 7.57767 10.00000 047 10 6.96433 6.96433 10.00000 50 10 7.58320 7.58320 10.00000 50 20 6.98660 6.98661 10.00000 40 20 7.58866 7.58867 10.00000 40 30 7.00779 7.00779 10.00000 30 30 7.59406 7.59406 10.00000 30 40 7.02800 7.02800 10.00000 20 40 7.59939 7.59939 10.00000 20 50 7.04730 7.04730 10.00000 10 50 7.60465 7.60466 10.00000 10 4 7.06579 7.06579 10.00000 56 14 7.60985 7.60986 10.00000 046 10 7.08351 7.08352 10.00000 50 10 7.61499 7.61500 10.00000 50 20 7.10055 7.10055 10.00000 40 20 7.62007 7.62008 10.00000 40 30 7.11694 7.11694 10.00000 30 30 7.62509 7.62510 10.00000 30 40 7.13273 7.13273 10.00000 20 40 7.63006 7.63006 10.00000 20 50 7.14797 7.14797 10.00000 10 50 7.63496 7.63497 10.00000 10 5 7.16270 7.16270 10.00000 55 15 7.63982 7.63982 10.00000 045 10 7.17694 7.17694 10.00000 50 10 7.64461 7.64462 10.00000 50 20 7.19072 7.19073 10.00000 40 20 7.64936 7.64937 10.00000 40 30 7.20409 7.20409 10.00000 30 30 7.65406 7.65406 10.00000 30 40 7.21705 7.21705 10.00000 20 40 7.65870 7.65871 10.00000 20 50 7.22964 7.22964 10.00000 10 50 7.66330 7.66330 10.00000 10 6 7.24188 7.24188 10.00000 54 16 7.66784 7.66785 10.00000 044 10 7.25378 7.25378 10.00000 50 10 7.67235 7.67235 10.00000 50 20 7.26536 7.26536 10.00000 40 20 7.67680 7.67680 10.00000 40 30 7.27664 7.27664 10.00000 30 30 7.68121 7.68121 10.00000 30 40 7.28763 7.28764 10.00000 20 40 7.68557 7.68558 9.99999 20 50 7.29836 7.29836 10.00000 10 50 7.68989 7.68990 9.99999 10 7 7.30882 7.30882 10.00000 53 17 7.69417 7.69418 9.99999 043 10 7.31904 7.31904 10.00000 50 10 7.69841 7.69842 9.99999 50 20 7.32903 7.32903 10.00000 40 20 7.70261 7.70261 9.99999 40 30 7.33879 7.33879 10.00000 30 30 7.70676 7.70677 9.99999 30 40 7.34833 7.34833 10.00000 20 40 7.71088 7.71088 9.99999 20 50 7.35767 7.35767 10.00000 10 50 7.71496 7.71496 9.99999 10 8 7.36682 7.36682 10.00000 52 18 7.71900 7.71900 9.99999 042 10 7.37577 7.37577 10.00000 50 10 7.72300 7.72301 9.99999 50 20 7.38454 7.38455 10.00000 40 20 7.72697 7.72697 9.99999 40 30 7.39314 7.39315 10.00000 30 30 7.73090 7.73090 9.99999 30 40 7.40158 7.40158 10.00000 20 40 7.73479 7.73480 9.99999 20 50 7.40985 7.40985 10.00000 10 50 7.73865 7.73866 9.99999 10 9 7.41797 7.41797 1000000 51 19 7.74248 7.74248 9.99999 041 10 7.42594 7.42594 10.00000 50 10 7.74627 7.74628 9.99999 50 20 7.43376 7.43376 10.00000 40 20 7.75003 7.75004 9.99999 40 30 7.44145 7.44145 10.00000 30 30 7.75376 7.75377 9.99999 30 40 7.44900 7.44900 10.00000 20 40 7.75745 7.75746 9.99999 20 50 7.45643 7.45643 10.00000 10 50 7.76112 7.76113 9.99999 10 10 7.46373 7.46373 10.00000 5O 20 7.76475 7.76476 9.99999 04O / // L cos L cot L sin // / / // L cos L cot L sin // / 89 81 / // L sin L tan L cos // / / // L sin L tan L cos // / 2O 7.76475 7.76476 9.99999 4O 3O 7.94084 7.94086 9.99998 03O 10 7.76836 7.76837 9.99999 50 10 7.94325 7.94326 9.99998 50 20 7.77 193 7.77 194 9.99 999 40 20 7.94564 7.94566 9.99998 40 30 7.77548 7.77549 9.99999 30 30 7.94802 7.94804 9.99998 30 40 7.77899 7.77900 9.99999 20 40 7.95039 7.95040 9.99998 20 50 7.78248 7.78249 9.99999 10 50 7.95 274 7.95 276 9.99 998 10 21 7.78594 7.78595 9.99999 39 31 7.95508 7.95510 9.99998 029 10 7.78938 7.78938 9.99999 50 10 7.95 741 7.95 743 9.99 998 50 20 7.79278 7.79279 9.99999 40 20 7.95973 7.95974 9.99998 40 30 7.79616 7.79617 9.99999 30 30 7.96203 7.96205 9.99998 30 40 7.79952 7.79952 9.99999 20 40 7.96432 7.96434 9.99998 20 50 7.80284 7.80285 9.99999 10 50 /.96660 7.96662 9.99998 10 22 7.80615 7.80615 9.99999 38 32 7.96887 7.96889 9.99998 028 10 7.80942 7.80943 9.99999 50 10 7.97113 7.97114 9.99998 50 20 7.81 268 7.81 269 9.99 999 40 20 7.97337 7.97339 9.99998 40 30 7.81591 7.81591 9.99999 30 30 7.97560 7.97562 9.99998 30 40 7.81911 7.81912 9.99999 20 40 7.97782 7.97784 9.99998 20 50 7.82229 7.82230 9.99999 10 50 7.98003 7.98005 9.99998 10 23 7.82545 7.82546 9.99999 37 33 7.98223 7.98225 9.99998 027 10 7.82859 7.82860 9.99999 50 10 7.98442 7.98444 9.99998 50 20 7.83170 7.83171 9.99999 40 20 7.98660 7.98662 9.99998 40 30 7.83479 7.83480 9.99999 30 30 7.98876 7.98878 9.99998 30 40 7.83786 7.83787 9.99999 20 40 7.99092 7.99094 9.99998 20 50 7.84091 7.84092 9.99999 10 50 7.99306 7.99308 9.99998 10 24 7.84393 .7.84394 9.99999 36 34 7.99520 7.99522 9.99998 026 10 7.84694 7.84695 9.99999 50 10 7.99732 7.99734 9.99998 50 20 7.84992 7.84993 9.99999 40 20 7.99943 7.99946 9.99998 40 30 7.85289 7.85290 9.99999 30 30 8.00154 8.00156 9.99998 30 40 7.85583 7.85584 9.99999 20 40 8.00363 8.00365 9.99998 20 50 7.85876 7.85877 9.99999 10 50 8.00571 8.00574 9.99998 10 25 7.86166 7.86167 9.99999 35 35 8.00779 8.00781 9.99998 025 10 7.86455 7.86456 9.99999 50 10 8.00985 8.00987 9.99998 50 20 7.86741 7.86743 9.99999 40 20 8.01 190 8.01 193 9.99 998 40 30 7.87026 7.87027 9.99999 30 30 8.01395 8.01397 9.99998 30 40 7.87309 787310 9.99999 20 40 8.01 598 8.01 600 9.99 998 20 50 7.87590 7.87591 9.99999 10 50 8.01801 S.01'803 9.99998 10 26 7.87870 7.87871 9.99999 34 36 8.02002 8.02004 9.99998 024 10 7.88147 7.88148 9.99999 50 10 8.02203 8.02205 9.99998 50 20 7.88423 7.88424 9.99999 40 20 8.02402 8.02405 9.99998 40 30 7.88697 7.88698 9.99999 30 30 8.02601 8.02604 9.99998 30 40 7.88969 7.88970 9.99999 20 40 8.02799 8.02801 9.99998 20 50 7.89240 7.89241 9.99999 10 50 8.02996 8.02998 9.99998 10 27 7.89509 7.89510 9.99999 33 37 8.03 192 8.03 194 9.99 997 023 10 7.89776 7.89777 9.99999 50 10 8.03387 8.03390 9.99997 50 20 7.90041 7.90043 9.99999 40 20 8.03581 8.03584 9.99997 40 30 7.90305 7.90307 9.99999 30 30 8 03 775 8.03 777 9.99 997 30 40 7.90568 7.90569 9.99999 20 40 8.03967 8.03970 9.99997 20 50 7.90829 7.90830 9.99999 10 50 8.04 159 8.04 162 9.99 997 10 28 7.91088 7.91089 9.99999 32 38 8.04350 8.04353 9.99997 022 10 7.91346 7.91347 9.99999 50 10 804540 8.04543 9.99997 50 20 7.91602 7.91603 9.99999 40 20 8.04729 8.04732 9.99997 40 30 7.91857 7.91858 9.99999 30 30 8.04918 8.04921 9.99997 30 40 7.92110 7.92111 9.99998 20 40 8.05 105 8.05 108 9.99 997 20 50 7.92362 7.92363 9.99998 10 50 8.05292 8.05295 9.99997 1(5 29 7.92612 7.92613 9.99998 31 39 8.05 478 8.05 481 9.99 997 021 10 7.92861 7.92862 9.99998 50 10 8.05663 8.05666 9.99997 50 20 7.93108 7.93110 9.99998 40 20 8.05848 8.05851 9.99997 40 30 7.93354 7.93356 9.99998 30 30 8.06031 8.06034 9.99997 30 40 7.93599 7.93601 9.99998 20 40 8.06214 8.06217 9.99997 20 50 7.93842 7.93844 9.99998 10 50 8.06396 8.06399 9.99997 10 3O 7.94084 7.94086 9.99998 3O 4O 8.06578 8.06581 9.99997 02O / // L cos L cot L sin // / / // L cos L cot L sin // / 89 / // L sin L tan L cos // / / // L sin L tan L cos // / 4O 8.06578 8.06581 9.99997 2O 5O 8.16268 8.16273 9.99995 01O 10 806758 8.06761 9.99997 50 10 8.16413 8.16417 9.99995 50 20 8.06938 8.06941 9.99997 40 20 8.16557 8.16561 9.99995 40 30 8.07117 8.07120 9.99997 30 30 8.16700 8.16705 9.99995 30 40 8.07295 8.07298 9.99997 20 40 8.16843 8.168-18 9.99995 20 50 8.07473 8.07476 9.99997 10 50 8.16986 8.16991 9.99995 10 41 8.07650 8.07653 9.99997 19 51 8.17128 8.17133 9.99995 9 10 8.07826 8.07829 9.99997 50 10 8.17270 8.17275 9.99995 50 20 8.08002 8.08005 9.99997 40 20 8.17411 8.17416 9.99995 40 30 8.08176 8.08180 9.99997 30 30 8.17552 8.17557 9.99995 30 40 8.08350 8.08354 9.99997 20 40 8.17692 8.17697 9.99995 20 50 8.08524 8.08527 9.99997 10 50 8.17832 8.17837 9.99995 10 42 8.08696 8.08700 9.99997 18 52 8.17971 8.17976 9.99995 8 10 8.08868 8.08872 9.99997 50 10 8.18110 8.18115 9.96995 50 20 8.09040 8.09043 9.99997 40 20 8.18249 8.18254 9.99995 40 30 8.09210 8.09214 9.99997 30 30 8.18387 8.18392 9.99995 30 40 8.09 380 8.09384 9.99997 20 40 8.18524 8.18530 999995 20 50 8.09550 8.09553 9.99997 10 50 8.18662 8.18667 9.99995 10 43 8.09718 8.09722 9.99997 17 53 8.18798 8.18804 9.99995 7 10 8.09886 8.09890 9.99997 50 10 8.18935 8.18940 9.99995 50 20 8.10054 8.10057 9.99997 40 20 8.19071 8.19076 9.99995 40 30 8.10220 8.10224 9.99997 30 30 8.19206 8.19211 9.99995 30 40 8.10386 8.10390 9.99996 20 40 8.19341 8.19347 9.99995 20 50 8.10552 8.10555 9.99996 10 50 8.19476 8.19481 9.99995 10 44 8.10717 8.10720 9.99996 16 54 8.19610 8.19616 9.99995 6 10 8.10881 8.10884 9.99996 50 10 8.19744 8.19749 9.99995 50 20 8.11044 8.11018 9.99996 40 20 8.19877 8.19883 9,99995 40 30 811207 8.11 211 9.99996 30 30 8.20010 8.20016 9-99995 30 40 8.11370 8.11373 9.99996 20 40 8.20143 8.20149 9.99995 20 50 8.11531 8.11535 9.99996 10 50 8.20275 8.20281 9.99994 10 45 8.11693 8.11696 9.99996 15 55 8.20407 8.20413 9.99994 5 10 8.11853 8.11857 9.99996 50 10 8.20 538 8.20 544 9.99 994 50 20 8.12013 8.12017 9.99996 40 20 8.20669 820675 9.99994 40 30 8.12172 8.12 176 9.99996 30 30 8.20800 8.20806 9.99994 30 40 8.12331 812335 9.99996 20 40 8.20930 8.20936 9.99994 20 50 8.12489 8.12*493 9.99996 10 50 8.21060 8.21066 9.99994 10 46 8.12647 8.12651 9.99996 14 56 8.21189 8.21 195 9.99994 4 10 8.12804 8.12808 9.99996 50 10 8.21319 8.21324 9.09994 50 20 8.12961 8.12965 9.99996 40 20 8.21 447 8.21 453 9.99 994 40 30 8.13117 8.13 121 9.99996 30 30 8.21 576 8.21581 9.99994 30 40 8.13272 8.13276 9.99996 20 40 8.21 703 8.21 709 9.99 994 20 50 8.13427 8.13431 9.99996 10 50 8.21831 8.21 837 9.99994 10 47 8.13581 8.13585 9.99996 13 57 8.21958 8.21964 9.99994 3 10 8.13735 8.13 739 9.99996 50 10 8.22085 8.22091 9.99994 50 20 8.13888 8.13892 9.99996 40 20 8.22211 8.22217 9.99994 40 30 8.14041 8.14045 9.99996 30 30 822337 8.22343 9.99994 30 40 8.14193 8.14197 9.99996 20 40 822463 8.22-169 9.99994 20 50 8.14344 8.143-18 9.99996 10 50 8.22588 8.22595 9.99994 10 48 8.14495 8.14500 9.99996 12 58 8.22713 8.22720 9.99994 2 10 8.14646 8.14650 9.99996 50 10 8.22838 8.22844 9.99994 50 20 8.14796 8.14800 9.99996 40 20 8.22962 8.22968 9.99994 40 30 814945 8.14950 9.99996 30 30 8.23086 8.23092 9.99994 30 40 8.15094 8.15099 9.99996 20 40 8.23210 8.23216 9.99994 20 50 8.15243 8.15247 9.99996 10 50 8.23333 8.23339 9.99994 10 49 8.15391 8.15395 9.99996 11 59 8.23456 8.23462 999994 1 10 8.15538 8.15543 9.99996 50 10 8.23578 8.23585 9.99994 50 20 8.15685 8.15690 9.99996 40 20 8.23 700 8.23 707 9.99 994 40 30 8.15832 8.15836 9.99995 30 30 8.23822 8.23829 9.99993 30 40 8.15978 8.15982 9.99995 20 40 8.23944 8.23950 9.99993 20 50 8.16123 8.16128 9.99995 10 50 8.24065 8.24071 9.99993 10 50 8.16268 8.16273 9.99995 1C 6O 8.24186 8.24192 9.99993 O / // L cos L cot L sin // / / // L cos L cot L sin // / 89' 33 / // L sin L tan L cos // / / // L sin L tan L cos // / O 8.24 186 8.24 192 9.99 993 6O 1O 8.30879 8.30888 9.99991 05O 10 8.24306 8.24313 9.99993 50 10 8.30983 8.30992 9.99991 50 20 8.24426 8.24433 9.99993 40 20 8.31086 8.31095 9.99991 40 30 824546 8.24553 9.99993 30 30 8.31188 8.31198 9.99991 30 40 8.24665 824672 9.99993 20 40 8.31291 8.31300 9.99991 20 50 8.24785 8.24791 9.99993 10 50 8.31393 8.31403 9.99991 10 1 8.24903 8.24910 9.99993 59 11 8.31495 8.31505 9.99991 049 10 8.25022 8.25029 9.99993 50 10 8.31597 8.31606 9.99991 50 20 8.25 140 8.25 147 9.99 993 40 20 8.31699 8.31 708 9.99991 40 30 8.25258 8.25265 9.99993 30 30 8.31800 8.31809 9.99991 30 40 .8.25375 8.25382 9.99993 20 40 8.31901 8.31911 9.99991 20 50 8.25493 8.25500 9.99993 10 50 8.32002 8.32012 9.99991 10 2 8.25609 8.25616 9.99993 58 12 8.32103 8.32112 9.99990 048 10 8.25 726 8.25 733 9.99 993 50 10 8.32203 8.32213 9.99990 50 20 8.25842 8.25849 9.99993 40 20 8.32303 8.32313 9.99990 40 30 8.25958 8.25965 9.99993 30 30 8.32403 8.32413 9.99990 30 40 8.26074 8.26081 9.99993 20 40 8.32503 8.32513 9.99990 20 50 8.26189 8.26196 9.99993 10 50 8.32602 8.32612 9.99990 10 3 8.26304 8.26312 9.99993 57 13 8.32702 8.32711 9.99990 047 10 8.26419 826426 9.99993 50 10 8.32801 8.32811 9.99990 50 20 8.26553 8.26541 9.99993 40 20 8.32899 8.32909 9.99990 40 30 8.26648 8.26655 9.99993 30 30 8.32998 8.33008 9.99990 30 40 8.26761 8.26769 9.99993 20 40 8.33096 8.33106 9.99990 20 50 8.26875 8.26882 9.99993 10 50 8.33195 8.33205 9.99990 10 4 8.26988 8.26996 9.99992 56 14 8.33292 8.33302 9.99990 046 10 8.27101 8.27109 9.99992 50 10 8.33390 8.33400 9.99990 50 20 8.27214 8.27221 9.99992 40 20 8.33488 8.33498 9.99990 40 30 8.27326 8.27334 9.99992 30 30 8.33585 8.33595 9.99990 30 40 8.27438 8.27446 9.99992 20 40 8.33682 8.33692 9.99990 20 50 8.27550 8.27558 9.99992 10 50 8.33779 8.33789 9.99990 10 5 8.27661 8.27669 9.99992 55 15 8.33875 8.33886 9.99990 045 10 8.27773 8.27780 9.99992 50 10 8.33972 8.33982 9.99990 50 20 8.27883 8.27891 9.99992 40 20 8.34086 8.34078 9.99990 40 30 8.27994 8.28002 9.99992 30 30 8.34164 8.34174 9.99990 30 40 8.28104 828112 9.99992 20 40 8.34260 8.34270 9.99989 20 50 8.28215 8.28223 9.99992 10 50 8.34355 8.34366 9.99989 10 6 8.28324 8.28332 9.99992 54 16 8.34450 8.34461 9.99989 044 10 8.28434 8.28442 9.99992 50 10 8.34546 8.34556 9.99989 50 20 8.28543 8.28551 9.99992 40 20 8.34640 8.34651 9.99989 40 30 8.28652 8.28660 9.99992 30 30 8.34735 8.34746 9.99989 30 40 8.28761 8.28769 9.99992 20 40 8.34830 8.34840 9.99989 20 50 8.2S869 8.28877 9.99992 10 50 8.34924 8.34935 9.99989 10 7 8.28977 8.28986 9.99992 53 17 8.35 018 8.35 029 9.99 989 043 10 8.29085 8.29094 9.99992 50 10 8.35112 8.35 123 9.99989 50 20 8.29093 8.29201 9.99992 40 20 8.35206 8.35217 9.99989 40 30 8.29300 8.29309 9.99992 30 30 8.35 299 8.35 310 9.99 989 30 40 8.29407 8.29416 9.99992 20 40 8.35 392 8.35 403 9.99 989 20 50 8.29514 8.29523 9.99992 10 50 8.35 485 8.35 497 9.99 989 10 8 8.29621 829629 9.99992 52 18 8.35578 8.35590 9.99989 042 10 8.29727 8.29736 9.99991 50 10 8.35671 8.35682 9.99989 50 20 8.29833 8.29842 9.99991 40 20 8.35764 8.35 775 9.99989 40 30 8.29939 8.29947 9.99991 30 30 8.35 856 8.35 867 9.99 989 30 40 8.30044 8.30053 9.99991 20 40 8.35948 8.35959 9.99989 20 -50 8.30150 8.30158 9.99991 10 50 8.36040 8.36051 9.99989 10 9 8.30255 8.30263 9.99991 51 19 8.36131 8.36143 9.99989 041 10 8.30359 8.30368 9.99991 50 10 8.36223 8.36235 9.99988 50 20 8.30464 8.30473 9.99991 40 20 8.36314 8.36326 9.99988 40 30 8.30568 8.30577 9.99991 30 30 8.36405 8.36417 9.99988 30 40 8.30672 8.30681 9.99991 20 40 836496 8.36508 9.99988 20 50 8.30776 8.30785 9.99991 50 8.36587 8.36599 9.99988 10 10 8.30879 8.30888 9.99991 *0 5O 2O 8.36678 8.36689 9.99988 04O / // L cos L cot L sin // / / // L cos L cot L sin // / 88 C 34 / // L sin L tan L cos // / / // L sin L tan L cos // / 2O 8.36678 8.36689 9.99988 40 3O 8.41792 8.41807 9.99985 03O 10 8.36768 8.36780 9.99 988 50 10 8.41872 8.41887 9.99985 50 20 8.36858 8.36870 9.99988 40 20 8.41952 8.41967 9.99985 40 30 8.36948 8.36960 9.99988 30 30 8.42032 8.42048 9.99985 30 40 8.37038 8.37050 9.99988 20 40 8.42112 8.42127 9.99985 20 50 8.37128 8.37140 9.99988 10 50 8.42192 8.42207 9.99985 10 21 8.37217 8.37229 9.99988 39 31 8.42272 8.42287 9.99985 029 10 8.37306 8.37318 9.99988 50 10 8.42351 8.42366 9.99985 50 20 8.37395 8.37408 9.99988 40 20 8.42430 8.42446 9.99985 40 30 8.37484 8.37497 9.99988 30 30 8.42510 8.42525 9.99985 30 40 8.37573 8.37585 9.99988 20 40 8.42589 8.42406 9.99985. 20 50 8.37662 8.37674 9.99988 10 50 8.42667 8.42683 9.99985 10 22 8.37750 8.37762 9.99988 38 32 8.42746 8.42762 9.99984 028 10 8.37838 8.37850 9.99988 50 10 8.42825 8.42840 9.96984 50 20 8.37926 8.37938 9.99988 40 20 8.42903 8.42919 9.99984 40 30 8.38014 8.38026 9.99987 30 30 8.42 982 8.42997 9.99984 30 40 8.38101 8.38114 9.99987 20 40 8.43060 8.43075 9.99984 20 50 8.38189 8.33202 9.99987 10 50 8.43 138 8.43 154 9.99 984 10 23 8.38276 8.38289 9.99987 37 33 8.43 216 8.43 232 9.99 984 027 10 8.38363 8.38376 9.99987 50 10 8.43293 8.43309 9.99984 50 20 8.38450 8.38463 9.99987 40 20 8.43371 8.43387 9.99984 40 30 8.38537 8.38550 9.99987 30 30 8.43448 8.43464 9.99984 30 40 8.38624 838636 9.99987 20 40 8.43526 8.43542 9.99984 20 50 8.38710 8.38723 9.99987 10 50 8.43603 8.43619 9.99984 10 24 8.38796 8.38809 9.99987 36 34 8.43680 8.43696 9.99984 026 10 8.38882 8.38895 9.99987 50 10 8.43757 8.43773 9.99984 50 20 8.38968 8.38981 9.99987 40 20 8.43834 8.43850 9.99984 40 30 8.39054 8.39067 9.99987 30 30 8.43910 8.43927 9.99984 30 40 8.39139 8.39153 9.99987 20 40 8.43987 8.44003 9.99984 20 50 8.39225 8.39238 9.99987 10 50 8.44063 8.44080 9.99983 10 25 8.39310 8.39323 9.99987 35 35 8.44139 8.44156 9.99983 025 10 8.39395 8.39408 9.99987 50 10 8.44216 8.44232 9.99983 50 20 8.39480 8.39493 9.99987 40 20 8.44292 8.44308 9.99983 40 30 8.39565 8.39587 9.99987 30 30 8.44367 8.44384 9.99983 30 40 8.39649 8.39663 9.99987 20 40 8.44443 8.44460 9.99983 20 50 8.39734 8.39747 9.99986 10 50 8.44519 8.44536 9.99983 10 26 8.39818 8.39832 9.99986 34 36 8.44594 8.44611 9.99983 024 10 8.39902 8.39916 9.99986 50 10 8.44669 8.44686 9.99 983 50 20 8.39986 8.40000 9.99986 40 20 8.44745 8.44762 9.99983 40 30 8.40070 8.40083 9.99986 30 30 8.44820 8.44837 9.99983 30 40 8.40153 8.40167 9.99986 20 40 8.44895 8.44912 9.99983 20 50 8.40237 8.40251 9.99986 10 50 8.44969 8.44987 9.99983 10 27 8.40320 8.40334 9.99986 33 37 8.45044 8.45061 9.99983 023 10 8.40403 8.40417 9.99986 50 10 8.45119 8.45 136 9.99983 50 20 8.40486 8.40500 9.99986 40 20 8.45 193 8.45 210 9.99 983 40 30 8.40569 8.40583 9.99986 30 30 8.45267 8.45285 9.99983 30 40 8.40651 8.40665 9.99986 20 40 8.45341 8.45359 9.99982 20 50 8.40734 8.40748 9.99986 10 50 8.45415 8.45433 9.99982 10 28 8.40816 8.40830 9.99986 32 38 8.45489 8.45507 9.99982 022 10 8.40898 8.40913 9.99986 50 10 8.45563 8.45581 9.99982 50 20 8.40 980 8.40995 9.99986 40 20 8.45637 8.45655 9.99982 40 30 8.41062 8.41077 9.99986 30 30 8.45 710 8.45 728 9.99 982 30 40 841144 8.41158 9.99986 20 40 8.45 784 8.45 802 9.99 982 20 50 8.41225 8.41240 9.99986 10 50 8.45857 8.45875 9.99982 10 29 8.41307 8.41321 9.99985 31 39 8.45930 8.45948 9.99982 021 10 8.41388 8.41403 9.99985 50 10 8.46003 8.46021 9.99982 50 20 8.41469 8.41484 9.99985 40 20 8.46076 8.46094 9.99982 40 30 8.41550 8.41565 9.99985 30 30 8.46149 8.46167 9.99982 30 40 8.41631 8.41646 9.99985 20 40 8.46222 8.46240 9.99982 20 50 8.41711 8.41726 9.99985 10 50 8.46294 8.46312 9.99982 10 3O 8.41 792 8.41 807 9.99 985 30 4O 8.46366 8.46385 9.99932 020 / // L cos L cot L sin // / / // L cos L cot L sin // / 1 i 88' / // L sin L tan L cos // / / // L sin L tan L cos // / 40 8.46366 8.46385 9.99982 2O 5O 8.50504 8.50527 9.99978 010 10 8.46439 8.46457 9.99982 50 10 8.50570 8.50593 9.99978 50 20 8.46511 8.46529 9.99982 40 20 8.50636 8.50658 9.99978 40 30 8.46583 8.46602 9.99981 30 30 8.50701 8.50724 9.99978 30 40 8.46655 8.46674 9.99981 20 40 8.50767 8.50789 9.99977 20 50 8.46727 8.46745 9.99981 10 50 8.50832 8.50855 9.99977 10 41 8.46799 8.46817 9.99981 19 51 8.50897 8.50920 9.99977 9 > 10 8.46870 8.46889 9.99981 50 10 8.50963 8.50985 9.99977 50 20 8.46942 8.46960 9.99981 40 20 8.51028 8.51050 9.99977 40 30 8.47013 8.47032 9.99981 30 30 8.51092 8.51015 9.99977 30 40 8.47084 8.47103 9.99981 20 40 8.51157 8.51180 9-99977 20 50 8.47155 8.47174 9.99981 10 50 8.51222 8.51245 9.99977 10 42 8.47226 8.47245 9.99981 18 52 8.51287 8.51310 9.99977 8 10 8.47297 8.47316 9.99981 50 10 8.51351 8.51374 9.99977 50 20 8.47368 847387 9.99981 40 20 8.51416 8.51439 9.99977 40 30 8.47439 8.47458 9.99981 30 30 8.51480 851 503 9.99977 30 40 8.47509 8.47528 9.99981 20 40 8.51544 8.51568 9.99977 20 50 8.47580 8.47599 9.99981 10 50 8.51609 8.51632 9.99-977 10 43 8.47650 8.47669 9.99981 17 53 8.51673 8.51696 9.99977 7 10 8.47720 8.47740 9.99980 50 10 8.51737 8.51760 9.99976 50 20 8.47790 8.47810 9.99980 40 20 8.51801 8.51824 9.99976 40 30 8.47860 8.47880 9.99980 30 30 8.51864 8.51888 9.99976 30 40 8.47930 8.47950 9.99980 20 40 8.51928 8.51952 9.99976 20 50 8.48000 8.48020 9.99980 10 50 8.51992 8.52015 9.99976 10 44 8.48096 8.48090 9.99980 16 54 8.52055 8.52079 9.99976 6 10 8.48139 8.48159 9.99980 50 10 8.52119 8.52143 9.99976 50 20 8.48208 8.48228 9.99980 40 20 8.52 182 8.52 206 9.99 976 40 30 8.48278 8.48298 9.99980 30 30 8.52245 8.52269 9.99976 30 40 8.48347 8.48367 9.99980 20 40 8.52308 8.52332 9.99976 20 50 8.48416 8.48436 9.99980 10 50 8.52371 8.52396 9.99976 10 45 8.48485 8.48505 9.99980 15 55 8.52434 8.52459 9.99976 5 10 8.48554 8.48574 9.99980 50 10 8.52497 8.52522 9.99976 50 20 8.48622 8.48643 9.99980 40 20 8.52560 8.52584 9.99976 40 30 8.48691 8.48711 9.99980 30 30 8.52623 8.52647 9.99975 30 40 8.48760 8.48780 9.99979 20 40 8.52685 8.52710 9.99975 20 50 8.48828 8.48849 9.99979 10 50 8.52748 8.52772 9.99975 10 46 8.48896 8.48917 9.99979 14 56 8.52810 8.52835 9.99975 4 10 8.48965 8.48985 9.99979 50 10 8.52872 8.52897 9.99975 50 20 8.49033 8.49053 9.99979 40 20 8.52935 8.52960 9.99975 40 30 8.49101 8.49121 9.99979 30 30 8.52997 8.53022 9.99975 30 40 8.49169 8.49189 9.99979 20 40 8.53059 8.53084 9.99975 20 50 8.49236 8.49257 9.99979 10 50 8.53121 8.53146 9.99975 10 47 8.49304 8.49325 9.99979 13 57 8.53183 8.53208 9.99975 3 10 8.49372 8.49393 9.99979 50 10 8.53245 8.53270 9.99975 50 20 8.49439 8.49460 9.99979 40 20 8.53306 8.53332 9.99975 40 30 8.49506 8.49528 9.99979 30 30 8.53368 8.53393 9.99975 30 40 8.49574 8.49595 9.99979 20 40 8.53 429 8.53 455 9.99 975 20 50 8.49641 8.49662 9.99979 10 50 8.53491 8.53516 9.99974 10 48 8.49708 8.49729 9.99979 12 58 8.53552 8.53578 9.99974 2 10 8.49775 8.49796 9.99979 50 10 8.53614 8.53639 9.99974 50 20 8.49842 8.49863 9.99978 40 20 8.53675 8.53700 9.99974 40 30 8.49908 8.49930 9.99978 30 30 8.53736 8.53762 9.99974 30 40 8.49975 8.49997 9.99978 20 40 8.53797 8.53823 9.99974 20 50 8.50042 8.50063 9.99978 10 50 8.53858 8.53884 9.99974 10 49 8.50108 850130 9.99978 11 59 8.53919 8.53945 9.99974 1 10 8.50174 8.50196 9.99978 50 10 8.53979 8.54005 9.99974 50 20 8.50241 8.50263 9.99978 40 20 8.54040 8.54066 9.99974 40 30 850307 8.50329 9.99978 30 30 8.54101 8.54127 9.99974 30 40 8.50373 8.50395 9.99978 20 40 8.54161 8.54187 9.99974 20 50 8.50439 8.50461 9.99978 10 50 8.54222 8.54248 9.99974 10 5O 8.50504 8.50527 9.99978 1O 6O 8.54282 8.54308 9.99974 O / // L cos L cot L sin // / / // L cos L cot L sin // / 36 / SLsin SLtan llLcot 9Lcos / o .24186 .24192 .75808 .99993 60 1 .24903 .24910 .75090 .99993 59 2 .25609 .25616 .74384 .99993 58 3 .26304 .26312 .73688 .99993 57 4 .26988 .26996 .73004 .99992 56 5 .27661 .27669 .72331 .99992 55 6 .28324 .28332 .71668 .99992 54 7 .28977 .28986 .71014 .99992 53 8 .29621 .29629 .70371 .99992 52 9 .30255 .30263 .69737 .99991 51 1C .30879 .30888 .69112 .99991 50 11 .31495 .31505 .68495 .99991 49 12 .32103 .32112 .67888 .99990 48 13 .32702 .32711 .67289 .99990 47 14 .33292 .33302 .66698 .99990 46 15 .33875 .33886 .66114 .99990 45 16 .34450 .34461 .65539 .99989 44 17 .35018 .35029 .64971 .99989 43 18 .35578 .35590 .64410 .99989 42 19 .36131 .36143 .63857 .99989 41 20 .36678 .36689 .63311 .99988 40 21 .37217 .37229 .62771 .99988 39 22 .37750 .37762 .62238 .99988 38 23 .38276 .38289 .61711 .99987 37 24 .38796 .38809 .61191 .99987 36 25 .39310 .39323 .60677 .99987 35 26 .39818 .39832 .60168 .99986 34 27 .40320 .40334 .59666 .99986 33 28 .40816 .40830 .59170 .99986 32 29 .41307 .41321 .58679 .99985 31 3O .41792 .41807 .58193 .99985 30 31 .42272 .42287 .57713 .99985 29 32 .42746 .42762 .57238 .99984 28 33 .43216 .43232 .56768 .99984 27 34 .43680 .43696 .56304 .99984 26 35 .44139 .44156 .55844 .99983 25 36 .44594 .44611 .55389 .99983 24 37 .45044 .45061 .54939 .99983 23 38 .45489 .45507 .54493 .99982 22 39 .45930 .45948 .54052 ,99982 21 40 .46366 .46385 .53615 .99982 2O 41 .46799 .46817 .53183 .99981 19 42 .47226 .47245 .52755 .99981 18 43 .47650 .47669 .52331 .99981 17 44 .48069 .48089 .51911 .99980 16 45 .48485 .48505 .51495 .99980 15 46 .48896 .48917 .51083 .99979 14 47 .49304 .49325 .50675 .99979 13 48 .49708 .49729 .50271 .99979 12 49 .50108 .50130 .49870 .99978 11 50 .50504 .50527 .49473 .99978 1O 51 .50897 .50920 .49080 .99977 9 52 .51287 .51310 .48690 .99977 8 53 .51673 .51696 .48304 .99977 7 54 .52055 .52079 .47921 .99976 6 55 .52434 .52459 .47541 .99976 5 56 .52810 .52835 .47165 .99975 4 57 .53183 .53208 .46792 .99975 3 58 .53552 .53578 .46422 .99974 2 59 .53919 .53945 .46055 .99974 1 60 .54282 .54308 .45692 .99974 O / SLcos SLcot ULtan 9Lsin / / SLsin SLtan 11 L cot 9Lcos / O .54282 .54308 .45692 .99974 60 1 .54642 .54669 .45331 .99973 59 2 .54999 .55027 .44973 .99973 58 3 .55354 .55382 .44618 .99972 57 4 .55705 .55734 .44266 .99972 56 5 .56054 .56083 .43917 .99971 55 6 .56400 .56429 .43571 .99971 54 7 .56743 .56773 .43227 .99970 53 8 .57084 .57114 .42886 .99970 52 9 .57421 .57452 .42548 .99969 51 10 .57757 .57788 .42212 .99969 r>o 11 .58089 .58121 .41879 .99968 49 12 .58419 .58451 .41549 .99968 48 13 .58747 .58779 .41221 .99967 47 14 .59072 .59105 .40895 .99967 46 15 .59395 .59428 .40572 .99967 45 16 .59715 .59749 .40251 .99966 44 17 .60033 .60068 .39932 .99966 43 18 .60349 .60384 .39616 .99965 42 19 .60662 .60698 .39302 .99964 41 20 .60973 .61009 .38991 .99964 4O 21 .61282 .61319 .38681 .99963 39 22 .61589 .61626 .38374 .99963 38 23 .61894 .61931 .38069 .99962 37 24 .62196 .62234 .37766 .99962 36 25 .62497 .62535 .37465 .99961 35 26 .62795 .62834 .37166 .99961 34 27 .63091 .63131 .36869 .99960 33 28 .63385 .63426 .36574 .99960 32 29 .63678 .63718 .36282 .99959 31 30 .63968 .64009 .35991 .99959 30 31 .64256 .64298 .35702 .99958 29 32 .64543 .64585 .35415 .99958 28 33 .64827 .64870 .35130 .99957 27 34 .65110 .65154 .34846 .99956 26 35 .65391 .65435 .34565 .99956 25 36 .65670 .65715 .34285 .99955 24 37 .65947 .65993 .34007 .99955 23 38 .66223 .66269 .33731 .99954 22 39 .66497 .66543 .33457 .99954 21 4O .66769 .66816 .33184 .99953 2O 41 .67039 .67087 .32913 .99952 19 42 .67308 .67356 .32644 .99952 18 43 .67575 .67624 .32376 .99951 17 44 .67841 .67890 .32110 .99951 16 45 .68104 .68154 .31846 .99950 15 46 .68367 .68417 .31583 .99949 14 47 .68627 .68678 .31322 .99949 13 48 .68886 .68938 .31062 .99948 12 49 .69144 .69196 .30804 .99948 11 50 .69400 .69453 .30547 .99947 10 51 .69654 .69708 .30292 .99946 9 52 .69907 .69962 .30038 .99946 8 53 .70159 .70214 .29786 .99945 7 54 .70409 .70465 .29535 .99944 6 55 .70658 .70714 .29286 .99944 5 56 .70905 .70962 .29038 .99943 4 57 .71151 .71208 .28792 .99942 3 58 .71395 .71453 .28547 .99942 2 59 .71638 .71697 .28303 .99941 1 60 .71880 .71940 .28060 .99940 O / SLcos SLcot ULtan 9Lsin / 88 C 87' 37 / SLsin SLtan llLcot 9Lcos / o .71880 .71940 .28060 .99940 60 1 .72120 .72181 .27819 .99940 59 2 .72359 .72420 .27580 .99939 58 3 .72597 .72659 .27341 .99938 57 4 .72834 .72896 .27104 .99938 56 5 .73069 .73132 .26868 .99937 55 * 6 .73303 .73366 .26634 .99936 54 7 .73535 .73600 .26400 .99936 53 8 .73767 .73832 .26168 .99935 52 9 .73997 .74063 .25937 .99934 51 10 .74226 .74292 .25708 .99934 50 11 .74454 .74521 .25479 .99933 49 12 .74680 .74748 .25252 .99932 48 13 .74906 .74974 .25026 .99932 47 14 .75130 .75199 .24801 .99931 46 IS .75353 .75423 .24577 .99930 45 16 .75575 .75645 .24355 .99929 44 17 .75795 .75867 .24133 .99929 43 18 .76015 .76087 .23913 .99928 42 19 .76234 .76306 .23694 .99927 41 20 .76451 .76525 .23475 .99926 4O 21 .76667 .76742 .23258 .99926 39 22 .76883 .76958 .23042 .99925 38 23 .77097 .77173 .22827 .99924 37 24 .77310 .77387 .22613 .99923 36 25 .77522 .77600 .22400 .99923 35 26 .77733 .77811 .22189 .99922 34 27 .77943 .78022 .21978 .99921 33 28 .78152 .78232 .21768 .99920 32 29 .78360 .78441 .21559 .99920 31 30 .78568 .78649 .21351 .99919 30 31 .78774 .78855 .21145 .99918 29 32 .78979 .79061 .20939. .99917 28 33 .79183 .79266 .20734 .99917 27 34 .79386 .79470 .20*530 .99916 26 35 .79588 .79673 .20327 .99915 25 36 .79789 .79875 .20125 .99914 24 37 .79990 .80076 .19924 .99913 23 38 .80189 .80277 .19723 .99913 22 39 .80388 .80476 .19524 .99912 21 4O .80585 .80674 .19326 .99911 2O 41 .80782 .80872 .19128 .99910 19 [ 42 .80978 .81068 .18932 .99909 18 43 .81173 .81264 .18736 .99909 17 44 .81367 .81459 .18541 .99908 16 45 .81560 .81653 .18347 .99907 15 46 .81752 .81846 .18154 .99906 14 47 .81944 .82038 .17962 .99905 13 48 .82134 .82230 .17770 .99904 12 49 .82324 .82420 .17580 .99904 11 50 .82513 .82610 .17390 .99903 1O 51 .82701 .82799 .17201 .99902 9 52 .82888 .82987 .17013 .'99901 8 53 .83075 .83175 .16825 .99900 7 54 .83261 .83361 .16639 .99899 6 55 .83446 .83547 .16453 .99898 5 56 .83630 .83732 .16268 .99898 4 57 .83813 .83916 .16084 .99897 3 58 .83996 .84100 .15900 .99896 2 59 .84177 .84282 .15718 .99895 1 52. .84358 .84464 .15536 .99894 / SLcos SLcot llLtan 9Lsin / / SLsin SLtan 11 Loot DLcos / O .84 358 .84 464 .15 536 .99894 6O 1 .84 539 .84646 .15354 .99893 59 2 .84718 .84 826 .15 174 .99 892 58 3 .84 897 .85006 .14994 .99891 57 4 .85 075 .85 185 .14815 .99 891 56 5 .85 252 .85 363 .14637 .99 890 55 6 .85 429 .85 540 .14460 .99889 54 7 .85 605 .85 717 .14283 .99888 53 8 .85 780 .85 893 .14 107 .99887 52 9 .85 955 .86069 .13931 .99886 51 1O .86 128 .86243 .13757 .99885 50 11 .86301 .86417 .13583 .99884 49 12 .86474 .86591 .13409 .99883 48 13 .86645 .86 763 .13237 .99882 47 14 .86816 .86935 .13065 .99881 46 15 .86987 .87 106 .12 894 .99880 45 16 .87156 .87277 .12 723 .99 879 44 17 .87325 .87447 .12 553 .99 879 43 18 .87494 .87616 .12384 .99 878 42 19 .87661 .87 785 .12215 .99 877 41 2O .87 829 .87953 .12047 .99876 40 21 .87995 .88 120 .11880 .99875 39 22 .88 161 .88287 .11713 .99874 38 23 .88326 .88453 .11547 .99 873 37 24 .88490 .88618 .11382 .99872 36 25 .88654 .88 783 .11217 .99871 35 26 .88817 .88948 .11052 .99 870 34 27 .88980 .89111 .10889 .99869 33 28 .89 142 89274 .10726 .99868 32 29 .89304 .89437 .10563 .99867 31 3O .89464 .89 598 .10402 .99866 30 31 .89625 .89 760 .10240 .99865 29 32 .89 784 .89920 .10080 .99864 28 33 .89943 .90080 .09920 .99863 27 34 .90 102 .90 240 .09 760 .99862 26 35 .90260 .90399 .09601 .99861 25 36 .90417 .90557 .09443 .99860 24 37 .90574 .90715 .09 285 .99859 23 38 .90 730 .90872 .09 128 .99858 22 39 .90885 .91 029 .08971 .99857 21 4O .91 040 .91 185 .08815 .99856 2O 41 .91 195 .91 340 .08660 .99855 19 42 .91 349 .91 495 .08 505 .99854 18 43 .91 502 .91 650 .08350 .99853 17 44 .91655 .91 803 .08 197 .99852 16 45 .91 807 .91957 .08043 .99851 15 46 .91 959 .92110 .07 890 .99850 14 47 .92110 .92 262 .07 738 .99848 13 48 .92 261 .92414 .07 586 .99847 12 49 .92411 .92 565 .07 435 .99846 11 5O .92 561 .92716 .07 284 .99845 10 51 .92710 .92 866 .07 134 .99844 9 52 .92 859 .93 016 .06984 .99843 8 53 .93 007 .93 165 .06835 .99842 7 54 .93 154 .93313 .06687 .99841 6 55 .93301 .93 462 .06538 .99840 5 56 .93 448 .93 609 .06391 .99839 4 57 .93 594 .93 756 .06244 .99838 3 58 .93 740 .93 903 .06097 .99837 2 59 .93 885 .94 049 .05951 .99836 1 60 .94030 .94 195 .05 805 .99834 O / SLcos SLcot 11 L tan 9Lsin / 86 85 C 38 / SLsin SLtan llLcot 9Lcos / o .94030 .94195 .05805 .99834 6O 1 .94174 .94340 .05660 .99833 59 2 .94317 .94485 .05515 .99832 58 3 .94461 .94630 .05370 .99831 57 4 .94603 .94773 .05227 .99830 56 5 .94746 .94917 .05083 .99829 55 6 .94887 .95060 .049-10 .99828 54 7 .95029 .95202 .04798 .99827 53 8 .95170 .95344 .04656 .99825 52 9 .95310 .95486 .04514 .99824 51 ID .95450 .95627 .04373 .99823 50 11 .95589 .95767 .04233 .99822 49 12 .95728 .95908 .04092 .99821 48 13 .95867 .96047 .03953 .99820 47 14 .96005 .96187 .03813 .99819 46 15 .96143 .96325 .03675 .99817 45 16 .96280 .96464 .03536 .99816 44 17 .96417 .96602 .03398 .99815 43 18 .96553 .96739 .03261 .99814 42 19 .96689 .96877 .03123 .99813 41 20 .96825 .97013 .02987 .99812 40 21 .96960 .97150 .02850 .99810 39 22 .97095 .97285 .02715 .99809 38 23 .97229 .97421 .02579 .99808 37 24 .97363 .97556 .02444 .99807 36 25 .97496 .97691 .02309 .99806 35 26 .97629 .97825 .02175 .99804 34 27 .97762 .97959 .02041 .99803 33 28 .97894 .98092 .01908 .99802 32 29 .98026 .98225 .01775 .99801 31 30 .98157 .98358 .01642 .99800 30 31 .98288 .98490 .01510 .99798 29 32 .98419 .98622 .01378 .99797 28 33 .98549 .98753 .01247 .99796 27 34 .98679 .98884 .01116 .99795 26 35 .98808 .99015 .00985 .99793 25 36 .98937 .99145 .00855 .99792 24 37 .99066 .99275 .00725 .99791 23 38 .99194 .99405 .00595 .99790 22 39 .99322 .99534 .00466 .99788 21 40 .99450 .99662 .00338 .99787 20 41 .99577 .99791 .00209 .99786 19 42 .99704 .99919 .00081 .99785 18 43 .99830 .00046 .99954 .99783 17 44 .99956 .00174 .99826 .99782 16 45 .00082 .00301 .99699 .99781 15 46 .00207 .00427 .99573 .99780 14 47 .00332 .00553 .99447 .99778 13 48 .00456 .00679 .99321 .99777 12 49 .00581 .00805 .99195 .99776 11 5O .00704 .00930 .99070 .99775 1O 51 .00828 .01055 .98945 .99773 9 52 .00951 .01179 .98821 .99772 8 53 .01074 .01303 .98697 .99771 7 54 .01196 .01427 .98573 .99769 6 55 .01318 .01550 .98450 .99768 5 56 .01440 .01673 .98327 .99767 4 57 .01561 .01796 .98204 .99765 3 58 .01682 .01918 .98082 .99764 2 59 .01803 .02040 .97960 .99763 1 60 .01923 .02162 .97838 .99761 O / 9 L cos 9 L cot 1O L tan 9 L sin / / 9Lsin 9LtanlOLcot 9Lcos / .01923 .02162 .97838 .99761 6O 1 .02043 .02283 .97717 .99760 59 2 .02163 .02404 .97596 .99759 58 3 .02283 .02525 .97475 .99757 57 4 .02402 .02645 .97355 .99756 56 5 .02520 .02766 .97234 .99755 55 6 .02639 .02885 .97115 .99753 54 7 .02757 .03005 .96995 .99752 53 8 .02874 .03124 .96876 .99751 52 9 .02992 .03242 .96758 .99749 51 1O .03109 .03361 .96639 .99748 5O 11 .03226 .03479 .96521 .99747 49 12 .03342 .03597 .96403 .99745 48 13 .03458 .03714 .96286 .99744 47 14 .03574 .03832 .96168 .99742 46 15 .03690 .03948 .96052 .99741 45 16 .03805 .04065 .95935 .99740 44 17 .03920 .04181 .95819 .99738 43 18 .04034 .04297 .95703 .99737 42 19 .04149 .04413 .95587 .99736 41 2O .04262 .04528 .95472 .99734 40 21 .04376 .04643 .95357 .99733 39 22 .04490 .04758 .95242 .99731 38 23 .04603 .04873 .95127 .99730 37 24 .04715 .04987 .95013 .99728 36 25 .04828 .05101 .94899 .99727 35 26 .04940 .05214 .94786 .99726 34 27 .05052 .05328 .94672 .99724 33 28 .05164 .05441 .94559 .99723 32 29 .05275 .05553 .94447 .99721 31 3O .05386 .05666 .94334 .99720 3O 31 .05497 .05778 .94222 .99718 29 32 .05607 .05890 .94110 .99717 28 33 .05717 .06002 .93998 .99716 27 34 .05827 .06113 .93887 .99714 26 35 .05937 .06224 .93776 .99713 25 36 .06046 .06335 .93665 .99711 24 37 .06155 .06445 .93555 .99710 23 38 .06264 .06556 .93444 .99708 22 39 .06372 .06666 .93334 .99707 21 4O .06481 .06775 .93225 .99705 2O 41 .06589 .06885 .93115 .99704 19 42 .06696 .06994 .93006 .99702 18 43 .06804 .07103 .92897 .99701 17 44 .06911 .07211 .92789 .99699 16 45 .07018 .07320 .92680 .99698 15 46 .07124 .07428 .92572 .99696 14 47 .07231 .07536 .92464 .99695 13 48 .07337 .07643 .92357 .99693 12 49 .07442 .07751 .92249 .99692 11 5O .07548 .07858 .92142 .99690 1O 51 .07653 .07964 .92036 .99689 9 52 .07758 .08071 .91929 .99687 8 53 .07863 .08177 .91823 .99686 7 54 .07968 .08283 .91717 .99684 6 55 .08072 .08389 .91611 .99683 5 56 .08176 .08495 .91505 .99681 4 57 .08280 .08600 .91400 .99680 3 58 .08383 .08705 .91295 .99678 2 59 .08486 .08810 .91190 .99677 1 6O .08589 .08914 .91086 .99675 O 7 9 L cos 9 L cot 1O L tan 9 L sin / 84 C 83< 7 8 C 39 / 9Lsin 9Ltan 10 L cot 9Lcos / .08589 .08914 .91086 .99675 6O 1 .08692 .09019 .90981 .99674 59 2 .08795 .09123 .90877 .99672 58 3 .08897 .09227 .90773 .99670 57 4 .08999 .09330 .90670 .99669 56 5 .09101 .09434 .90566 .99667 55 6 .09202 .09537 .90463 .99666 54 7 .09304 .09640 .90360 .99664 53 8 .09405 .09742 .90258 .99663 52 9 .09506 .09845 .90155 .99661 51 1C .09606 .09947 .90053 .99659 50 11 .09707 .10049 .89951 .99658 49 12 .09807 .10150 .89850 .99656 48 13 .09907 .10252 .89748 .99655 47 14 .10006 .10353 .89647 .99653 46 15 .10106 .10454 .89546 .99651 45 16 .10205 .10555 .89445 .99650 44 17 .10304 .10656 .89344 .99648 43 18 .10402 .10756 .89244 .99647 42 19 .10501 .10856 .89144 .99645 41 2O .10599 .10956 .89044 .99643 4O 21 .10697 .11056 .88944 .99642 39 22 .10795 .11155 .88845 .99640 38 23 .10893 ..11254 .88746 .99638 37 24 .10990 .11353 .88647 .99637 36 25 .11087 .11452 .88548 .99635 35 26 -.11184 .11551 .88449 .99633 34 27 .11281 .11649 .88351 .99632 33 28 .11377 .11747 .88253 .99630 32 29 .11474 .11845 .88155 .99629 31 30 .11570 .11943 .88057 .99627 30 31 .11666 .12040 .87960 .99625 29 32 .11761 .12138 .87862 .99624 28 33 .11857 .12235 .87765 ,.99622 27 34 .11952 .12332 .87668 .99620 26 35 .12047 .12428 .87572 .99618 25 36 .12142 .12525 .87475 .99617 24 37 .12236 .12621 .87379 .99615 23 38 .12331 .12717 .87283 .99613 22 39 .12425 .12813 .87187 .99612 21 40 .12519 .12909 .87091 .99610 2O 41 .12612 .13004 .86996 .99608 19 42 .12706 .13099 .86901 .99607 18 43 .12799 .13194 .86806 .99605 17 44 .12892 .13289 .86711 .99603 16 45 .12985 .13384 .86616 .99601 15 46 .13078 .13478 .86522 .99600 14 47 .13171 .13573 .86427 .99598 13 48 .13263 .13667 .86333 .99596 12 49 .13355 .13761 .86239 .99595 11 50 .13447 .13854 .86146 .99593 10 51 .13539 .13948 .86052 .99591 9 52 .13630 .14041 .85959 .99589 8 53 .13722 .14134 .85866 .99588 7 54 .13813 .14227 .85773 .99586 6 55 .13904 .14320 .85680 .99584 5 56 .13994 .14412 .85588 .99582 4 57 .14085 .14504 .85496 .99581 3 58 .14175 .14597 .85403 .99579 2 59 .14266 .14688 .85312 .99577 1 6O .14356 .14780 .85220 .99575 / 9 L cos 9 L cot 10 L tan 9 L sin / / 9Lsin 9Ltan lOLcot 9Lcos / o .14356 .14780 .85220 .99575 60 1 .14445 .14872 .85128 .99574 59. 2 .14535 .14963 .85037 .99572 58 3 .14624 .15054 .84946 .99570 57 4 .14714 .15145 .84855 .99568 56 5 .14803 .15236 .84764 .99566 55 6 .14891 .15327 .84673 .99565 54 7 .14980 .15417 .84583 .99563 53 8 .15069 .15508 .84492 .99561 52 9 .15157 .15598 .84402 .99559 51 1O .15245 .15688 .84312 .99557 50 11 .15333 .15777 .84223 .99556 49 12 .15421 .15867 .84133 .99554 48 13 .15508 .15956 .84044 .99552 47 14 .15596 .16046 .83954 .99550 46 15 .15683 .16135 .83865 .99548 45 16 .15770 .16224 .83776 .99546 44 17 .15857 .16312 .83688 .99545 43 18 .15944 .16401 .83599 .99543 42 19 .16030 .16489 .83511 .99541 41 2O .16116 .16577 .83423 .99539 40 21 .16203 .16665 .83335 .99537 39 22 .16289 .16753 .83247 .99535 38 23 .16374 .16841 .83159 .99533 37 24 .16460 .16928 .83072 .99532 36 25 .16545 .17016 .82984 .99530 35 26 .16631 .17103 .82897 .99528 34 27 .16716 .17190 .82810 .99526 33 28 .16801 .17277 .82723 .99524 32 29 .16886 .17363 .82637 .99522 31 3O .16970 .17450 .82550 .99520 30 31 .17055 .17536 .82464 .99518 29 32 .17139 .17622 .82378 .99517 28 33 .17223 .17708 .82292 .99515 27 34 .17307 .17794 .82206 .99513 26 35 .17391 .17880 .82120 .99511 25 36 .17474 .17965 .82035 .99509 24 37 .17558 .18051 .81949 .99507 23 38 .17641 .18136 .81864 .99505 22 39 .17724 .18221 .81779 .99503 21 40 .17807 .18306 .81694 .99501 2O 41 .17800 .18391 .81609 .99499 19 42 .17973 .18475 .81525 .99497 18 43 .18055 .18560 .81440 .99495 17 44 .28137 .18644 .81356 .99494 16 45 .18220 .18728 .81272 .99492 15 46 .18302 .18812 .81188 .99490 14 47 .18383 .18896 .81104 .99488 13 48 .18465 .18979 .81021 .99486 12 49 .18547 .19063 .80937 .99484 11 50 .18628 .19146 .80854 .99482 10 51 .18709 .19229 .80771 .99480 9 52 .18790 .19312 .80688 .99478 8 53 .18871 .19395 .80605 .99476 7 54 .18952 .19478 .80522 .99474 6 55 .19033 .19561 .80439 .99472 5 56 .19113 .19643 .80357 .99470 4 57 .19193 .19725 .80275 .99468 3 58 .19273 .19807 .80193 .99466 2 59 .19353 .19889 .80111 .99464 1 60 .19433 .19971 .80029 .99462 O / 9 L cos 9 L cot 1O L tan 9 L sin / 81 40 10 / 9Lsin 9Ltan lOLcot 9Lcos / o .19433 .19971 .80029 .99462 6O I .19513 .20053 .79947 .99460 59 2 .19592 .20134 .79866 .99458 58 3 .19672 .20216 .79784 .99456 57 4 .19751 .20297 .79703 .99454 56 5 .19830 .20378 .79622 .99452 55 6 .19909 .20459 .79541 .99450 54 7 .19988 .20540 .79460 .99448 53 8 .20067 .20621 .79379 .99446 52 9 .20145 .20701 .79299 .99444 51 1O .20223 .20782 .79218 .99442 50 11 .20302 .20862 .79138 .99440 49 12 .20380 .20942 .79058 .99438 48 13 .20458 .21022 .78978 .99436 47 14 .20535 .21102 .78898 .99434 46 15 .20613 .21182 .78818 .99432 45 16 .20691 .21 261 .78739 .99429 44 17 .20768 .21341 .78659 .99427 43 18 .20845 .21420 .78580 .99425 42 19 .20922 .21499 .78501 .99423 41 20 .20999 .21578 .78422 .99421 4O 21 .21076 .21657 .78343 .99419 39 22 .21153 .21736 .78264 .99417 38 23 .21229 .21814 .78186 .99415 37 24 .21306 .21893 .78107 .99413 36 25 .21382 .21971 .78029 .99411 35 26 .21458 .22049 .77951 .99409 34 27 .21534 .22127 .77873 .99407 33 28 .21610 .22205 .77795 .99404 32 29 .21685 .22283 .77717 .99402 31 3O .21761 .22361 .77639 .99400 3O 31 .21836 .22438 .77562 .99398 29 32 .21912 .22516 .77484 .993.96 28 33 .21987 .22593 .77407 .99394 27 34 .22062 .22670 .77330 .99392 26 35 .22137 .22747 .77253 .99390 25 36 .22211 .22824 .77176 .99388 24 37 .22286 .22901 .77099 .99385 23 38 .22361 .22977 .77023 .99383 22 39 .22435 .23054 .76946 .99381 21 40 .22509 .23130 .76870 .99379 20 41 .22583 .23206 .76794 .99377 19 42 .22657 .23283 .76717 .99375 18 43 .22731 .23359 .76641 .99372 17 44 .22805 .23435 .76565 .99370 16 45 .22878 .23510 .76490 .99368 15 46 .22952 .23586 .76414 .99366 14 47 .23025 .23661 .76339 .99364 13 48 .23098 .23737 .76263 .99362 12 49 .23171 .23812 .76188 .99359 11 5O .23244 .23887 .76113 .99357 10 51 .23317 .23962 .76038 .99355 9 52 .23390 .24037 -.75963 .99353 8 53 .23462 .24112 .75888 .99351 7 54 .23535 .24186 .75814 .99348 6 55 .23607 .24261 .75739 .99346 5 56 .23679 .24335 .75665 .99344 4 57 .23752 .24410 .75590 .99342 3 58 .23823 .24484 .75516 .99340 2 59 .23895 .24558 .75442 .99337 1 60 .23967 .24632 .75368 .99335 O / 9Lcos 9Lcot lOLtan 9Lsin / / 9Lsin 9Ltan 1O L cot 9 Lcos / O .23 967 .24 632 .75 368 .99335 750 1 .24 039 .24 706 .75 294 .99 333 59 2 .24110 .24 779 .75 221 .99331 58 3 .24181 .24 853 .75 147 .99328 57 4 .24253 .24926 .75074 .99 326 56 5 .24324 .25000 .75000 .99324 55 6 .24 395 .25 073 .74 927 .99322 54 7 .24466 .25 146 .74 854 .99319 53 8 .24 536 .25 219 .74 781 .99317 52 9 .24607 .25 292 .74 708 .99315 51 10 .24677 .25 365 .74 635 .99313 5O 11 .24 748 .25 437 .74563 .99310 49 12 .24818 .25510 .74490 .99308 48 13 .24 888 .25 582 .74418 .99306 47 14 .24958 .25 655 .74345 .99304 46 15 .25 028 .25 727 .74273 .99301 45 16 .25 098 .25 799 .74 201 .99299 44 17 .25 168 .25 871 .74129 .99 297 43 18 .25 237 .25 943 .74057 .99 294 i . 42 19 .25307 .26015 .73985 .99 292 41 2O .25 376 .26086 .73914 .99290 4O 21 .25 445 .26158 .73842 .99288 39 22 .25514 .26 229 .73 771 .99 285 38 23 .25 583 .26301 .73699 .99 283 37 24 .25 652 .26372 .73 628 .99281 36 25 .25 721 .26443 .73557 .99278 35 26 .25 790 .26514 .73486 .99 276 34 27 .25 858 .26 585 .73415 .99274 33 28 .25927 .26655 .73 345 .99271 32 29 .25 995 .26 726 .73 274 .99269 31 3O .26063 .26 797 .73 203 .99 267 30 31 .26131 .26867 .73133 .99 264 29 32 .26 199 .26937 .73 063 .99262 28 33 .26267 ..27008 .72992 .99260 27 34 .26335 .27078 .72 922 .99257 26 35 .26403 .27148 .72852 .99255 25 36 .26470 .27218 .72 782 .99252 24 37 .26 538 .27 288 .72712 .99 250 23 38 .26605 .27357 .72643 .99 248 22 39 .26672 .27427 .72573 .99 245 21 40 .26 739 .27496 .72 504 .99243 2O 41 .26806 .27 566 .72434 .99 241 19 42 .26 873 .27635 .72365 .99 238 18 43 .26940 .27704 .72 296 .99 236 17 44 .27007 .27773 .72227 .99 233 16 45 .27073 .27 842 .72155 .99231 15 46 .27140 .27911 .72 089 .99 229 14 47 .27 206 .27 980 .72020 .99 226 13 48 .27273 .28049 .71951 .99 224 12 49 .27339 .28117 .71883 .99221 11 50 .27405 .28 186 .71814 .99219 10 51 .27471 .28254 .71 746 .99217 9 52 .27537 .28323 .71677 .99214 8 53 .27 602 .28391 .71 609 .99212 7 54 .27668 .28459 .71 541 .99 209 6 55 .27 734 .28 527 .71473 .99207 5 56 .27 799 .28 595 .71405 .99 204 4 57 .27 864 .28 662 .71338 .99 202 3 58 .27930 .28 730 .71270 .99 200 2 59 .27995 .28 798 .71 202 .99 197 1 60 .28060 .28 865 .71 135 .99 195 O / 9 Lcos 9Lcot lOLtan 9Lsin / 80 C 79 ir 41 / 9 L sin 9 L tan 1O L cot 9 L cos / o .28060 .28865 .71135 .99195 6O I .28125 .28933 .71067 .99192 59 2 .28190 .29000 .71000 .99190 58 3 .28254 .29067 .70933 .99187 57 4 .28319 .29134 .70866 .99185 56 5 .28384 .29201 .70799 .99182 55 6 .28448 .29268 .70732 .99180 54 7 .28512 .29335 .70665 .99177 53 8 .28577 .29402 .70598 .99175 52 9 .28641 .29468 .70532 .99172 51 1C .28705 .29535 .70465 .99170 50 11 .28769 .29601 .70399 .99167 49 12 .28833 .29668 .70332 .99165 48 13 .28896 .29734 .70266 .99162 47 . 14 .28960 .29800 .70200 .99160 46 15 .29024 .29866 .70134 .99157 45 16 .29087 .29932 .70068 .99155 44 17 .29150 .29998 .70002 .99152 43 18 .29214 .30064 .69936 .99150 42 19 .29277 .30130 .69870 .99147 41 2O .29340 .30195 .69805 .99145 40 21 .29403 .30261 .69739 .99142 39 22 .29466 .30326 .69674 .99140 38 23 .29529 .30391 .69609 .99137 37 I 24 .29591 .30457 .69543 .99135 36 25 .29654 .30522 .69478 .99132 35 26 .29716 .30587 .69413 .99130 34 27 .29779 .30652 .69348 .99127 33 28 .29841 .30717 .69283 .99124 32 29 .29903 .30782 .69218 .99122 31 30 .29966 .30 846 '.69 154 .99119 30 31 .30028 .30911 .69089 .99117 29 32 .30090 .30975 .69025 .99114 28 33 .30151 .31040 .68960 .99112 27 34 .30213 .31104 .68896 .99109 26 35 .30275 .31168 .68832 .99106 25 36 .30336 .31233 .68767 .99104 24 37 .30398 .31297 .68703 .99101 23 38 .30459 .31361 .68639 .99099 22 39 .30521 .31425 .68575 .99096 21 40 .30582 .31489 .68511 .99093 2O 41 .30643 .31552 .68448 .99091 19 42 .30704 .31616 .68384 .99088 18 43 .30765 .31679 .68321 .99086 17 44 .30826 .31743 .68257 .99083 16 45 .30887 .31806 .68194 .99080 15 46 .30947 .31870 .68130 .99078 14 47 .31008 .31933 .68067 .99075 13 48 .31068 .31996 .68004 .99072 12 49 .31129 .32059 .67941 .99070 11 50 .31189 .32122 .67878 .99067 1O 51 .31250 .32185 .67815 .99064 9 52 .31310 .32248 .67752 .99062 8 53 .31370 .32311 .67689 .99059 7 54 .31430 .32373 .67627 .99056 6 55 .31490 .32436 .67564 .99054 5 56 .31549 .32498 .67502 .99051 4 57 .31609 .32561 .67439 .99048 3 58 .31669 .32623 .67377 .99046 2 59 .31728 .32685 .67315 .99043 1 6O .31788 .32747 .67253 .99040 O / 9 L cos 9 L cot 1O L tan 9 L sin / / 9Lsin 9Ltan 1O L cot 9Lcos -L- O .31 788 .32 747 .67 253 .99040 ou CO 1 .31847 .32810 .67 190 .99038 V 2 .31907 .32872 .67 128 .99035 58 3 .31 966 .32933 .67 067 .99032 57 4 .32025 .32995 .67005 .99030 56 5 .32084 .33057 .66943 .99027 55 6 .32 143 .33 119 .66881 .99024 54 7 .32 202 .33 180 .66 820 .99022 53 CO 8 .32 261 .33 242 .66 758 .99019 OL 9 .32319 .33303 .66697 .99016 51 1.O .32378 .33 365 .66635 .99013 50 zin 11 .32437 .33 426 .66574 .99011 *ty 12 .32495 .33487 .66513 .99008 48 13 .32553 .33548 .66452 .99005 47 14 .32612 .33609 .66391 .99002 46 15 .32670 .33 670 .66330 .99000 45 16 .32 728 .33 731 .66 269 .98997 44 17 .32 786 .33 792 .66208 .98994 43 18 .32 844 .33 853 .66 147 .98991 42 19 .32902 .33913 .66087 .98989 41 20 .32960 .33974 .66026 .98986 4O 21 .33 018 .34 034 .65 966 .98983 39 '22 .33 075 .34095 .65 905 .98980 38 1 r- 23 .33 133 .34 155 .65 845 .98978 o/ ox; 24 .33 190 .34215 .65 785 .98975 36 25 .33 248 .34 276 .65 724 .98972 35 26 .33 305 .34336 .65 664 .98969 34 27 .33 362 .34 396 .65 604 .98967 33 28 .33 420 .34456 .65 544 .98964 32 29 .33 477 .34516 .65 484 .98961 31 30 .33 534 .34576 .65 424 .98958 30 31 .33 591 .34 635 .65 365 .98955 29 32 .33647 .34695 .65 305 .98953 28 33 .33 704 .34 755 .65 245 .98950 27 34 .33 761 .34814 .65 186 .98947 26 35 .33 818 .34874 .65 126 .98944 25 36 .33 874 .34 933 .65 067 .98941 24 37 .33931 .34992 .65 008 .98938 23 38 .33 987 .35051 .649-19 .98 936 22 39 .34043 .35 111 .64 889 .98933 21 40 .34 100 .35 170 .64830 .98930 2O 41 .34 156 .35 229 ,64771 .98927 19 42 .34212 .35 288 .64712 .98924 18 43 .34 268 .35 347 .64653 .98921 17 44 .34324 .35 405 .64595 .98919 16 45 .34 380 .35 464 .64 536 .98916 15 46 .34 436 .35 523 .64477 .98913 14 47 .34491 .35 581 .64419 .98910 13 48 .34 547 .35 640 .64360 .9890? 12 49 .34602 .35 698 .64302 .98904 11 50 .34658 .35 757 .64 243 .98901 1O 51 .34 713 .35815 .64 185 .98898 9 52 .34 769 .35 873 .64127 .98896 8 53 .34 824 .35931 .64069 .98893 7 54 .34 879 .35989 .64011 .98890 6 55 .34934 .36047 .63 953 .98887 5 56 .34989 .36 105 .63 895 .98884 4 57 .35 044 .36 163 .63 837 .98 881 3 58 .35 099 .36221 .63 779 .98878 2 59 .35 154 .36279 .63 721 .98 875 1 60 .35 209 .36336 .63 664 .98 872 / 9Lccs 9Lcot 1O L tan 9Lsin / 78 C 77 C 42 13 14 C / 9 L sin 9 L tan 1O L cot 9 L cos / o .35209 .36336 .63664 .98872 60 1 .35263 .36394 .63606 .98869 59 2 .35318 .36452 .63548 .98867 58 3 .35373 .36509 .63491 .98864 57 4 .35427 .36566 .63434 .98861 56 5 .35481 .36624 .63376 .98858 55 6 .35536 .36681 .63319 .98855 54 7 .35590 .36738 .63262 .98852 53 8 .35644 .36795 .63205 .98849 52 9 .35698 .36852 .63148 .98846 51 1C .35752 .36909 .63091 .98843 5O 11 .35806 .36966 .63034 .98840 49 12 .35860 .37023 .62977 .98837 48 13 .35914 .37080 .62920 .98834 47 14 .35968 .37137 .62863 .98831 46 15 .36022 .37193 .62807 .98828 45 16 .36075 .37250 .62750 .98825 44 17 .36129 .37306 .62694 .98822 43 18 .36182 .37363 .62637 .98819 42 19 .36236 .37419 .62581 .98816 41 20 .36289 .37476 .62524 .98813 40 21 .36342 .37532 .62468 .98810 39 22 .36395 .37588 .62412 .98807 38 23 .36449 .37644 .62356 .98804 37 24 .36502 .37700 .62300 .98801 36 25 .36555 .37756 .62244 .98798 35 26 .36608 .37812 .62188 .98795 34 27 .36660 .37868 .62132 .98792 33 28 .36713 .37924 .62076 .98789 32 29 .36766 .37980 .62020 .98786 31 30 .36819 .38035 .61965 .98783 30 31 .36871 .38091 .61909 .98780 29 32 .36924 .38147 .61853 .98777 28 33 .36976 .38202 .61798 .98774 27 34 .37028 .38257 .61743 .98771 26 35 .37081 .38313 .61687 .98768 25 36 .37133 .38368 .61632 .98765 24 37 .37185 .38423 .61577 .98762 23 38 .37237 .38479 .61521. .98759 22 39 .37289 .38534 .61466 .98756 21 4O .37341 .38589 .61411 .98753 2O 41 .37393 .38644 .61356 .98750 19 42 .37445 .38699 .61301 .98746 18 43 .37497 .38754 .61246 .98743 17 44 .37549 .38808 .61192 .98740 16 45 .37600 .38863 .61137 .98737 15 46 .37652. .38918 .61082 .98734 14 47 .37703 .38972 .61028 .98731 13 48 .37755 .39027 .60973 .98728 12 49 .37806 .39082 .60918 .98725 11 50 .37858 .39136 .60864 .98722 10 51 .37909 .39190 .60810 .98719 9 52 .37960 .39245 .60755 .98715 8 53 .38011 .39299 .60701 .98712 7 54 .38062 .39353 .60647 .98709 6 55 .38113 .39407 .60593 .98706 5 56 .38164 .39461 .60539 '.98703 4 57 .38215 .39515 .60485 .98700 3 58 .38266 .39569 .60431 .98697 2 59 .38317 .39623 .60377 .98694 1 60 .38368 .39677 .60323 .98690 O / 9Lcos 9LcotlOLtan9Lsin / / 9 L sin 9 L tan 1O L cot 9 L cos / ~o .38368 .39677 .60323 .98690 6O 1 .38418 .39731 .60269 .98687 59 2 .38469 .39785 .60215 .98684 58 3 .38519 .39838 .60162 .98681 57 4 .38570 .39892 .60108 .98678 56 5 .38620 .39945 .60055 .98675 55 6 .38670 .39999 .60001 .98671 54 7 .38721 .40052 .59948 .98668 53 8 .38771 .40106 .59894 .98665 52 9 .38821 .40159 .59841 .98662 51 10 .38871 .40212 .59788 .98659 5O 11 .38921 .40266 .59734 .98656 49 12 .38971 .40319 .59681 .98652 48 13 .39021 .40372 .59628 .98649 47 14 .39071 .40425 .59575 .98646 46 15 .39121 .40478 .59522 .98643 45 16 .39170 .40531 .59469 .98640 44 17 .39220 .40584 .59416 .98636 43 18 .39270 .40636 .59364 .98633 42 19 .39319 .40689 .59311 .98630 41 20 .39369 .40742 .59258 .98627 40 21 .39418 .40795 .59205 .98623 39 22 .39467 .40847 .59153 .98620 38 23 .39517 .40900 .59100 .98617 37 24 .39566 .40952 .59048 .98614 36 25 .39615 .41005 .58995 .98610 35 26 .39.664 .41057 .58943 .98607 34 27 .39713 .41109 .58891 .98604 33 28 .39762 .41161 .58839 .98601 32 29 .39811 .41214 .58786 .98597 31 3O .39860 .41266 .58734 .98594 30 31 .39909 .41318 .58682 .98591 29 32 .39958 .41370 .58630 .98588 28 33 .40006 .41422 .58578 .98584 27 34 .40055 .41474 .58526 .98581 26 35 .40103 .41526 .58474 .98578 25 36 .40152 .41578 .58422 .98574 24 37 .40200 .41629 .58371 .98571 23 38 .40249 .41681 .58319 .98568 22 39 .40297 .41733 .58267 .98565 21 4O .40346 .41784 .58216 .98561 20 41 .40394 .41836 .58164 .98558 19 42 .40442 .41887 .58113 .98555 18 43 .40490 .41939 .58061 .98551 17 44 .40538 .41990 .58010 .98548 16 45 .40586 .42041 .57959 .98545 15 46 .40634 .42093 .57907 .98541 14 47 .40682 .42144 .57856 .98538 13 48 .40730 .42195 .57805 .98535 12 49 .40778 .42246 .57754 .98531 11 50 .40825 .42297 .57703 .98528 1O 51 .40873 .42348 .57652 .98525 9 52 .40921 .42399 .57601 .98521 8 53 .40968 .42450 .57550 .98518 7 54 .41016 .42501 .57499 .98515 6 55 .41063 .42552 .57448 .98511 5 56 .41111 .42603 .57397 .98508 4 57 .41158 .42653 .57347 .98505 3 58 .41205 .42704 .57296 .98501 2 59 .41252 .42755 .57245 .98498 1 60 .41300 .42805 .57195 .98494 O / 9Lcos 9LcotlOLtan 9Lsin / 76' 75 C 15' 16 43 / 9Lsin 9Ltan lOLcot 9Lcos / o .41300 .42805 .57195 .98494 60 I .41347 .42856 .57144 .98491 59 2 .41394 .42906 .57094 .98488 58 3 .41441 .42957 .57043 .98484 57 4 .41488 .43007 .56993 .98481 56 5 .41535 .43057 .56943 .98477 55 6 .41582 .43108 .56892 .98474 54 7 .41628 .43158 .56842 .98471 53 8 .41675 .43208 .56792 .98467 52 9 .41722 .43258 .56742 .98464 51 10 .41768 .43308 .56692 .98460 50 11 ATSIS .43358 .56642 .98457 49 12 .41861 .43408 .56592 .98453 48 13 .41908 .43458 .56542 .98450 47 14 .41954 .43508 .56492 .98447 46 15 .42001 .43558 .56442 .98443 45 16 .42047 .43607 .56393 .98440 44 17 .42093 .43657 .56343 .98436 43 18 .42140 .43707 .56293 .98433 42 19 .42186 .43756 .56244 .98429 41 2O .42232 .43806 .56194 .98426 4O 21 .42278 .43855 .56145 .98422 39 22 .42324 .43905 .56095 .98419 38 23 .42370 .43954 .56046 .98415 37 24 .42416 .44004 .55996 .98412 36 25 .42461 .44053 .55947 .98409 35 26 .42507 .44102 .55898 .98405 34 27 .42553 .44151 .55849 .98402 33 28 .42599 .44201 .55799 .98398 32 29 .42644 .44250 .55750 .98395 31 3O .42690 .44299 .55701 .98391 30 31 .42735 .44348 .55652 .98388 29 32 .42781 .44397 .55603 .98384 28 33 .42-826 .44446 .55554 .98381 27 34 .42872 .44495 .55505 .98377 26 35 .42917 .44544 .55456 .98373 25 36 .42962 .44592 .55408 .98370 24 37 .43008 .44641 .55359 .98366 23 38 .43053 .44690 .55310 .98363 22 39 .43098 .44738 .55262 .98359 21 4O .43143 .44787 .55213 .98356 2O 41 .43188 .44836 .55164 .98352 19 42 .43233 .44884 .55116 .98349 18 43 .43278 .44933 .55067 .98345 17 44 .43323 .44981 .55019 .98342 16 45 .43367 .45029 .54971 .98338 15 46 .43412 .45078 .54922 .98334 14 47 .43457 .45126 .54874 .98331 13 48 .43502 .45174 .54826 .98327 12 49 .43546 .45222 .54778 .98324 11 50 .43591 .45271 .54729 .98320 10 51 .43635 .45319 .54681 .98317 9 52 .43680 .45367 .54633 .98313 8 53 .43724 .45415 .54585 .98309 7 54 .43769 .45463 .54537 .98306 6 55 .43813 .45511 .54489 .98302 5 56 .43857 .45559 .54441 .98299 4 57 .43901 .45606 .54394 .98295 3 58 .43946 .45654 .54346 .98291 2 59 .43990 .45702 .54298 .98288 1 6O .44034 .45750 .54250 .98284 O / 9 L cos 9 L cot 10 L tan 9 L sin / / 9 Lain 9Ltan 10 L cot 9Lcos / .44034 .45 750 .54250 .98284 60 1 .44 078 .45 797 .54203 .98 281 59 2 .44 122 .45 845 .54 155 .98277 58 3 .44 166 .45 892 .54 108 .98 273 57 4 .44 210 .45 940 .54060 .98 270 56 5 .44 253 .45 987 .54013 .98266 55 6 .44 297 .46035 .53 965 .98 262 54 7 .44341 .46082 .53918 .98 259 53 8 .44385 .46 130 .53870 .98255 52 9 .44428 .46 177 .53823 .98251 51 1O .44472 .46224 .53 776 .98248 50 11 .44516 .46271 .53 729 .98244 49 12 .44559 .46319 .53681 .98240 48 13 .44602 .46366 .53 634 .98237 47 14 .44646 .46413 .53587 .98233 46 15 .44 689 .46 460 .53 540 .98 229 45 16 .44 733 .46 507 .53 493 .98 226 44 17 .44 776 .46554 .53 446 .98 222 43 18 .44819 .46601 .53 399 .98 218 42 19 .44862 .46648 .53 352 .98215 41 2O .44905 .46694 .53 306 .98211 40 21 .44 948 .46741 .53259 .98 207 39 22 .44992 .46 788 .53212 .98 204 38 23 .45 035 .46 835 .53165 .98 200 37 24 .45 077 .46881 .53119 .98 196 36 25 .45 120 .46928 .53072 .98 192 35 26 .45 163 .46975 .53 025 .98 189 34 27 .45 206 .47021 .52979 .98 185 33 28 .45 249 .47068 .52932 .98181 32 29 .45 292 .47 114 .52886 .98177 31 30 .45 334 .47 160 .52840 .98174 30 31 .45377 .47 207 .52 793 .98170 29 32 .45 419 .47 253 .52 747 .98 166 28 33 .45 462 .47 299 .52701 .98 162 27 34 .45 504 .47346 .52654 .98159 26 35 .45 547 .47 392 .52608 .98 155 25 36 .45 589 .47438 .52562 .98151 24 37 .45 632 .47 484 .52516 .98 147 23 38 .45 674 .47530 .52470 .98 144 22 39 .45 716 .47576 .52424 .98 140 21 40 .45 758 .47 622 .52378 .98 136 2O 41 .45 801 .47 668 .52332 .98 132 19 42 .45 843 .47714 .52 286 .98 129 18 43 .45 885 .47 760 .52 240 .98 125 17 44 .45 927 .47806 .52 194 .98 121 16 45 .45969 .47 852 .52 148 .98117 15 46 .46011 .47 897 .52 103 .98113 14 47 .46053 .47 943 .52057 .98110 13 48 .46095 .47 989 .52011 .98 106 12 49 .46 136 .48035 .51965 .98 102 11 50 .46178 .48080 .51920 .98098 1O 51 .46 220 .48 126 .51874 .98094 9 52 .46262 .48 171 .51829 .98090 8 53 .46303 .48217 .51783 .98087 7 54 .46345 .48 262 .51 738 .98083 6 55 .46386 .48307 .51693 .98079 5 56 .46428 .48353 .51647 .98075 4 57 .46469 .48398 .51 602 .98071 3 58 .46511 .48443 .51557 .98067 2 59 .46552 .48489 .51511 .98063 1 60 .46594 .48 534 .51466 .98060 O / 9Lcos 9Lcot 1O L tan 91 sin / 73 44 17 18 C / 9Lsin 9Ltan lOLcot 9Lcos / o .46594 .48534 .51466 .98060 6O 1 .46635 .48579 .51421 .98056 59 2 .46676 .48624 .51376 .98052 58 3 .46717 .48669 .51331 .98048 57 4 .46758 .48714 .51286 .98044 56 5 .46800 .48759 .51241 .98040 55 6 .46841 .48804 .51196 .98036 54 7 .46882 .48849 .51151 .98032 53 8 .46923 .48894 .51106 .98029 52 9 .46964 .48939 .51061 .98025 51 10 .47005 .48984 .51016 .98021 5O 11 .47045 .49029 .50971 .98017 49 12 .47086 .49073 .50927 .98013 48 13 .47127 .49118 .50882 .98009 47 14 .47168 .49163 .50837 .98005 46 15 .47209 .49207 .50793 .98001 45 16 .47249 .49252 .50748 .97997 44 17 .47290 .49296 .50704 .97993 43 18 .47330 .49341 .50659 .97989 42 19 .47371 .49385 .50615 .97986 41 20 .47411 .49430 .50570 .97982 40 21 .47452 .49474 .50526 .97978 39 22 .47492 .49519 .50481 .97974 38 23 .47533 .49563 .50437 .97970 37 24 .47573 .49607 .50393 .97966 36 25 .47613 .49652 .50348 .97962 35 26 .47654 .49696 .50304 .97958 34 27 .47694 .49740 .50260 .97954 33 28 .47734 .49784 .50216 .97950 32 29 .47774 .49828 .50172 .97946 31 30 .47814 .49872 .50128 .97942 30 31 .47854 .49916 .50084 .97938 29 32 .47894 .49960 .50040 .97934 28 33 .47934 .50004 .49996 .97930 27 34 .47974 .50048 .49952 .97926 26 35 .48014 .50092 .49908 .97922 25 36 .48054 .50136 .49864 .97918 24 37 .48094 .50180 .49820 .97914 23 38 .48133 .50223 .49777 .97910 22 39 .48173 .50267 .49733 .97906 21 4O .48213 .50311 .49689 .97902 2O 41 .48252 .50355 .49645 .97898 19 42 .48292 .50398 .49602 .97894 18 43 .48332 .50442 .49558 .97890 17 44 .48371 .50485 .49515 .97886 16 45 .48411 .50529 .49471 .97882 15 46 .48450 .50572 .49428 .97878 14 47 .48490 .50616 .49384 .97874 13 48 .48529 .50659 .49341 .97870 12 49 .48568 .50703 .49297 .97866 11 50 .48607 .50746 .49254 .97861 1O 51 .48647 .50789 .49211 .97857 9 52 .48686 .50833 .49167 .97853 8 53 .48725 .50876 .49124 .97849 7 54 .48764 .50919 .49081 .97845 6 55 .48803 .50962 .49038 .97841 5 56 .4S8H2 .51005 .48995 .97837 4 57 .48881 .51048 .48952 .97833 3 58 .48920 .51092 .48908 .97829 2 59 .48959 .51135 .48865 .97825 1 60 .48998 .51 178 .48822 .97821 / 9Lcos 9Lcot lOLtan 9Lsin / / 9Lsin 9Ltan 10 L cot 9Lcos / O .48998 .51178 .48822 .97821 6O 1 .49037 .51221 .48779 .97817 59 2 .49076 .51264 .48736 .97812 58 3 .49115 .51306 .48694 .97808 57 4 .49153 .51349 .48651 .97804 56 5 .49192 .51392 .48608 .97800 55 6 .49231 .51435 .48565 .97796 54 7 .49269 .51478 .48522 .97792 53 8 .49308 .51520 .48480 .97788 52 9 .49347 .51563 .48437 .97784 51 1O .49385 .51606 .48394 .97779 5O 11 .49424 .51648 .48352 .97775 49 12 .49462 .51691 .48309 .97771 48 13 .49500 .51734 .48266 .97767 47 14 .49539 .51776 .48224 .97763 46 15 .49577 .51819 .48181 .97759 45 16 .49615 .51861 .48139 .97754 44 17 .49654 .51903 .48097 .97750 43 18 .49692 .51946 .48054 .97746 42 19 .49730 .51988 .48012 .97742 41 2O .49768 .52031 .47969 .97738 40 21 .49806 .52073 .47927 .97734 39 22 .49844 .52115 .47885 .97729 38 - 23 .49882 .52157 .47843 .97725 37 24 .49920 .52200 .47800 .97721 36 25 .49958 .52242 .47758 .97717 35 26 .49996 .52284 .47716 .97713 34 27 .50034 .52326 .47674 .97708 33 28 .50072 .52368 .47632 .97704 32 29 .50110 .52410 .47590 .97700 31 30 .50148 .52452 .47548 .97696 3O 31 .50185, .52494 .47506 .97691 29 32 .50223 .52536 .47464 .97687 28 33 .50261 .52578 .47422 .97683 27 34 .50298 .52620 .47380 .97679 26 35 .50336 .52661 .47339 .97674 25 36 .50374 .52703 .47297 .97670 24 37 .50411 .52745 .47255 .97666 23 38 .50449 .52787 .47213 .97662 22 39 .50486 .52829 .47171 .97657 21 4O .50523 .52870 .47130 .97653 20 41 .50561 .52912 .47088 .97649 19 42 .50598 .52953 .47047 .97645 18 43 .50635 .52995 .47005 .97540 17 44 .50673 .53037 .46963 .97636 16 45 .50710 .53078 .46922 .97632 15 46 .50747 .53120 .46880 .97628 14 47 .50784 .53161 .46839 .97623 13 48 .50821 .53202 .46798 .97619 12 49 .50858 .53244 .46756 .97615 11 50 .50896 .53285 .46715 .97610 10 51 .50933 .53327 .46673 .97606 9 52 .50970 .53368 .46632 .97602 8 53 .51007 .53409 .46591 .97597 7 54 .51043 .53450 .46550 .97593 6 55 .51080 .53492 .46508 .97589 5 56 .51117 .53533 .46467 .97584 4 57 .51154 .53574 .46426 .97580 3 58 .51191 .53615 .46385 .97576 2 59 .51227 .53656 .46344 .97571 1 60 .51264 .53697 .46303 .97567 O / 9 L cos 9 L cot 1O L tan 9 L sin / 72 C 71 19 20 C 45 / 9 L sin 9 L tan 1O L cot 9 L cos / o .51264 .53697^.46303 .97567 6O 1 .51301 .53738 .46262 .97563 59 2 .51338 .53779 .46221 .97558 58 3 .51374 .53820 .46180 .97554 57 4 .51411 .53861 .46139 .97550 56 5 .51447 .53902 .46098 .97545 55 6 .51484 .53943 .46057 .97541 54 7 .51520 .53984 .46016 .97536 53 8 .51557 .54025 .45975 .97532 52 9 .51593 .54065 .45935 .97528 51 1O .51629 .54106 .45894 .97523 5O 11 .51666 .54147 .45853 .97519 49 12 .51702 .54187 .45813 .97515 48 13 .51738 .54228 .45772 .97510 47 14 .51774 .54269 .45731 .97506 46 15 .51811 .54309 .45691 97501 45 16 .51847 .54350 .45650 .97497 44 17 .51883 .54390 .45610 .97492 43 18 .51919 .54431 .45569 .97488 42 19 .51955 .54471 .45529 .97484 41 2O .51991 .54512 .45488 .97479 4O 21 .52027 .54552 .45448 .97475, 39 22 .52063 .54593 .45407 .97470 38 23 .52099 .54633 .45367 .97466 37 24 .52135 .54673 .45327 .97461 36 25 .52171 .54714 .45286 .97457 35 26 .52207 .54754 .45246 .97453 34 27 .52242 .54794 .45206 .97448 33 28 .52 278 .54 835 .45 165 .97 444 32 29 .52314 .54875 .45125 .97439 31 3O .52350 .54915 .45085 .97435 3Q 31 .52385 .54955 .45045 .97430 29 32 .52421 .54995 .45005 .97426 28 33 .52456 .55035 .44965 .97421 27 34 .52492 .55075 .44925 .97417 26 35 .52527 .55115 .44885 .97412 25 36 .52563 .55155 .44845 .97408 24 37 .52598 .55195 .44805 .97403 23 *3S .52634 .55235 .44765 .97399 22 39 .52669 .55275 .44725 .97394 21 4O .52705 .55315 .44685 .97390 2O 41 .52740 .55355 .44645 .97385 19 42 .52775 .55395 .44605 .97381 18 43 .52811 .55434 .44566 .97376 17 44 .52846 .55474 .44526 .97372 16 45 .52881 .55514 .44486 .97367 15 46 .52916 .55554 .44446 .97363 14 47 .52951 .55593 .44407 .97358 13 48 .52986 .55633 .44367 .97353 12 49 .53021 .55673 .44327 .97349 11 5O .53056 .55712 .44288 .97344 1O 51 .53092 .55752 .44248 .97340 9 52 .53126 .55791 .44209 .97335 8 53 .53161 .55831 .44169 .97331 7 54 .53196 .55870 .44130 .97326 6 55 .53231 .55910 .44090 .97322 5 56 .53266 .55949 .44051 .97317 4 57 .53301 .55989 .44011 .97312 3 58 .53336 .56028 .43972 .97308 2 59 .53370 .56067 .43933 .97303 1 60 .53405 .56107 .43893 .97299 O / 9Lcos 9Lcot lOLtan 9Lsin / / 9Lsin 9Ltan lOLcot 9Lcos / O .53405 .56107 .43893 .97299 6O 1 .53440 .56146 .43854 .97294 59 2 .53475 .56185 .43815 .97289 58 3 .53509 .56224 .43776 .97285 57 4 .53544 .56264 .43736 .97280 56 5 .53578 .56303 .43697 .97276 55 6 .53613 .56342 .43658 .97271 54 7 .53647 .56381 .43619 .97266 53 8 .53682 .56420 .43580 .97262 52 9 .53716 .56459 .43541 '97257 51 1O .53751 .56498 .43502 .97252 5O 11 .53785 .56537 .43463 .97248 49 12 .53819 .56576 .43424 .97243 48 13 .53854 .56615 .43385 .97238 47 14 .53888 .56654 .43346 .97234 46 15 .53922 .56693 .43307 .97.229 45 16 .53957 .56732 .43268 .97224 44 17 .53991 .56771 .43229 .97220 43 18 .54025 .56810 .43190 .97215 42 19 .54059 .56849 .43151 .97210 41 20 .54093 .56887 .43113 .97206 4O 21 .54127 .56926 .43074 .97201 39 22 .54161 .56965 .43035 .97196 38 23 .54195 .57004 .42996 .97192 37 24 .54229 .57042 .42958 .97187 36 25 .54263 .57081 .42919 .97182 35 26 .54297 .57120 .42880 .97178 34 27 .54331 .57158 .42842 .97173 33 28 .54365 .57197 .42803 .97168 32 1 29 .54399 .$7235 .42765 .97163 31 30 .54433 .57274 .42726 .97159 30 31 .54466 .57312 .42688 .97154 29 32 .54500 .57351 .42649 .97149 28 33 .54534 .57389 .42611 .97145 27 34 .54567 .57428 .42572 .97140 26 35 .54601 .57466 .42534 .97135 25 36 .54635 .57504 .42496 .97130 24 37 .54668 .57543 .42457 .97126 23 38 .54702 .57581 .42419 .97121 22 39 .54735 .57619 .42381 .97116 21 4O .54769 .57658 .42342 .97111 2O 41 .54802 .57696 .42304 .97107 19 42 .54836 .57734 .42266 .97102 18 43 .54869 .57772 .42228 .97097 17 44 .54903 .57810 .42190 .97092 16 45 .54936 .57849 .42151 .97087 15 46 .54969 .57887 .42113 .97083 14 47 .55003 .57925 .42075 .97078 13 48 .55036 .57963 .42037 .97073 12 49 .55069 .58001 .41999 .97068 11 50 .55102 .58039 .41961 .97063 10 51 .55136 .58077 .41923 .97059 9 52 .55169 .58115 .41885 .97054 8 53 .55202 .58153 .41847 .97049 7 54 .55235 .58191 .41809 .97044 6 55 .55268 .58229 .41771 .97039 5 56 .55301 .58267 .41733 .97035 4 57 .55334 .58304 .41696 .97030 3 58 .55367 .58342 .41658 .97025 2 59 .55400 .58380 .41620 .97020 1 60 .55433 .58418 .41582 .97015 O / 9Lcos 9 Loot lOLtan 9Lsin / 70 C 69 22' / 9Lsin 9Ltan 1O L cot 9 Lcos / o .55433 .58418 .41 582 .97015 6O 1 .55 466 .58455 .41545 .97010 59 2 .55 499 .58493 .41507 .97 005 58 3 .55 532 .58531 .41 469 .97 001 57 4 .55 564 .58 569 .41431 .96996 56 5 .55 597 .58606 .41 394 .96991 55 6 .55 630 .58644 .41 356 .96986 54 7 .55 663 .58681 .41319 .96981 53 8 .55 695 .58719 .41 281 .96976 52 9 .55 728 .58757 .41 243 .96971 51 10 .55 761 .58 794 .41 206 .96966 5O 11 .55 793 .58832 .41 168 .96962 49 12 .55 826 .58869 .41 131 .96957 48 13 .55 858 .58907 .41 093 .96952 47 14 .55 891 ,58944 .41 056 .96947 46 15 .55.923 .58981 .41 019 .96942 45 16 .55 956 .59019 .40981 .96937 44 17 .55 988 .59056 .40944 .96932 43 18 .56021 .59094 .40906 .96927 42 19 .56053 .59131 .40869 .96922 41 20 .56085 .59168 .40832 .96917 40 21 .56118 .59205 .40 795 .96912 39 22 .56150 .59243 .40757 .96907 38 23 .56182 .59280 .40 720 .96903 37 24 .56215 .59317 .40683 .96898 36 25 .56247 .59354 .40646 .96893 35 26 .56279 .59391 .40 609 .96888 34 27 .56311 .59429 .40571 .96883 33 28 .56343 .59466 .40534 .96878 32 29 .56375 .59503 .40497 .96873 31 30 .56408 .59540 .40460 .96868 30 31 .56440 .59577 .40423 .96 863 29 32 .56472 .59614 .40386 .96 858 28 33 .56504 .59651 .40349 .96853 27 34 .56536 .59688 .40312 .96848 26 35 .56568 .59725 .40275 .96843 25 36 .56 599 .59 762 .40 238 .96838 24 37 .56631 .59 799 .40 201 .96833 23 38 .56663 .59835 .40 165 .96828 22 39 .56695 .59872 .40 128 .96823 21 40 .56727 .59909 .40091 .96818 20 41 .56759 .59946 .40054 .96813 19 42 .56 790 .59983 .40017 .96808 18 43 .56822 .60019 .39981 .96803 17 44 .56854 .60056 .39944 .96 798 16 45 .56886 .60093 .39907 .96793 15 46 .56917 .60 130 .39870 .96 788 14 47 .56949 .60 166 .39834 .96 783 13 48 .56980 .60 203 .39 797 .96778 12 49 .57012 .60240 .39 760 .96772 11 50 .57044 .60276 .39 724 .96 767 10 51 .57075 .60313 .39687 .96 762 9 52 .57107 .60349 .39651 .96757 8 53 .57138 .60386 .39614 .96752 7 54 .57169 .60422 .39578 .96 747 . 6 55 .57201 .60459 .39 541 .96 742 5 56 .57232 .60495 .39505 .96737 4 "57 .57264 .60532 .39468 .96732 3 58 .57295 .60568 .39432 .96 727 2 59 .57326 .60605 .39395 .96 722 1 60 .57358 .60641 .39359 .96 717 O / 9Lcos 9LcotlOLtan9Lsin / / 9Lsin 9Ltan 1O L cot 9 Lcos / O .57358 .60641 .39359 .96717 6O 1 .57389 .60677 .39323 .96711 59 2 .57420 .60 714 .39286 .96 706 58 3 .57451 .60 750 .39250 .96 701 57 4 .57482 .60786 .39 214 .96696 56 5 .57514 .60823 .39177 .96691 55 6 .57545 .60859 .39 141 .96686 54 7 .57576 .60895 .39 105 .96681 53 8 .57607 .60931 .39069 .96676 52 9 .57638 .60967 .39033 .96670 51 1O .57669 .61 004 .38996 .96665 50 11 .57700 .61 040 .38960 .96660 49 12 .57731 .61 076 .38924 .96655 48 13 .57762 .61] 12 .38 888 .96650 47 14 .57793 .61 148 .38852 .96645 46 15 .57824 .61 184 .38816 .96640 45 16 .57855 .61 220 .38 780 .96634 44 17 .57885 .61 256 .38 744 .96629 43 18 .57916 .61 292 .38 708 .96624 42 19 .57947 .61 328 .38672 .96619 41 20 .57978 .61 364 .38636 .96614 40 21 .58008 .61 400 .38 600 .96608 39 22 .58039 .61436 .38 564 .96603 38 23 .58070 .61 472 .38528 .96 598 37 24 .58101 .61 508 .38492 .96593 36 25 .58131 .61 544 .38456 .96 588 35 26 .58 162 .61 579 .38421 .96 582 34 27 .58 192 .61 615 .38385 .96577 33 28 .58223 .61651 .38349 .96572 32 29 .58253 .61 687 .38313 .96567 31 .30 .58284 .61 722 .38278 .96562 30 31 .58314 .61 758 .38 242 .96556 29 32 ,58345 .61 794 .38 206 .96551 28 33 .58375 .61 830 .38 170 .96546 27 34 .58406 .61 865 .38 135 .96541 26 35 .58436 .61901 .38099 .96535 25 36 .58467 .61936 .38064 .96530 24 37 .58497 .61 972 .38028 .96 525 23 38 .58527 .62008 .37992 .96 520 22 39 .58557 .62043 .37957 .96514 21 4O .58588 .62079 .37921 .96509 2O 41 .58618 .62 114 .37886 .96504 19 42 .58648 .62 150 .37850 .96498 18 43 .58678 .62185 .37815 .96493 17 44 .58 709 .62221 .37779 .96488 16 45 .58739 .62256 .37 744 .96483 15 46 .58769 .62 292 .37 708 .96477 14 47 .58 799 .62327 .37673 .96472 13 48 .58829 .62 362 .37638 .96467 12 49 .58859 .62398 .37602 .96461 11 5O .58889 .62433 .37567 .96456 10 51 .58919 .62 468 .37532 .96451 9 52 .58949 .62 504 .37496 .96445 8 53 .58979 .62 539 .37461 .96440 7 54 .59009 .62 574 .37426 .96435 6 55 .59039 .62 609 .37391 .96429 5 56 .59069 .62645 .37355 .96424 4 57 .59098 .62680 .37320 .96419 3 58 .59128 .62715 .37 285 .96413 2 59 .59158 .62 750 .37 250 .96408 1 60 .59188 .62 785 .37215 .96403 / 9 Lcos 9 L cot 1O L tan 9 L sin / 68' 67 23' 47 / 9 L sin 9 L tan 1O L cot 9 L cos / .59188 .62785 .37215 .96403 6O 1 .59218 .62820 .37180 .96397 59 2 .59247 .62855 .37145 .96392 58 3 .59277 .62890 .37110 .96387 57 4 .59307 .62926 .37074 .96381 56 5 .59336 .62961 .37039 .96376 55 6 .59366 .62996 .37004 .96370 54 7 .59396 .63031 .36969 .96365 53 8 .59425 .63066 .36934 .96360 52 9 .59455 .63101 -36899 .96354 51 1C .59484 .63135 .36865 .96349 50 11 .59514 .63170 .36830 .96343 49 12 .59543 .63205 .36795 .96338 48 13 .59573 .63240 .36760 .96333 47 14 .59602 .63275 .36725 .96327 46 15 .59632 .63310 .36690 .96322 45 16 .59661 .63345 .36655 .96316 44 17 .59690 .63379 .36621 .96311 43 18 .59720 .63414 .36586 .96305 42 19 .59749 .63449 .36551 .96300 41 2O .59778 .63484 .36516 .96294 40 21 .59808 .63519 .36481 .96289 39 22 .59837 .63553 .36447 .96284 38 23 .59866 .63588 .36412 .96278 37 24 .59895 .63623 .36377 .96273 36 25 .59924 .63657 .36343 .96267 35 26 .59954 .63692 .36308 .96262 34 27 .59983 .63726 .36274 .96256 33 28 .60012 .63761 .36239 .96251 32 29 .60041 .63796 .36204 .96245 31 30 .60070 .63830 .36170 .96240 30 31 .60099 .63865 .36135 .96234 29 32 .60128 .63899 .36101 .96229 28 33 .60157 .63934 .36066 .96223 27 34 .60186 .63968 .36032 .96218 26 35 .60215 .64003 .35997 .96212 25 36 .60244 .64037 .35963 .96207 24 37 .60273 .64072 .35928 .96201 23 38 .60302 .64106 .35894 .96196 22 39 .60331 .64140 .35860 .96190 21 4O .60359 .64175 .35825 .96185 2O 41 .60388 .64209 .35791 .96179 19 42 .60417 .64243 .35757 .96174 18 43 .60446 .64278 .35722 .96168 17 44 .60474 .64312 .35688 .96162 16 45 .60503 .64346 .35654 .96157 15 46 .60532 .64381 .35619 .96151 14 47 .60561 .64415 .35585 .96146 13 48 .60589 .64449 .35551 .96140 12 49 .60618 .64483 .35517 .96135 11 50 .60646 .64517 .35483 .96129 1O 51 .60675 .64552 .35448 .96123 9 52 .60704 .64586 .35414 .96118 8 53 .60732 .64620 .35380 .96112 7 54 .60761 .64654 .35346 .96107 6 55 .60789 .64688 .35312 .96101 5 56 .60818 .64722 .35278 .96095 4 57 .60846 .64756 .35244 .96090 3 58 .60875 .64790 .35210 .96084 2 59 .60903 .64824 .35176 .96079 1 60 .60931 .64858 .35142 .96073 O / 9Lcos 91 cot lOLtan 9Lsin / / 9Lsin 9Ltan lOLcot 9Lcos / O .60931 .64858 .35 142 .96073 6O 1 .60960 .64892 .35 108 .96067 59 2 .60988 .64 926 .35 074 .96062 58 3 .61016 .64960 .35 040 .96056 57 4 .61 045 .64994 .35 006 .96050 56 5 .61 073 .65 028 .34972 .96045 55 6 .61 101 .65 062 .34938 .96039 54 7 .61 129 .65096 .34904 96034 53 8 .61 158 .65 130 .3^870 .96028 52 9 .61 186 .65 164 .34836 .96022 51 1O .61 214 .65 197 .34803 .96017 5O 11 .61 242 .65 231 .34 769 .96011 49 12 .61 270 .65 265 .34735 .96005 48 13 .61 298 .65 299 .34 701 .96000 47 14 .61 326 .65333 .34667 .95 994 46 15 .61 354 .65 366 .34634 .95988 45 16 .61 382 .65 400 .34600 .95 982 44 17 .61411 .65 434 .34 566 .95 977 43 18 .61 438 .65 467 .34 533 .95971 42 19 .61 466 .65 501 .34499 .95 965 41 2O .61 494 .65 535 .34465 .95 960 40 21 .61 522 .65 568 .34 432 .95 954 39 22 .61 550 .65 602 .34398 .95 948 38 23 .61 578 .65 636 .34364 .95 942 37 24 .61 606 .6T669 .34331 .95937 36 25 .61 634 .65 703 .34297 .95931 35 26 .61 662 .65 736 .34 264 .95 925 34 27 .61 689 .65 770 .34 230 .95920 33 28 .61 717 .65 803 .34 197 .95 914 32 29 .61 745 .65 837 .34163 .95 908 31 30 .61 773 .65 870 .34 130 .95 902 30 31 .61 800 .65 904 .34096 .95 897 29 32 .61 828 .65937 .34063 .95 891 28 33 .61 856 .65 971 .34 029 .95 885 27 34 .61 883 .66004 .33 996 .95 879 26 35 .61911 .66038 .33 962 .95 873 25 35 .61 939 .66071 .33 929 .95 868 24 37 .61 966 .66104 .33 896 .95 862 23 38 .61 994 .66138 .33862 .95 856 22 39 .62021 .66171 .33 829 .95 850 21 40 .62049 .66204 .33 796 .95 844 2O 41 .62 076 .66238 .33 762 .95 839 19 42 .62 104 .66271 .33 729 .95 833 18 43, .62 131 .66304 .33 696 .95 827 17 ^ 44 .62 159 .66337 .33663 .95 821 16 45 .62 186 .66371 .33 629 .95 815 15 46 .62 214 .66404 .33 596 .95 810 14 47 .62 241 .66437 .33 563 .95 804 13 48 .62 268 .66470 .33 530 .95 798 12 49 .62 296 .66503 .33497 .95 792 11 5O .62323 .66537 .33 463 .95 786 1O 51 .62 350 .66570 .33 430 .95 780 9 52 .62377 .66603 .33397 .95 775 8 53 .62 405 .66636 .33 364 .95 769 7 54 .62432 .66669 .33331 .95 763 6 55 .62 459 .66 702 .33 298 .95 757 5 56 .62 486 .66 735 .33 265 .95 751 4 57 .62513 .66 768 .33 232 .95 745 3 58 .62 541 .66801 .33 199 .95 739 2 59 .62 568 .66834 .33 166 .95 733 ' 1 60 .62595 .66867 .33 133 .95 728 O / 9Lcos 9 Loot 10 L tan 9 L sin / 66 65 C 48 25 26 / DLsin 9Ltan lOLcot 9Lcos / O .62595 .66867 .33133 .95728 6O I .62622 .66900 .33100 .95722 59 2 .62649 .66933 .33067 .95716 58 3 .62676 .66966 .33034 .95710 57 4 .62703 .66999 .33001 .95704 56 5 .62730 .67032 .32968 .95698 55 6 .62757 .67065 .32935 .95692 54 7 .62784 .67098 .32902 .95686 53 8 .62811 .67131 .32869 .95680 52 9 .62838 .67163 .32837 .95674 51 10 .62865 .67196 .32804 .95668 50 11 .62892 .67229 .32771 .95663 49 12 .62918 .67262 .32738 .95657 48 13 .62945 .67295 .32705 .95651 47 14 .62972 .67327 .32673 .95645 46 15 .62999 .67360 .32640 .95639 45 16 .63026 .67393 .32607 .95633 44 17 .63052 .67426 .32574 .95627 43 18 .63079 .67458 .32542 .95621 42 19 .63106 .67491 .32509 .95615 41 2O .63133 .67524 .32476 .95609 40 21 .63159 .67556 .32444 .95603 39 22 .63186 .67589 .32411 .95597 38 23 .63213 .67622 .32378 .95591 37 24 .63239 .67654 .32346 .95585 36 25 .63266 .67687 .32313 .95579 35 26 .63292 .67719 .32281 .95573 34 27 .63319 .67752 .32248 .95567 33 28 .63345 .67785 .32215 .95561 32 29 .63372 .67817 .32183 .95555 31 30 .63398 .67850 .32150 .95549 30 31 .63425 .67882 .32118 .95543 29 32 .63451 .67915 .32085 .95537 28 33 .63478 .67947 .32053 .95531 27 34 .63504 .67980 .32020 .95525 26 35 .63531 .68012 .31988 .95519 25 36 .63557 .68044 .31956 .95513 '24 37 .63583 .68077 .31923 .95507 23 38 .63610 .68109 .31891 .95500 22 39 .63636 .68142 .31858 .95494 21 40 .63662 .68174 .31826 .95488 2O 41 .63689 .68206 .31794 .95482 19 42 .63715 .68239 .31761 .95476 18 43 .63741 .68271 .31729 .95470 17 44 .63767 .68303 .31697 .95464 16 45 .63794 .68336 .31664 .95458 15 46 .63820 .68368 .31632 .95452 14 47 .63846 .68400 .31600 .95446 13 48 .63872 .68432 .31568 .95440 12 49 .63898 .68465 .31535 .95434 11 5O .63924 .68497 .31503 .95427 10 51 .63950 .68529 .31471 .95421 9 52 .63976 .68561 .31439 .95415 8 53 .64002 .68593 .31407 .95409 7 54 .64028 .68626 .31374 .95403 6 55 .64054 .68658 .31342 .95397 5 56 .64080 .68690 .31310 .95391 4 57 .64106 .68722 .31278 .95384 3 58 .64132 .68754 .31246 .95378 2 59 .64158 .68786 .31214 .95372 1 6O .64184 .68818 .31182 .95366 O / 9 L cos 9 L cot 10 L tan 9 L sin / / 9Lsin 9Ltan 1O L cot 9 L cos / O .64 184 .68818 .31 182 .95 366 6O 1 .64 210 .68850 .31150 .95 360 59 2 .64 236 .68882 .31118 .95 354 58 3 .64 262 .68914 .31086 .95 348 57 4 .64 288 .68946 .31 054 .95 341 56 5 .64313 .68978 .31022 .95 335 55 6 .64339 .69010 .30990 .95 329 54 7 .64365 .69042 .30958 .95 323 53 8 .64391 .69074 .30926 .95317 52 9 .64417 .69 106 .30894 .95310 51 10 .64442 .69 138 .30862 .95 304 5O 11 .64 468 .69 170 .30830 .95 298 49 12 .64494 .69 202 .30 798 .95 292 48 13 .64519 .69 234 .30 766 .95 286 47 14 .64545 .69266 .30 734 .95 279 46 15 .64571 .69 298 .30 702 .95 273 45 16 .64 596 .69329 .30671 .95 267 44 17 .64622 .69361 .30639 .95 261 43 18 .64647 .69393 .30607 .95 254 42 19 .64673 .69425 .30575 .95 248 41 2O .64698 .69457 .30 543 .95 242 4O 21 .64 724 .69488 .30512 .95 236 39 22 .64 749 .69520 .30480 .95 229 38 23 .64 775 .69552 .30448 .95 223 37 24 .64800 .69 584 .30416 .95217 36 25 .64826 .69615 .30385 .95211 35 26 .64851 .69647 .30353 .95 204 34 27 .64877 .69 679 .30321 .95 198 33 28 .64902 69710 .30290 .95 192 32 29 .64927 .69 742 .30258 .95 185 31 3O .64953 .69 774 .30226 .95 179 3O 31 .64978 .69805 .30 195 .95 173 29 32 .65 003 .69837 .30163 .95 167 28 33 .65 029 .69868 .30 132 .95 160 27 34 .65 054 .69900 .30 100 .95 154 26 35 .65 079 .69932 .30068 .95 148 25 36 .65 104 .69963 .30037 .95141 24 37 .65 130 .69995 .30005 .95 135 23 38 .65 155 .70026 .29974 .95 129 22 39 .65 180 .70058 .29942 .95 122 21 40 .65 205 .70089 .29911 .95116 20 41 .65 230 .70121 .29879 .95 110 19 42 .65 255 .70152 .29 848 .95 103 18 43 .65 281 .70184 .29816 .95 097 17 44 .65 306 .70215 .29 785 .95 090 16 45 .65331 .70247 .29 753 .95 084 15 46 .65 356 .70278 .29 722 .95 078 14 47 .65 381 .70309 .29691 .95 071 13 48 .65 406 .70341 .29 659 .95 065 12 49 .65431 .70372 .29628 .95 059 11 50 .65 456 .70404 .29596 .95 052 1O 51 .65 481 .70435 .29565 .95 046 9 52 .65 506 .70466 .29 534 .95 039 8 53 .65 531 .70498 .29 502 .95 033 7 54 .65 556 .70 529 .29471 .95 027 6 55 .65 580 .70560 .29440 .95 020 5 56 .65 605 .70 592 .29408 .95 014 4 57 .65 630 .70623 .29377 .95 007 3 58 .65 655 .70654 .29346 .95 001 2 59 .65 680 .70685 .29315 .94995 1 60 .65 705 .70717 .29283 .94988 O / 9Lcos 9Lcot 10 L tan 9Lsin / 64 C 63 C 27'- 28 C 49 / 9 L sin 9 L tan 10 L cot 9 L cos / o .65705 .70717 .29283 .94988 00 1 .65729 .70748 .29252 .94982 59 2 .65754 .70779 .29221 .94975 58 3 .65779 .70810 .29190 .94969 57 4 .65804 .70841 .29159 .94962 56 5 .65828 .70873 .29127 .94956 55 6 .65853 .70904 .29096 .94949 54 7 .65878 .70935 .29065 .94943 53 8 .65902 .70966 .29034 .94936 52 9 .65927 .70997 .29003 .94930 51 1C .65952 .71028 .28972 .94923 50 11 .65976 .71059 .28941 .94917 49 12 .66001 .71090 .28910 .94911 48 13 .66025 .71121 .28879 .94904 47 14 .66050 .71153 .28847 .94898 46 15 .66075 .71184 .28816 .94891 45 16 .66099 .71215 .28785 .94885 44 17 .66124 .71246 .28754 .94878 43 18 .66148 .71277 .28723 .94871 42 19 .66173 .71308 .28692 .94865 41 20 .66197 .71339 .28661 .94858 4O 21 .66221 .71370 .28630 .94852 39 22 .66246 .71401 .28599 .94845 38 23 .66270 .71431 .28569 .94839 37 24 .66295 .71462 .28538 .94832 36 25 .66319 .71493 .28507 .94826 35 26 .66343 .71524 .28476 .94819 34 27 .66368 .71555 .28445 .94813 33 28 .66392 .71586 .28414 .94806 32 29 .66416 .71617 .28383 .94799 31 SO .66441 .71648 .28352 .94793 3O 3H.66465 .71679 .28321 .94786 29 32 .66489 .71709 .28291 .94780 28 33 .66513 .71740 .28260 .94773 27 34 .66537 .71771 .28229 .94767 26 35 .66562 .71802 .28198 .94760 25 36 .66586 .71833 .28167 .94753 24 37 .66610 .71863 .28137 .94747 23 38 .66634 .71894 .28106 .94740 22 39 .66658 .71925 .28075 .94734 21 4O .66682 .71955 .28045 .94727 2O 41 .66706 .71986 .28014 .94720 19 42 .66731 .72017 .27983 .94714 18 43 .66755 .72048 .27952 .94707 17 44 .66779 .72078 .27922 .94700 16 45 .66803 .72109 .27891 .94694 15 46 .66827 .72140 .27860 .94687 14 47 .66851 .72170 .27830 .94680 13 48 .66875 .72201 .27799 .94674 12 49 .66899 .72231 .27769 .94667 11 50 .66922 .72262 .27738 .94660 1O 51 .66946 .72293 .27707 .94654 9 52 .66970 .72323 .27677 .94647 8 53 .66994 .72354 .27646 .94640 7 54 .67018 .72384 .27616 .94634 6 55 .67042 .72415 .27585 .94627 5 56 .67066 .72445 .27555 .94620 4 57 .67090 .72476 .27524 .94614 3 58 .67113 .72506 .27494 .94607 2 59 .67137 .72537 .27463 .94600 1 6O .67161 .72567 .27433 .94593 O / 9Lcos 9Lcot 10 L tan 9Lsin / / 9Lsin 9Ltan lOLcot 9Lcos / .67161 .72567 .27433 .94593 60 1 .67185 .72598 .27402 .94587 59 2 .67208 .72628 .27372 .94580 58 3 .67232 .72659 .27341 .94573 57 4 .67256 .72689 .27311 .94567 56 5 .67280 .72720 .27280 .'94560 55 6 .67303 .72750 .27250 .94553 54 7 .67327 .72780 .27220 .94546 53 8 .67350 .72811 .27189 .94540 52 9 .67374 .72841 .27159 .94533 51 1O .67398 .72872 .27128 .94526 5O 11 .67421 .72902 .27098 .94519 49 12 .67445 .72932 .27068 .94513 48 13 .67468 .72963 .27037 .94506 47 14 .67492 .72993 .27007 .94499 46 15 .67515 .73023 .26977 .94492 45 16 .67539 .73054 .26946 .94485 44 17 .67562 .73084 .26916 .94479 43 18 .67586 .73114 .26886 .94472 42 19 .67609 .73144 .26856 .94465 41 2O .67633 .73175 .26825 .94458 4O 21 .67656 .73205 .26795 .94451 39 22 .67680 .73235 .26765 .94445 38 23 .67703 .73265 .26735 .94438 37 24 .67726 .73295 .26705 .94431 36 25 .67750 .73326 .26674 .94424 35 26 .67773 .73356 .26644 .94417 34 27 .67796 .73386 .26614 .94410 33 28 .67820 .73416 .26584 .94404 32 29 .67843 .73446 .26554 .94397 31 3O .67866 .73476 .26524 .94390 30 31 .67890 .73507 .26493 .94383 29 32 .67913 .73537 .26463 .94376 28 33 .67936 .73567 .26433 .94369 27 34 -.67959 .73597 .26403 .94362 26 35 .67982 .73627 .26373 .94355 25 36 .68006 .73657 .26343 .94349 24 37 .68029 .73687 .26313 .94342 23 38 .68052 .73717 .26283 .94335 22 39 .68075 .73747 .26253 .94328 21 40 .68098 .73777 .26223 .94321 2O 41 .68121 .73807 .26193 .94314 19 42 .68144 .73837 .26163 .94307 18 43 .68167 .73867 .26133 .94300 17 44 .68190 .73897 .26103 .94293 16 45 .68213 .73927 .26073 .94286 15 46 .63237 .73957 .26043 .94279 14 47 .68260 .73987 .26013 .94273 13 48 .68283 .74017 .25983 .94266 12 49 .68305 .74047 .25953 .94259 11 5O .68328 .74077 .25923 .94252 10 51 .68351 .74107 .25893 .94245 9 52 .68374 .74137 .25863 .94238 8 53 .68397 .74166 .25834 .94231 7 54 .68420 .74196 .25804 .94224 6 55 .68443 .74226 .25774 .94217 5 56 .68466 .74256 .25744 .94210 4 57 .68489 .74286 .25714 .94203 3 58 .68512 .74316 .25684 .94196 2 59 .68534 .74345 .25655 .94189 1 6O .68557 .74375 .25625 .94182 O / 9Lcos 9Lcot lOLtan 9Lsin / 62' 61 50 29 30< / 9 L sin 9 L tan 1O L cot 9 L cos / .68557 .74375 .25625 .94182 60 1 .68580 .74405 .25595 .94175 59 2 .68603 .74435 .25565 .94168 58 3 .68625 .74465 .25535 .94161 57 4 .68648 .74494 .25506 .94154 56 5 .68671 .74524 .25476 .94147 55 6 .68694 .74554 .25446 .94140 54 7 .68716 .74583 .25417 .94133 53 8 .68739 .74613 .25387 .94126 52 9 .68762 .74643 .25357 .94119 51 10 .68784 .74673 .25327 .94112 50 11 .68807 .74702 .25298 .94105 49 12 .68829 .74732 .25268 .94098 48 13 .68852 .74762 .25238 .94090 47 14 .68875 .74791 .25209 .94083 46 15 .68897 .74821 .25179 .94076 45 16 .68920 .74851 .25149 .94069 44 17 .68942 .74880 .25120 .94062 43 18 .68965 .74910 .25090 .94055 42 19 .68987 .74939 .25061 .94048 41 20 .69010 .74969 .25031 .94041 4O 21 .69032 .74998 .25002 .94034 39 22 .69055 .75028 .24972 .94027 38 23 .69077 .75058 .24942 .94020 37 24 .69100 .75087 .24913 .94012 36 25 .69122 .75117 .24883 .94005 35 26 .69144 .75146 .24854 .93998 34 27 .69167 .75176 .24824 .93991 33 28 .69189 .75205 .24795 .93984 32 29 .69212 .75235 .24765 .93977 31 30 .69234 .75264 .24736 .93970 30 31 .69256 .75294 .24706 .93963 29 32 .69279 .75323 .24677 .93955 28 33 .69301 .75353 .24647 .93948 27 34 .69323 .75382 .24618 .93941 26 35 .69345 .75411 .24589 .93934 25 36 .69368 .75441 .24559 .93927 24 37 .69390 .75470 .24530 .93920 23 38 .69412 .75500 .24500 .93912 22 39 .69434 .75529 .24471 .93905 21 40 .69456 .75558 .24442 .93898 20 41 .69479 .75588 .24412 .93891 19 42 .69501 .75617 .24383 .93884 18 43 .69523 .75647 .24353 .93876 17 44 .69545 .75676 .24324 .93869 16 45 .69567 .75705 .24295 .93862 15 46 .69589 .75735 .24265 .93855 14 47 .69611 .75764 .24236 .93847 13 48 .69633 .75793 .24207 .93840 12 49 .69655 .75822 .24178 .93833 11 5O .69677 .75852 .24148 .93826 1O 51 .69699 .75881 .24119 .93819 9 52 .69721 .75910 .24090 .93811 8 53 .69743 .75939 .24061 .93804 7 54 .69765 .75969 .24031 .93797 6 55 .69787 .75998 .24002 .93789 5 56 .69809 .76027 .23973 .93782 4 57 .69831 .76056 .23944 .93775 3 58 .69853 .76086 .23914 .93768 2 59 .69875 .76115 .23885 .93760 1 6O '.69897 .76144 .23856 .93753 O / 9 L cos 9 L cot 1O L tan 9 L sin / / 9 L sin 9 L tan 1O L cot 9 L cos / O .69897 .76144 .23856 .93753 6O 1 .69919 .76173 .23827 .93746 59 2 .69941 .76202 .23798 .93738 58 3 .69963 .76231 .23769 .93731 57 4 .69984 .76261 .23739 .93724 56 5 .70006 .76290 .23710 .93717 55 6 .70028 .76319 .23681 .93709 54 7 .70050 .76348 .23652 .93702 53 8 .70072 .76377 .23623 .93695 52 9 .70093 .76406 .23594 .93687 51 1O .70115 .76435 .23565 .93680 5O 11 .70137 .76464 .23536 .93673 49 12 .70159 .76493 .23507 .93665 48 13 .70180 .76522 .23478 .93658 47 14 .70202 .76551 .23449 .93650 46 15 .70224 .76580 .23420 .93643 45 16 .70245 .76609 .23391 .93636 44 17 .70267 .76639 .23361 .93628 43 18 .70288 .76668 .23332 .93621 42 19 .70310 .76697 .23303 .93614 41 20 .70332 .76725 .23275 .93606 40 21 .70353 .76754 .23246 .93599 39 22 .70375 .76783 .23217 .93591 36 23 .70396 .76812 .23188 .93584 37 24 .70418 .76841 .23159 .93577 36 25 .70439 .76870 .23130 .93569 35 26 .70461 .76899 .23101 .93562 34 27 .70482 .76928 .23072 .93554 33 28 .70504 .76957 .23043 .93547 32 29 .70525 .76986 .23014 .93539 31 30 .70547 .77015 .22985 .93532 30 31 .70568 .77044 .22956 .93525 29 32 .70590 .77073 .22927 .93517 28 33 .70611 .77101 .22899 .93510 27 34 .70633 .77130 .22870 .93502 26 35 .70654 .77159 .22841 .93495 25 36 .70675 .77188 .22812 .93487 24 37 .70697 .77217 .22783 .93480 23 38 .70718 .77246 .22754 .93472 22 39 .70739 .77274 .22726 .93465 21 4O .70761 .77303 .22697 .93457 2O 41 .70782 .77332 .22668 .93450 19 42 .70803 .77361 .22639 .93442 18 43 .70824 .77390 .22610 .93435 17 44 .70846 .77418 .22582 .93427 16 45 .70867 .77447 .22553 .93420 15 46 .70888 .77476 .22524 .93412 14 47 .70909 .77505 .22495 .93405 13 48 .70931 .77533 .22467 .93397 12 49 .70952 .77562 .22438 .93390 11 50 .70973 .77591 .22409 .93382 1O 51 .70994 .77619 .22381 .93375 9 52 .71015 .77648 .22352 .93367 8 53 .71036 .77677 .22323 .93360 7 54 .71058 .77706 .22294 .93352 6 55 .71079 .77734 .22266 .93344 5 56 .71100 .77763 .22237 .93337 4 57 .71121 .77791 .22209 .93329 3 58 .71142 .77820 .22180 .93322 2 59 .71163 .77849 .22151 .93314 1 60 .71184 .77877 .22123 .93307 O / 9 L cos 9 L cot 1O L tan 9 L sin / 60 59 32' 51 / 9Lsin 9Ltan lOLcot9Lcos / o .71184 .77877 .22123 .93307 7*0^ 1 .71205 .77906 .22094 .93299 59 2 .71226 .77935 .22065 .93291 58 3 .71247 .77963 .22037 .93284 57 4 .71268 .77992 .22008 .93276 56 5 .71289 .78020 .21980 .93269 55 6 .71310 .78049 .21951 .93261 54 7 .71331 .78077 .21923 .93253 53 8 .71352 .78106 .21894 .93246 52 9 .71373 .78135 .21865 .93238 51 1O .71393 .78163 .21837 .93230 5O 11 .71414 .78192 .21808 .93223 49 12 .71435 .78220 .21780 .93215 48 13 .71456 .78249 .21751 .93207 47 14 .71477 .78277 .21723 .93200 46 15 .71498 .78306 .21694 .93192 45 16 .71519 .78334 .21666 .93184 44 17 .71539 .78363 .21637 .93177 43 18 .71560 .78391 .21609 .93169 42 19 .71581 .78419 .21581 .93161 41 20 .71602 .78448 .21552 .93154 40 21 .71622 .78476 .21524 .93146 39 22 .71643 .78505 .21495 .93138 38 23 .71664 .78533 .21467 .93131 37 24 .71685 .78562 .21438 .93123 36 25 .71705 .78590 .21410 .93115 35 26 .71726 .78618 .21382 .93108 34 27 .71747 .78647 .21353 .93100 33 28 .71767 .78675 .21325 .93092 32 29 .71788 .78704 .21296 .93084 31 3O .71809 .78732 .21268 .93077 3O 31 .71829 .78760 .21240 .93069 29 32 .71850 .78789 .21211 .93061 28 33 .71870 .78817 .21183 .93053 27 34 .71891 .78845 .21155 .93046 26 35 .71911 .78874 .21126 .93038 25 36 .71932 .78902 .21098 .93030 24 37 .71952 .78930 .21070 .93022 23 38 .71973 .78959 .21041 .93014 22 39 .71994 .78987 .21013 .93007 21 40 .72014 .79015 .20985 .92999 2O 41 .72034 .79043 .20957 .92991 19 42 .72055 .79072 .20928 .92983 18 43 .72075 .79100 .20900 .92976 17 44 .72096 .79128 .20872 .92968 16 45 .72116 .79156 .20844 .92960 15 46 .72137 .79185 .20815 .92952 14 47 .72157 .79213 .20787 .92944 13 48 .72177 .79241 .20759 .92936 12 49 .72198 .79269 .20731 .92929 11 50 .72218 .79297 .20703 .92921 1O 51 .72238 .79326 .20674 .92913 9 52 .72259 .79354 .20646 .92905 8 53 .72279 .79382 .20618 .92897 7 54 .72299 .79410 .20590 .92889 6 55 .72320 .79438 .20562 .92881 5 56 .72340 .79466 .20534 .92874 4 57 .72360 .79495 .20505 .92866 3 58 .72381 .79523 .20477 .92858 2 59 .72401 .79551 .20449 .92850 1 60 .72421 .79579 .20421 .92842 / 9Lcos 9Lcot lOLtan 9Lsin / / 9Lsin 9Ltan lOLcot 9Lcos / O .72421 .79579 .20421 .92842 6O 1 .72441 .79607 .20393 .92834 59 2 .72461 .79635 .20365 .92826 58 3 .72482 .79663 .20337 .92818 57 4 .72502 .79691 .20309 .92810 56 5 .72522 .79719 .20281 .92803 55 6 .72542 .79747 .20253 .92795 54 7 .72562 .79776 .20224 .92787 53 8 .72582 .79804 .20196 .92779 52 9 .72602 .79832 .20168 .92771 51 10 .72 622 .79 860 .20 140 ,.92 763 5O 11 .72643 .79888 .20112 .92755 49 12 .72663 .79916 .20084 .92747 48 13 .72683 .79944 .20056 .92739 47 14 .72703 .79972 .20028 .92731 46 15 .72723 .80000 .20000 .92723 45 16 .72743 .80028 .19972 .92715 44 17 .72763 .80056 .19944 .92707 43 18 .72783 .80084 .19916 .92699 42 19 .72803 .80112 .19888 .92691 41 2O .72823 .80140 .19860 .92683 40 21 .72843 .80168 .19832 .92675 39 22 .72863 .80195 .19805 .92667 38 23 .72883 .80223 .19777 .92659 37 24 .72902 .80251 .19749 .92651 36 25 .72922 .80279 .19721 .92643 35 26 .72942 .80307 .19693 .92635 34 27 .72962 .80335 .19665 .92627 33 28 .72982 .80363 .19637 .92619 32 29 .73002 .80391 .19609 .92611 31 3O .73022 .80419 .19581 .92603 3O 31 .73041 .80447 .19553 .92595 29 32 .73061 .80474 .19526 .92587 28 33 .73081 .80502 .19498 .92579 27 34 .73101 .80530 .19470 .92571 26 35 .73121 .80558 .19442 .92563 25 36 .73140 .80586 .19414 .92555 24 37 .73160 .80614 .19386 .92546 23 38 .73180 .80642 .19358 .92538 22 39 .73200 .80669 .19331 .92530 21 4O .73219 .80697 .19303 .92522 2O 41 .73239 .80725 .19275 .92514 19 42 .73259 .80753 .19247 .92506 18 43 .73278 .80781 .19219 .92498 17 44 .73298 .80808 .19192 .92490 16 45 .73318 .80836 .19164 .92482 15 46 .73337 .80864 .19136 .92473 14 47 .73357 .80892 .19108 .92465 13 48 .73377 .80919 .19081 .92457 12 49 .73396 .80947 .19053 .92449 11 5O .73416 .80975 .19025 .92441 1O 51 .73435 .81003 .18997 .92433 9 52 .73455 .81030 .18970 .92425 8 53 .73474 .81058 .18942 .92416 7 54 .73494 .81086 .18914 .92408 6 55 .73513 .81113 .18887 .92400 5 56 .73533 .81141 .18859 .92392 4 57 .73552 .81169 .18831 .92384 3 58 .73572 .81196 .18804 .92376 2 59 .73591 .81224 .18776 .92367 1 60 .73611 .81252 .18748 .92359 O / 9 L cos 9 L cot 1O L tan 9 L sin / 58 C 57 C 52 33 34' / 9Lsin 9Ltan lOLcot 9Lcos / o .73611 .81252 .18748 .92359 60 1 .73630 .81279 .18721 .92351 59 2 .73650 .81307 .18693 .92313 58 3 .73669 .81335 .18665 .92335 57 4 .73689 .81362 .18638 .92326 56 5 .73708 .81390 .18610 .92318 55 6 .73727 .81418 .18582 .92310 54 7 .73747 .81445 .18555 .92302 53 8 .73766 .81473 .18527 .92293 52 9 .73785 .81500 .18500 .92285 51 10 .73805 .81528 .18472 .92277 5O 11 .73824 .81556 .18444 .92269 49 12 .73843 .81583 .18417 .92260 48 13 .73863 .81611 .18389 .92252 47 14 .73882 .81638 .18362 .92244 46 15 .73901 .81666 .18334 .92235 45 16 .73921 .81693 .18307 .92227 44 17 .73940 .81721 .18279 .92219 43 18 .73959 .81748 .18252 .92211 42 19 .73978 .81776 .18224 .92202 41 20 .73997 .81803 .18197 .92194 4O 21 .74017 .81831 .18169 .92186 39 22 .74036 .81858 .18142 .92177 38 23 .74055 .81886 .18114 .92169 37 24 .74074 .81913 .18087 .92161 36 25 .74093 .81941 .18059 .92152 35 26 .74113 .81968 .18032 .92144 34 27 .74132 .81996 .18004 .92136 33 28 .74151 .82023 .17977 .92127 32 29 .74170 .82051 .17949 .92119 31 3O .74189 .82078 .17922 .92111 30 31 .74208 .82106 .17894 .92102 29 32 .74227 .82133 .17867 .92094 28 33 .74246 .82161 .17839 .92086 27 34 .74265 .82188 .17812 .92077 26 35 .74284 .82215 .17785 .92069 25 36 .74303 .82243 .17757 .92060 24 37 .74322 .82270 .17730 .92052 23 38 .74341 .82298 .17702 .92044 22 39 .74360 .82325 .17675 .92035 21 4O .74379 .82352 .17648 .92027 20 41 .74398 .82380 .17620 .92018 19 42 .74417 .82407 .17593 .92010 18 43 .74436 .82435 .17565 .92002 17 44 .74455 .82462 .17538 .91993 16 45 .74474 .82489 .17511 .91985 15 46 .74493 .82517 .17483 .91976 14 47 .74512 .82544 .17456 .91968 13 48 .74531 .82571 .17429 .91959 12 49 .74549 .82599 .17401 .91951 11 5O .74568 .82626 .17374 .91942 1O 51 .74587 .82653 .17347 .91934 9 52 .74606 .82681 .17319 .91925 8 53 .74625 .82708 .17292 .91917 7 54 .74644 .82735 .17265 .91908 6| 55 .74662 .82762 .17238 .91900 5 56 .74681 .82790 .17210 .91891 4 57 .74700 .82817 .17183 .91883 3 58 .74719 .82844 .17156 .91874 2 59 .74737 .82871 .17129 .91866 1 60 .74756 .82899 .17101 .91857 O / 9Lcos 9LcotlOLtan9Lsin / / 9 L sin 9 L tan 1O L cot 9 L cos / O .74756 .82899 .17101 .91857' 60 1 .74775 .82926 .17074 .91849 59 2 .74794 .82953 .17047 .91840 58 3 .74812 .82980 .17020 .91832 57 4 .74831 .83008 .16992 .91823 56 5 .74850 .83035 .16965 .91815 55 6 .74868 .83062 .16938 .91806 54 7 .74887 .83089 .16911 .91798 53 8 .74906 .83117 .16883 .91789 52 9 .74924 .83144 .16856 .91781 51 1O .749-13 .83171 .16829 .91772 50 11 .74961 .83198 .16802 .91763 49 12 .74980 .83225 .16775 .91755 48 13 .74999 .83252 .16748 .91746 47 14 .75017 .83280 .16720 .91738 46 15 .75036 .83307 .16693 .91729 45 16 .75054 .83334 .16666 .91720 44 17 .75073 .83361 .16639 .91712 43 18 .75091 .83388 .16612 .91703 42 19 .75110 .83415 .16585 .91695 41 2O .75128 .83442 .16558 .91686 4O 21 .75 147 .83470 .16530 .91677 39 22 .75165 .83497 .16503 .91669 38 23 .75184 .83524.16476 .91660 37 24 .75202 .83551 .16449 .91651 36 25 .75221 .83578 .16422 .91643 35 26 .75239 .83605 .16395 .91634 34 27 .75258 .83632 .16368 .91625 33 28 .75276 .83659 .16341 .91617 32 29 .7-5294 .83686 .16314 .91608 31 30 .75313 .83713 .16287 .91599 30 31 .75331 .83740 .16260 .91591 29 32 .75350 .83768 .16232 .91582 28 33 .75368 .83795 .16205 .91573 27 34 .75386 .83822 .16178 .91565 26 35 .75405 .83849 .16151 .91556 25 36 .75423 .83876 .16124 .91547 24 37 .75441 .83903 .16097 .91538 23 38 .75459 .83930 .16070 .91530 22 39 .75478 .83957 .16043 .91521 21 4O .75496 .83984 .16016 .91512 2O 41 .75514 .84011 .15989 .91504 19 42 .75533 .84038 .15962 .91495 18 43 .75551 .84065 .15935 .91486 17 44 .75569 .84092 .15908 .91477 16 45 .75587 .84119 .15881 .91469 15 46 .75605 .84146 .15854 .91460 14 47 .75624 .84173 .15827 .91451 13 48 .75642 .84200 .15800 .91442 12 49 .75660 .84227 .15773 .91433 11 50 .75678 .84254 .15746 .91425 1OI 51 .75696 .84280 .15720 .91416 9 52 .75714 .84307 .15693 .91407 8 53 .75733 .84334 .15666 .91398 7 54 .75751 .84361 .15639 .91389 6 55 .75769 .84388 .15612 .91381 5 56 .75787 .84415 .15585 .91372 4 57 .75805 .84442 .15558 .91363 3 58 .75823 .84469 .15531 .91354 2 59 .75841 .84496 .15504 .91345 1 60 .75859 .84523 .15477 .91336 O ~7 9 L cos 9 L cot 1O L tan 9 L sin / 56 55 35 36< 53 / 9Lsin 9Ltan 1O L cot 9 Lcos / .75 859 .84 523 .15477 .91 336 6O 1 .75 877 .84550 .15450 .91 328 59 2 .75 895 .84 576 .15 424 .91319 58 3 .75913 .84 603 .15397 .91 310 57 4 .75931 .84630 .15370 .91 301 56 5 .75949 .84657 .15343 .91 292 55 6 .75 967 .84684 .15316 .91 283 54 7 .75985 .84711 .15 289 .91 274 53 8 .76003 .84 738 .15262 .91 266 52 9 .76021 .84764 .15 236 .91 257 51 10 .76039 .84 791 .15 209 .91 248 50 11 .76057 .84818 .15 182 .91 239 49 12 .76075 .84 845 .15 155 .91 230 48 13 .76093 .84 872 .15 128 .91 221 47 14 .76111 .84 899 .15101 .91212 46 15 .76 129 .84925 .15075 .91 203 45 16 .76 146 .84952 .15048 .91 194 44 17 .76 164 .84979 .15021 .91 185 43 18 .76 182 .85 006 .14994 .91 176 42 19 .76 200 .85033 .14967 .91 167 41 2O .76218 .85 059 .14941 .91 158 40 21 .76 236 .85 086 .14914 .91 149 39 22 .76253 .85 113 .14887 .91 141 38 23 .76271 .85 140 .14860 .91 132 37 24 .76 289 .85 166 .14834 .91 123 36 25 .76307 .85 193 .14807 .91114 35 26 .76324 .85 220 .14780 .91 105 34 27 .76342 .85 247 .14753 .91 096 33 28 .76360 .85 273 .14 727 .91087 32 29 .76378 .85 300 .14 700 .91 078 31 30 .76395 .85 327 .14673 .91 069 30 31 .76413 .85 354 .14646 .91 060 29 32 .76431 .85 380 .14620 .91051 28 33 .76448 .85 407 .14593 .91 042 27 34 .76466 .85 434 .14 566 .91033 26 35 .76484 .85 460 .14540 .91 023 25 36 .76501 .85 487 .14513 .91014 24 37 .76519 .85 514 .14486 .91 005 23 38 .76537 .85 540 .14460 .90996 22 39 .76554 .85 567 .14433 .90987 21 4O .76572 .85 594 .14406 .90978 2O 41 .76 590 .85 620 .14380 .90969 19 42 .76607 .85 647 .14353 .90960 18 43 .76625 .85 674 .14326 .90951 17 44 .76642 .85 700 .14300 .90942 16 45 .76660 .85 727 .14273 .90933 15 46 .76677 .85 754 .14246 .90924 14 47 .76695 .85 780 .14220 .90915 13 48 .76712 .85 807 .14 193 .90906 12 49 .76 730 .85 834 .14 166 .90896 11 50 .76 747 .85 860 .14 140 .90887 1O 51 .76 765 .85 887 .14113 .90878 9 52 .76 782 .85 913 .14087 .90869 8 53 .76800 .85 940 .14060 .90860 7 54 .76817 .85 967 .14033 .90851 6 55 .76835 .85 993 .14007 .90842 5 56 .76852 .86020 .13 980 .90832 4 57 .76870 .86046 .13954 .90823 3 58 .76887 .86073 .13927 .90814 2 59 .76904 .86 103 .13900 .90805 1 60 .76922 .86 126 .13874 .90 796 O / 9 Lcos 9Lcot 10 L tan 9Lsin / / 91 sin 9Ltan lOLcot 9 Lcos t O .76922 .86126 .13874 .90 796 60 1 .76939 .86153 .13847 .90787 59 2 .76957 .86179 .13821 .90777 58 3 .76974 .86 206 .13 794 .90 768 57 4 .76991 .86232 .13 768 .90759 56 5 .77009 .86259 .13 741 .90 750 55 6 .77026 .86285 .13 715 .90 741 54 7 .77043 .86312 .13688 .90 731 53 8 .77061 .86338 .13662 .90 722 52 9 .77078 .86365 .13635 .90713 51 1O .77 095 .86392 .13608 .90 704 5O 11 .77112 .86418 .13 582 .90694 49 12 .77 130 .86445 .13 555 .90685 48 13 .77 147 .86471 .13 529 .90676 47 14 .77 164 .86498 .13 502 .90667 46 15 .77181 .86524 .13476 .90657 45 16 .77 199 .86551 .13449 .90648 44 17 .77216 .86577 .13 423 .90639 43 18 .77233 .86603 .13397 .90630 42 19 .77250 .86630 .13370 .90620 41 2O .77 268 .86656 .13 344 .90611 40 21 .77285 .86683 .13317 .90602 39 22 .77302 .86 709 .13291 .90592 38 23 .77319 .86 736 .13 264 .90 583 37 24 .77336 .86762 .13 238 .90574 36 25 .77353 .86 789 .13211 .90565 35 26 .77370 .86815 .13 185 .90555 34 27 .77387 .86842 .13158 .90 546 33 28 .77405 .86868 .13 132 .90537 32 29 .77422 .86894 .13 106 .90527 31 30 .77439 .86921 .13079 .90518 30 31 .77456 .86947 .13053 .90 509 29 32 .77473 .86974 .13026 .90499 28 33 .77 490 .87000 .13000 .90490 27 34 .77507 .87027 .12973 .90480 26 35 .77524 .87053 .12947 .90471 25 36 .77541 .87079 .12921 .90462 24 37 .77558 .87 106 .12 894 .90452 23 38 .77575 .87 132 .12868 .90443 22 39 .77 592 .87 158 .12842 .90434 21 4O .77609 .87 185 .12815 .90424 2O 41 .77 626 .87211 .12 789 .90415 19 42 .77 643 .87 238 .12 762 .90405 18 43 .77660 .87264 .12 736 .90396 17 44 .77677 .87 290 .12710 .90386 16 45 .77694 .87317 .12683 .90377 15 46 .77711 .87343 .12657 .90368 14 47 .77 728 .87 369 .12631 .90358 13 48 .77 744 .87 396 .12604 .90349 12 49 .77761 .87422 .12578 .90339 11 5O .77 778 .87448 .12552 .90330 10 51 .77 795 .87475 .12525 .90320 9 52 .77812 .87501 .12499 .90311 8 53 .77829 .87 527 .12473 .90301 7 54 .77 846 .87554 .12 446 .90292 6 55 .77 862 .87 580 .12420 .90282 5 56 .77879 .87 606 .12394 .90273 4 57 .77896 .87 633 .12367 .90263 3 58 .77913 .87659 .12341 .90254 2 59 .77930 .87685 .12315 .90244 1 60 .77946 .87711 .12289 .90235 O / 9 Lcos 9Lcot 10 L tan 9Lsin / 54 C 53' 54 37 38 / 9Lsin 9Ltan lOLcot 9 Lcos / o .77946 .87711 .12289 .90 235 60 1 .77963 .87 738 .12262 .90 225 59 2 .77980 .87 764 .12236 .90216 58 3 .77997 .87 790 .12210 .90 206 57 4 .78013 .87817 .12183 .90 197 56 5 .78030 .87 843 .12157 .90187 55 6 .78047 .87 869 .12131 .90178 54 7 .78063 .87895 .12105 .90 168 53 8 .78080 .87922 .12078 .90 159 52 9 .78097 .87948 .12052 .90 149 51 1O .78113 .87974 .12026 .90 139 50 11 .78 130 .88000 .12000 .90 130 49 12 .78 147 .88027 .11973 .90 120 48 13 .78 163 .88053 .11947 .90111 47 14 .78 180 .88079 .11921 .90 101 46 15 .78 197 .88 105 .11895 .90091 45 16 .78213 .88131 .11869 .90082 44 17 .78 230 .88 158 .11 842 .90072 43 18 .78 246 .88 184 .11816 .90063 42 19 .78 263 .88210 .11790 .90053 41 20 .78280 .88236 .11764 .90043 40 21 .78 296 .88 262 .11738 .90034 39 22 .78313 .88 289 .11711 .90024 38 23 .78329 .88315 .11685 .90014 37 24 .78346 .88341 .11659 .90005 36 25 .78362 .88367 .11633 .89995 35 26 .78379 .88393 .11607 .89985 34 27 .78 395 .88420 .11580 .89976 33 28 .78412 .88446 .11554 .89966 32 29 .78428 .88472 .11528 .89956 31 30 .78445 .88498 .11502 .89947 3O 31 .78461 .88 524 .11476 .89937 29 32 .78478 .88550 .11450 .89927 28 33 .78494 .88577 .11423 .89918 27 34 .78510 .88603 .11397 .89908 26 35 .78527 .88629 .11371 .89898 25 36 .78543 .88655 .11345 .89888 24 37 .78560 .88681 .11319 .89879 23 38 .78576 .88 707 .11293 .89869 22 39 .78 592 .88 733 .11267 .89859 21 40 .78609 .88759 .11241 .89849 2O 41 .78625 .88 786 .11214 .89840 19 42 .78642 .88812 .11188 .89830 18 43 .78658 .88838 .11162 .89 820 17 44 .78674 .88864 .11136 .89810 16 45 .78691 .88890 .11110 .89801 15 46 .78 707 .88916 .11084 .89791 14 47 .78 723 .88942 .11058 .89781 13 48 .78739 .88968 .11032 .89771 12 49 .78756 .88994 .11 006 .89 761 11 5O .78 772 .89020 .10980 .89 752 10 51 .78 788 .89046 .10954 .89 742 9 52 .78805 .89073 .10927 .89 732 8 53 .78821 .89099 .10901 .89 722 7 54 .78837 .89 125 .10875 .89712 6 55 .78853 .89151 .10849 .89 702 5 56 .78869 .89177 .10823 .88693 4 57 .78 886 .89 203 .10797 .89683 3 58 .78902 .89 229 .10771 .89673 2 59 .78918 .89255 .10745 .89663 1 60 .78934 .89281 .10719 .89653 / 9 Lcos 9Lcot 1O L tan 9 L sin / / 9 L sin 9 L tan 1O L cot 9 L cos / .78934 .89281 .10719 .89653 60 1 .78950 .89307 .10693 .89643 59 2 .78967 .89333 .10667 .89633 58 3 .78983 .89359 .10641 .89624 57 4 .78999 .89385 .10615 .89614 56 5 .79015 .89411 .10589 .89604 55 6 .79031 .89437 .10563 .89594 54 7 .79047 .89463 .10537 .89584 53 8 .79063 .89489 .10511 .89574 52 9 .79079 .89515 .10485 .89564 51 1O .79095 .89541 .10459 .89554 5O 11 .79111 .89567 .10433 .89544 49 12 .79128 .89593 .10407 .89534 48 13 .79144 .89619 .10381 .89524 47 14 .79160 .89645 .10355 .89514 46 15 .79176 .89671 .10329 .89504 45 16 .79192 .89697 .10303 .89495 44 17 .79208 .89723 .10277 .89485 43 18 .79224 .89749 .10251 .89475 42 19 .79240 .89775 .10225 .89465 41 20 .79256 .89801 .10199 .89455 4O 21 .79272 .89827 .10173 .89445 39 22 .79288 .89853 .10147 .89435 38 23 .79304 .89879 .10121 .89425 37 24 .79319 .89905 .10095 .89415 36 25 .79335 .89931 .10069 .89405 35 26 .79351 .89957 .10043 .89395 34 27 .79367 .89983 .10017 .89385 33 28 .79383 .90009 .09991 .89375 32 29 .79399 .90035 .09965 .89364 31 3O .79415 .90061 .09939 .89354 3O 31 .79431 .90086 .09914 .89344 29 32 .79447 .90112 .09888 .89334 28 33 .79463 .90138 .09862 .89324 27 34 .79478 .90164 .09836 .89314 26 35 .79494 .90190 .09810 .89304 25 36 .79510 .90216 .09784 .89294 24 37 .79526 .90242 .09758 .89284 23 38 .79542 .90268 .09732 .89274 22 39 .79558 .90294 .09706 .89264 21 4O .79573 .90320 .09680 .89254 2O 41 .79589 .90346 .09654 .89244 19 42 .79605 .90371 .09629 .89233 18 43 .79621 .90397 .09603 .89223 17 44 .79636 .90423 .09577 .89213 16 45 .79652 .90449 .09551 .89203 15 46 .79668 .90475 .09525 .89193 14 47 .79684 .90501 .09499 .89183 13 48 .79699 .90527 .09473 .89173 12 49 .79715 .90553 .09447 .89162 11 50 .79731 .90578 .09422 .89152 1O 51 .79746 .90604 .09396 .89142 9 52 .79762 .90630 .09370 .89132 8 53 .79778 .90656 .09344 .89122 7 54 .79793 .90682 .09318 .89112 6 55 .79809 .90708 .09292 .89101 5 56 .79825 .90734 .09266 .89091 4 57 .79840 .90759 .09241 .89081 3 58 .79856 .90785 .09215 .89071 2 59 .79872 .90811 .09189 .89060 1 60 .79887 .90837 .09163 .89050 / 9 Lcos 9Lcot lOLtan 9Lsin / 52 C 51 C 39 40' 55 / 9 L sin 9 L tan 1O L cot 9 L cos / o .79887 .90837 .09163 .89050 BO 1 .79903 .90863 .09137 .89040 59 2 .79918 .90889 .09111 .89030 58 3 .79934 .90914 .09086 .89020 57 4 .79950 .90940 .09060 .89009 56 5 .79965 .90966 .09034 .88999 55 6 .79981 .90992 .09008 .88989 54 7 .79996 .91018 .08982 .88978 53 8 .80012 .91043 .08957 .88968 52 9 .80027 .91069 .08931 .88958 51 10 .80043 .91095 .08905 .88948 50 11 .80058 .91121 .08879 .88937 49 12 .80074 .91147 .08853 .88927 48 13 .80089 .91172 .08828 .88917 47 14 .80105 .91198 .08802 .88906 46 15 .80120 .91224 .08776 .88896 45 16 .80136 .91250 .08750 .88886 44 17 .80151 .91276 .08724 .88875 43 18 .80166 .91301 .08699 .88865 42 19 .80182 .91327 .08673 .88855 41 2O .80197 .91353 .08647 .88844 4O 21 .80213 .91379 .08621 .88834 39 22 .80228 .91404 .08596 .88824 38 23 .80244 .91430 .08570 .88813 37 24 .80259 .91456 .08544 .88803 36 25 .80274 .91482 .08518 .88793 35 26 .80290 .91507 .08493 .88782 34 27 .80305 .91533 .08467 .88772 33 28 .80320 .91559 .08441 .88761 32 29 .80336 .91585 .08415 .88751 31 30 .80351 .91610 .08390 .88741 3O 31 .80366 .91636 .08364 .88730 29 32 .80382 .91662 .08338 .88720 28 33 .80397 .91688 .08312 .88709 27 34 .80412 .91713 .08287 .88699 26 35 .80428 .91739 .08261 .88688 25 36 .80443 .91765 .08235 .88678 24 37 .80458 .91791 .08209 .88668 23 38 .80473 .91816 .08184 .88657 22 39 .80489 .91842 .08158 .88647 21 4O .80504 .91868 .08132 .88636 20 41 .80519 .91893 .08107 .88626 19 42 .80534 .91919 .08081 .88615 18 43 .80550 .91945 .08055 .88605 17 44 .80565 .91971 .08029 .88594 16 45 .80580 .91996 .08004 .88584 15 46 .80595 .92022 .07978 .88573 14 47 .80610 .92048 .07952 .88563 13 48 .80625 .92073 .07927 .88552 12 49 .80641 .92099 .07901 .88542 11 50 .80656 .92125 .07875 .88531 10 51 .80671 .92150 .07850 .88521 9 52 .80686 .92176 .07824 .88510 8 53 .80701 .92202 .07798 .88499 7 54 .80716 .92227 .07773 .88489 6 55 .80731 .92253 .07747 .88478 5 56 .80746 .92279 .07721 .88468 4 57 .80762 .92304 .07696 .88457 3 58 .80777 .92330 .07670 .88447 2 59 .80792 .92356 .07644 .88436 1 60 .80807 .92381 .07619 .88425 O / 9Lcos 9Lcot lOLtan 9Lsin / / 9Lsin 9Ltan 10 L cot 9Lcos / O .80807 .92381 .07 619 .88425 60 1 .80822 .92407 .07 593 .88415 59 2 .80837 .92433 .07 567 .88404 58 3 .80852 .92458 .07 542 .88394 57 4 .80867 .92484 .07516 .88383 56 5 .80882 .92510 .07 490 .88372 55 6 .80897 .92 535 .07 465 .88362 54 7 .80912 .92561 .07 439 .88351 53 8 .80927 .92 587 .07413 .88340 52 9 .80942 .92612 .07388 .88330 51 10 .80957 .92638 .07 362 .88319 50 11 .80972 .92663 .07 337 .88308 49 12 .80987 .92 689 .07311 .88 298 48 13 .81 002 .92 715 .07 285 .88 287 47 14 .81017 .92 740 .07 260 .88276 46 15 .81032 .92 766 .07 234 .88 266 45 16 .81 047 .92 792 .07 208 .88255 44 17 .81061 .92817 .07 183 .88 244 43 18 .81076 .92 843 .07157 .88 234 42 19 .81 091 .92 868 .07 132 .88223 41 2O .81 106 .92894 .07 106 .88212 40 21 .81 121 .92920 .07080 .88 201 39 22 .81 136 .92 945 .07055 .88 191 38 23 .81151 .92971 .07029 .88 180 37 24 .81 166 .92996 .07004 .88 169 36 25 .81 180 .93022 .06978 .88 158 35 26 .81 195 .93048 .06952 .88 148 34 27 .81210 .93073 .06927 .88137 33 28 .81 225 .93099 .06901 .88126 32 29 .81 240 .93 124 .06876 .88115 31 3O .81 254 .93 150 .06850 .88 105 3O 31 .81269 .93175 .06825 .88094 29 32 .81 284 .93 201 .06 799 .88083 28 33 .81 299 .93 227 .06 773 .88072 27 34 .81314 .93252 .06748 .88061 26 35 .81328 .93 278 .06 722 .88051 25 36 .81 343 .93 303 .06697 .88040 24 37 .81358 .93329 .06671 .88029 23 38 .81372 .93 354 .06646 .88018 22 39 .81387 .93380 .06620 .88007 21 4O .81 402 .93 406 .06594 .87996 20 41 .81417 .93431 .06569 .87985 19 42 .81431 .93457 .06543 .87975 18 43 .81 446 .93 482 .06518 .87964 17 44 .81461 .93 508 .06492 .87953 16 45 .81 475 .93 533 .06467 .87942 15 46 .81 490 .93 559 .06441 .87931 14 47 .81 505 .93 584 .06416 .87 920 13 48 .81519 .93610 .06390 .87 909 12 49 .81 534 .93 636 .06364 .87 898 11 50 .81 549 .93 661 .06339 .87 887 1O 51 .81 563 .93 687 .06313 .87877 9 52 .81578 .93 712 .06288 .87 866 8 53 .81 592 .93 738 .06262 .87855 7 54 .81 607 .93 763 .06237 .87844 6 55 .81622 .93 789 .06211 .87 833 5 56 .81636 .93 814 .06 186 .87822 4 57 .81651 .93 840 .06 160 .87811 3 58 .81 665 .93 865 .06 135 .87 800 2 59 .81680 .93 891 .06 109 .87 789 1 60 .81 694 .93 916 .06084 .87 778 r 9Lcos 9Lcot lOLtan 9Lsin ' 50 C 49' 41 C 42 C / 9Lsin 9Ltan lOLcot 9Lcos / o .81694 .93916 .06084 .87778 GO 1 .81709 .93942 .06058 .87767 59 2 .81723 .93967 .06033 .87756 58 3 .81738 .93993 .06007 .87745 57 4 .81752 .94018 .05982 .87734 56 5 .81767 .94044 .05956 .87723 55 6 .81781 .94069 .05931 .87712 54 *7 .81796 .94095 .05905 .87701 53 8 .81810 .94120 .05880 .87690 52 9 .81825 .94146 .05854 .87679 51 10 .81839 .94171 .05829 .87668 50 11 .81854 .94197 .05803 .87657 49 12 .81868 .94222 .05778 .87646 48 13 .81882 .94248 .05752 .87635 47 14 .81897 .94273 .05727 .87624 46 15 .81911 .94299 .05701 .87613 45 16 .81926 .94324 .05676 .87601 44 17 .81940 .94350 .05650 .87590 43 18 .81955 .94375 .05625 .87579 42 19 .81969 .94401 .05599 .87568 41 20 .81983 .94426 .05574 .87557 40 21 .81998 .94452 .05548 .87546 39 22 .82012 .94477 .05523 .87535 38 23 .82026 .94503 .05497 .87524 37 24 .82041 .94528 .05472 .87513 36 25 .82055 .94554 .05446 .87501 35 26 .82069 .94579 .05421 .87490 34 27 .82084 .94604 .05396 .87479 33 28 .82098 .94630 .05370 .87468 32 29 .82112 .94655 .05345 .87457 31 3O .82126 .94681 .05319 .87446 3O 31 .82141 .94706 .05294 .87434 29 32 .82155 .94732 .05268 .87423 28 33 .82169 .94757 .05243 .87412 27 34 .82184 .94783 .05217 .87401 26 35 .82198 .94808 .05192 .87390 25 36 .82212 .94834 .05166 .87378 24 37 .82226 .94859 .05141 .87367 23 38 .82240 .94884 .05116 .87356 22 39 .82255 .94910 .05090 .87345 21 4O .82269 .94935 .05065 .87334 20 41 .82283 .94961 .05039 .87322 19 42 .82297 .94986 .05014 .87311 18 43 .82311 .95012 .04988 .87300 17 44 .82326 .95037 .04963 .87288 16 45 .82340 .95062 .04938 .87277 15 46 .82354 .95088 .04912 .87266 14 47 .82368 .95113 .04887 .87255 13 48 .82382 .95139 .04861 .87243 12 49 .82396 .95164 .04836 .87232 11 5O .82410 .95190 .04810 .87221 1O 51 .82424 .95215 .04785 .87209 9 52 .82439 .95240 .04760 .87198 8 53 .82453 .95266 .04734 .87187 7 54 .82467 .95291 .04709 .87175 6 55 .82481 .95317 .04683 .87164 5 56 .82495 .95342 .04658 .87153 4 57 .82509 .95368 .04632 .87141 3 58 .82523 .95393 .04607 .87130 2 59 .82537 .95418 .04582 .87119 1 60 .82551 .95444 .04556 .87107 / 9Lcos 9LcotlOLtan9Lsin / / 9Lsin 9Ltan lOLcot 9 Lcos |_/_ O .82551 .95 444 .04 556 .87 107 GO 1 .82 565 .95 469 .04531 .87 096 59 2 .82579 .95 495 .04 505 .87085 58 3 .82 593 .95 520 .04480 .87073 57 4 .82607 .95 545 .04 455 .87062 56 5 .82621 .95 571 .04 429 .87050 55 6 .82635 .95 596 .04 404 .87 039 54 7 .82649 .95 622 .04378 .87028 53 8 .82 663 .95647 .04353 .87016 52 9 .82677 .95 672 .04328 .87 005 51 1O .82691 .95 698 .04 302 .86993 50 11 .82 705 .95 723 .04 277 .86982 i 49 12 .82719 .95 748 .04 252 .86970 48 13 .82 733 .95 774 .04 226 .86959 47 14 .82 747 .95 799 .04 201 .86947 46 15 .82 761 .95 825 .04 175 .86936 45 16 .82775 .95 850 .04 150 .86924 44 17 .82 788 .95 875 .04 125 .86913 43 18 .82 802 .95 901 .04 099 .86902 42 19 .82816 .95 926 .04074 .86890 41 20 .82830 .95952 .04048 .86879 4O 21 .82 844 .95 977 .04023 .86867 39 22 .82858 .96002 .03 998 .86855 38 23 .82872 .96028 .03 972 .86844 37 24 .82885 .96 053 .03947 .86832 36 25 .82 899 .96078 .03 922 .86821 35 26 .82913 .96 104 .03 896 .86809 34 27 .82927 .96129 .03871 .86 798 33 28 .82941 .96 155 .03 845 .86 786 32 29 .82955 .96 180 .03 820 .86 775 31 30 .82968 .96205 .03 795 .86 763 30 31 .82982 .96231 .03 769 .86752 29 32 .82996 .96256 .03 744 .86 740 28 33 .83 010 .96 281 .03 719 .86 728 27 34 .83 023 .96307 .03 693 .86717 26 35 .83 037 .96332 .03 668 .86705 25 36 .83051 .96357 .03 643 .86694 24 37 .83 065 .96383 .03617 .86682 23 38 .83 078 .96408 .03 592 .86670 22 39 .83 092 .96433 .03 567 .86659 21 40 .83 106 .96459 .03 541 .86647 20 41 .83120 .96484 .03516 .86635 19 42 .83 133 .96510 .03 490 .86624 18 43 .83 147 .96535 .03465 .86612 17 44 .83 161 .96560 .03 440 .86600 16 45 .83 174 .96 586 .03 414 .86589 15 46 .83 188 .96611 .03389 .86577 14 47 .83 202 .96636 .03 364 .86 565 13 48 .83215 .96662 .03 338 .86554 12 49 .83 229 .96687 .03313 .86542 11 50 .83 242 .96712 .03 288 .86530 1O 51 .83 256 .96 738 .03 262 .86518 9 52 .83 270 .96 763 .03 237 .86 507 8 53 .83 283 .96 788 .03 212 .86495 7 54 .83 297 .96814 .03 186 .86483 6 55 .83310 .96839 .03 161 .86472 5 56 .83 324 .96864 .03 136 .86460 4 57 .83338 .96890 .03 110 .86448 3 58 .83351 .96915 .03085 .86436 2 59 .83365 .96940 .03 060 .86425 1 6O .83 378 .96966 .03034 .86413 / 9Lcos 9LcotlOLtan 9 L sin / 48' 47 43' 44 C 57 / 9 L sin 9 L tan 1O L cot 9 L cos / o .83378 .96966 .03034 .86413 W 1 .83392 .96991 .03009 .86401 59 2 .83405 .97016 .02984 .86389 58 3 .83419 .97042 .02958 .86377 57 4 .83.432 .97067 .02933 .86366 56 5 .83446 .97092 .02908 .86354 55 6 .83459 .97118 .02882 .86342 54 7 .83473 .97143 .02857 .86330 53 8 .83486 .97168 .02832 .86318 52 9 .83500 .97193 .02807 .86306 51 1C .83513 .97219 .02781 .86295 50 11 .83527 .97244 .02756 .86283 49 12 .83540 .97269 .02731 .86771 48 13 .83554 .97295 .02705 .86259 47 14 .83567 .97320 .02680 .86247 46 IS .83581 .97345 .02655 .86235 45 16 .83594 .97371 .02629 .86223 44 17 .83608 .97396 .02604 .86211 43 18 .83621 .97421 .02579 .86200 42 19 .83634 .97447 .02553 .86188 41 2O .83648 .97472 .02528 .86176 40 21 .83661 .97497 .02503 .86164 39 22 .83674 .97523 .02477 .86152 38 23 .83688 .97548 .02452 .86140 37 24 .83701 .97573 .02427 .86128 36 25 .83715 .97598 .02402 .86116 35 26 .83728 .97624 .02376 .86104 34 27 .83741 .97649 .02351 .86092 33 28 .83755 .97674 .02326 .86080 32 29 .83768 .97700 .02300 .86068 31 30 .83781 .97725 .02275 .86056 30 31 .83795 .97750 .02250 .86044 29 32 .83808 .97776 .02224 .86032 28 33 .83821 .97801 .02199 .86020 27 34 .83834 .97826 .02174 .86008 26 35 .83848 .97851 .02149 .85996 25 36 .83861 .97877 .02123 .85984 24 37 .83874 .97902 .02098 .85972 23 38 .83887 .97927 .02073 .85960 22 39 .83901 .97953 .02047 .85948 21 4O .83914 .97978 .02022 .85936 2O 41 .83927 .98003 .01997 .85924 19 42 .83940 .98029 .01971 .85912 18 43 .88954 .98054 .01946 .85900 17 44 .83967 .98079 .01921 .85888 16 45 .83980 .98104 .01896 .85876 15 46 .83993 .98130 .01870 .85864 14 47 .84006 .98155 .01845 .85851 13 48 .84020 .98180 .01820 .85839 12 49 .84033 .98206 .01794 .85827 11 50 .84046 .98231 .01769 .85815 1O 51 .84059 .98256 .01744 .85803 9 52 .84072 .98281 .01719 .85791 8 53 .84085 .98307 .01693 .85779 7 54 .84098 .98332 .01668 .85766 6 55 .84112 .98357 .01643 .85754 5 56 .84125 .98383 .01617 .85742 4 57 .84138 .98408 .01592 .85730 3 58 .84151 .98433 .01567 .85718 2 59 .84164 .98458 .01542 .85706 1 60 .84177 .98484 .01516 .85693 / 9Lcos 9Lcot 10 L tan 9Lsin / / 9 L sin 9 L tan 1O L cot 9 L cos / ~cT .84177 .98484 .01516 .85693 6O i .84190 .98509 .01491 .85681 59 2 .84203 .98534 .01466 .85669 58 3 .84216 .98560 .01440 .85657 57 4 .84229 .98585 .01415 .85645 56 5 .84242 .98610 .01390 .85632 55 6 .84255 .98635 .01365 .85620 54 .7 .84269 .98661 .01339 .85608 53 8 .84282 .98686 .01314 .85596 52 9 .84295 .98711 .01289 .85583 51 1O .84308 .98737 .01263 .85571 50 11 .84321 .98762 .01238 .85559 49 12 .84334 .98787 .01213 .85547 48 13 .84347 .98812 .01188 .85534 47 14 .84360 .98838 .01162 .85522 46 15 .84373 .98863 .01137 .85510 45 16 .84385 .98888 .01112 .85497 44 17 .84398 .98913 .01087 .85485 43 18 .84411 .98939 .01061 .85473 42 19 .84424 .98964 .01036 .85460 41 2O .84437 .98989 .01011 .85448 4O 21 .84450 .99015 .00985 .85436 39 22 .84463 .99040 .00960 .85423 38 23 .84476 .99065 .00935 .85411 37 24 .84489 .99090 .00910 .85399 36 25 .84502 .99116 .00884 .85386 35 26 .84515 .99141 .00859 .85374 34 27 .84528 .99166 .00834 .85361 33 28 .84540 .99191 .00809 .85349 32 29 .84553 .99217 .00783 .85337 31 30 .84566 .99242 .00758 .85324 3O 31 .84579 .99267 .00733 .85312 29 32 .84592 .99293 .00707 .85299 28 33 .84605 .99318 .00682 .85287 27 34 .84618 .99343 .00657 .85274 26 35 .84630 .99368 .00632 .85262 25 36 .84643 .99394 .00606 .85250 24 37 .84656 .99419 .00581 .85237 23 38 .84669 .99444 .00556 .85225 22 39 .84682 .99469 .00531 .85212 21 40 .84694 .99495 .00505 .85200 20 41 .84707 .99520 .00480 .85187 19 42 .84720 .99545 .00455 .85175 18 43 .84733 .99570 .00430 .85162 17 44 .84745 .99596 .00404 .85150 16 45 .84758 .99621 .00379 .85137 15 46 .84771 .99646 .00354 .85125 14 47 .84784 .99672 .00328 .85112 13 48 .84796 .99697 .00303 .85100 12 49 .84809 .99722 .00278 .85087 11 50 .84822 .99747 .00253 .85074 1O 51 .84835 .99773 .00227 .85062 9 52 .84847 .99798 .00202 .85049 8 53 .84860 .99823 .00177 .85037 7 54 .84873 .99848 .00152 .85024 6 55 .84885 .99874 .00126 .85012 5 56 .84898 .99899 .00101 .84999 4 57 .84911 .99924 .00076 .84986 3 58 .84923 .99949 .00051 .84974 2 59 .84936 .99975 .00025 .84961 1 6O .84949 .00000 .00000 .S49J2_. O / 9 L cos 1O L cot 1O L tan 9~lTsiii / 46' 45< 58 TABLE IV NATURAL FUNCTIONS / sin tan cot cos / o .00000 .00000 co 1.0000 60 1 .00029 .00029 3437.7 1.0000 59 2 .00058 .00058 1718.9 1.0000 58 3 .00087 .00087 1145.9 1.0000 57 4 .00116 .00116 859.44 1.0000 56 5 .00 145 .00 145 687.55 1.0000 55 6 .00 175 .00175 572.96 1.0000 54 7 .00204 .00204 491.11 1.0000 53 8 .00233 .00233 429.72 1.0000 52 9 .00262 .00262 381.97 1.0000 51 1C .00291 .00291 343.77 1.0000 5O 11 .00320 .00320 312.52 .99999 49 12 .00349 .00349 286.48 .99999 48 13 .00378 .00378 264.44 .99999 47 14 .00407 .00407 245.55 .99999 46 15 .00436 .00436 229.18 .99999 45 16 .00465 .00465 214.86 .99999 44 17 .00495 .00495 202.22 .99999 43 18 .00524 .00524 190.98 .99999 42 19 .00553 .00553 180.93 .99998 41 20 .00 582 -.00 582 171.89 .99 998 4O 21 .00611 .00611 163.70 .99998 39 22 .00640 .00640 156.26 .99998 38 23 .00669 .00669 149.47 .99998 37 24 .00698 .00698 143.24 .99998 36 25 .00727 .00727 137.51 .99997 35 26 .00756 .00756 132.22 .99997 34 27 .00785 .00785 127.32 .99997 33 28 .00814 .00815 122.77 .99997 32 29 .00844 .00844 118.54 .99996 31 30 .00873 .00873 114.59 .99996 3O 31 .00902 .00902 110.89 .99996 29 32 .00931 .00931 107.43 .99996 28 33 .00960 .00960 104.17 .99995 27 34 .00989 .00989 101.11 .99995 26 35 .01018 .01018 98.218 .99995 25 36 .01047 .01047 95.489 .99995 24 37 .01 076 .01 076 92.908 .99 994 23 38 .01105 .01 105 90.463 .99994 22 39 .01 134 .01 135 88.144 .99994 21 40 .01 164 .01 164 85.940 .99993 20 41 .01 193 .01 193 83.844 .99 993 19 42 .01222 .01222 81.847 .99993 18 43 .01251 .01251 79.943 .99992 17 44 .01280 .01280 78.126 .99992 16 45 .01 309 .01 309 76.390 .99 991 15 46 .01338 .01338 74.729 .99991 14 47 .01367 .01367 73.139 .99991 13 48 .01396 .01396 71.615 .99990 12 49 .01425 .01425 70.153 .99990 11 50 .01 454 .01 455 68.750 .99 989 10 51 .01483 .01484 67.402 .99989 9 52 .01513 .01 513 66.105 .99989 8 53 .01542 .01 542 64.858 .99988 7 54 .01571 .01571 63.657 .99988 6 55 .01 600 .01 600 62.499 .99987 5 56 .01629 .01629 61.383 .99987 4 57 .01 658 .01 658 60.306 .99 986 3 58 .01 687 .01 687 59.266 .99 986 2 59 .01716 .01 716 58.261 .99985 1 6O .01 745 .01 746 57.290 .99985 O / cos cot tan sin / 89 1 / sin tan cot cos / O .01 745 .01 746 57.290 .99 985 00 1 .01774 .01775 56.351 .99984 59 2 .01803 .01804 55.442 .99984 58 3 .01832 .01833 54.561 .99983 57 4 .01862 .01862 53.709 .99983 56 5 .01891 .01891 52.882 .99982 55 6 .01920 .01 920 52.081 .99982 54 7 .01949 .01949 51.303 .99981 53 8 .01978 .01978 50.549 .99980 52 9 .02007 .02007 49.816 .99980 51 1O .02036 .02036 49.104 .99979 50 11 .02065 .02066 48.412 .99979 49 12 .02094 .02095 47.740 .99978 48 13 .02123 .02124 47.085 .99977 47 14 .02152 .02153 46.449 .99977 46 15 .02181 .02182 45.829 .99976 45 16 .02211 .02211 45.226 .99976 44 17 .02 240 .02 240 44.639 .99 975 43 18 .02 269 .02 269 44.066 .99 974 42 19 .02298 .02298 43.508 .99974 41 2O .02327 .02328 42.964 .99973 40 21 .02356 .02357 42.433 .99972 39 22 .02385 .02386 41.916 .99972 38 23 .02414 .02415 41.411 .99971 37 24 .02443 .02444 40.917 .99970 36 25 .02 472 .02 473 40.436 .99 969 35 26 .02501 .02502 39.965 .99969 34 27 .02530 .02531 39.506 .99968 33 28 .02560 .02560 39.057 .99967 32 29 .02589 .02589 38.618 .99966 31 3O .02618 .02619 38.188 .99966 30 31 .02647 .02648 37.769 .99965 29 32 .02676 .02677 37.358 .99964 28 33 .02 705 .02 706 36.956 .99 963 27 34 .02 734 .02 735 36.563 .99 963 26 35 .02763 .02764 36.178 .99962 25 36 .02792 .02793 35.801 .99961 24 37 .02821 .02822 35.431 .99960 23 38 .02850 .02851 35.070 .99959 22 39 .02879 .02881 34.715 .99959 21 40 .02908 .02910 34.368 .99958 20 41 .02938 .02939 34.027 .99957 19 42 .02967 .02968 33.694 .99956 18 43 .02996 .02997 33.366 .99955 17 44 .03 025 .03 026 33.045 .99 954 16 45 .03054 .03055 32.730 .99953 15 46 .03083 .03084 32.421 .99952 14 47 .03112 .03114 32.118 .99952 13 48 .03141 .03143 31.821 .99951 12 49 .03 170 .03 172 31.528 .99950 11 50 .03199 .03201 31.242 .99949 10 51 .03 228 .03 230 30.960 .99 948 9 52 .03257 .03259 30.683 .99947 8 53 .03 286 .03 288 30.412 .99 946 7 54 .03316 .03317 30.145 .99945 6 55 .03345 .03346 29.882 .99944 5 56 .03374 .03376 29.624 .99943 4 57 .03403 .03405 29.371 .99942 3 58 .03432 .03434 29.122 .99941 2 59 .03461 .03463 28.877 .99940 1 GO .03 490 .03 492 28.636 .99 939 O / cos cot tan sin / 88 NATURAL FUNCTIONS 59 2 / sin tan cot cos / .03490 .03492 28.636 .99939 60 1 .03519 .03521 28.399 .99938 59 2 .03548 .03550 28.166 .99937 58 3 .03577 .03579 27.937 .99936 57 4 .03 606 .03 609 27.712 .99 935 56 5 .03 635 .03 638 27.490 .99 934 55 6 .03664 .03667 27.271 .99933 54 7 .03693 .03696 27.057 .99932 53 8 .03723 .03725 26.845 .99931 52 9 .03 752 .03 754 26.637 .99 930 51 1C .03 781 .03 783 26.432 .99 929 5O 11 .03810 .03812 26.230 .99927 49 12 .03839 .03842 26.031 .99926 48 13 .03868 .03871 25.835 .99925 47 14 .03897 .03900 25.642 .99924 46 15 .03 926 .03 929 25.452 .99 923 45 16 .03955 .03958 25.264 .99922 44 17 .03984 .03987 25.080 .99921 43 18 .04013 .04016 24.898 .99919 42 19 .04042 .04046 24.719 .99918 41 2O .04071 .04075 24.542 .99917 4O 21 .04 100 .04 104 24.368 .99 916 39 22 .04129 .04133 24.196 .99915 38 23 .04 159 .04 162 24.026 .99913 37 24 .04188 .04191 23.859 .99912 36 25 .04217 .04220 23.695 .99911 35 26 .04 246 .04 250 23.532 .99 910 34 27 .04 275 .04 279 23.372 .99 909 33 28 .04304 .04308 23.214 .99907 32 29 .04333 .04337 23.058 .99906 31 3O .04362 .04366 22.904 .99905 30 31 .04391 .04395 22.752 .99904 29 32 .04420 .04424 22.602 .99902 28 33 .04449 .04454 22.454 .99901 27 34 .04478 .04483 22.308 .99900 26 35 .04507 .04512 22.164 .99898 25 36 .04536 .04541 22.022 .99897 24 37 .04565 .04570 21.881 .99896 23 38 .04 594 .04 599 21.743 .99 894 22 39 .04623 .04628 21.606 .99893 21 40 .04653 .04658 21.470 .99892 2O 41 .04682 .04687 21.337 .99890 19 42 .04711 .04716 21.205 .99889 18 43 .04 740 .04 745 21.075 .99 888 17 44 .04 769 .04 774 20.946 .99 886 16 45 .04 798 .04 803 20.819 .99 885 15 46 .04827 .04833 20.693 .99883 14 47 .04856 .04862 20.569 .99882 13 48 .04885 .04891 20.446 .99881 12 49 .04914 .04920 20.325 .99879 11 5O .04943 .04949 20.206 .99878 1O 51 .04 972 .04 978 20.087 .99 876 9 52 .05 001 .05 007 19.970 .99 875 8 53 .05 030 .05 037 19.855 .99 873 7 54 .05 059 .05 066 19.740 .99 872 6 55 i .05 OSS .05 095 19.627 .99 870 5 56 .05117 .05124 19.516 .99869 4 57 .05 146 .05 153 19.405 .99867 3 58 .05 175 .05 182 19.296 .99866 2 59 .05 205 .05 212 19.188 .99 864 1 60 .05 234 .05 241 19.081 .99 863 / cos cot tan sin / 87 3 / sin tan cot cos / .05 234 .05 241 19.081 .99 863 6O 1 .05 263 .05 270 18.976 .99 861 59 2 .05 292 .05 299 18.871 .99 860 58 3 .05 321 .05 328 18.768 .99 858 57 4 .05350 .05357 18.666 .99857 56 5 .05 379 .05 387 18.564 .99 855 55 6 .05408 .05416 18.464 .99854 54 7 .05437 .05445 18.366 .99852 53 8 .05466 .05474 18.268 .99851 52 9 .05495 .05503 18.171 .99849 51 1O .05 524 .05 533 18.075 .99 847 50 11 .05 553 .05 562 17.980 .99 846 49 12 .05582 .05591 17.886 .99844 48 13 .05611 .05620 17.793 .99842 47 14 .05 640 .05 649 17.702 .99 841 46 15 .05669 .05678 17.611 .99839 45 16 .05.698 ,05 708 17.521 a83& 44 17 .05727 .05737 17.431 .99836 43 18 .05 756 .05 766 17.343 .99834 42 19 .05785 .05795 17.256 .99833 41 20 .05 814 .05 824 17.169 .99 831 4O 21 .05 844 .05 854 17.084 .99 829 39 22 .05 873 .05 883 16.999 .99 827 38 23 .05902 .05912 16.915 .99826 37 24 .05931 .05941 16.832 .99824 36 25 .05 960 .05 970 16.750 .99 822 35 26 .05989 .05999 16.668 .99821 34 27 .06018 .06029 16.587 .99819 33 28 .06047 .06058 16.507 .99817 32 29 .06076 .06087 16.428 .99815 31 30 .06105 .06116 16.350 .99813 3O 31 .06 134 .06 145 16.272 .99 812 29 32 .06163 .06175 16.195 .99810 28 33 .06192 .06204 16.119 .99808 27 34 .06221 .06233 16.043 .99806 26 35 .06250 .06262 15.969 .99804 25 36 .06279 .06291 15.895 .99803 24 37 .06308 .06321 15.821 .99801 23 38 .06337 .06350 15.748 .99799 22 39 .06 366 .06 379 15.676 .99 797 21 4O .06395 .06408 15.605 .99795 20 41 .06424 .06438 15.534 .99793 19 42 .06453 .06467 15.464 .99792 18 43 .06482 .06496 15.394 .99790 17 44 .06511 .06525 15.325 .99788 16 45 .06 540 .06 554 15.257 .99 786 15 46 .06569 .06584 15.189 .99784 14 47 .06 59S .06 613 15.122 .99 782 13 48 .06627 .06642 15.056 .99780 12 49 .06 656 .06 671 14.990 .99 778 11 50 .06685 .06700 14.924 .99776 10 51 .06714 .06730 14.860 .99774 9 52 .06743 .06759 14.795 .99772 8 53 .06 773 .06 788 14.732 .99 770 7 54 .06802 .06817 14.669 .99768 6 55 .06831 .06847 14.606 .99766 5 56 .06 860 .06 876 14.544 .99 764 4 57 .06889 .06905 14.482 .99762 3 58 .06918 .06934 14.421 .99760 2 59 .06947 .06963 14.361 .99758 1 6O .06 976 .06 993 14.301 .99 756 O / cos cot tan sin / 86 60 NATURAL FUNCTIONS 4 / sin tan cot cos / o .06976 .06993 14.301 .99756 6O 1 .07005 .07022 14.241 .99754 59 2 .07034 .07051 14.182 .99752 58 3 .07063 .07080 14.124 .99750 57 4 .07 092 .07 110 14.065 .99 748 56 5 .07 121 .07 139 14.008 .99 746 55 6 .07 150 .07 168 13.951 .99 744 54 7 .07 179 .07 197 13.894 .99 742 53 8 .07208 .07227 13.838 .99740 52 9 .07 237 .07 256 13.782 .99 738 51 10 .07 266 .07 285 13.727 .99 736 50 11 .07295 .07314 13.672 .99734 49 12 .07324 .07344 13.617 .99731 48 13 .07353 .07373 13.563 .99729 47 14 .07382 .07402 13.510 .99727 46 15 .07411 .07431 13.457 .99725 45 16 .07 440 .07 461 13.404 .99 723 44 17 .07 469 .07 490 13.352 .99 721 43 18 .07498 .07519 13.300 .99719 42 19 .07527 .07548 13.248 .99716 41 20 .07556 .07578 13.197 .99714 4O 21 .07585 .07607 13.146 .99712 39 22 .07 614 .07 636 13.096 .99 710 38 23 .07 643 .07 665 13.046 .99 708 37 24 .07 672 .07 695 12.996 .99 705 36 25 .07 701 .07 724 12.947 .99 703 35 26 .07 730 .07 753 12.898 .99 701 34 27 .07 759 .07 782 12.850 .99 699 33 28 .07 788 .07 812 12 801 .99 696 32 29 .07817 .07841 12.754 .99694 31 30 .07846 .07870 12.706 .99692 3O 31 .07 875 .07 899 12.659 .99 689 29 32 .07904 .07929 12.612 .99687 28 33 .07933 .07958 12.566 .99685 27 34 .07962 .07987 12.520 .99683 26 35 .07991 .08017 12.474 .99680 25 36 .08020 .08046 12.429 .99678 24 37 .080*9 .08075 12.384 .99676 23 38 .08 078 .08 104 12.339 .99 673 22 39 .08107 .08134 12.295 .99671 21 40 .08136 .08163 12.251 .99668 20 41 .08 165 .08 192 12.207 .99 666 19 42 .08 194 .08 221 12.163 .99 664 18 43 .08223 .08251 12.120 .99661 17 44 .08252 .08280 12.077 .99659 16 45 .08 281 .08 309 12.035 .99 657 15 46 .08310 .08339 11.992 .99654 14 47 .08339 .08368 11.950 .99652 13 48 .08368 .08397 11.909 .99649 12 49 .08397 .08427 11.867 .99647 11 50 .08426 .08456 11.826 .99644 1O 51 .08455 .08485 11.785 .99642 9 52 .08484 .08514 11.745 .99639 8 53 .08513 .08544 11.705 .99637 7 54 .08542 .08573 11.664 .99635 6 55 .08571 .08602 11.625 .99632 5 56 .08600 .08632 11.585 .99630 4 57 .08629 .08661 11.546 .99627 3 58 .08658 .08690 11.507 .99625 2 59 .08687 .08720 11.468 .99622 1 60 .08716 .08749 11.430 .99619 / cos cot tan sin / 85 5 / sin tan cot cos / O .08 716 .08 749 11.430 .99 619 TK> 1 .08745 .08778 11.392 .99' 617 59 2 .08774 .08807 11.354 .99614 58 3 .08803 .08837 11.316 .99612 57 4 .08831 .08866 11.279 .99609 56 5 .08860 .08895 11.242 .99607 55 6 .08889 .08925 11.205 .99604 54 7 .08918 .08954 11.168 .99602 .53 8 .08947 .08983 11.132 .99599 52 9 .08976 .09013 11.095 .99596 51 10 .09005 .09042 11.059 .99594 5O 11 .09034 .09071 11.024 .99591 49 12 .09 063 .09 101 10.988 .99 588 48 13 .09092 .09130 10.953 .99586 47 14 .09 121 .09 159 10.918 .99 583 46 15 .09150 .09189 10.883 .99580 45 16 .09179 .09218 10.848 .99578 44 17 .09208 .09247 10.814 .99575 43 18 .09237 .09277 10.780 .99572 42 19 .09 266 .09 306 10.746 .99 570 41 20 .09295 .09335 10.712 .99567 40 21 .09324 .09365 10.678 .99564 39 22 .09353 .09394 10.645 .99562 38 23 .09382 .09423 10.612 .99559 37 24 .09411 .09453 10.579 .99556 36 25 .09440 .09482 10.546 .99553 35 26 .09469 .09511 10.514 .99551 34 27 .09498 .09541 10.481 .99548 33 28 .09527 .09570 10.449 .99545 32 29 .09556 .09600 10.417 .99542 31 30 .09585 .09629 10.385 .99540 30 31 .09614 .09658 10.354 .99537 29 32 .09642 .09688 10.322 .99534 28 33 .09671 .09717 10.291 .99531 27 34 .09 700 .09 746 10.260 .99 528 26 35 .09 729 .09 776 10.229 .99 526 25 36 .09758 .09805 10.199 .99523 24 37 .09 787 .09 834 10.168 .99 520 23 38 .09816 .09864 10.138 .99517 22 39 .09845 .09893 10.108 .99514 21 40 .09874 .09923 10.078 .99511 20 41 .09903 .09952 10.048 .99508 19 42 .09932 .09981 10.019 .99506 18 43 .09961 .10011 9.9893 .99503 17 44 .09990 .10040 9.9601 .99500 16 45 .10019 .10069 9.9310 .99497 15 46 .10048 .10099 9.9021 .99494 14 47 .10077 .10128 9.8734 .99491 13 48 .10106 .10158 9.8448 .99488 12 49 .10 135 .10 187 9.8164 .99 485 11 50 .10164 .10216 9.7882 .99482 1O 51 .10 192 .10 246 9.7601 .99 479 9 52 .10221 .10275 9.7322 .99476 8 53 .10250 .10305 9.7044 .99473 7 54 .10279 .10334 9.6768 .99470 6 55 .10308 .10363 9.6493 .99467 5 56 .10337 .10393 9.6220 .99464 4 57 .10366 .10422 9.5949 .99461 3 58 .10395 .10452 9.5679 .99458 2 59 .10424 .10481 9.5411 .99455 1 6O .10453 .10510 9.5144 .99452 O / cos cot tan sin / 84 NATURAL FUNCTIONS 61 6 D / sin tan cot cos / o .10453 .10510 9.5144 .99452 GO 1 .10482 JO 540 9.4878 .99449 59 2 .10511 .10569 9.46L4 .99446 58 3 .10540 .10599 9.4352 .99443 57 4 .10569 .10628 9.4090 .99440 56 5 .10597 .10657 9.3S31 .99437 55 6 .10626 .10687 9.3572 .99434 54 7 .106S5 .10716 93315 .99431 53 8 .10681 .10746 9.3060 .99428 52 9 .10713 .10775 9.2806 .99-124 51 10 .10742 .10805 9.2553 .99421 50 11 .10771 .10834 9.2302 .99418 49 12 .10800 .10863 9.2052 .99415 48 13 .10829 .10893 9.1803 .99412 47 14 .10858 .10922 9.1555 .99409 46 15 .10837 .10952 9.1309 .99406 45 16 .10916 .10981 9.1065 .99402 44 17 .10945 .11011 9.0821 .99399 43 18 .10973 .11040 9.0579 .99396 42 19 .11002 .11070 9.0338 .99393 41 20 .11031 .11099 9.0098 .99390 4O 21 .11050 .11 128 8.9860 .99386 39 22 .11089 .11 158 8.9623 .99383 38 23 .11 118 .11 187 8.9387 .99380 37 24 .11 147 .11217 8.9152 .99377 36 25 .11176 .11246 8.8919 .99374 35 26 .11205 .11276 8.8686 .99370 34 27 .11234 .11305 8.8455 .99367 33 28 .11263 .11335 8.8225 .99364 32 29 .11291 .11364 8.7996 .99360 31 30 .11 320 .11394 8.7769 .99357 30 31 .11349 .11423 8.7542 .99354 29 32 .11378 .11452 8.7317 .99351 28 33 .11407 .11482 8.7093 .99347 27 34 .11436 .11511 8.6870 .99344 26 35 .11465 .11541 8.6648 .99341 25 36 .11494 .11570 8.6427 .99337 24 37 .11523 .11600 8.6208 .99334 23 38 .11552 .11629 8.5989 .99331 22 39 .11580 .11659 8.5772 .99327 21 4O .11609 .11688 8.5555 .99324 2O 41 .11 638 .11 718 8.5340 .99320 19 42 .11667 .11 747 8.5126 .99317 18 43 .11696 .11 777 8.4913 .99314 17 44 .11725 .11806 8.4701 .99310 16 45 .11 754 .11 836 8.4490 .99307 15 46 .11 783 .11865 8.4280 .99303 14 47 '.11812 .11895 8.4071 .99300 13 48 .11840 .11924 8.3863 .99297 12 49 .11869 .11954 8.3656 .99293 11 50 .11898 .11983 8.3450 .99290 1O 51 .11927 .12013 8.3245 .99286 9 52 .11956 .12042 8.3041 .99283 8 53 .11985 .12072 8.2838 .99279 7 54 .12014 .12101 8.2636 .99276 6 55 .12043 .12131 8.2434 .99272 5 56 .12071 .12160 8.2234 .99269 4 57 .12 100 .12 190 8.2035 .99265 3 58 .12 129 .12 219 8.1837 .99 262 2 59 .12 158 .12249 8.1640 .99258 1 6O .12187 .12278 8.1443 .99255 O / cos cot tan sin / 83 7 / sin tan cot cos / .12187 .12278 8.1443 .99255 60 1 .12216 .12308 8.1248 .99251 59 2 .12245 .12338 8.1054 .99248 58 3 .12274 .12367 8.0860 .99244 57 4 .12302 .12397 8.0667 .99240 56 5 .12331 .12426 8.0476 .99237 55 6 .12360 .12456 80285 .99233 54 7 .12389 .12485 8.C095 .99230 53 8 .12418 .12515 7.9906 .99226 52 9 .12447 .12544 7.9718 .99222 51 1O .12476 .12574 7.9530 .99219 50 11 .12501 .12603 7.9344 .99215 49 12 .12533 .12633 7.9158 .99211 48 13 .12562 .12662 7.8973 .99208 47 14 .12591 .12692 7.8789 .99204 46 15 .12 620 .12 722 7.8606 .99 200 45 16 .12649 .12751 7.8424 .99197 44 17 .12678 .12781 7.8243 .99193 43 18 .12 706 .12 810 7.8062 .99 189 42 19 .12735 .12840 7.7882 .99186 41 2O .12764 .12869 7.7704 .99182 4O 21 .12793 .12899 7.7525 .99178 39 22 .12822 .12929 7.7348 .99175 38 23 .12851 .12958 7.7171 .99171 37 24 .12880 .12988 7.6996 .99167 36 25 .12908 .13017 7.6821 .99163 35 26 .12937 .13047 7.6647 .99160 34 27 .12966 .13076 7.6473 .99156 33 28 .12 995 .13 106 7.6301 .99 152 32 29 .13024 .13136 7.6129 .99148 31 30 .13053 .13165 7.5958 .99144 30 31 .13081 .13 195 7.5787 .99141 29 32 .13110 .13224 7.5618 .99137 28 33 .13 139 .13 254 7.5449 .99 133 27 34 .13 168 .13 284 7.5281 .99 129 26 35 .13197 .13313 7.5113 .99125 25 36 .13226 .13343 7.4947 .99122 24 37 .13254 .13372 7.4781 .99118 23 38 .13283 .13402 7.4615 .99114 22 39 .13 312 .13 432 7.4451 .99 110 21 4O .13341 .13461 7.4287 .99106 20 41 .13370 .13491 7.4124 .99102 19 42 .13 399 .13 521 7.3962 .99 098 18 43 .13427 .13550 7.3300 .99094 17 44 .13456 .13580 7.3639 .99091 16 45 .13485 .13609 7.3479 .99087 15 46 .13514 .13639 7.3319 .99083 14 47 .13543 .13669 7.3160 .99079 13 48 .13 572 .13 698 7.3002 .99 075 12 49 .13 600 .13 728 7.2844 .99 071 11 50 .13629 .13758 7.2687 .99067 1O 51 .13658 .13 787 7.2531 .99063 9 52 .13687 .13817 7.2375 .99059 8 53 .13716 .13846 7.2220 .99055 7 54 .13744 .13876 7.2066 .99051 6 55 .13773 .13906 7.1912 .99047 5 56 .13802 .13935 7.1759 .99043 4 57 .13831 .13965 7.1607 .99039 3 58 .13860 .13995 7.1455 .99035 2 59 .13 889 .14 024 7.1304 .99 031 1 60 .13917 .14054 7.1154 .99027 O / cos cot tan sin / 82 62 NATURAL FUNCTIONS 8 / sin tan cot cos / o .13917 .14054 7.1154 .99027 6O 1 .13946 .14084 7.1004 .99023 59 2 13975 .14113 7.0855 .99019 58 3 .14004 .14143 7.0706 .99015 57 4 .14033 .14173 7.0558 .99011 56 5 .14061 .14202 7.0410 .99006 55 6 .14 090 .14 232 7.0264 .99 002 54 7 .14119 .14262 7.0117 .98998 53 8 .14 148 .14291 6.9972 .98994 52 9 .14177 .14321 6.9827 .98990 51 1O .14205 .14351 6.9682 .98986 5O 11 .14234 .14381 6.9538 .98982 49 12 .14263 .14410 6.9395 .98978 48 13 .14 292 .14 440 6.9252 .98 973 47 14 .14320 .14470 6.9110 .98969 46 15 .14349 .14499 6.8969 .98965 45 16 .14 378 .14 529 6.8828 .98 961 44 17 .14407 .14559 6.8687 .98957 43 18 .14 436 .14 588 6.8548 .98 953 42 19 .14464 .14618 6.8408 .98948 41 2O .14493 .14648 6.8269 .98944 4O 21 .14522 .14678 6.8131 .98940 39 22 .14551 .14707 6.7994 .98936 38 23 .14580 .14737 6.7856 .98931 37 24 .14 608 .14 767 6.7720 .98 927 36 25 .14637 .14796 6.7584 .98923 35 26 .14666 .14826 6.7448 .98919 34 27 .14695 .14856 6.7313 .98914 33 28 .14723 .14886 6.7179 .98910 32 29 .14752 .14915 6.7045 .98906 31 30 .14781 .14945 6.6912 .98902 30 31 .14810 .14975 6.6779 .98897 29 32 .14 838 .15 005 6.6646 .98 893 28 33 .14867 .15034 6.6514 .98889 27 34 .14 896 .15 064 6.6383 .98 884 26 35 .14925 .15094 6.6252 .98880 25 36 .14954 .15 124 6.6122 .98876 24 37 .14 982 .15 153 6.5992 .98 871 23 38 .15011 .15 183 6.5863 .98867 22 39 .15 040 .15 213 6.5734 .98 863 21 40 .15 069 .15 243 6.5606 .98 858 20 41 .15097 .15272 6.5478 .98854 19 42 .15 126 .15 302 6.5350 .98 849 18 43 .15 155 .15 332 6.5223 .98 845 17 44 .15 184 .15 362 6.5097 .98 841 16 45 .15212 .15391 6.4971 .98836 15 46 .15 241 .15 421 6.4846 .98 832 14 47 .15270 .15451 6.4721 .98827 13 48 .15 299 .15 481 6.4596 .98 823 12 49 .15327 .15511 6.4472 .98818 11 50 .15 356 .15 540 6.4348 .98 814 10 51 .15385 .15570 6.4225 .98809 9 52 .15 414 .15 600 6.4103 .98 805 8 53 .15442 .15630 6.3980 .98800 7 54 .15471 .]5660 6.3859 .98796 6 55 .15 500 .15 689 6.3737 .98 791 5 56 .15 529 .15 719 6.3617 .98 787 4 57 .15557 .15749 6.3496 .98782 3 58 .15 586 .15 779 6.3376 .98 778 2 59 .15 615 .15 809 6.3257 .98 773 1 6O .15 643 .15 838 6.3138 .98 769 O / cos cot tan sin / 81 9 / sin tan cot cos / O .15643 .15838 6.3138 .98769 60^ 1 .15 672 .15 868 6.3019 .98 764 59 2 .15 701 .15898 6.2901 .98760 58 3 .15 730 .15928 6.2783 .98755 57 4 .15 758 .15958 6.2666 .98751 56 5 .15 787 .15 988 6.2549 .98 746 55 6 .15816 .16017 6.2432 .98741 54 7 .15845 .16047 6.2316 .98737 53 8 .15873 .16077 6.2200 .98732 52 9 .15 902 .16 107 6.2085 .98 728 51 10 .15931 .16137 6.1970 .98723 50 11 .15 959 .16 167 6.1856 .98 718 49 12 .15988 .16196 6.1742 .98714 48 13 .16017 .16226 6.1628 .98709 47 14 .16046 .16256 6.1515 .98704 46 15 .16074 .16286 6.1402 .98700 45 16 .16103 .16316 6.1290 .98695 44 17 .16132 .16346 6.1178 .98690 43 18 .16 160 .16 376 6.1066 .98 686 42 19 .16189 .16405 6.0955 .98681 41 20 .16218 .16435 6.0844 .98676 40 21 .16246 .16465 6.0734 .98671 39 22 .16275 .16495 6.0624 .98667 38 23 .16304 .16525 6.0514 .98662 37 24 .16333 .16555 6.0405 .98657 36 25 .16361 .16585-6.0296 .98652 35 26 .16390 .16615 6.0188 .98648 34 27 .16419 .16645 6.0080 .98643 33 28 .16447 .16674 5.9972 .98638 32 29 .16476 .16704 5.9865 .98633 31 30 .16505 .16734 5.9758 .98629 30 31 .16533 .16764 5.9651 .98624 29 32 .16562 .16794 5.9545 .98619 28 33 .16591 .16824 5.9439 .98614 27 34 .16620 .16854 5.9333 .98609 26 35 .16648 .16884 5.9228 .98604 25 36 .16677 .16914 5.9124 .98600 24 37 .16706 .16944 5.9019 .98595 23 38 .16734 .16974 5.8915 .98590 22 39 .16763 .17004 5.8811 .98585 21 40 .16792 .17033 5.8708 .98580 2O 41 ..16820 .17063 5.8605 .98575 19 42 .16849 .17093 5.8502 .98570 18 43 .16878 .17123 5.8400 .98565 17 44 .16906 .17153 5.8298 .98561 16 45 .16935 .17183 5.8197 .98556 15 46 .16964 .17213 5.8095 .98551 14 47 .16992 .17243 5.7994 .98546 13 48 .17021 .17273 5.7894 .98541 12 49 .17050 .17303 5.7794 .98536 11 5O .17078 .17333 5.7694 .98531 10 51 .17107 .17363 5.7594 .98526 9 52 .17136 .17393 5.7495 .98521 8 53 .17164 .17423 5.7396 .98516 7 54 .17193 .17453 5.7297 .98511 6 55 .17222 .17483 5.7199 .98506 5 56 .17250 .17513 5.7101 .98501 4 57 .17279 .17543 5.7004 .98496 3 58 .17308 .17573 5.6906 .98491 2 59 .17336 .17603 5.6809 .98486 1 60 .17365 .17633 5.6713 .98481 O / cos cot tan sin / 80 NATURAL FUNCTIONS 63 10 / sin tan cot cos / o .17365 .17633 5.6713 .98481 6O 1 .17393 .17663 5.6617 .98476 59 2 .17422 .17693 5.6521 .98471 58 3 .17451 .17723 5.6425 .98466 57 4 .17479 .17753 5.6329 .98461 56 5 .17 508 .17 783 5.6234 .98 455 55 6 .17537 .17813 5.6140 .98450 54 7 .17565 .17843 5.6045 .98445 53 8 .17594 .17873 5.5951 .98440 52 9 .17623 .17903 5.5857 .98435 51 1O .17651 .17933 5.5764 .98430 50 11 .17680 .17963 5.5671 .98425 49 12 .17 708 .17 993 5.5578 .98 420 48 13 .17737 .18023 5.5485 .98414 47 14 .17766 .18053 5.5393 .98409 46 15 .17794 .18083 5.5301 .98404 45 16 .17823 .18113 5.5209 .98399 44 17 .17852 .18143 5.5118 .98394 43 18 .17880 .18173 5.5026 .98389 42 19 .17909 .18203 5.4936 .98383 41 20 .17937 .18233 5.4845 .98378 40 21 .17 966 .18 263 5.4755 .98 373 39 22 .17 995 .18 293 5.4665 .98 368 38 23 .18023 .18323 5.4575 .98362 37 24 .18052 .18353 5.4486 .98357 36 25 .18081 .18384 5.4397 .98352 35 26 .18 109 .18 414 5.4308 .98 347 34 27 .18 138 .18 444 5.4219 .98 341 33 28 .18166 .18474 5.4131 .98336 32 29 .18195 .18504 5.4043 .98331 31 '30 .18 224 .18 534 5.3955 .98 325 3O 31 .18 252 .18 564 5.3868 .98 320 29 32 .18281 .18594 5.3781 .98315 28 33 .18309 .18624 5.3694 .98310 27 34 .18 338 .18 654 5.3607 .98 304 26 35 .18367 .18684 5.3521 .98299 25 36 .18395 .18714 5.3435 .98294 24 37 .18 424 .18 745 5.3349 .98 288 23 38 .18452 .18775 5.3263 .98283 22 39 .18481 .18805 5.3178 .98277 21 40 .18 509- .18 835 5.3093 .98272 2O 41 .18 538 .18 865 5.3008 .98 267 19 42 .18 567 .18 895 5.2924 .98 261 18 43 .18 595 .18 925 5.2839 .98 256 17 44 .18624 .18955 5.2755 .98250 16 45 .18652 .18986 5.2672 .98245 15 46 .18681 .19016 5.2588 .98240 14 47 .13710 .19046 5.2505 .98234 13 48 .18738 .19076 5.2422 .98229 12 49 .18 767 .19 106 5.2339 .98 223 11 50 .18 795 .19 136 5.2257 .98 218 1O 51 .18 824 .19 166 5.2174 .98 212 9 52 .18 852 .19 197 5.2092 .98 207 8 53 .18881 .19227 5.2011 .98201 7 54 .18910 .19257 5.1929 .98196 6 55 .18938 .19287 5.1848 .98190 5 56 .18967 .19317 5.1767 .98185 4 57 .18995 .19347 5.1686 .98179 3 58 .19024 .19378 5.1606 .98174 2 59 .19052 .19408 5.1526 .98168 1 60 .19 081 .19 438 5.1446 .98 163 / cos cot tan sin / 79 11 / sin tan cot cos / .19081 .19438 5.1446 .98163 60 1 .19109 .19468 5.1366 .98157 59 2 .19138 .19498 5.1286 .98152 58 3 .19 167 .19 529 5.1207 .98 146 57 4 .19195 .19559 5.1128 .98140 56 5 .19 224 .19 589 5.1049 .98 135 55 6 .19 252 .19 619 5.0970 .98 129 54 7 .19281 .19649 5.0892 .98124 53 8 .19309 .19680 5.0814 .98118 52 9 .19 338 .19 710 5.0736 .98 112 51 1O .19 366 .19 740 5.0658 .98 107 5O 11 .19395 .19770 5.0581 .98101 49 12 .19423 .19801 5.0504 .98096 48 13 .19452 .19831 5.0427 .98090 47 14 .19481 .19861 5.0350 .98084 46 15 .19 509 .19 891 5.0273 .98 079 45 16 .19538 .19921 5.0197 .98073 44 17 .19566 .19952 5.0121 .98067 43 18 .19595 .19982 5.0045 .98061 42 19 .19 623 .20 012 4.9969 .98 056 41 2O .19652 .20042 4.9894 .98050 4O 21 .19680 .20073 4.9819 .98044 39 22 .19709 .20103 4.9744 .98039 38 23 .19 737 .20 133 4.9669 .98 033 37 24 .19766 .20164 4.9594 .98027 36 25 .19794 .20194 4.9520 .98021 35 26 .19 823 .20 224 4.9446 .98 016 34 27 .19851 .20254 4.9372 .98010 33 28 .19 880 .20 285 4.9298 .98 004 32 29 .19908 .20315 4.9225 .97998 31 30 .19937 .20345 4.9152 .97992 3O 31 .19 965 .20 376 4.9078 .97 987 29 32 .19994 .20406 4.9006 .97981 28 33 .20022 .20436 4.8933 .97975 27 34 .20051 .20466 4.8860 .97969 26 35 .20079 .20497 4.8788 .97963 25 36 .20 108 .20 527 4.8716 .97 958 24 37 .20 136 .20 557 4.8644 .97 952 23 38 .20 165 .20 588 4.8573 .97 946 22 39 .20 193 .20 618 4.8501 .97 940 21 4O .2022? .20648 4.8430 .97934 2O 41 .20 250 .20 679 4.8359 .97 928 19 42 .20 279 .20 709 4.8288 .97 922 18 43 .20307 .20739 4.8218 .97916 17 44 .20 336 .20 770 4.8147 .97 910 16 45 .20364 .20800 4.8077 .97905 15 46 .20393 .20830 4.8007 .97899 14 47 .20421 .20861 4.7937 .97893 13 48 .20450 .20891 4.7867 .97887 12 49 .20478 .20921 4.7798 .97881 11 5O .20 507 .20 952 4.7729 .97 875 1O 51 .20 535 .20 982 4.7659 .97 869 9 52 .20 563 .21 013 4.7591 .97 863 8 53 .20 592 .21 043 4.7522 .97 857 7 54 .20 620 .21 073 4.7453 .97 851 6 55 .20649 .21 104 4.7385 .97845 5 56 .20677 .21 134 4.7317 .97839 4 57 .20706 .21 164 4.7249 .97833 3 58 .20 734 .21 195 4.7181 .97 827 2 59 .20763 .21225 4.7114 .97821 1 60 .20 791 .21 256 4.7046 .97 815 O / cos cot tan sin / 78 64 NATURAL FUNCTIONS 12 / sin tan cot cos / o .20 791 .21 256 4.7046 .97 815 60^ 1 .20 820 .21 286 4.6979 .97 809 59 2 .20848 .21316 4.6912 .97803 58 3 .20 877 .21 347 4.6845 .97 797 57 4 .20905 .21377 4.6779 .97791 56 5 .20 933 .21 408 4.6712 .97 784 55 6 .20 962 .21 438 4.6646 .97 778 54 7 .20 990 .21 469 4.6580 .97 772 53 8 .21 019 .21 499 4.6514 .97 766 52 9 .21 047 .21 529 4.6448 .97 760 51 1O .21 076 .21 560 4.6382 .97 754 5O 11 .21 104 .21 590 4.6317 .97 748 49 .12 .21 132 .21 621 4.6252 .97 742 48 13 .21 161 .21 651 4.6187 .97 735 47 14 .21 189 .21 682 4.6122 .97 729 46 15 .21 218 .21 712 4.6057 .97 723 45 16 .21 246 .21 743 4.5993 .97 717 44 17 .21 275 .21 773 4.5928 .97 711 43 18 .21 303 .21 804 4.5864 .97 705 42 19 .21331 .21834 4.5800 .97698 41 20 .21 360 .21 864 4.5736 .97 692 4O 21 .21 388 .21 895 4.5673 .97 686 39 22 .21 417 .21 925 4.5609 .97 680 38 23 .21445 .21956 4.5546 .97673 37 24 .21 474 .21 986 4.5483 .97 667 36 25 .21502 .22017 4.5420 .97661 35 26 .21 530 .22 047 4.5357 .97 655 34 27 .21 559 .22 078 4.5294 .97 648 33 28 .21 587 .22 108 4.5232 .97 642 32 29 .21616 .22139 4.5169 .97636 31 30 .21644 .22 169 4.5107 .97630 30 31 .21 "57^.22 200 4.5045 .97623 29 32 .21701 .22231 4.4983 .97617 28 33 .21 729 .22261 4.4922 .97611 27 34 .21 758 .22 292 4.4860 .97 604 26 35 .21 786 .22 322 4.4799 .97 598 25 36 .21 814 .22 353 4.4737 .97 592 24 37 .21 843 .22 383 4.4676 .97 585 23 38 .21871 .22414 4.4615 .97579 22 39 .21 899 .22 444 4.4555 .97 573 21 4O .21928 .22475 4.4494 .97566 20 41 .21 956 .22 505 4.4434 .97 560 19 42 .21 985 .22 536 4.4373 .97 553 18 43 .22013 .22567 4.4313 .97547 17 44 .22 041 .22 597 4.4253 .97 541 16 45 .22 070 .22 628 4.4194 .97 534 15 46 .22 098 .22 658 4.4134 .97 528 14 47 .22 126 .22 689 4.4075 .97 521 13 48 .22155 .22719 4.4015 .97515 12 49 .22 183 .22 750 4.3956 .97 508 11 50 .22 212 .22 781 4.3897 .97 502 1O 51 .22240 .22811 4.3838 .97496 9 52 .22 268 .22 842 4.3779 .97 489 8 53 .22297 .22872 4.3721 .97483 7 54 .22' 325 .22903 4.3662 .97476 6 55 .22 353 .22 934 4.3604 .97 470 5 56 .22 382 .22 964 4.3546 .97 463 4 57 .22410 .22995 4.3488 .97457 3 58 .22 438 .23 026 4.3430 .97 450 2 59 .22 467 .23 056 4.3372 .97 444 1 6O .22 495 .23 087 4.3315 .97 437 / cos cot tan sin / 77 13 / sin tan cot cos / O .22495 .23087 4.3315 .97437 60 1 .22523 .23117 4.3257 .97430 59 2 .22 552 .23 148 4.3200 .97 424 58 3 .22580 .23179 4.3143 .97417 57 4 .22 608 .23 209 4.3086 .97 411 56 5 .22 637 .23 240 4.3029 .97 404 55 6 .22 665 .23 271 4.2972 .97 398 54 7 .22 693 .23 301 4.2916 .97 391 53 8 .22 722 .23 332 4.2859 .97 384 52 9 .22 750 .23 363 4.2803 .97 378 51 10 .22 778 .23 393 4.2747 .97 371 5O 11 .22 807 .23 424 4.2691 .97 365 49 12 .22 835 .23 455 4.2635. .97 358 48 13 .22863 .23485 4.2580 .97351 47 14 .22892 .23 516 4.2524 .97345 46 - 15 .22 920 .23 547 4.2468 .97 338 45 16 .22948 .23578 4.2413 .97331 44 17 .22 977 .23 608 4.2358 .97 325 43 18 .23 005 .23 639 4.2303 .97 318 42 19 .23033 .23670 4.2248 .97311 41 2O .23 062 .23 700 4.2193 .97 304 4O 21 .23 090 .23 731 4.2139 .97 298 39 22 .23 118 .23 762 4.2084 .97291 38 23 .23 146 .23 793 4.2030 .97 284 37 21 .23 175 .23 823 4.1976 .97 278 36 25 .23 203 .23 854 4.1922 .97 271 35 26 .23 231 .23 885 4.1868 .97 264 34 27 .23 260 .23 916 4.1814 .97 257 33 28 .23 288 .23 946 4.1760 .97 251 32 29 .23 316 .23 977 4.1706 .97 244 31 3O .23345 .24008 4.1653 .97237 3O 31 .23373 .24039 4.1600 .97230 29 32 .23 401 .24 069 4.1547 .97 223 28 33 .23 429 .24 100 4.1493 .97 217 27 34 .23 458 .24 131 4.1441 .97 210 26 35 .23 486 .24 162 4.1388 .97 203 25 36 .23 514 .24 193 4.1335 .97 196 24 37 .23 542 .24 223 4.1282 .97 189 23 38 .23 571 .24 254 4.1230 .97 182 22 39 .23 599 .24 285 4.1178 .97 176 21 4O .23627 .24316 4.1126 .97169 2O 41 .23 656 .24 347 4.1074 .97 162 19 42 .23 684 .24 377 4.1022 .97 155 18 43 .23 712 .24 408 4.0970 .97 148 17 44 .23 740 .24 439 4.0918 .97 141 16 45 .23 769 .24 470 4.0867 .97 134 15 46 .23797 .24 501 4.0815 .97127 14 47 .23 825 .24 532 4.0764 .97 120 13 48 .23 853 .24 562 4.0713 .97 113 12 49 .23 882 .24 593 4.0662 .97 106 11 50 .23910 .24624 4.0611 .97100 10 51 .23 938 .24 655 4.0560 .97 093 9 52 .23 966 .24 686 4.0509 .97 086 8 53 .23 995 .24 717 4.0459 .97 079 7 54 .24 023 .24 747 4.0408 .97 072 6 55 .24051 .24778 4.0358 .97065 5 56 .24 079 .24 809 4.0308 .97 058 4 57 .24 108 .24 840 4.0257 .97 051 3 58 .24 136 .24 871 4.0207 .97 044 2 59 .24 164 .24 902 4.0158 .97 037 1 60 .24192 .24933 4.0108 .97030 O / cos cot tan sin / 76 NATURAL FUNCTIONS 65 14 / sin tan cot cos / o .24 192 .24 933 4.0108 .97 030 6O 1 .24 220 .24 964 4.0058 .97 023 59 2 .24249 .24995 4.0009 .97015 58 3 .24 277 .25 026 3.9959 .97 008 57 4 .24 305 .25 056 3.9910 .97 001 56 5 .24 333 .25 087 3.9861 .96 994 55 6 .24362 .25 118 3.9812 .96987 54 7 .24 390 .25 149 3.9763 .96 980 53 8 .24 418 .25 180 3.9714 .96973 52 9 .24446 .25211 3.9665 .96966 51 10 .24 474 .25 242 3.9617 .96 959 5O 11 .24 503 .25 273 3.9568 .96 952 49 12 .24 531 .25 304 3 9520 .96 945 48 13 .24559 .25335 3.9471 .96937 47 14 .24587 .25*366 3.9423 .96930 46 15 .24 615 .25 397 3.9375 .96 923 45 16 .24644 .25428 3.9327 .96916 44 17. .24 672 .25 459 3.9279 .96 909 43 18 .24 700 .25 490 3.9232 .96 902 42 19 .24 728 .25 521 3.9184 .96 894 41 2O .24756 .25552 3.9136 .96887 4O 21 .24 784 .25 583 3.9089 .96 880 39 22 .24 813 .25 614 3.9042 .96 873 38 23 .24841 .25645 3.8995 .96866 37 24 .24 869 .25 676 3.8947 .96 858 36 25 .24897 .25707 3.8900 .96851 35 26 .24 925 .25 738 3.8854 .96 844 34 27 .24 954 .25 769 3.8807 .96 837 33 | 28 .24982 .25800 3.8760 .96829 32 29 .25010 .25831 3.8714 .96822 31 3D .25038 .25862 3.8667 .96815 30 31 .25066 .25893 3.8621 .96807 29 32 .25094 .25924 3.8575 .96800 28 33 .25 122 .25 955 3.8528 .96 793 27 34 .25 151 .25 986 3.8482 .96 786 26 35 .25179 .26017 3.8436 .96778 25 36 .25 207 .26 048 3.8391 .96 771 24 37 .25 235 .26 079 3.8345 .96 764 23 38 .25 263 .26 110 3.8299 .96 756 22 39 .25 291 .26 141 3.8254 .96 749 21 4O .25 320 .26 172 3.8208 .96 742 2O 41 .25348 .26203 3.8163 .96734 19 42 .25376 .26235 3.8118 .96727 18 43 .25 404 .26 266 3.8073 .96 719 17 44 .25432 .26297 3.8028 .96712 16 45 .25 460 .26 328 3.7983 .96 705 15 46 .25 488 .26 359 3.7938 .96 697 14 47 .25516 .26390 3.7893 .96690 13 48 .25 545 .26 421 3.7848 .96 682 12 49 .25573 .26452 3.7804 .96675 11 50 .25 601 .26 483 3.7760 .96 667 10 51 .25629 .26515 3.7715 .96660 9 52 .25657 .26546 3.7671 .96653 8 53 .25685 .26577 3.7627 .96645 7 54 .25713 .26608 3.7583 .96638 6 55 .25741 .26639 3.7539 .96630 5 56 .25 769 .26 670 3.7495 .96 623 4 57 .25798 .26701 3.7451 .96615 3 58 .25 826 .26 733 3.7408 .96 608 2 59 .25 854 .26 764 3.7364 .96 600 1 60 .25882 .26795 3.7321 .96593 O / cos cot tan sin / 75 15 / sin tan cot cos / O .25 882 .26 795 3.7321 .96 593 60 1 .25 910 .26 826 3.7277 .96 585 59 2 .25 938 .26 857 3.7234 .96 578 58 3 .25 966 .26 888 3.7191 .96 570 57 4 .25 994 .26 920 3.7148 .96 562 56 5 .26022 .26951 3.7105 .96555 55 6 .26050 .26982 3.7062 .96547 54 7 .26079 .27013 3.7019 .96540 53 8 .26 107 .27 044 3.6976 .96 532 52 9 .26 135 .27 076 3.6933 .96 524 51 1O .26163 .27107 3.6891 .96517 5O 11 .26 191 .27 138 3.6848 .96 509 49 12 .26 219 .27 169 3.6806 .96 502 48 13 .26247 .27201 3.6764 .96494 47 14 .26 275 .27 232 3.6722 .96 486 46 15 .26303 .27263 3.6680 .96479 45 16 .26331 .27294 3.6638 .96471 44 17 .26 359 .27 326 3.6596 .96 463 43 18 .26387 .27357 3.6554 .96456 42 19 .26415 .27388 3.6512 .96448 41 2O .26443 .27419 3.6470 .96440 4O 21 .26471 .27451 3.6429 .96433 39 22 .26500 .27482 3.6387 .96425 38 23 .26528 .27513 3.6346 .96417 37 24 .26556 .27545 3.6305 .96410 36 25 .26584 .27576 3.6264 .96402 35 26 .26612 .27607 3.6222 .96394 34 27 .26 640 .27 638 3.6181 .96 386 33 28 .26 668 .27 670 3.6140 .96 379 32 29 .26 696 .27 701 3.6100 .96 371 31 30 .26 724 .27 732 3.6059 .96 363 3O 31 .26 752 .27 764 3.6018 .96 355 29 32 .26 780 .27 795 3.5978 .96 347 28 33 .26 808 .27 826 3.5937 .96 340 27 34 .26 836 .27 858 3.5897 .96 332 26 35 .26 864 .27 889 3.5856 .96 324 25 36 .26892 .27921 3.5816 .96316 24 37 .26920 .27952 3.5776 .96308 23 38 .26948 .27983 3.5736 .96301 22 39 .26976 .28015 3.5696 .96293 21 4O .27004 .28046 3.5656 .96285 2O 41 .27032 .28077 3.5616 .96277 19 42 .27 060 .28 109 3.5576 .96 269 18 43 .27 088 .28 140 3.5536 .96 261 17 44 .27 116 .28 172 3.5497 .96 253 16 45 .27 144 .28 203 3.5457 .96 246 15 46 .27 172 .28 234 3.5418 .96 238 14 47 .27 200 .28 266 3.5379 .96 230 13 48 .27 228 .28 297 3.5339 .96 222 12 49 .27256 .28329 3.5300 .96214 11 50 .27284 .28360 3.5261 .96206 10 51 .27312 .28391 3.5222 .96198 9 52 .27340 .28423 3.5183 .96190 8 53 .27 368 .28 454 3.5144 .96 182 7 54 .27396 .28486 3.5105 .96174 6 55 .27424 .28517 3.5067 .96166 5 56 .27 452 .28 549 3.5028 .96 158 4 57 .27480 .28580 3.4989 .96150 3 58 .27508 .28612 3.4951 .96142 2 59 .27536 .28643 3.4912 .96134 1 6O .27564 .28675 3.4874 .96126 O / cos cot tan sin / 74 66 NATURAL FUNCTIONS 16 / sin tan cot cos / |O .27 564 .28 675 3.4874 .96 126 6O 1 .27592 .28706 3.4836 .96118 59 2 .27620 .28738 3.4798 .96110 58 3 .27 648 .28 769 3.4760 .96 102 57 4 .27676'.28801 3.4722 .96094 56 5 .27 704 .28 832 3.4684 .96 086 55 6 .27731 .28864 3.4646 .96078 54 7 .27759 .28895 3.4608 .96970 53 8 .27787 .28927 3.4570 .96062 52 9 .27815 .28958 3.4533 .96954 51 10 .27843 .28990 3.4495 .96046 5O 11 .27871 .29021 3.4458 .96037 49 12 .27899 .29053 3.4420 .96029 48 13 .27927 .29084 3.4383 .96021 47 14 .27955 .29116 3.4346 .96013 46 15 .27 983 .29 147 3.4308 .96 005 45 16 .28 Oil .29 179 3.4271 .95 997 44 17 .28 039 .29 210 3.4234 .95 989 43 18 .28 067 .29 242 3.4197 .95 981 42 19 .28095 .29274 3.4160 .95972 41 20 .28 123 .29 305 3.4124 .95 964 40 21 .28150 .29337 3.4087 .95956 39 22 .28178 .29368 3.4050 .95948 38 23 .28206 .29400 3.4014 .95940 37 24 .28234 .29432 3.3977 .95931 36 25 .28 262 .29 463 3.3941 .95 923 35 26 .28 290 .29 495 3.3904 .95 915 34 27 .28 318 .29 526 3.3868 .95 907 33 28 .28346 .29558 3.3832 .95898 32 29 .28 374 .29 590 3.3796 .95 890 31 30 .28402 .29621 3.3759 .95882 30 31 .28429 .29653 3.3723 .95874 29 32 .28457 .29685 3.3687 .95865 28 33 .28485 .29716 3.3652 .95857 27 34 .28513 .29748 3.3616 .95849 26 35 .28 541 .29 780 3.3580 .95 841 25 36 .28569 .29811 3.3544 .95832 24 37 .28 597 .29 843 3.3509 .95 824 23 38 .28625 .29875 3.3473 .95816 22 39 .28 652 .29 906 3.3438 .95 807 21 4O .28 680 .29 938 3.3402 .95 799 20 41 .28708 .29970 3.3367 .95791 19 42 .28 736 .30 001 3.3332 .95 782 18 43 .28 764 .30 033 3.3297 .95 774 17 44 .28 792 .30 065 3.3261 .95 766 16 45 .28820 .30097 3.3226 .95757 15 46 .28 847 .30 128 3.3191 .95 749 14 47 .28875 .30160 3.3156 .95740 13 48 .28903 .30192 3.3122 .95732 12 49 .28931 .30224 3.3087 .95724 11 50 .28 959 .30 255 3.3052 .95 715 1O 51 .28 987 .30 287 3.3017 .95 707 9 52 .29015 .30319 3.2983 .95698 8 53 .29042 .30351 3.2948 .95690 7 54 .29 070 .30 382 3.2914 .95 681 6 55 .29 098 .30 414 3.2879 .95 673 5 56 .29 126 .30 446 3.2845 .95 664 4 1 57 .29154 .30478 3.2811 .95656 3 58 .29 182 .30 509 3.2777 .95 647 2 59 .29 209 .30 541 3.2743 .95 639 1 6O .29237 .30573 3.2709 .95630 O / cos cot tan sin / 73 17 / sin tan cot cos / O .29237 .30573 3.27Q9 .95630 GO 1 .29265 .30605 3.2675 .95622 59 2 .29293 .30637 3.2641 .95613 58 3 .29321 .30669 3.2607 .95605 57 4 .29348 .30700 3.2573 .95596 56 5 .29376 .30732 3.2539 .95588 55 6 .29404 .30764 3.2506 .95579 54 7 .29432 .30796 3.2472 .95571 53 8 .29460 .30828 3.2438 .95562 52 9 .29487 .30860 3.2405 .95554 51 1O .29515 .30891 3.2371 .95545 50 11 .29543 .30923 3.2338 .95536 49 12 .29571 .30955 3.2305 .95528 48 13 .29599 .30987 3.2272 .95519 47 14 .29626 .31019 3.2238 .95511 46 15 .29654 .31051 3.2205 .95502 45 16 .29682 .31083 3.2172 .95493 44 17 .29710 .31115 3.2139 .95485 43 18 .29 737 .31 147 3.2106 .95 476 42 19 .29765 .31 178 3.2073 .95467 41 20 .29 793 .31 210 3.2041 .95 459 40 21 .29 821 .31 242 3.2008 .95 450 39 22 .29849 .31274 3.1975 .95441 38 23 .29876 .31306 3.1943 .95433 37 24 .29904 .31338 3.1910 .95424 36 25 .29932 .31370 3.1878 .95415 35 26 .29960 .31402 3.1845 .95407 34 27 .29987 .31434 3.1813 .95398 33 28 .30015 .31466 3.1780 .95389 32 29 .30043 .31498 3.1748 .95380 31 30 .30071 .31530 3.1716 .95372 30 31 .30098 .31562 3.1684 .95363 29 32 .30 126 .31 594 3.1652 .95 354 28 33 .30 154 .31 626 3.1620 .95 345 27 34 .30 182 .31 658 3.1588 .95 337 26 35 .30 209 .31 690 3.1556 .95 328 25 36 .30 237 .31 722 3.1524 .95 319 24 37 .30 265 .31 754 3.1492 .95 310 23 38 .30292 .31786 3.1460 .95301 22 39 .30 320 .31 818 3.1429 .95 293 21 40 .30348 .31850 3.1397 .95284 2O 41 .30376 .31882 3.1366 .95275 19 42 .30403 .31914 3.1334 .95266 18 43 .30431 .31946 3.1303 .95257 17 44 .30459 .31978 3.1271 .95248 16 45 .30486 .32010 3.1240 .95240 15 46 .30514 .32042 3.1209 .95231 14 47 .30542 .32074 3.1178 .95222 13 48 .30570 .32106 3.1146 .95213 12 49 .30597 .32139 3.1115 .95204 11 50 -.30625 .32171 3.1084 .95 195 10 51 .30653 .32203 3.1053 .95 186 9 52 .30680 .32235 3.1022 .95 177 8 53 .30 708 .32 267 3.0991 .95 168 7 54 .30 736 .32 299 3.0961 .95 159 6 55 .30763 .32331 3.0930 .95150 5 56 .30 791 .32 363 3.0S99 .95 142 4 57 .30 819 .32 396 3.0868 .95 133 3 58 .30 846 .32 428 3.0838 .95 124 2 59 .30874 .32460 3.0807 .95 115 1 6O .30902 .32492 3.0777 .95 106 O / cos cot tan sin / 72 NATURAL FUNCTIONS 6T 18 / sin tan cot cos / O .30 902 .32 492 3.0777 .95 106 6O 1 .30 929 .32 524 3.0746 .95 097 59 2 .30957 .32556 3.0716 .95088 58 3 .30985 .32588 3.0686 .95079 57 4 .31012 .32621 3.0655 .95070 56 5 .31 040 .32 653 3.0625 .95 061 55 6 .31068 .32685 3.0595 .95052 54 7 .31 095 .32 717 3.0565 .95 043 53 8 .31 123 .32 749 3.0535 .95 033 52 9 .31151 .32782 3.0505 .95024 51 10 .31 178 .32 814 3.0475 .95 015 50 11 .31 206 .32 846 3.0445 .95 006 49 12 .31233 .32878 3.0415 .94997 48 13 .31261 .32911 3.0385 .94988 47 14 .31289 .32943 3.0356 .94979 46 15 .31316 .32975 3.0326 .94970 45 >16 .31 344 .33 007 3.0296 .94 961 44 17 .31 372 .33 040 3.0267 .94 952 43 18 .31399 .33072 3.0237 .94943 42 19 .31 427 .33 104 3.0208 .94 933 41 2O .31 454 .33 136 3.0178 .94924 40 21 .31482 .33169 3.0149 .94915 39 22 .31510 .33201 3.0120 .94906 38 23 .31 537 .33 233 3.0090 .94 897 37 24 .31565 .33266 3.0061 .94888 36 25 .31593 .33298 3.0032 .94878 35 26 .31620 .33330 3.0003 .94869 34 27 .31648 .33363 2.9974 .94860 33 28 .31675 .33395 2.9945 .94851 32 29 r. 31 703 .33427 2.9916 .94842 31 30 \31730 .33460 2.9887 .94832 30 31 .31 758 .33 492 2.9858 .94 823 29 32 .31 786 .33 524 2.9829 .94 814 28 33 .31 813 .33 557 2.9800 .94 805 27 34 .31 841 .33 589 2.9772 .94 795 26 35 .31 868 .33 621 2.9743 .94 786 25 36 .31 896 .33 654 2.9714 .94 777 24 37 .31923 .33686 2.9686 .94768 23 38 .31951 .33718 2.9657 .94758 22 39 .31 979 .33 751 2.9629 .94 749 21 4O .32 006 .33 783 2.9600 .94 740 2O 41 .32 034 .33 816 2.9572 .94 730 19 42 .32 061 .33 848 2.9544 .94 721 18 43 .32089 .33881 2.9515 .94712 17 44 .32116 .33913 2.9487 .94702 16 45 .32144 .33945 2.9459 .94693 15 46 .32 171 .33 978 2.9431 .94 684 14 47 .32 199 .34 010 2.9403 .94 674 13 48 .32227 .34043 2.9375 .94665 12 49 .32 254 .34 075 2.9347 .94 656 11 50 .32282 .34108 2.9319 .94646 1O 51 .32 309 .34 140 2.9291 .94 637 9 52 .32 337 .34 173 2.9263 .94 627 8 53 .32364 .34205 2.9235 .94618 7 54 .32392 .34238 2.9208 .94609 6 55 .32 419 .34 270 2.9180 .94 599 5 56 .32447 .34303 2.9152 .94590 4 57 .32474 .34335 2.9125 .94580 3 58 .32502 .34368 2.9097 .94571 2 59 .32 529 .34 400 2.9070 .94 561 1 6O .32557 .34433 2.9042 .94552 / cos cot tan sin / 71 19 / sin tan cot cos / o .32557 .34433 2.9042 .94552 6O 1 .32 584 .34 465 2.9015 .94 542 59 2 .32612 .34498 2.8987 .94533 58 3 .32 639 .34 530 2.8960 .94 523 57 4 .32667 .34563 2.8933 .94514 56 5 .32 694 .34 596 2.8905 .94 504 55 6 .32 722 .34 628 2.8878 .94 495 54 7 .32749 .34661 2.8851 .94485 53 8 .32 777 .34 693 2.8824 .94 476 52 9 .32 804 .34 726 2.8797 .94 466 51 10 .32832 .34758 2.8770 .94457 5O 11 .32859 .34791 2.8743 .94447 49 12 .32887 .34824 2.8716 .94438 48 13 .32914 .34856 2.8689 .94428 47 14 .32942 .34889 2.8662 .94418 46 15 .32969 .34922 2.8636 .94409 45 16 .32997 .34954 2.8609 .94399 44 17 .33 024 .34 987 2.8582 .94 390 43 18 .33051 .35020 2.8556 .94380 42 19 .33 079 .35 052 2.8529 .94 370 41 20 .33 106 .35 085 2.8502 .94 361 4O 21 .33134 .35118 2.8476 .94351 39 22 .33 161 .35 150 2.8449 .94 342 38 23 .33 189 .35 183 2.8423 .94 332 37 24 .33216 .35216 2.8397 .94322 36 25 .33 244 .35 248 2.8370 .94 313 35 26 .33 271 .35 281 2.8344 .94 303 34 27 .33298 .35314 2.8318 .94293 33 28 .33 326 .35 346 2.8291 .94 284 32 29 .33 353 .35 379 2.8265 .94 274 31 3O .33 381 .35 412 2.8239 .94 264 30 31 .33 408 .35 445 2.8213 .94 254 29 32 .33436 .35477 2.8187 .94245 28 33 .33463 .35510 2.8161 .94235 27 34 .33 490 .35 543 2.8135 .94 225 26 35 .33 518 .35 576 2.8109 .94 215 25 36 .33 545 .35 608 2.8083 .94 206 24 37 .33 573 .35 641 2.8057 .94 196 23 38 .33 600 .35 674 2.8032 .94 186 22 39 .33 627 .35 707 2.8006 .94 176 21 4O .33 655 .35 740 2.7980 .94 167 2O 41 .33 682 .35 772 2.7955 .94 157 19 42 .33 710 .35 805 2.7929 .94 147 18 43 .33 737 .35 838 2.7903 .94 137 17 44 .33 764 .35 871 2.7878 .94 127 16 45 .33 792 .35 904 2.7852 .94 118 15 46 .33 819 .35 937 2.7827 .94 108 14 47 .33846 .35969 2.7801 .94098 13 48 .33 874 .36 002 2.7776 .94 088 12 49 .33901 .36035 2.7751 .94078 11 50 .33 929 .36 068 2.7725 .94 068 1O 51 .33 956 .36 101 2.7700 .94 058 9 52 .33 983 .36 134 2.7675 .94 049 8 53 .34011 .36167 2.7650 .94039 7 54 .34 038 .36 199 2.7625 .94 029 6 55 .34065 .36232 2.7600 .94019 5 56 .34093 .36265 2.7575 .94009 4 57 .34120 .36298 2.7550 .93999 3 58 .34147 .36331 2.7525 .93989 2 59 .34 175 .36 364 2.7500 .93 979 1 6O .34 202 .36 397 2.7475 .93 969 / cos cot tan sin / 70 68 NATURAL FUNCTIONS 20 / sin tan cot cos / o .34202 .36397 2.7475 .93969 6O 1 .34 229 .36 430 2.7450 .93 959 59 2 .34 257 .36 463' 2.7425 .93 949 58 3 .34 284 .36 496 2.7400 .93 939 57 4 .34311 .36529 2.7376 .93929 56 5 .34339 .36562 2.7351 .93919 55 6 .34 366 .36 595 2.7326 .93 909 54 7 .34 393 .36 628 2.7302 .93 899 53 8 .34 421 .36 661 2.7277 .93 889 52 9 .34 448 .36 694 2.7253 .93 879 51 10 .34475 .36727 2.7228 .93869 50 11 .34 503 .36 760 2.7204 .93 859 49 12 .34 530 .36 793 2.7179 .93 849 48 13 .34 557 .36 826 2.7155 .93 839 47 14 .34584 .36859 2.7130 .93829 46 15 .34612 .36892 2.7106 .93819 45 16 .34 639 .36 925 2.7082 .93 809 44 17 .34666 .36958 2.7058 .93799 43 18 .34 694 .36 991 2.7034 .93 789 42 19 .34 721 .37 024 2.7009 .93 779 41 20 .34748 .37057 2.6985 .93769 4O 21 .34.775 .37090 2.6961 .93759 39 22 .34 803 .37 123 2.6937 .93 748 38 23 .34830 .37157 2.6913 .93738 37 24 .34 857 .37 190 2.6889 .93 728 36 25 .34 884 .37 223 2.6865 .93 718 35 26 .34 912 .37 256 2.6841 .93 708 34 27 .34 939 .37 289 2.6818 .93 698 33 28 .34 966 .37 322 2.6794 .93 688 32 29 .34 993 .37 355 2.6770 .93 677 31 30 .35 021 .37 388 2.6746 .93 667 30 31 .35048 .37422 2.6723 .93657 29 32 .35 075 '.37 455 2.6699 .93 647 28 33 .35 102 .37 488 2.6675 .93 637 27 34 .35 130 .37 521 2.6652 .93 626 26 35 .35157 .37554 2.6628 .93616 25 36 .35 184 .37 588 2.6605 .93 606 24 37 .35211 .37621 2.6581 .93596 23 38 .35 239 .37 654 2.6558 .93 585 22 39 .35 266 .37 687 2.6534 .93 575 21 40 .35 293 .37 720 2.6511 .93 565 20 41 .35 320 .37 754 2.6488 .93 555 19 42 ;35 347 .37 787 2.6464 .93 544 18 43 .35 375 .37 820 2.6441 .93 534 17 44 .35402 .37853 2.6418 .93524 16 45 .35429 .37887 2.6395 .93514 15 46 .35 456 .37 920 2.6371 .93 503 14 47 .35 484 .37 953 2.6348 .93 493 13 48 .35 511 .37 986 2.6325 .93 483 12 49 .35 538 .38 020 2.6302 .93 472 11 50 .35565 .38053 2.6279 ;93 462 1O 51 .35 592 .38 086 2.6256 .93 452 9 52 .35 619 .38 120 2.6233 .93 441 8 53 35647 .38153 2.6210 .93431 7 54 .35 674 .38 186 2.6187 .93 420 6 55 .35 701 .38 220 2.6165 .93 410 5 56 .35 728 .38 253 2.6142 .93 400 4 57 .35 755 .38 286 2.6119 .93 389 3 58 .35 782 .38320 2.6096 .93379 2 59 .35 810 .38 353 2.6074 .93 368 1 6O .35837 .38386 2.6051 .93358 O / cos cot tan sin / 69 21 / sin tan cot cos / O .35837 .38386 2.6051 .93358 6O 1 .35 864 .38 420 2.6028 .93 348 59 2 .35891 .38453 2.6006 .93337 58 3 .35 918 .38 487 2.5983 .93 327 57 4 .35 945 .38 520 2.5961 .93 316 56 5 .35 973 .38 553 2.5938 .93 306 55 6 .36 000 .38 587 2.5916 .93 295 54 7 .36 027 .38 620 2.5893 .93 285 53 8 .36054 .38654 2.5871 .93274 52 9 .36 081 .38 687 2.5848 .93 264 51 1O .36 108 .38 721 2.5826 .93 253 5O 11 .36 135 .38 754 2.5804 .93 243 49 12 .36 162 .38 787 2.5782 .93 232 48 13 .36 190 .38 821 2.5759 .93 222 47 14 .36217 .38854 2.5737 .93211 46 15 .36 244 .38 888 2.5715 .93 201 45 16 .36 271 .38 921 2.5693 .93 190 44 17 .36 298 .38 955 2.5671 .93 180 43 18 .36 325 .38 988 2.5649 .93 169 42 19 .36 352 .39 022 2.5627 .93 159 41 20 .36 379 .39 055 2.5605 .93 148 4O 21 .36406 .39089 2.5583 .93137 39 22 .36 434 .39 122 2.5561 .93 127 38 23 .36461 .39156 2.5539 .93 116 37 24 .36488 .39190 2.5517 .93106 36 25 .36515 .39223 2.5495 .93095 35 26 .36542 .39257 2.5473 .93084 34 27 .36 569 .39 290 2.5452 .93 074 33 28 .36 596 .39 324 2.5430 .93 063 32 29 .36623 .39357 2.5408 .93052 31 30 .36 650 .39 391 2.5386 .93 042 30 31 .36677 .39425 2.5365 .93031 29 32 .36 704 .39 458 2.5343 .93 020 28 33 .36731 .39492 2.5322 .93010 27 34 .36 758 .39 526 2.5300 .92 999 26 35 .36785 .39559 2.5279 .92988 25 36 .36812 .39593 2.5257 .92978 24 37 .36 839 .39 626 2.5236 .92 967 23 38 .36867 .39660 2.5214 .92956 22 39 .36894 .39694 2.5193 .92945 21 4O .36921 .39727 2.5172 .92935 2O 41 .36948 .39761 2.5150 .92924 19 42 .36975 .39795 2.5129 .92913 18 43 .37002 .39829 2.5108 .92902 17 44 .37029 .39862 2.5086 .92892 16 45 .37 056 .39 896 2.5065 .92 881 15 46 .37 083 .39 930 2.5044 .92 870 14 47 .37110 .39963 2.50Z3 .92859 13 48 .37 137 .39 997 2.5002 .92 849 12 49 .37 164 .40 031 2.4981 .92 838 11 5O .37191 .40065 2.4960 .92827 10 51 .37218 .40098 2.4939 .92816 9 52 .37 245 .40 132 2.4918 .92 805 8 53 .37 272 .40 166 2.4897 .92 794 7 54 .37 299 .40 200 2.4876 .92 784 6 55 .37 326 .40 234 2.4855 .92 773 5 56 .37 353 .40 267 2.4834 .92 762 4 57 .37380 .40301 2.4813 .92751 3 58 .37407 .40335 2.4792 .92740 2 59 .37 434 .40 369 2.4772 .92 729 1 6O .37461 .40403 2.4751 .92718 / cos cot tan sin / 68 NATURAL FUNCTIONS 69 22 / sin tan cot cos / o .37461 .40403 2.4751 .92718 60 1 .37 488 .40 436 2.4730 .92 707 59 2 .37515 .40470 2.4709 .92697 58 3 .37 542 .40 504 2.4689 .92 686 57 4 .37 569 .40 538 2.4668 .92 675 56 5 .37595 .40572 2.4648 .92664 55 6 .37622 .40606 2.4627 .92653 54 7 .37649 .40640 2.4606 .92642 53 8 .37676 .40674 2.4586 .92631 52 9 .37 703 .40 707 2.4566 .92 620 51 1C .37 730 .40 741 2.4545 .92 609 50 11 .37757 .40775 2.4525 .92598 49 12 .37 784 .40 809 2.4504 .92 587 48 13 .37811 .40843 2.4484 .92576 47 14 .37 838 .40 877 2.4464 .92 565 46 15 .37865 .40911 2.4443 .92554 45 16 .37892 .40945 2.4423 .92543 44 17 .37 919 .40 979 2.4403 .92 532 43 18 .37 946 .41 013 2.4383 .92 521 42 19 .37973 .41047 2.4362 .92510 41 20 .37 999 .41 081 2.4342 .92 499 4O 21 .38 026 .41 115 2.4322 .92 488 39 22 .38 053 .41 149 2.4302 .92 477 38 23 .38 080 .41 183 2.4282 .92 466 37 24 .38 107 .41 217 2.4262 .92 455 36 25 .38134 .41251 2.4242 .92444 35 26 .38161 .41285 2.4222 .92432 34 27 .38 188 .41 319 2.4202 .92 421 33 28 .38215 .41353 2.4182 .92410 32 29 .38 241 .41 387 2.4162 .92 399 31 30 .38268 .41421 2.4142 .92388 30 31 .38295 .41455 2.4122 .92377 29 32 .38 322 .41 490 2.4102 .92 366 28 33 .38349 .41524 2.4083 .92355 27 34 .38 376 .41 558 2.4063 .92 343 26 35 .38 403 .41 592 2.4043 .92 332 25 36 .38430 .41626 2.4023 .92321 24 37 .38 456 .41 660 2.4004 .92 310 23 38 .38 483 .41 694 2.3984 .92 299 22 39 .38 510 .41 728 2.3964 .92 287 21 40 .38 537 .41 763 2.3945 .92 276 2O 41 .38 564 .41 797 2.3925 .92 265 19 42 .38 591 .41 831 2.3906 .92 254 18 43 .38 617 .41 865 2.3886 .92 243 17 44 .38644 .41899 2.3867 .92231 16 45 .38671 .41933 2.3847 .92220 15 46 .38 698 .41 968 2.3828 .92 209 14 47 .38 725 .42 002 2.3808 .92 198 13 48 .38 752 .42 036 2.3789 .92 186 12 49 .38 778 .42 070 2.3770 .92 175 11 5O .38 805 .42 105 2.3750 .92 164 1O 51 .38 832 .42 139 2.3731 .92 152 9 52 .38 859 .42 173 2.3712 .92 141 8 53 .38 886 .42 207 2.3693 .92 130 7 54 .38912 .42242 2.3673 .92119 6 55 .38 939 .42 276 2.3654 .92 107 5 56 .38966 .42310 2.3635 .92096 4 57 .38993 .42345 2.3616 .92085 3 58 .39020 .42379 2.3597 .92073 2 59 .39046 .42413 2.3578 .92062 1 6O .39073 .42447 2.3559 .92050 O / cos cot tan sin / 67 23 / sin tan cot cos / .39073 .42447 2.3559 .92050 60 1 .39 100 .42 482 2.3539 .92 039 59 2 .39 127 .42 516 2.3520 .92 028 58 3 .39153 .42551 2.3501 .92016 57 4 .39 180 .42 585 2.3483 .92 005 56 5 .39207 .42619 2.3464 .91994 55 6 .39 234 .42 654 2.3445 .91 982 54 7 .39 260 .42 688 2.3426 .91 971 53 8 .39 287 .42 722 2.3407 .91 959 52 9 .39 314 .42 757 2.3388 .91 948 51 10 .39 341 .42 791 2.3369 .91 936 5O 11 .39 367 .42 826 2.3351 .91 925 49 12 .39 394 .42 860 2.3332 .91 914 48 13 .39421 .42894 2.3313 .91902 47 14 .39 448 .42 929 2.3294 .91 891 46 15 .39474 .42963 2.3276 .91879 45 16 .39 501 .42 998 2.3257 .91 868 44 17 .39 528 .43 032 2.3238 .91 856 43 18 .39 555 .43 067 2.3220 .91 845 42 19 .39 581 .43 101 2.3201 .91 833 41 2O .39 608 .43 136 2.3183 .91 822 4O 21 .39635 .43170 2.3164 .91*810 39 22 .39 661 .43 205 2.3146 .91 799 38 23 .39 688 .43 239 2.3127 .91 787 37 24 .39 715 .43 274 2.3109 .91 775 36 25 .39 741 .43 308 2.3090 .91 764 35 26 .39 768 .43 343 2.3072 .91 752 34 27 .39 795 .43 378 2.3053 .91 741 33 28 .39 822 .43 412 2.3035 .91 729 32 29 .39 848 .43 447 2.3017 .91 718 31 30 .39 875 .43 481 2.2998 .91 706 30 31 .39902 .43516 2.2980 .91694 29 32 .39 928 .43 550 2.2962 .91 683 28 33 .39 955 .43 585 2.2944 .91 671 27 34 .39 982 .43 620 2.2925 .91 660 26 35 .40 008 .43 654 2.2907 .91 648 25 36 .40 035 .43 689 2.2889 .91 636 24 37 .40 062 .43 724 2.2871 .91 625 23 38 .40 088 .43 758 2.2853 .91 613 22 39 .40 115 .43 793 2.2835 .91 601 21 4O .40 141 .43 828 2.2817 .91 590 2O 41 .40 168 .43 862 2.2799 .91 578 19 42 .40 195 .43 897 2.2781 .91 566 18 43 .40 221 .43 932 2.2763 .91 555 17 44 .40 2H8 .43 966 2.2745 .91 543 16 45 .40 275 .44 001 2.2727 .91 531 15 46 .40 301 .44 036 2.2709 .91 519 14 47 .40 328 .44 071 2.2691 .91 508 13 48 .40355 .44" 105 2.2673 .91496 12 49 .40381 .44140 2.2655 .91484 11 50 .40408 .44175 2.2637 .91472 1O 51 .40434 .44210 2.2620 .91461 9 52 .40461 .44244 2.2602 .91449 8 53 .40 488 .44 279 2.2584 .91 437 7 54 .40514 .44314 2.2566 .91425 6 55 .40 541 .44 349 2.2549 .91 414 5 56 .40567 .44384 2.2531 .91402 4 57 .40594 .44418 2.2513 .91390 3 58 .40621 .44453 2.2496 .91378 2 59 .40 647 .44 488 2.2478 .91 366 1 60 .406.74 .44 523 2.2460 .91 355 / cos cot tan sin / 66 70 NATURAL FUNCTIONS 24 / sin tan cot cos / o .40674 .44523 2.2460 .91355 6O 1 .40 700 .44 558 2.2443 .91 343 59 2 .40727 .44593 2.2425 .91331 58 3 .40753 .44627 2.2408 .91319 57 4 .40 780 .44 662 2.2390 .91 307 56 5 .40 806 .44 697 2.2373 .91 295 55 6 .40 833 .44 732 2.2355 .91 283 54 7 .40 860 .44 767 2.2338 .91 272 53 8 .40 886 .44 802 2.2320 .91 260 52 9 .40913 .44837 2.2303 .91248 51 1C .40 939 .44 872 2.2286 .91 236 5O 11 .40 966 .44 907 2.2268 .91 224 49 12 .40992 .44942 2.2251- .91212 48 13 .41019 .44977 2.2234 .91200 47 14 .41 045 .45 012 2.2216 .91 1S8 46 15 .41 072 .45 047 2.2199 .91 176 45 16 .41 098 .45 082 2.2182 .91 164 44 17 .41 125 .45 117 2.2165 .91 152 43 18 .41 151 .45 152 2.2148 .91 140 42 19 .41 178 .45 187 2.2130 .91 128 41 2O .41204 .45222 2.2113 .91116 40 21 .41 231 .45 257 2.2096 .91 104 39 22 .41 257 .45 292 2.2079 .91 092 38 23 .41 284 .45 327 2.2062 .91 080 37 24 .41 310 .45 362 2.2045 .91 068 36 25 .41 337 .45 397 2.2028 .91 056 35 26 .41 363 .45 432 2.2011 .91 044 34 27 .41 390 .45 467 2.1994 .91 032 33 28 .41 416 .45 502 2.1977 .91 020 32 29 .41 443 .45 538 2.1960 .91 008 31 30 .41469 .45573 2.1943 .90996 30 31 .41 496 .45 608 2.1926 .90984 29 32 .41 522 .45 643 2.1909 .90 972 28 33 .41549 .45678 2.1892 .90960 27 34 .41575 .45 713 2.1876 .90948 26 35 .41602 .45748 2.1859 .90936 25 36 .41 628 .45 784 2.1842 .90 924 24 37 .41655 .45819 2.1825 .90911 23 38 .41 681 .45 854 2.1808 .90 899 22 39 .41 707 .45 889 2.1792 .90 887 21 40 .41 734 .45 924 2.1775 .90 875 20 41 .41 760 .45 960 2.1758 .90 863 19 42 .41787 .45995 2.1742 .90851 18 43 .41813 .46030 2.1725 .90839 17 44 .41 840 .46 065 2.1708 .90 826 16 45 .41 866 .46 101 2.1692 .90 814 15 46 .41 892 .46 136 2.1675 .90 802 14 47 .41 919 .46 171 2.1659 .90 790 13 48 .41 945 .46 206 2.1642 .90 778 12 49 .41 972 .46 242 2.1625 .90 766 11 5O .41998 .46277 2.1609 .90753 10 51 .42 024 .46 312 2.1592 .90 741 9 52 .42051 .46348 2.1576 .90729 8 53 .42077 .46383 2.1560 .90717 7 54 .42104 .46418 2.1543 .90704 6 55 .42 130 .46 454 2.1527 .90 692 5 56 .42156 .46489 2.1510 .90680 4 57 .42 183 .46 525 2.1494 .90 668 3 58 .42 209 .46 560 2.1478 .90 655 2 59 .42235 .46595 2.1461 .90643 1 60 .42262 .46631 2.1445 .90631 / cos cot tan sin / 65 25 / sin tan cot cos / .42262 .46631 2.1445 .90631 60 1 .42 288 .46 666 2.1429 .90 618 59 2 .42315 .46702 2.1413 .90606 58 3 .42341 .46737 2.1396 .90594^ 57 4 .42367 .46772 2.1380 .90582 56 5 .42 394 .46 808 2.1364 .90 569 55 6 .42420 .46843 2.1348 .90557 54 7 .42 446 .46 879 2.1332 .90 545 53 * 8 .42473 .46914 2.1315 .90532 52 % 9 .42499 .46950 2.1299 .90520 51 10 .42525 .46985 2.1283 .90507 50 11 .42552 .47021 2.1267 .90495 49 12 .42578 .47056 2.1251 .90483 48 13 .42604 .47092 2.1235 .90470 47 14 .42631 .47128 2.1219 .90458 46 15 .42657 .47163 2.1203 .90446 45 16 .42683 .47199 2.1187 .90433 44 17 .42709 .47234 2.1171 .90421 43 18 .42736 .47270 2.1155 .90408 42 19 .42762 .47305 2.1139 .90396 41 2O .42788 .47341 2.1123 .90383 40 21 .42815 .47377 2.1107 .90371 39 22 .42841 .47412 2.1092 .90358 38 23 .42867 .47448 2.1076 .90346 37 24 .42 894 .47 483 2.1060 .90 334 36 25 .42920 .47519 2.1044 .90321 35 26 .42946 .47555 2.1028 .90309 34 27 .42 972 .47 590 2.1013 .90 296 33 28 .42999 .47626 2.0997 .90284 32 29 .43025 .47662 2.0981 .90271 31 3O .43051 .47698 2.0965 .90259 3O 31 .43 077 .47 733 2.0950 .90 246 29 32 .43 104 .47 769 2.0934 .90 233 28 33 .43 130 .47 805 2.091S .90 221 27 34 .43 156 .47 840 2.0903 .90 208 26 35 .43 182 .47 876 2.0887 .90 196 25 36 .43 209 .47 912 2.0872 .90 183 24 37 .43235 .47948 2.0856 .90171 23 38 .43261 .47984 2.0840 .90158 22 39 .43 287 .48 019 2.0825 .90 146 21 40 .43313 .48055 2.0809 .90133 2O 41 .43 340 .48 091 2.0794 .90 120 19 42 .43 366 .48 127 2.0778 .90 108 18 43 .43 392 .48 163 2.0763 .90 095 17 44 .43 418 .48 198 2.0748 .90 082 16 45 .43445 .48234 2.0732 .90070 15 46 .43471 .48270 2.0717 .90057 14 47 .43497 .48306 2.0701 .90045 13 48 .43 523 .48 342 2.0686 .90 032 12 49 .43549 .48378 2.0671 .90019 11 50 .43575 .48414 2.0655 .90007 1O 51 .43 602 .48 450 2.0640 .89 994 9 52 .43628 .48486 2.0625 .89981 8 53 .43 654 .48 521 2.0609 .89 968 7 54 .43680 .48557 2.0594 .89956 6 55 .43 706 .48 593 2.0579 .89 943 5 56 .43 733 .48 629 2.0564 .89 930 4 57 .43 759 .48 665 2.0549 .89 918 3 58 .43 785 .48 701 2.0533 .89 905 2 59 .43 811 .48 737 2.0518 .89 892 1 6O .43 837 .48 773 2.0503 .89 879 / cos cot tan sin / 64 NATURAL FUNCTIONS 71 26 / sin tan cot cos / .43837 .48773 2.0503 .89879 60 1 .43863 .48809 2.0488 .89867 59 2 .43 889 .48 845 2.0473 .89 854 58 3 .43916 .48881 2.0458 .89841 57 4 .43942 .48917 2.0443 .89828 56 5 .43968 .48953 2.0428 .89816 55 6 .43 994 .48 989 2.0413 .89 803 54 7 .44020 .49026 2.0398 .89790 53 8 .44 046 .49 062 2.0383 .89 777 52 9 .44 072 .49 098 2.0368 .89 764 51 1O .44098 .49134 2.0353 .89752 5O 11 .44 124 .49 170 2.0338 .89 739 49 12 .44151 .49206 2.0323 .89726 48 13 .44177 .49242 2.0308 .89713 47 14 .44203 .49278 2.0293 .89700 46 15 .44229 .49315 2.0278 .89687 45 16 .44255 .49351 2.0263 .89674 44 17 .44281 .49387 2.0248 .89662 43 18 .44307 .49423 2.0233 .89649 42 19 .44333 .49459 2.0219 .89636 41 2O .44359 .49495 2.0204 .89623 4O 21 .44385 .49532 2.0189 .89610 39 22 .44411 .49568 2.0174 .89597 38 23 .44437 .49604 2.0160 .89584 37 24 .44 464 .49 640 2.0145 .89 571 36 25 .44 490 .49 677 2.0130 .89 558 35 26 .44 516 .49 713 2.0115 .89 545 34 27 .44 542 .49 749 2.0101 .89 532 33 28 .44568 .49786 2.0086 .89519 32 29 .44594 .49822 2.0072 .89506 31 '3O .44620 .49858 2.0057 .89493 30 31 .44646 .49894 2.0042 .89480 29 32 .44672 .49931 2.0028 .89467 28 33 .44698 .49967 2.0013 .89454 27 34 .44 724 .50 004 1.9999 .89 441 26 35 .44750 .50040 1.9981- .89428 25 36 .44 776 .50 076 1.9970 .89 415 24 37 .44802 .50113 1.9955 .89402 23 38 .44 828 .50 149 1.9941 .89 389 22 39 .44 854 .50 185 1.9926 .89 376 21 40 .44880 .50222 1.9912 .89363 20 41 .44906 .50258 1.9897 .89350 19 42 .44932 .50295 1.9883 .89337 18 43 .44958 .50331 1.9868 .89324 17 44 .44984 .50368 1.9854 .89311 16 45 .45010 .50404 1.9840 .89298 15 46 .45036 .50441 1.9825 .89285 14 47 .45062 .50477 1.9811 .89272 13 48 .45 088 .50 514 1.9797 .89 259 12 49 .45114 .50550 1.9782 .89245 11 5O .45 140 .50 587 1.9768 .89 232 10 51 .45 166 .50 623 1.9754 .89 219 9 52 .45192 .50660 1.9740 .89206 8 53 .45218 .50696 1.9725 .89193 7 54 .45243 .50733 1.9711 .89180 6 55 .45269 .50769 1.9697 .89167 5 56 .45 295 .50 806 1.9683 .89 153 4 57 .45321 .50843 1.9669 .89140 3 58 .45347 .50879 1.9654 .89127 2 59 .45 373 .50 916 1.9640 .89 114 1 60 .45399 .50953 1.9626 .89101 O / cos cot tan sin / 63 27 / sin tan cot cos f .45 399 .50 953 1.9626 .89 101 60 1 .45425 .50989 1.9612 .89087 59 2 .45451 .51026 1.9598 .89074 58 3 .45477 .51063 1.9584 .89061 57 4 .45503 .51099 1.9570 .89048 56 5 .45529 .51136 1.9556 .89035 55 6 .45554 .51173 1.9542 .89021 54 '7 .45580 .51209 1.9528 .89008 53 8 .45606 .51246 1.9514 .88995 52 9 .45 632 .51 283 1.9500 .88 981 51 1O .45658 .51319 1.9486 .88968 5O 11 .45 684 .51 356 1.9472 .88 955 49 12 .45 7JO .51393 1.9458 .88942 48 13 .45736 .51430 1.9444 .88928 47 14 .45762 .51467 1.9430 .88915 46 15 .45 787 .51 503 1.9416 .88902 45 16 .45813 .51 540 1.9402 .88888 44 17 .45839 .51577 1.9388 .88875 43 18 .45865 .51 614 1.9375 .88862 42 19 .45891 .51651 1.9361 .88848 41 20 .45917 .51688 1.9347 .88835 4O 21 .45942 .51 724 1.9333 .88822 39 22 .45968 .51 761 1.9319 .88808 38 23 .45 994 .51 798 1.9306 .88 795 37 24 .46020 .51 835 1.9292 .88782 36 25 .46046 .51872 1.9278 .88768 35 26 .46072 .51909 1.9265 .88755 34 27 .46097 .51946 1.9251 .88741 33 28 .46123 .51983 1.9237 .88728 32 29 .46149 .52020 1.9223 .88715 31 30 .46 175 .52 057 1.9210 .88 701 3O 31 .46201 .52094 1.9196 .88688 29 32 .46226 .52131 1.9183 .88674 28 33 .46252 .52168 1.9169 .88661 27 34 .46278 .52205 1.9155 .88647 26 35 .46304 .52242 1.9142 .88634 25 36 .46330 .52279 1.9128 .88620 24 37 .46355 .52316 1.9115 .88607 23 38 .46381 .52353 1.9101 .88593 22 39 .46407 .52390 1.9088 .88580 21 40 .46433 .52427 1.9074 .88566 2O 41 .46458 .52464 1.9061 .88553 19 42 .46484 .52501 1.9047 .88539 18 43 .46 510 .52 538 1.9034 .88 526 17 44 .46536 .52575 1.9020 .88512 16 45 .46 561 .52 613 1.9007 .88 499 15 46 .46 587 .52 650 1.8993 .88 485 14 47 .46 613 .52 687 1.89SO .88 472 13 48 .46 639 .52 724 1.8967 .88 458 12 49 .46664 .52761 1.8953 .88445 11 50 .46690 .52798 1.8940 .88431 1O 51 .46716 .52836 1.8927 .88417 9 52 .46742 .52873 1.8913 .88404 8 53 .46767 .52910 1.8900 .88390 71 54 .46793 .52947 1.8887 .88377 6 55 .46819 .52985 1.8873 .88363 5 56 .46 844 .53 022 1.8860 .88 349 4 57 .46 870 .53 059 1.8847 .88 336 3 58 .46 896 .53 096 1.8834 .88 322 2 59 .46921 .53 134 1.8820 .88308 1 60 .46947 .53171 1.SS07 .88 295 / cos cot tan sin / 62 72 NATURAL FUNCTIONS 28 / sin tan cot cos / o .46947 .53171 1.8S07 .88295 60 1 .46973 .53208 1.8794 .88281 59 2 .46999 .53246 1.8781 .88267 58 3 .47 024 .53 283 1.8768 .88 254 57 4 .47050 .53320 1.8755 .88240 56 5 .47 076 .53 358 1.8741 .88 226 55 6 .47 101 .53 395 1.8728 .88 213 54 7 .47 127 .53 432 1.8715 .88 199 53 8 .47 153 .53470 1.8702 .88 185 52 9 .47 178 .53 507 1.8689 .88 172 51 10 .47 204 .53 545 1.8676 .88 158 5O 11 .47 229 .53 582 1.8663 .88 144 49 12 .47 255 .53 620 1.8650 .88 130 48 13 .47 281 .53 657 1.8637 .88 117 47 14 .47 306 .53 694 1.8624 .88 103 46 15 .47332 .53732 1.8611 .88089 45 16 .47 358 .53 769 1.8598 .88 075 44 17 .47 383 .53 807 1.8585 .88 062 43 18 .47409 .53844 1.8572 .88048 42 19 .47434 .53882 1.8559 .88034 41 2O .47 460 .53 920 1.8546 .88 020 40 21 .47486 .53957 1.8533 .88006 39 22 .47511 .53995 1.8520 .87993 38 23 .47 537 .54 032 1.8507 .87 979 37 24 .47562 .54070 1.8495 .87965 36 25 .47588 .54307 1.8482 .87951 35 26 .47 614 .54 145 1.8469 .87 937 34 27 .47 639 .54 183 1.8456 .87 923 33 28 .47665 .54220 18443 .87909 32 29 .47 690 .54 258 1.8430 .87 896 31 3O .47 716 .54 296 1.8418 .87 882 3O 31 .47741 .54333 1.8405 .87868 29 32 .47 767 .54 371 1.8392 .87 854 28 33 .47 793 .54 409 1.8379 .87 840 27 34 .47 818 .54 446 1.8367 .87 826 26 35 .47844 .54484 1.8354 .87812 25 36 .47 869 .54 522 1.8341 .87 798 24 37 .47 895 .54 560 1.8329 .87 784 23 38 .47 920 .54 597 1.8316 .87 770 22 39 .47 946 .54 635 1.8303 .87 756 21 4O .47971 .54673 1.8291 .87743 2O 41 .47 997 .54 711 1.8278 .87 729 19 42 .48022 .54748 1.8265 .87715 18 43 .48048 .54786 1.8253 .87701 17 44 .48073 .54824 1.8240 .87687 16 45 .48099 .54862 1.8228 .87673 15 46 .48124 .54900 1.8215 .87659 14 47 .48150 .54938 1.8202 .87645 13 48 .48175 .54975 1.8190 .87631 12 49 .48201 .55013 1.8177 .87617 11 50 .48226 .55051 1.8165 .87603 1O 51 .48 252 .55 089 1.8152 .87 589 9 52 .48 277 .55 127 1.8140 .87 575 8 53 .48 303 .55 165 1.8127 .87 561 7 54 .48328 .55203 1.8115 .87546 6 55 .48 354 .55 241 1.8103 .87 532 5 56 .48379 .55279 1.8090 .87518 4 57 .48405 .55317 1.8078 .87504 3 58 .48430 .55355 1.8065 .87490 2 59 .48 456 .55 393 1.8053 .87 476 1 GO .48481 .55431 1.8040 .87462 / cos cot tan sin / 61 29 / sin tan cot cos / .48481 .55431 1.8040 .87462 6O 1 .48506 .55469 1.8028 .87448 59 2 .48 532 .55 507 1.8016 .87 434 58 3 .48 557 .55 545 1.8003 .87 420 57 4 .48 583 .55 583 1.7991 .87 406 56 5 .48608 .55621 1.7979 .87391 55 6 .48634 .55659 1.7966 .87377 54 7 .48 659 .55 697 1.7954 .87 363 53 8 .48684 .55 736 1.7942 .87349 52 9 .48710 .55774 1.7930 .87335 51 1O .48735 .55812 1.7917 .87321 50 11 .48 761 .55 850 1.7905 .87 306 49 12 .48786 .55888 1.7893 .87292 48 13 .48811 .55926 1.7881 .87278 47 14 .48837 .55964 1.7868 .87264 46 15 .48862 .56003 1.7856 .87250 45 16 .48888 .56041 1.7844 .87235 44 17 .48913 .56079 .7832 .87221 43 18 .48938 .56117 .7820 .87207 42 19 .48 964 .56 156 .7808 .87 193 41 2O .48 989 .56 194 .7796 .87 178 4O 21 .49 014 .56 232 .7783 .87 164 " 39 22 .49 040 .56 270 .7771 .87 150 38 23 .49065 .56309 .7759 .87136 37 24 .49 090 .56 347 .7747 .87 121 36 25 .49 116 .56 385 .7735 .87 107 35 26 .49141 .56424 .7723 .87093 34 27 .49166 .56462 .7711 .87079 33 28 .49192 .56 501 .7699 .87 064 32. 29 .49 217 .56 539 .7687 .87 05.0 31 30 .49242 .56577 .7675 .87036 3O 31 .49268 .56616 .7663 .87021 29 32 .49293 .56654 .7651 .87007 28 33 .49318 .56693 .7639 .86993 27 34 .49344 .56731 .7627 .86978 26 35 .49369 .56769 1.7615 .86964 25 36 .49394 .56808 1.7603 .86949 24 37 .49419 .56846 1.7591 .86935 23 38 .49445 .56885 1.7579 .86921 22 39 .49470 .56923 1.7567 .86906 21 40 .49495 .56962 1.7556 .86892 2O 41 .49521 .57000 1.7544 .86878 19 42 .49546 .57039 1.7532 .86863 18 43 .49 571 .57 078 1.7520 .86 849 17 44 .49596 .57116 1.7508 .86834 16 45 .49 622 .57 155 .7496 .86 820 15 46 .49647 .57 193 .7485 .86805 14 47 .49 672 .57 232 .7473 .86 791 13 48 .49 697 .57 271 .7461 .86 777 12 49 .49 723 .57 309 .7449 .86 762 11 50 .49 748 .57 348 .7437 .86 748 1O 51 .49 773 .57 386 .7426 .86 733 9 52 .49 798 .57 425 1.7414 .86 719 8 53 .49824 .57464 1.7402 .86704 7 54 .49849 .57503 1.7391 .86690 6 55 .49874 .57541 1.7379 .86675 5 56 .49899 .57580 1.7367 .86661 4 57 .49 924 .57 619 1.7355 .86 646 3 58 .49950 .57657 1.7344 .86632 2 59 .49975 .57696 1.7332 .86617 1 60 .50000 .57735 1.7321 .86603 O / cos cot tan sin / 60 NATURAL FUNCTIONS 73 30 / sin tan cot cos / o .50000 .57735 1.7321 .86603 6O 1 .50025 .57774 1.7309 .86588 59 2 .50050 .57813 1.7297 .86573 58 3 .50076 .57851 1.7286 .86559 57 4 .50 101 .57 890 1.7274 .86 544 56 5 .50 126 .57 929 1.7262 .86 530 55 6 .50151 .57968 1.7251 .86515 54 7 .50176 .58007 1.7239 .86501 53 8 .50201 .58046 1.7228 .86486 52 9 .50227 .58085 1.7216 .86471 51 10 .50252 .58124 1.7205 .86457 50 11 .50277 .58162 1.7193 .86442 49 12 .50302 .58201 1.7182 .86427 48 13 .50327 .58240 1.7170 .86413 47 14 .50352 .58279 1.7159 .86398 46 15 .50377 .58318 1.7147 .86384 45 16 .50403 .58357 1.7136 .86369 44 17 .50428 .58396 1.7124 .86354 43 18 .50453 .58435 1.7113 .86340 42 ! 19 .50478 .58474 1.7102 .86325 41 2O .50503 .58513 1.7090 .86310 40 21 .50 528 .58 552 1.7079 .86 295 39 22 .50553 .58591 1.7067 .86281 38 23 .50578 .58631 1.7056 .86266 37 24 .50603 .58670 1.7045 .86251 36 25 .50628 .58709 1.7033 .86237 35 26 .50654 .58748 1.7022 .86222 34 27 .50679 .58787 1.7011 .86207 33 28 .50704 .58826 1.6999 .86192 32 29 .50729 .58865 1.6988 .86178 31 3O .50 754 .58 905 1.6977 .86 163 30 31 .50779 .58944 1.6965 .86148 29 32 .50804 .58983 1.6954 .86133 28 33 .50829 .59022 1.6943 .86119 27 34 .50854 .59061 1.6932 .86104 26 35 .50879 .59101 1.6920 .86089 25 36 .50904 .59140 1.6909 .86074 24 37 .50929 .59179 1.6898 .86059 23 38 .50954 .59218 1.6887 .86045 22 39 .50979 .59258 1.6875 .86030 21 4O .51004 .59297 1.6864 .86015 2O 41 .51029 .59336 1.6853 .86000 19 42 .51054 .59376 1.6S42 .85985 18 43 .51079 .59415 1.6831 .85970 17 44 .51104 .59454 1.6820 .85956 16 45 .51 129 .59494 1.6808 .85941 15 46 .51 154 .59533 1.6797 .85926 14 47 .51 179 .59573 1.67S6 .85911 13 48 .51204 .59612 1.6775 .85896 12 49 .51229 .59651 1.6764 .85881 11 50 .51254 .59691 1.6753 .85866 10 51 .51279 .59730 1.6742 .85851 9 52 .51304 .59770 1.6731 .85836 8 53 .51329 .59809 1.6720 .85821 7 54 .51354 .59849 1.6709 .85806 6 55 .51379 .59888 1.6698 .85792 5 56 .51404 .59928 1.6687 .85777 4 57 .51429 .59967 1.6676 .85762 3 58 .51454 .60007 1.6665 .85747 2 59 .51479 .60046 1.6654 .85732 1 60 .51504 .60086 1.6643 .85717 O / cos cot tan sin / 59 31 / sin tan cot cos / .51504 .60086 1.6643 .85717 60 1 .51 529 .60 126 1.6632 .85 702 59 2 .51 554 .60 165 1.6621 .85 687 58 3 .51579 .60205 1.6610 .85672 57 4 51604 .60245 1.6599 .85657 56 5 .51 628 .60 284 1.6588 .85 642 55 6 .51653 .60324 1.6577 .85627 54 '7 .51678 .60364 1.6566 .85612 53 8 .51703 .60403 1.6555 .85597 52 9 .51728 .60443 1.6545 .85582 51 1O .51753 .60483 1.6534 .85567 50 11 .51 778 .60522 1.6523 .85551 49 12 .51803 .60562 1.6512 .85536 48 13 .51828 .60602 1.6501 .85521 47 14 .51852 .60642 1.6490 .85506 46 15 .51877 .60681 1.6479 .85491 45 16 .51902 .60721 1.6469 .85476 44 17 .51927 .60761 1.6458 .85461 43 18 .51952 .60801 1.6447 .85446 42 19 .51977 .60841 1.6436 .85431 41 2O .52 002 .60 881 1.6426 .85 416 40 21 .52026 .60921 1.6415 .85401 39 22 .52051 .60960 1.6404 .85385 38 23 .52076 .61000 1.6393 .85370 37 24 .52 101 .61 040 1.6383 .85 355 36 25 .52 126 .61 080 1.6372 .85 340 35 26 .52151 .61 120 1.6361 .85325 34 27 .52175 .61160 1.6351 .85310 33 28 .52 200 .61 200 1.6340 .85 294 32 29 .52 225 .61 240 1.6329 .85 279 31 30 .52 250 .61 280 1.6319 .85 264 30 31 .52 275 .61 320 1.6308 .85 249 29 32 .52 299 .61 360 1.6297 .85 234 28 33 .52 324 .61 400 1.6287 .85 218 27 31 .52 349 .61 440 1.6276 .85 203 26 35 .52374 .61480 1.6265 .85188 25 36 .52 399 .61 520 1.6255 .85 173 24 37 .52 423 .61 561 1.6244 .85 157 23 38 .52 448 .61 601 1.6234 .85 142 22 39 .52473 .61641 1.6223 .85 127 21 4O .52498 .61681 1.6212 .85112 2O 41 .52 522 .61 721 1.6202 .85 096 19 42 .57 547 .61 761 1.6191 .85 081 18 43 .52572 .61801 1.6181 .85066 17 44 .52 597 .61 842 1.6170 .85 051 16 45 .52 621 .61 882 1.6160 .85 035 15 46 .52 646 .61 922 1.6149 .85 020 14 47 .52671 .61962 1.6139 .85005 13 48 .52696 .62003 1.6128 .84989 12 49 .52720 .62043 1.6118 .84974 11 5O .52745 .62083 1.6107 .84959 1O 51 .52770 .62 124 1.6097 .84943 9 52 .52 794 .62 164 1.6087 .84 928 8 53 .52 819 .62 204 1.6076 .84 913 7 54 .52 844 .62 245 1.6066 .84 897 6 55 .52 869 .62 285 1.6055 .84 882 5 56 .52893 .62325 1.6045 .84866 4 57 .52918 .62366 1.6034 .84851 3 58 .52 943 .62 406 1.6024 .84 836 2 59 .52967 .62446 1.6014 .84820 1 60 .52992 .62487 1.6003 .84805 / cos cot tan sin / 58 74 NATURAL FUNCTIONS 32 / sin tan cot cos / ~0 .52992 .62487 1.6003 .84805 6O 1 .53017 .62527 1.5993 .84789 59 2 .53041 .62568 1.5983 .84774 58 3 .53 066 .62 608 1.5972 .84 759 57 4 .53 091 .62 649 1.5962 .84 743 56 5 .53 115 .62 689 1.5952 .84 728 55 6 .53 140 .62 730 1.5941 .84 712 54 7 .53164 .62770 1.5931 .84697 53 8 .53189 .62811 1.5921 .84681 52 9 .53214 .62852 1.5911 .84666 51 10 .53 238 .62 892 1.5900 .84 650 50 11 .53 263 .62 933 1.5890 .84 635 49 12 .53288 .62973 1.5880 .84619 48 13 .53312 .63014 1.5869 .84604 47 14 .53337 .63055 1.5859 .84588 46 15 .53 361 .63 095 1.5849 .84 573 45 16 .53 386 .63 136 1.5839 .84 557 44 17 .53411 .63177 1.5829 .84542 43 18 .53435 .63217 1.5818 .84526 42 19 53460 .63258 1.5808 .84511 41 2O .53 484 .63 299 1.5798 .84 495 40 21 .53 509 .63 340 1.5788 .84 480 39 22 .53534 .63380 1.5778 .84464 38 23 .53558 .63421 1.5768 .84448 37 24 .53583 .63462 1.5757 .84433 36 25 .53607 .63503 1.5747 .84417 35 26 .53632 .63544 1.5737 .84402 34 27 .53 656 .63 584 1.5727 .84 386 33 28 .53681 .63625 1.5717 .84370 32 29 .53705 .63666 1.5707 .84355 31 3D .53 730 .63 707 1.5697 .84 339 30 31 .53 754 .63 748 1.5687 .84 324 29 32 .53 779 .63 789 1.5677 .84 308 28 33 .53 804 .63 830 1.5667 .84 292 27 34 .53 828 .63 871 1.5657 .84 277 26 35 .53853 .63912 1.5647 .84261 25 36 .53 877 .63 953 L5637 .84 245 24 37 .53902 .63994 1.5627 .84230 23 38 .53926 .64035 1.5617 .84214 22 39 .53951 .64076 1.5607 .84198 21 4O .53975 .64117 1.5597 .84182 20 41 .54 000 .64 158 1.5587 .84 167 19 42 .54024 .64199 1.5577 .84151 18 43 .54 049 .64 240 1.5567 .84 135 17 44 .54073 .64281 1.5557 .84120 16 45 .54097 .64322 1.5547 .84104 15 46 .54122 .64363 1.5537 .84088 14 47 .54146 .64404 1.5527 .84072 13 48 .54171 .64446 1.5517 .84057 12 49 .54 195 .64 487 1.5507 .84 041 11 50 .54220 .64528 1.5497 .84025 10 51 .54244 .64569 1.5487 .84009 9 52 .54269 .64610 1.5477 .83994 8 53 .54293 .64652 1.5468 .83978 7 54 .54317 .64693 1.5458 .83962 6 55 .54342 .64734 1.5448 .83946 5 56 .54 366 .64 775 1.5438 .83 930 4 57 .54 391 .64 817 1.5428 .83 915 3 58 .54415 .64858 1.5418 .83899 2 59 .54 440 .64 899 1.5408 .83 883 1 6O .54 464 .64 941 1.5399 .83 867 O / cos cot tan sin / 57 33 / sin tan cot cos / .54464 .64941 1.5399 .83867 6O 1 .54488 .64982 1.5389 .83851 59 2 .54 513 .65 024 1.5379 .83 835 58 3 .54 537 .65 065 1.5369 .83 819 57 4 .54 561 .65 106 1.5359 .83 804 56 5 .54 586 .65 148 1.5350 .83 788 55 6 .54610 .65 189 1.5340 .83772 54 7 .54 635 .65 231 1.5330 .83 756 53 8 .54 659 .65 272 1.5320 .83 740 52 9 .54683 .65314 1.5311 .83724 51 10 .54 708 .65 355 1.5301 .83 708 50 11 .54 732 .65 397 1.5291 .83 692 49 12 .54 756 .65 438 1.5282 .83 676 48 13 .54781 .65480 1.5272 .83660 47 14 .54805 .65521 1.5262 .83645 46 15 .54 829 .65 563 1.5253 .83 629 45 16 .54 854 .65 604 1.5243 .83 613 44 17 .54 878 .65 646 1.5233 .83 597 43 18 .54 902 .65 688 1.5224 .83 581 42 19 .54 927 .65 729 1.5214 .83 565 41 2O .54951 .65771 1.5204 .83549 40 21 .54975 .65813 1.5195 .83533 39 22 .54999 .65854 1.5185 .83517 38 23 .55 024 .65 896 1.5175 .83 501 37 24 .55 048 .65 938 1.5166 .83 485 36 25 .55072 .65980 1.5156 .83469 35 26 .55097 .66021 1.5147 .83453 34 27 .55121 .66063 1.5137 .83437 33 28 .55-145 .66105 1.5127 .83421 32 29 .55 169 .66 147 1.5118 .83 405 31 30 .55 194 .66 189 1.5108 .83 389 30 31 .55 218 .66 230 1.5099 .83 373 29 32 .55 242 .66 272 1.5089 .83 356 28 33 .55266 .66314 1.5080 .83340 27 34 .55291 .66356 1.5070 .83324 26 35 .55315 .66398 1.5061 .83308 25 36 .55339 .66440 1.5051 .83292 24 37 .55 363 .66 482 1.5042 .83 276 23 38 .55 388 .66 524 1.5032 .83 260 22 39 .55412 .66566 1.5023 .83244 21 40 .55 436 .66 608 1.5013 .83 228 20 41 .55460 .66650 1.5004 .83212 19 42 .55484 .66692 1.4994 .83195 18 43 .55 509 .66 734 1.4985 .83 179 17 44 .55 533 .66 776 1.4975 .83 163 16 45 .55557 .66818 1.4966 .83147 15 46 .55581 .66860 1.4957 .83131 14 47 .55605 .66902 1.4947 .83115 13 48 .55630 .66944 1.4938 .83098 12 49 .55654 .66986 1.4928 .83082 11 50 .55 678 .67 028 1.4919 .83 066 10 51 .55 702 .67 071 1.4910 .83 050 9 52 .55726 .67113 1.4900 .83034 8 53 .55750 .67155 1.4891 .83017 7 54 .55 775 .67 197 1.4882 .83 001 6 55 .55 799 .67 239 1.4872 .82 985 5 56 .55823 .67282 1.4863 .82969 4 57 .55847 .67324 1.4854 .82953 3 58 .55871 .67366 1.4844 .82936 2 59 .55 895 .67 409 1.4835 .82 920 1 60 .55919 .67451 1.4826 .82904 / cos cot tan sin / 56 NATURAL FUNCTIONS 75 34 / sin tan cot cos / o .55919 .67451 1.4826 .82904 6O 1 .55 943 .67 493 1.4816 .82 887 59 2 .55968 .67536 1.4807 .82871 58 3 .55992 .67578 1.4798 .82855 57 4 .56016 .67620 1.4788 .82839 56 5 .56 040 .67 663 1.4779 .82 822 55 6 .56064 .67705 1.4770 .82806 54 7 .56088 .67748 1.4761 .82790 53 8 .56112 .67790 1.4751 .82773 52 9 .56 136 .67 832 1.4742 .82 757 51 10 .56 160 .67 875 1.4733 .82 741 50 11 .56 184 .67 917 1.4724 .82 724 49 12 .56 208 .67 960 1.4715 .82 708 48 13 .56232 .68002 1.4705 .82692 47 14 .56 256 .68 045 1.4696 .82 675 46 15 .56280 .68088 1.4687 .82659 45 16 .56 305 .68 130 1.4678 .82 643 44 17 .56 329 .68 173 1.4669 .82 626 43 18 .56353 .68215 1.4659 .82610 42 19 .56377 .68258 1.4650 .82593 41 2O .56401 .68301 1.4641 .82577 40 21 .56425 .68343 1.4632 .82561 39 22 .56 449 .68 386 1.4623 .82 544 38 23 .56473 .68429 1.4614 .82528 37 24 .56497 .68471 1.4605 .82511 36 25 .56521 .68514 1.4596 .82495 35 26 .56545 .68557 1.4586 .82478 34 27 .56569 .68600 1.4577 .82462 33 28 .56593 .68642 1.4568 .82446 32 29 .56617 .68685 1.4559 .82429 31 30 .56641 .68728 1.4550 .82413 3O 31 .56665 .68771 1.4541 .82396 29 32 .56689 .68814 1.4532 .82380 28 33 .56713 .68857 1.4523 .82363 27 34 .56736 .68900 1.4514 .82347 26 35 .56760 .68942 1.4505 .82330 25 36 .56784 .68985 1.4496 .82314 24 37 .56808 .69028 1.4487 .82297 23 38 .56832 .69071 1.4478 .82281 22 39 .56856 .69114 1.4469 .82264 21 4O .56880 .69157 1.4460 .82248 2O 41 .56904 .69200 1.4451 .82231 19 42 .56928 .69243 1.4442 .82214 18 43 .56952 .69286 1.4433 .82198 17 44 .56 976 .69 329 1.4424 .82 181 16 45 .57 000 .69 372 1.4415 .82 165 15 46 .57024 .69416 1.4406 .82148 14 47 .57047 .69459 1.4397 .82132 13 48 .57071 .69502 1.4388 .82115 12 49 .57095 .69545 1.4379 .82098 11 50 .57119 .69588 1.4370 .82082 1O 51 .57143 .69631 1.4361 .82065 9 52 .57167 .69675 1.4352 .82048 8 53 .57191 .69718 1.4344 .82032 7 54 .57215 .69761 1.4335 .82015 6 55 .57238 .69804 1.4326 .81999 5 56 .57262 .69847 1.4317 .81982 4 57 .57286 .69891 1.4308 .81965 3 58 .57310 .69934 1.4299 .81949 2 59 .57334 .69977 1.4290 .81932 1 6O .57358 .70021 1.4281 .81915 O / cos cot tan sin / 55 35 / sin tan cot cos / O .57358 .70021 1.4281 .81915 60 1 .57381 .70064 1.4273 .81899 59 2 .57405 .70107 1.4264 .81882 58 3 .57429 .70151 1.4255 .81865 57 4 .57 453 .70 194 1.4246 .81 848 56 5 .57 477 .70 238 1.4237 .81 832 55 6 .57 501 .70 281 1.4229 .81 815 54 7 .57 524 .70 325 1.4220 .81 798 53 8 .57548 .70368 1.4211 .81782 52 9 .57572 .70412 1.4202 .81765 51 10 .57596 .70455 1.4193 .81748 50 11 .57619 .70499 1.4185 .81731 49 12 .57643 .70542 1.4176 .81714 48 13 .57667 .70586 1.4167 .81698 47 14 .57691 .70629 1.4158 .81681 46 15 .57715 .70673 1.4150 .81664 45 16 .57738 .70717 1.4141 .81647 44 17 .57 762 .70 760 1.4132 .81 631 43 18 .57786 .70804 1.4124 .81614 42 19 .57810 .70848 1.4115 .81597 41 20 .57833 .70891 1.4106 .81580 40 21 .57857 .70935 1.4097 .81563 39 22 .57881 .70979 1.4089 .81546 38 23 .57904 .71023 1.4080 .81530 37 24 .57 928 .71 066 1.4071 .81 513 36 25 .57952 .71110 1.4063 .81496 35 26 .57976 .71154 1.4054 .81479 34 27 .57999 .71198 1.4045 .81462 33 28 .58 023 .71 242 1.4037 .81 445 32 29 .58 047 .71 285 1.4028 .81 428 31 3O .58070 .71329 1.4019 .81412 30 31 .58 094 .71 373 1.4011 .81 395 29 32 .58 118 .71 417 1.4002 .81 378 28 33 .58 141 .71 461 1.3994 .81 361 27 34 .58 165 .71 505 1.3985 .81 344 26 35 .58189 .71549 1.3976 .81327 25 36 .58212 .71593 1.3968 .81310 24 37 .58236 .71637 1.3959 .81293 23 38 .58260 .71681 1.3951 .81 276 22 39 .58 283 .71 725 1.3942 .81 259 21 4O .58307 .71769 1.3934 .81242 20 41 .58330 .71813 1.3925 .81225 19 42 .58354 .71857 1.3916 .81208 18 43 .58378 .71901 1.3908 .81191 17 44 .58401 .71946 1.3899 .81174 16 45 .58425 .71990 1.3891 .81157 15 46 .58 449 .72 034 1.3882 .81 140 14 47 .58472 .72078 1.3874 .81 123 13 48 .58 496 .72 122 1.3865 .81 106 12 49 .58 519 .72 167 1.3857 .81 089 11 5O .58543 .72211 1.3848 .81072 1O 51 .58567 .72255 1.3840 .81055 9 52 .58590 .72299 1.3831 .81038 8 53 .58614 .72344 1.3823 .81021 7 54 .58637 .72388 1.3814 .81004 6 55 .58661 .72432 1.3806 .80987 5 56 .58684 .72477 1.3798 .80970 4 57 .58708 .72521 1.3789 .80953 3 58 .58731 .72565 1.3781 .80936 2 59 .58755 .72610 1.3772 .80919 1 60 .58 779 .72 654 1.3764 .80 902 / cos cot tan sin / 54 76 NATURAL FUNCTIONS 36 / sin tan cot cos / ~o .58779 .72654 1.3764 .80902 6O i .58802 .72699 1.3755 .80885 59 2 .58 826 .72 743 1.3747 .80 867 58 3 .58 849 .72 788 1.3739 .80 850 57 4 .58 873 .72 832 1.3730 .80 833 56 5 .58896 .72877 1.3722 .80816 55 6 .58920 .72921 1.3713 .80799 54 7 .58943 .72966 1.3705 .80782 53 8 .58967 .73010 1.3697 .80765 52 9 .58 990 .73 055 1.3688 .80 748 51 10 .59014 .73100 1.3680 .80730 50 11 .59037 .73144 1.3672 .80713 49 12 .59 061 .73 189 1.3663 .80 696 48 13 .59084 .73234 1.3655 .80679 47 14 .59108 .73.278 1.3647 .80662 46 IS .59131 .73323 1.3638 .80644 45 16 .59154 .73368 1.3630 .80627 44 17 .59178 .73413 1.3622 .80610 43 18 .59201 .73457 1.3613 .80593 42 19 .59225 .73502 1.3605 .80576 41 2O .59248 .73' 547 1.3597 .80558 40 21 .59 272 .73 592 1.3588 .80 541 39 22 .59 295 .73 637 1.3580 .80 524 38 23 .59318 .73681 1.3572 .80507 37 24 .59 342 .73 726 1.3564 .80 489 36 25 .59365 .73771 1.3555 .80472 35 26 .59389 .73816 1.3547 .80455 34 27 .59412 .73861 1.3539 .80438 33 28 .59436 .73906 1.3531 .80420 32 29 .59459 .73951 1.3522 .80403 31 3O .59482 .73996 1.3514 .80386 30 31 .59506 .74041 1.3506 .80368 29 32 .59529 .74086 1.3498 .80351 28 33 .59552 .74131 1.3490 .80334 27 34 .59576 .74176 1.3481 .80316 26 35 .59599 .74221 1.3473 .80299 25 36 .59622 .74267 1.3465 .80282 24 37 .59646 .74312 1.3457 .80264 23 38 .59669 .74357 1.3449 .80247 22 39 .59693 .74402 1.3440 .80230 21 40 .59716 .74447 1.3432 .80212 2O 41 .59739 .74492 1.3424 .80195 19 42 .59763 .74538 1.3416 .80178 18 43 .59 786 .74 583 1.3408 .80 160 17 44 .59809 .74628 1.3400 .80143 16 45 .59832 .74674 1.3392 .80125 15 46 .59 856 .74 719 1.3384 .80 108 14 47 .59 879 .74 764 1.3375 .80 091 13 48 .59902 .74810 1.3367 .80073 12 49 .59926 .74855 1.3359 .80056 11 50 .59949 .74900 1.3351 .80038 1O 51 .59972 .749-16 1.3343 .80021 9 52 .59995 .74991 1.3335 .80003 8 53 .60019 .75037 1.3327 .79986 7 54 .60042 .75082 1.3319 .79968 6 55 .60065 .75128 1.3311 .79951 5 56 .60089 .75173 1.3303 .79934 4 57 .60112 .75219 1.3295 .79916 3 58 .60135 .75264 1.3287 .79899 2 59 .60158 .75310 1.3278 .79881 1 60 .60 182 .75 355 1.3270 .79 864 O / cos cot tan sin / 53 37 / sin tan cot cos / O .60182 .75355 1.3270 .79864 60 1 .60205 .75401 1.3262 .79846 59 2 .60228 .75447 1.3254 .79829 58 3 .60251 .75492 1.3246 .79811 57 4 .60 274 .75 538 1.3238 .79 793 .56 5 .60 298 .75 584 1.3230 .79 776 55 6 .60321 .75629 1.3222 .79758 54 7 .60344 .75675 1.3214 .79741 53 8 .60 367 .75 721 1.3206 .79 723 52 9 .60390 .75767 1.3198 .79706 51 1O .60414 .75812 1.3190 .79688 50 11 .60437 .75858 1.3182 .79671 49 12 .60460 .75904 1.3175 .79653 48 13 .60483 .75950 1.3167 .79635 47 14 .60506 .75996 1.3159 .79618 46 15 .60529 .76042 1.3151 .79600 45 16 .60553 .76088 1.3143 .79583 44 17 .60576 .76134 1.3135 .79565 43 18 .60599 .76180 1.3127 .79547 42 19 .60622 .76226 1.3119 .79530 41 2O .60645 .76272 1.3111 .79512 4O 21 .60668 .76318 1.3103 .79494 39 22 .60691 .76364 1.3095 .79477 38 23 .60714 .76410 1.3087 .79459 37 24 .60738 .76456 1.3079 .79441 36 25 .60761 .76502 1.3072 .79424 35 26 .60784 .76548 1.3064 .79406 34 27 .60807 .76594 1.3056 .79388 33 28 .60830 .76640 1.3048 .79371 32 29 .60853 .76686 1.3040 .79353 31 3O .60876 .76733 1.3032 .79335 30 31 .60899 .76779 1.3024 .79318 29 32 .60922 .76825 1.3017 .79300 28 33 .60945 .76871 1.3009 .79282 27 34 .60968 .76918 1.3001 .79264 26 35 .60991 .76964 1.2993 .79247 25 36 .61015 .77010 1.2985 .79229 24 37 .61038 .77057 1.2977 .79211 23 38 .61 061 .77 103 1.2970 .7? 193 22 39 .61 084 .77 149 1.2962 .79 176 21 4O .61 107 .77 196 1.2954 .79 158 20 41 .61 130 .77242 1.2946 .79140 19 42 .61 153 .77 289 1.2938 .79 122 18 43 .61 176 .77335 .2931 .79105 17 44 .61199 .77382 .2923 .79087 16 45 .61222 .77428 .2915 .79069 15 46 .61245 .77475 .2907 .79051 14 47 .61 268 .77 521 .2900 .79 033 13 48 .61 291 .77 568 .2892 .79 016 12 49 .61314 .77615 1.28S4 .78998 11 50 .61337 .77661 1.2876 .78980 1O 51 .61 360 .77 708 1.2869 .Z8_%4 9 52 .61383 .77754 1.2861 ./*94^ 8. 53 .61406 .77801 1.2853 .78926 7 54 .61 429 .77 848 1.2846 .78 90S 6 55 .61451 .77895 1.2838 .78891 5 56 .61 474 .77 941 1.2830 .78 873 4 57 .61497 .77988 1.2822 .78855 3 58 .61520 .78035 1.2815 .78837 2 59 .61543 .78082 1.2807 .78819 1 00 .61 566 .78 129 1.2799 .78 801 O / cos cot tan sin / 52 NATURAL FUNCTIONS 38 f sin tan cot cos / o .61566 .78129 1.2799 .78801 60 1 .61 589 .78 175 1.2792 .78 783 59 2 .61 612 .78 222 1.2784 .78 765 58 3 .61 635 .78 269 1.2776 .78 747 57 4 .61658 .78316 1.2769 .78729 56 5 .61681 .78363 1.2761 .78711 55 6 .61 704 .78 410 1.2753 .78 694 54 7 .61 726 .78 457 1.2746 .78 676 53 8 .61749 .78504 1.2738 .78658 52 9 ..61772 .78551 1.2731 .78640 51 1O .61 795 .78 598 1.2723 .78 622 50 11 .61818 .78645 1.2715 .78604 49 12 .61 841 .78 692 1.2708 .78 586 48 13 .61 864 .78 739 1.2700 .78 568 47 14 .61 887 .78 786 1.2693 .78 550 46 15 .61 909 .78 834 1.2685 .78 532 45 16 .61932 .78881 1.2677 .78514 44 17 .61 955 .78 928 1.2670 .78 496 43 18 .61 978 .78 975 1.2662 .78 478 42 19 .62001 .79022 1.2655 .78460 41 2O .62 024 .79 070 1.2647 .78 442 4O 21 .62046 .79117 1.2640 .78424 39 22 .62069 .79164 1.2632 .78405 38 23 .62092 .79212 1.2624 .78387 37 24 .62115 .79259 1.2617 .78369 36 25 .62138 .79306 1.2609 .78351 35 26 .62 160 .79 354 1.2602 .78 333 34 27 .62183 .79401 1.2594 .78315 33 28 .62 206 .79 449 1.2587 .78 297 32 29 .62229 .79496 1.2579 .78279 31 30 .62251 .79544 1.2572 .78261 30 31 .62 274 .79 591 1.2564 .78 243 29 32 .62297 .79639 1.2557 .78225 28 33 .62 320 .79 686 1.2549 .78 206 27 34 .62 342 .79 734 1.2542 .78 188 26 35 .62 365 .79 781 1.2534 .78 170 25 36 .62388 .79829 1.2527 .78152 24 37 .62411 .79877 1.2519 .78134 23 38 .62433 .79924 1.2512 .78116 22 39 .62456 .79972 1.2504 .78098 21 40 .62479 .80020 1.2497 .78079 2O 41 .62502 .80067 1.2489 .78061 19 42 .62524 .80115 1.2482 .78043 18 43 .62547 .80163 1.2475 .78025 17 44 .62570 .80211 1.2467 .78007 16 45 .62592 .80258 1.2460 .77988 15 46 .62615 .80306 1.2452 .77970 14 47 .62638 .80354 1.2445 .77952 13 48 .62660 .80402 1.2437 .77934 12 49 .62683 .80450 1.2430 .77916 11 50 .62 706 .80 498 1.2423 .77 897 1O 51 .62 728 .80 546 1.2415 .77 879 9 52 .62 751 .80 594 1.2408 .77 861 8 53 .62 774 .80 642 1.2401 .77 843 7 54 . .62 796 .80 690 1.2393 .77 824 6 55 .62819 .80738 1.23S6 .77806 5 56 .62 842 .80 786 1.2378 .77 788 4 57 .62864 .80834 1.2371 .77769 3 58 .62887 .80882 1.2364 .77751 2 59 .62909 .80930 1.2356 .77733 1 6O .62932 .80978 1.2349 .77715 O / cos cot tan sin / 51 39 / sin tan cot cos / O .62932 .80978 1.2349 .77715 60 1 .62955 .81027 1.2342 .77696 59 2 .62977 .81075 1.2334 .77678 58 3 .63000 .81123 1.2327 .77660 57 4 .63 022 .81 171 1.2320 .77 641 56 5 .63045 .81220 1.2312 .77623 55 6 .63 068 .81 268 1.2305 .77 605 54 7 .63 090 .81 316 1.2298 .77 586 53 8 .63 113 .81 364 1.2290 .77 568 52 9 .63 135 .81 413 1.2283 .77 550 51 10 .63 158 .81 461 1.2276 .77 531 50 11 .63180 .81510 1.2268 .77513 49 12 .63 203 .81 558 1.2261 .77 494 48 13 .63 225 .81 606 1.2254 .77 476 47 14 .63248 .81655 1.2247 .77458 46 15 .63 271 .81 703 1.2239 .77 439 45 16 .63 293 .81 752 1.2232 .77 421 44 17 .63 316 .81 800 1.2225 .77 402 43 18 .63 338 .81 849 1.2218 .77 384 42 19 .63 361 .81 898 1.2210 .77 366 41 20 .63 383 .81 946 1.2203 .77 347 40 21 .63 406 .81 995 1.2196 .77 329 39 22 .63 428 .82 044 1.2189 .77 310 38 23 .63451 .82092 1.2181 .77292 37 24 .63473 .82141 1.2174 .77273 36 25 .63 496 .82 190 1.2167 .77 255 35 26 .63 518 .82 238 1.2160 .77 236 34 27 .63 540 .82 287 1.2153 .77 218 33 28 .63 563 .82 336 1.2145 .77 199 32 29 .63 585 .82 385 1.2138 .77 181 31 30 .63608 .82434 1.2131 .77162 3O 31 .63 630 .82 483 1.2124 .77 144 29 32 .63653 .82531 1.2117 .77125 28 33 .63 675 .82 580 1.2109 .77 107 27 34 .63698 .82629 1.2102 .77088 26 35 .63 720 .82 678 1.2095' .77 070 25 36 .63742 .82727 1.2088 .77051 24 37 .63765 .82776 1.2081 .77033 23 38 .63 787 .82 825 1.2074 .77 014 22 39 .63 810 .82 874 1.2066 .76 996 21 4O .63832 .82923 1.2059 .76977 2O 41 .63854 .82972 1.2052 .76959 19 42 .63877 .83022 1.2045 .76940 18 43 .63 899 .83 071 1.2038 .76 921 17 44 .63 922 .83 120 1.2031 .76 903 16 45 .63944 .83169 1.2024 .76884 15 46 .63966 .83218 1.2017 .76866 14 47 .63989 .83268 1.2009 .76847 13 48 .64011 .83317 1.2002 .76828 12 49 .64 033 .83 366 1.1995 .76 810 11 50 .64056 .83415 1.1988 .76791 10 51 .64078 .83465 1.1981 .76772 9 52 .64 100 .83 514 1.1974 .76 754 8 53 .64 123 .83 564 1.1967 .76 735 7 54 .64145 .83613 1.1960 .76717 6 55 .64 167 .83 662 1.1953 .76 698 5 56 .64 190 .83 712 1.1946 .76 679 4 57 .64212 .83761 1.1939 .76661 3 58 .64234 .83811 1.1932 .76642 2 59 .64256 .83860 1.1925 .76623 1 6O .64279 .83910 1.1918 .76604 O / cos cot tan sin / 50 78 NATURAL FUNCTIONS 40 / sin tan cot cos / .64279 .83910 1.1918 .76604 60 1 .64301 .83960 1.1910 .76586 59 2 .64323 .84009 1.1903 .76567 58 3 .64346 .84059 1.1896 .76548 57 4 .64368 .84108 1.1889 .76530 56 5 .64390 .84158 1.18S2 .76511 55 6 .64412 .84208 1.1875 .76492 54 7 .64435 .84258 1.1868 .76473 53 8 .64457 .84307 1.1861 .76455 52 9 .64479 .84357 1.1854 .76436 51 1O .64501 .84407 1.1847 .76417 50 11 .64524 .84457 1.1840 .76398 49 12 .64546 .84507 1.1833 .76380 48 13 .64568 .84556 1.1826 .76361 47 14 .64590 .84606 1.1819 .76342 46 15 .64612 .84656 1.1812 .76323 -45 16 .64 635 .84 706 1.1806 .76 304 44 17 .64 657 .84 756 1.1799 .76 286 43 18 .64679 .84806 1.1792 .76267 42 19 .64701 .84856 1.1785 .76248 41 2O .64723 .81-906 1.1778 .76229 4O 21 .64746 .84956 .1771 .76210 39 22 .64 768 .85 006 1.1764 .76 192 38 23 .64790 .85057 .1757 .76173 37 24 .64 812 .85 107 .1750 .76 154 36 25 .64834 .85157 .1743 .76135 35 26 .64856 .85207 .1736 .76116 34 27 .64878 .85257 1.1729 .76097 33 28 .64901 .85308 1.1722 .76078 32 29 .64923 .85358 1.1715 .76059 31 3O .64945 .85408 1.1708 .76041 30 31 .64967 .85458 1.1702 .76022 29 32 .64 989 .85 509 1.1695 .76 003 28 33 .65011 .85559 1.1688 .75984 27 34 .65 033 .85 609 1.1681 .75 965 26 35 .65055 .85660 1.1674 .75946 25 36 .65 077 .85 710 1.1667 .75 927 24 37 .65 100 .85 761 1.1660 .75 908 23 38 .65 122 .85 811 1.1653 .75 889 22 39 .65 144 .85 862 1.1647 .75 870 21 4O .65 166 .85912 1.1640 .75851 20 41 .65 188 .85 963 1.1633 .75 832 19 42 .65 210 .86 014 1.1626 .75 813 18 43 .65 232 .86064 1.1619 .75 794 17 44 .65254 .86115 1.1612 .75775 16 45 .65 276 .86 166 1.1606 .75 756 15 46 .65 298 .86 216 1.1599 .75 738 14 47 .65 320 -86 267 1.1592 .75 719 13 48 .65342 .86318 1.1585 .75700 12 49 .65364 .86368 1.1578 .75680 11 5O .65386 .86419 1.1571 .75661 10 51 .65408 .86470 1.1565 .75642 9 52 65430 .86521 1.1558 .75623 8 53 .65452 .86572 1.1551 .75604 7 54 .65474 .86623 1.1544 .75585 , 6 55 .65496 .86674 1.1538 .75566 5 56 .65518 .86725 1.1531 .75547 4 57 .65 540 .86 776 1.1524 .75 528 3 58 .65562 .86827 1.1517 .75509 2 59 .65584 .86878 1.1510 .75490 1 6O .65606 .86929 1.1504 .75471 o / cos cot tan sin / 49 41 / sin tan cot cos / .65606 .86929 1.1504 .75471 60 1 .65 628 .86 980 1.1497 .75 452 59 2 .65650 .87031 1.1490 .75433 58 3 .65 672 .87 082 1.1483 .75 414 57 4 .65 694 .87 133 1.1477 .75 395 56 5 .65 716 .87 184 1.1470 .75 375 55 6 .65 738 .87 236 1.1463 .75 356 54 7 .65 759 .87 287 1.1456 .75 337 53 8 .65781 .87338 1.1450 .75318 52 9 .65803 .87389 1.1443 .75299 51 1O .65 825 .87 441 1.1436 .75 280 50 11 .65 847 .87 492 1.1430 .75 261 49 12 .65 869 .87 543 1.1423 .75 241 48 13 .65 891 .87 595 1.1416 .75 222 47. 14 .65913 .87646 1.1410 .75203 46 15 .65 935 .87 698 1.1403 .75 184 45 16 .65956 .87 749 1.1396 .75 165 44 17 .65 978 .87 801 1.1389 .75 146 43 18 .66000 .87852 1.1383 .75 126 42 19 .66022 .87904 1.1376 .75107 41 2O .66044 .87955 1.1369 .75088 4O 21 .66066 .88007 1.1363 .75069 39 22 .66088 .88059 1.1356 .75050 38 23 .66109 .88110 1.1349 .75030 37 24 .66131 .88162 1.1343 .75 QU^ 36 25 .66153 .88214 1.1336 .74992 35 26 .66175 .88265 1.1329 .74973 34 27 .66197 .88317 1.1323 .74953 33 28 .66218 .88369 1.1316 .74934 32 29 .66240 .88421 1.1310 .74915 31 30 .66262 .88473 1.1303 .74896 30 31 .66 284 .88 524 1.1296 .74 876 29 32 .66306 .88576 1.1290 .74857 28 33 .66327 .88628 1.1283 .74838 27 34 .66349 .88680 1.1276 .74818 26 35 .66371 .88732 1.1270 .74799 25 36 .66393 .88784 1.1263 .74780 24 37 .66414 .88836 1.1257 .74760 23 38 .66436 .88888 1.1250 .74741 22 39 .66458 .88940 1.1243 .74722 21 4O .66480 .88992 1.1237 .74703 2O 41 .66501 .89045 1.1230 .74683 19 42 .66523 .89097 1.1224 .74664 18 43 .66545 .89149 1.1217 .74644 17 44 .66566 .89201 1.1211 .74625 16 45 .66588 .89253 1.1204 .74606' 15 46 .66610 .89306 1.1197 .74586 14 47 .66632 .89358 1.1191 .74567 13 48 .66653 .89410 1.1184 .74548 12 49 .66675 .89463 1.1178 .74528 11 50 .66697 .89515 1.1171 .74509 1O 51 .66718 .89567 1.1165 .74489 9 52 .66 740 .89 620 1.1158 .74 470 8 53 .66762 .89672 1.1152 .74451 7 54 .66783 .89725 1.1145 .74431 6 55 .66805 .89777 1.1139 .74412 5 56 .66827 .89830 1.1132 .74392 4 57 .66848 .89883 1.1126 .74373 3 58 .66870 .89935 1.1119 .74353 2 59 .66891 .89988 1.1113 .74334 1 6O .66913 .90040 1.1106 .74314 O / cos cot tan sin / 48 NATURAL FUNCTIONS 79 42 / sin tan cot cos / o .66913 .90040 1.1106 .74314 60 1 .66935 .90093 1.1100 .74295 59 2 .66956 .90146 1.1093 .74276 58 3 .66 978 .90 199 1.1087 .74 256 57 4 .66999 .90251 1.1080 .74237 56 5 .67021 .90304 1.1074 .74217 55 6 .67043 .90357 1.1067 .74198 54 7 .67064 .90410 1.1061 .74178 53 8 .67086 .90463 1.1054 .74159 52 9 .67107 .90516 1.1048 .74139 51 1O .67 129 .90 569 1.1041 .74 120 50 11 .67151 .90621 1.1035 .74100 49 12 .67172 .90674 1.1028 .74080 48 13 .67194 .90727 1.1022 .74061 47 14 .67 215 .90 781 1.1016 .74 041 46 15 .67 237 .90 834 1.1009 .74 022 45 16 .67258 .90887 1.1003 .74002 44 17 .67 280 .90 940 1.0996 .73 983 43 18 .67301 .90993 1.0990 .73963 42 19 .67323 .91046 1.0983 .73944 ' 41 20 .67344 .91099 1.0977 .73924 4O 21 .67366 .91153 1.0971 .73904 39 22 .67 387 .91 206 1.0964 .73 885 38 23 .67409 .91259 1.0958 .73865 37 24 .67430 .91313 1.0951 .73846 36 25 .67 452 .91 366 1.0945 .73 826 35 26 .67473 .91419 1.0939 .73806 34 27 .67495 .91473 1.0932 .73787 33 28 .67516 .91526 1.0926 .73767 32 29 .67 538 .91 580 1.0919 .73 747 31 3O .67 559 .91 633 1.0913 .73 728 30 31 .67 580 .91 687 1.0907 .73 708 29 32 .67 602 .91 740 1.0900 .73 688 28 33 .67 623 .91 794 1.0894 .73 669 27 34 .67 645 '.91 847 1.0888 .73 649 26 35 .67 666 .91 901 1.0881 .73 629 25 36 .67 688 .91 955 1.0875 .73 610 24 37 .67709 .92008 1.0869 .73590 23 38 .67 730 .92 062 1.0862 .73 570 22 39 .67 752 .92 116 1.0856 -.73 551 21 40 .67 773 .92 170 1.0850 .73 531 2O 41 .67795 .92224 1.0843 .73511 19 42 .67 816 .92 277 1.0837 .73 491 18 43 .67837 .92331 1.0831 .73472 17 44 .67859 .92385 1.0824 .73452 ,16 45 .67880 .92439 1.0818 .73432 15 46 .67901 .92493 1.0812 .73413 14 47 .67923 .92547 1.0805 .73393 13 48 .67 944 .92 601 1.0799 .73 373 12 49 .67965 .92655 1.0793 .73353 11 50 .67 987 .92 709 1.0786 .73 333 1O 51 .68008 .92763 1.0780 .73314 9 52 .68029 .92817 1.0774 .73294 8 53 .68051 .92872 1.0768 .73274 7 54 .68 072 .92 926 1.0761 .73 254 6 55 .68093 .92980 1.0755 .73234 5 56 .68 115 .93 034 1.0749 .73 215 4 57 .68 136 .93 088 1.0742 .73 195 3 58 .68 157 .93 143 1.0736 .73 175 2 59 .68 179 .93 197 1.0730 .73 155 1 60^ .68 200 .93 252 1.0724 .73 135 O / cos cot tan sin / 47 43 / sin tan cot cos / .68200 .93252 1.0724 .73135 6O 1 .68221 .93306 1.0717 .73116 59 2 .68242 .93360 1.0711 .73096 58 3 .68 264 .93 415 1.0705 .73 076 57 4 .68285 .93469 1.0699 .73056 56 5 .68 306 .93 524 1.0692 .73 036 55 6 .68327 .93578 1.0686 .73016 54 7 .68 349 .93 633 1 .0680 .72 996 53 8 .68 370 .93 688 1.0674 .72 976 52 9 .68 391 .93 742 1.0668 .72 957 51 1O .68412 .93797 1.0661 .72937 50 11 .68434 .93852 1.0655 .72917 49 12 .68455 .93906 1.0649 .72897 48 13 .68476 .93961 1.0643 .72877 47 14 .68497 .94016 1.0637 .72857 46 15 .68518 .94071 1.0630 .72837 45 16 .68 539 .94 125 1.0624 .72 817 44 17 .68 561 .94 180 1.0618 .72 797 43 18 .68582 .94235 1.0612 .72777 42 19 .68603 .94290 1.0606 .72757 41 2.0 .68624 .94345 1.0599 .72737 40 21 .68645 .94400 1.0593 .72717 39 22 .68666 .94455 1.0587 .72697 38 23 .68688 .94510 1.0581 .72677 37 24 .68 709 .94 565 1.0575 .72 657 36 25 .68 730 .94 620 1.0569 .72 637 35 26 .68751 .94676 1.0562 .72617 34 27 .68 772 .94 731 1.0556 .72 597 33 28 .68 793 .94 786 1.0550 .72 577 32 29 .68814 .94841 1.0544 .72557 31 30 .68835 .94896 1.0538 .72537 30 31 .68 857 .94 952. 1.0532 .72 517 29 32 .68878 .95007 1.0526 .72497 28 33 .63_S.9_9 .95062 1.0519 .72477 27 34 .68920 .95118 1.0513 .72457 26 35 .68941 .95173 1.0507 .72437 25 36 .68962 .95229 1.0501 .72417 24 37 .68 983 .95 284 1.0495 .72 397 23 38 .69004 .95340 1.0489 .72377 22 39 .69025 .95395 1.0483 .72357 21 40 .69046 .95451 1.0477 .72337 2O 41 .69067 .95506 1.0470 .72317 19 42 .69 088 .95 562 1.0464 .72 297 18 43 .69 109 .95 618 1.0458 .72 277 17 44 .69 130 .95 673 1.0452 .72 257 16 45 .69 151 .95 729 1.0446 .72 236 15 46 .69172 .95785 1.0440 .72216 14 47 .69 193 .95 841 1.0434 .72 196 13 48 .69 214 .95 897 1.0428 .72 176 12 49 .69235 .95952 1.0422 .72156 11 50 .69256 .96008 1.0416 .72136 1O 51 .69 277 .96 064 1.0410 .72 116 9 52 .69 298 .96 120 1.0404 .72 095 8 53 .69319 .96176 1.0398 .72075 7 54 .69340 .96232 1.0392 .72.055 6 55 .69361 .96288 1.0385 .72035 5 56 .69382 .96344 1.0379 .72015 4 57 .69403 .96400 1.0373 .71995 3 58 .69424 .96*457 1.0367 .71974 2 59 .69445 .96513 1.0361 .71954 1 6O .69466 .96569 1.0355 .71934 O / cos cot tan sin / 46 80 NATURAL FUNCTIONS 44 / sin tan cot cos / o .69 466 .96 569 1.0355 .71 934 6O 1 .69487 .96625 1.0349 .71914 59 2 .69508 .96681 1.0343 .71894 58 3 .69 529 .96 738 .1.0337 .71 873 57 4 .69549 .96794 1.0331 .71853 56 5 .69 570 .96 850 1.0325 .71 833" 55 6 .69591 .96907 1.0319 .71813 54 7 .69612 .96963 1.0313 .71792 53 8 .69633 .97020 1.0307 .71772 52 9 .69654 .97076 1.0301 .71752 51 10 .69 675 .97 133 1.0295 .71 732 50 11 .69696 .97189 1.0289 .71711 49 12 .69717 .97246 1.0283 .71691 48 13 .69737 .97302 1.0277 .71671 47 14 .69758 .97359 1.0271 .71650 46 15 .69 779 .97 416 1.0265 .71 630 45 16 .69800 .97472 1.0259 .71610 44 17 .69821 .97529 1.0253 .71590 43 18 .69842 .97586 1.0247 .71569 42 19 .69862 .97643 1.0241 .71549 41 2O .69883 .97700 1.0235 .71529 4O 21 .69904 .97756 1.0230 .71508 39 22 .69925 .97813 1.0224 .71488 38 23 .69946 .97870 1.0218 .71468 37 24 .69966 .97927 1.0212 .71447 36 25 .69987 .97984 1.0206 .71427 35 26 .70008 .98041 1.0200 .71407 34 27 .70029 .98098 1.0194 .71386 33 28 .70049 .98155 1.01SS .71366 32 29 .70070 .98213 1.0182 .71345 31 30 .70091 .98270 1.0176 .71325 30 31 .70112 .98327 1.0170 .71305 29 32 .70 132 .98 384 1.0164 .71 284 28 33 .70153 .98441 1.0158 .71264 27 34 .70174 .98499 1.0152 .71243 26 35 .70195 .98556 1.0147 .71223 25 36 .70215 .98613 1.0141 .71203 24 37 .70 236 .98 671 1.0135 .71 182 23 38 .70257 .98728 1.0129 .71162 22 39 .70277 .98786 1.0123 .71141 21 40 .70298 .98843 1.0117 .71 121 2O 41 .70319 .98901 1.0111 .71100 19 42 .70339 .98958 1.0105 .71080 18 43 .70360 .99016 1.0099 .71059 17 44 .70381 .99073 1.0094 .71039 16 45 .70401 .99131 1.0088 .71019 15 46 .70422 .99189 1.0082 .70998 14 47 .70443 .99247 1.0076 .70978 13 48 .70463 .99304 1.0070 .70957 12 49 .70484 .99362 1.0064 .70937 11 5O .70505 .99420 1.0058 .70916 1O 51 .70525 .99478 1.0052 .70896 9 52 .70546 .99536 1.0047 .70875 8 53 .70567 .99594 1.0041 .70855 7 54 .70587 .99652 1.0035 .70834 6 55 .70608 .99710 1.0029 .70813 5 56 .70628 .99768 1.0023 .70793 4 57 .70649 .99826 1.0017 .70772 3 58 .70670 .99884 1.0012 .70752 2 59 .70690 .99942 1.0006 .70731 1 6O .70711 1.0000 1.0000 .70711 O / cos cot tan sin . / 45 n I 7 - 5"' 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. *EP $3 i960 rrcr 65 -4PM ^25- LD 21-100m-7,'40 (0936s) 7~1 ^ o^ ^v-^e 9 , 7- 918256 THE UNIVERSITY OF CALIFORNIA LIBRARY "'*,>.. < ^* .'-