11. (Sec+ sin 2 0) (sec † π = cot (+0) tan (+0). sin 15°, and cosec 30 cosec 18-cosec 542. 14. 1+ cos 2 A cos 2 B = 2 (sin 'A sin 2B + cos 'A cos 2B). INVERSE TRIGONOMETRICAL FUNCTIONS. 17. Let sin a=s, cos a=c, tan a = t, etc. . . . . (1); then a is an arc whose sine, cosine, tangent, are s, c, t, etc., and these are termed direct functions of the arc a. On the contrary, the arc a is termed an inverse function of s, c, or t, and is expressed by the index - 1 written above the symbol which indicates the function. Thus, instead of saying that a is the arc whose sine is s, etc., this relation is expressed by the notation, in which 1 merely expresses the connexion between the arc and its function in accordance with the preceding explanation. The relations then, απ sin-11, 45; = for it has been shown that sin 90° = 1, cos 60° = 4, tan 45° = 1. Again by (2), sin sin's sin a, but by (1), sin a = s, hence sin sin's = s; consequently sin and sin -1 indicate operations that mutually destroy each other, and so for other functions. The student will see the value of this notation when he comes to the Integral Calculus. CHANGING THE RADIUS IN TRIGONOMETRICAL EQUATIONS. 18. As has been noticed in a preceding part (the "Application,") when in applying algebra to the resolution of a problem of geometry or physics, we have not taken any of the quantities under consideration for the linear unit, the equation or equations which we obtain are necessarily homogeneous. In the preceding investigations, however, the radius has been assumed as the unit, and hence if we wish to transform the results into others in which the radius is some other quantity r, this quantity must be introduced as a multiplier or divisor, as the case may be, to render all the terms of the equation or equations of the same degree. Thus, (1), (2), (3) of single angles, to radius 1, become sin A+ cos 2A = r2, tan A sin A cot A cos A 3 to radius r. Again, in the equation sin 3 A-3 sin A+ 4 sin 'A=0, the first and second terms are of the first degree, but the remaining one is of the third; it is necessary, therefore, to render the equation homogeneous to multiply the first and second terms by r. Hence 7.2 sin 3 A · 37 siu A+ 4 sin 3A = 0. This method applies to all equations which involve direct functions only of the arcs or angles. In the case of inverse functions, we thus proceed: Let a and ẞ be the arcs which subtend an angle A° to radii 1 and r, then, 1:r::a: ẞ, hence ẞra, and a = β Consequently, if the expressions are to be transformed from radius r to radius 1, r a must be written for the arc ß, and if they are to be transВ formed to radius r, must be written for the arc a. r cot A+ cot B + cot C = cot A cot B cot C, tan A + tan B+tan C = tan A tan B tan C + sec A sec B sec C, sin 2 A+ sin 2 B + sin 2 C = 4 cos A cos B cos C. 4. If A + B + C = 180°, then 1= sin A+ sin B + sin C = 4 cos A cos B cos C, B sin C + 1, cos A+ cos B + cos C = 4 sin A sin 5. Find x from each of the equations: sin(x+a) = cos(x — α), sin(x+a) + cos (x + a) = sin (x − a) + cos ( x − α), sin a+sin(x - a) + sin (2 x + a) =sin(x+a) + sin(2x-a). 6. Prove that sin -13 8. Given sin + cos 0 a, and sin + cos = b, to find and . 9. Determine a from each of the equations tan a cot a = 4, and tan a + 3 cot a = 4. VOL. I. 10. Prove that if cos (AC) cos B = cos (A - B+C); tan A, tan B, and tan C are in harmonical progression. 11. Prove that if cos A = cos B cos C, then chd 72° = 2 cos 54°, and sec 72° = sec 60° + sec 36°. 15. Prove that vers 45° is an arithmetic mean between sec 60 ̊ and sec 225, and that cosec 150 is a geometric mean between cosec 105° and cosec 165°. ON THE USE OF THE TRIGONOMETRICAL TABLES. 19. It will now be necessary to give a description of the tables of sines, tangents, etc., the method of constructing such being reserved for a subsequent part. In the Table of Natural Sines, etc., the radius is unity, and therefore the sines and cosines of all angles are either unity or decimal fractions less than unity. The decimal point, however, is sometimes omitted. The same is the case with the tangents of angles less than 45 ̊. The Table of Log. Sines, etc., is formed by taking the logarithms of the numbers in the corresponding Table of Natural Sines, etc.; and, to avoid negative indices, 10 is added to each, so that log sin A = log nat sin of A+ 10, and so on. (A.) To find the sine, cosine, etc., of any angle less than 90° expressed by degrees and minutes. If the angle be less than 45°, find the degrees at the top, and the minutes on the left-hand side, of the page. In the same line with the minutes, and under the proper name at the top (sine, cosine, etc.), take out the number required. Thus the natural sine and tangent of 27° 16′ are 4581325 and 5154019 respectively. Also the log sine and tangent of the same are 9.6609911 and 9.7121461. For angles between 44° and 90°, find the degrees at the bottom, and the minutes on the right hand side, of the page. In the same line with the minutes, and above the proper name at the bottom, take out the number. Thus the natural cosine of 56' 7' is 5575036. (B.) To find the sine, cosine, etc., of any angle less than 90°, expressed by degrees, minutes, and seconds. Find the sine, etc., of the next less and next greater angles in the table, and take the difference of these; then because the difference of any two trigonometrical functions of two angles (except in extreme cases) is as the difference of the angles themselves when the difference of the angles is less than one minute, we have 60" given number of seconds :: diff. thus found correction for seconds. It must be kept in mind that the sine, tangent, secant, when the angle is under 90, increase as the angle increases; but the cosine, cotangent, and cosecant decrease as the angle increases. Hence the correction must be added in the case of the sine, tangent, and secant, but subtracted in the case of the cosine, cotangent, and cosecant. 1. Let it be required to find the sine and cosine of 59° 14′ 16′′. pro. pts. for 16" = sin 59° 14′ 16′′ = ·8592973 The log sines, cosines, etc., of angles and seconds, are found in the same way. to the nearest unit; and hence, if the greater than 5, we must add 1 to the last figure, as in the preceding examples. (C.) To find the angle corresponding to any given sine, cosine, etc., to the nearest second. If the function be a sine, tangent, or secant, find the next less to the given one; but if a cosine, cotangent, or cosecant, the next greater, and take out the corresponding number of degrees and minutes. Then having found the difference between the next less or next greater, as the case may be, and the given one, we have this proportion : tabular diff. diff. found :: 60" seconds, to be added to the degrees and minutes already found. This proportion will be obvious from what has been stated. If the "tabular diff." be not given in the tables, it will be found by taking the next less from the next greater in the tables. 2. Given sin A = ·5432107, to find A. sin A5432107 sin 32° 54′ (next less) = ·5431744 * Hutton's Tables are referred to, but other tables are used in a similar way. 2138 (tab. diff.): 1423 :: 60" 40" nearly. The method for log sines, etc., is exactly similar. Scholium.-For angles greater than 90°, and less than 180°, the functions of π A may be substituted for them, subject to the changes of sign as indicated in Art. 15. Thus sin Asin (180° — A), cos A 13 tan (180 - A), etc. Hence, sin 91° 11′ 12′′ cos 91° 11' 12" = = = sin (180 - cos 88° 48′ 48", etc. = - cos (180° - A), tan A 91° 11′ 12′′) = sin 88° 48′ 48′′, PROPERTIES OF PLANE TRIANGLES, WITH NUMERICAL EXAMPLES.* THE RIGHT-ANGLED TRIAngle. 20. Let A B C be a triangle, right-angled at C. Denote the angles by A, B, C, and the sides opposite to these, respectively, by a, b, c. Then by Art. 14, These relations comprise every trigonometrical property of the rightangled triangle. NUMERICAL EXAMPLES. 1. In the triangle ABC, right-angled at C, there are given BC = 142, and the angle A = 26° 17′ 19′′, to find the other parts. The angle B being the complement of A, we have also, and B=90 A 63° 42′ 41′′; = tan B, or ba tan B, c = a cosec A, by the preceding expressions (A) and (C). Calculation of b and c, by Natural Tangents, etc. (p. p. in the work means proportional parts.) Nat tan 63° 42′ = 2·0233462 A triangle consists of six parts, viz., three sides and three angles; and if three of these be given (one being a side), the triangle is completely defined. |
