69. The examinations thus far, have been limited to arcs which do not exceed 360°. It is easily shown, however, that the addition of 360° to any arc as x, will make no difference in its trigonometrical functions; for, such addition would terminate the arc at precisely the same point of the circumference. Hence, if C represent an entire circumference, or 360°, and n any whole number, we shall have, sin (C+x) = sin x; or sin (n x C + x) = sin x. The same is also true of the other functions. 70. It will further appear, that whatever be the value of an arc denoted by x, the sine may be expressed by that of an arc less than 180°. For, in the first place, we may subtract 360° from the arc x, as often as 360° is contained in it: then denoting the remainder by y, we Thus, all the cases are reduced to that in which the are whose functions we take, is less than 180°; and since we also know that, sin (90+ x)=sin (90 - x), they are ultimately reducible to the case of arcs between 0 and 90°. GENERAL FORMULAS. 71. To find the formula for the sine of the difference of two angles or arcs. Let ACB be a triangle. From the vertex C let fall the perpendicular CD, on the base AB, produced. Denote the exterior angle CBD by A B D a, and the angle CAB by b. Then, ABAD - DB. But (Art. 25), AD = AC cos b, and BD = BO cos CBD. But, since sin a = sin CBA, we have (Art. 21) But the angle C is equal to the difference between the angles a and b (Geom. B. I., P. 25, C. 6): hence, sin (a - b) that is, The sine of the difference of any two arcs or angles is equal to the sine of the first into the cosine of the second, minus the cosine of the first into the sine of the second. It is plain that the formula' is equally true in whichever quadrant the vertex of the angle C be placed: hence, the formula is true for all values, of the arcs a and b. 72. To find the formula for the sine of the sum of two angles substituting for b,-b, and recollecting (Art. 66) that, we shall have, after making the substitutions and combining the algebraic signs, sin (a + b) = sin a cos b + cos a sin b. (b) 73. To find the formula for the cosine of the sum of two angles or arcs. By formula (b) we have, sin (a + b) = sin a cos b + cos a sin b, substitute for a, 90° + a, and we have, sin [(90° + a) +b] = sin (90° + a) cos b + cos (90° + a) sin b. But, sin [90° + (a + b)] = cos (a + b) (Table II.): making the substitutions, we have, cos (a + b) = cos a cos b- sin a sin b. 74. To find the formula for the cosine of the difference between two angles or arcs. sin [90° — (a - b)] = sin (90° — a) cos b + cos (90° — a) sin b. But, — — making the substitutions, we have, cos (a - b) = cos a cos b + sin a sin b.. 75. To find the formula for the tangent of the sum of two sin a cos b + cos a sin b cos a cos b - sin a sin b' by (b) and (e), dividing both numerator and denominator by cos a cos ↳ 76. To find the tangent of the difference of two arcs. Dividing both numerator and denominator by cos a cos o, 77. The student will find no difficulty in deducing the 78. To find the sine of twice an arc, in functions of the arc By formula (6) sin (a + b) and radius. = sin a cos b + cos a sin b. Make a = b, and the formula becomes, 79. To find the cosine of twice an arc in functions of the are and radius. |