But the angle C is equal to the difference between the angles a and b (Geom. B. I., P. 25, C. 6): hence, 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 and recollecting (Art. 66) that, sin(x) - sin x and cos(x)cos x; and also that a-(-b) = a + b, 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, and, 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. By formula (b) we have, sin (a+b) = sin a cos b + cos a sin b. For a substitute 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.), sin (90° — a) = cos α, cos (90° - a) = sin a; making the substitutions, we have, cos (a - b) = cos a cos b + sin a sin b.. (d) 75. To find the formula for the tangent of the sum of two dividing both numerator and denominator by cos a cos b, 76. To find the tangent of the difference of two arcs. sin a cos b = cos a sin b (Table I). cos a cos b + sin a sin b' by (a) and (d). Dividing both numerator and denominator by cos a cos o, tan (a b) = tan a tan b (g) 77. The student will find no difficulty in deducing the following formulas. sin (a + b) 78. To find the sine of twice an arc, in functions of the are By formula (b) and radius. Make a = b, and the formula becomes, = sin a cos b + cos a sin b. 79. To find the cosine of twice an arc in functions of the arc By formula (c) and radius. cos (a + b) = cos a cos b- sin a sin b. By Table I., sin a 1 cos a; hence, by substitution, 80. To determine the tangent of twice or thrice a given arc in functions of the arc and radius. substituting the value of tan 2a, and reducing, we have, 81. To find the sine of half an arc in terms of the functions of the arc and radius. 82. To find the cosine of half a given are in terms of the functions of the arc and radius. By formula (1) cos 2a = 2 cos2 a - 1. For a, substitute a, and we have, 1 + cos a (p) 83. To find the tangent of half a given arc, in functions of the arc and radius. Divide formula (o) by (p), and we have, Multiplying both terms of the second member by √1 — cos a, Multiplying both terms by the denominator √1 + cos a, 84. The formulas of Articles 71, 72, 73, 74, furnish a great number of consequences; among which it will be enough to mention those of most frequent use. By adding and subtracting we obtain the four which follow, sin (a) sin (ab) = 2 sin a cos b, (2) (s) 1 and which serve to change a product of several sines or cosines into linear sines or cosines, that is, into sines and cosines multiplied only by constant quantities. 85. If in these formulas we put a + b = P, a which gives a = p + q, b = P = q we shall find 2 2 + sin 7 = 2 sin (p + q) cos (p − 2), ・ sin p + sin ・ 2 cos(p + q) cos (pg),.. 2 sin (p + q) sin (p − q), . (11) (*) |