formulæ (A) and (B), and inquire, what the signs of cos 2 sin must be, and which is the greatest: we can then deter mine whether cos 2 + sin is positive or negative, and so also, 2 whether cos - sin is positive or negative. Having thus 2 2 given their proper signs to the radicals, we can add and subtract the equations (A) and (B), and obtain the proper values 34. Writing for a in the forms for cos 2a, we obtain 35. Returning to the formulæ in Arts. 27, 28, sin (a + B) = sin a cos B + cos a sin ß, sin (a-B) = sin a cos ẞ - cos a sin ß, We get by addition and subtraction 2 sin a cos B = sin (a+B) + sin (a − ß), 2 sin a sin ẞ= cos (a — B) - cos (a+B), which forms enable us when we have the product of two sines or cosines to replace them by the sum or difference of two other sines and cosines. forms which enable us to pass from the sum or difference of two sines or cosines, to the product of two other sines or cosines. EXAMPLES. A. 1. Find the sine of 15o. sin 15o = sin (45o — 30°) = sin 45° cos 30° - cos 45° sin 30° √3 - 1 2. Find the Trigonometrical Ratios of 71°, 105°. = sin (a + B) sin (a — B) = sin2 a — sin2 ß. sin (a+B) sin (a – B) (sin a cós B + cos a sin ẞ) (sin a cos ẞ - cos a sin ẞ) =sin2 a cos2 B-cos2 a sin2 ß The other value of sin corresponds to 20+30=-270°, or =-54°. 6. If a+B+y= 180°, we have tan a+tan B+tan y = tan a tan ẞ tan y, tan (a + B) = tan (180° — y), tan a+tan B 1- tan a tanẞ or tan Y, 8. 9. sin (a-3) sin y+sin (B-y) sin a + sin (y- a) sin ẞ=0. sin B. 10. cos a + cos (a + 2B)=2 cos (a + B) cos B. 11. cos B cos (2x + B) = cos2 (a + B) - sin2 a. cos (a + B) sin a = 15. sin (30°+a) + sin (30°- a) = cos a. 18. cos2 (a-B) — sin2 (a + B) = cos 2x cos 26. 2 19. (sin - sin 4)2 + (cos — cos 4)2 = 2 vers (0 − 4). 20. cos (30-a) - cos (30+ a) = sin a. 21. cota + tan a = 2 cosec 2a. |