Find the values of the following, using the principal values of the angles. 16 tan2 0 + tan1 0 23. sin (a + B + y) = sin a cos ẞ cos y + cos a sin ẞ cos y + cos a cos ẞ sin y - sin a sin ẞ sin y. 24. cos (a + B + y) = cos a cos ẞ cos Y sin a sin ẞ cos Y cos a sin ẞ sin y. sin a cos ẞ sin y 72. Functions of an angle in terms of functions of twice the angle. By [20], cos 2 a - = we have sin a = ± 1-2 sin2 a. Solving this for sin a, That is, the sine of an angle is equal to the square root of one half of the quantity, one minus the cosine of twice the angle. Also by [20], cos 2 a = 2 cos2 a 1. Solving this for cos a, = and we have 1+ cos 0 COS 0 = ± 2 That is, the cosine of an angle is equal to the square root of one half of the quantity, one plus the cosine of twice the angle. The last two forms given in [24] may be obtained as follows: Multiplying numerator and denominator of V 1 COS 0 by 1. Given the functions of 45°; find the functions of 221°. Ans. sin 221° = √2 - √2, etc. 4. Having obtained the functions of 20°, 36°, and 72° from the tables, find by computation, sine, cosine, and tangent of 10°, 18°, and 36° respectively. 5. Find the value of sin ( cos1 }). Find the values of the three following expressions, using only the principal values of the inverse functions. 11. In [24], show why the sign is not necessary before cos 8 and sin 0 sin 0 1 + cos 0 73. To express the sum and difference of two like trigonometric functions as a product. In this article the following formulas are proved. [25] sin a + sin ẞ = 2 sin (a + B) cos (a — B). B = −2 sin (a + ẞ) sin (a − ẞ). [28] cosa cos ẞ The object of these four relations is to express sums and differences of functions as products. In this manner formulas can be made suitable for logarithmic computations. Proof of [25] and [26]. Let a = x + y and ẞxy. Subtracting (b) from (a) and substituting for x and y, sin ẞ = 2 cos x sin y = 2 cos(a + B) sin (a — B). In Exercises 1 to 4, find the values of the functions from the tables, carry out the indicated operations on each side of the equality sign, and compare results. 1. sin 60° + sin 40° = 2 sin 50° cos 10°. Solution. The right-hand member is best computed by logarithms. By [27], cos 70° + cos 30° = 2 cos(70° + 30°) cos (70° — 30°) = 2 cos 50° cos 20°. Proof. Express cot a and tan ẞ in terms of sine and cosine; then If a + B + r = 180°, prove the identities in the following problems. 17. sin a + sin ẞ + sin y = 4 sin (a + B) cosa cos B. 2 cosa cos B. sin a + sin ẞ+ sin y = 2 sin (a + B) 2 cosa cos B 4 sin (a + B) cosa cos B. 18. cos acos 8+ cos y = 4 cos (a + B) sina sin 18 +1. 19. cos 2a + cos 26+ cos 2 y -4 cos a cos ẞ cos y - 1. 20. sin 2a + sin 28 + sin 2y = 4 sin a sin ẞ sin y. = |