Let sin(x)= -4/5 in quadrant 4 and let tan(y)= -5/12 in quadrant 2

Question

Find sin (x-y)

Find cos (x+y)

Find cos (x/2)

Find tan (2x)

Find cos (2y)

Answer 1

First I make a sketch and fill in missing lengths with the Pythagorean Theorem, then tabulate the 3 major functions for each angle: sin x = -4/5 [given], cos x = 3/5, tanx = -4/3; sin y = 5/13, cos y = -12/13, tan y = -5/12 [given].

sin(x-y) = sin x cos y - cos x sin y = (-4/5)(-12/13) - (3/5)(5/13) = (48 - 15)/65 = 33/65.

cos(x+y) = cos x cos y - sin x sin y = (3/5)(-12/13) - (-4/5)(5/13) = (-36 + 20)/65 = -16/65.

cos(x/2) = ±[(1 + cos x)/2]^1/2 = ±[(1 + 3/5)/2 = (8/5)/2 = 8/10 = 4/5]^1/2 = - 2√5 /5, in quadrant 2.

tan(2x) = 2 tan x /(1 - tan² x) = 2(-4/3) /(1 - 16/9) = (-8/3)/(-7/9) = 8/3 • 9/7 = 24/7.

cos(2y) = [my choice of 3:] 1 - 2 sin² y = 1 - 2(5/13)² = 1 - 50/169 = 119/169.