If sin x + sin 2x = 1 and cos x + cos 2x = 0, find
x.

Solution

Given sin x + sin 2x = 1 and cos x + cos 2x = 0. Using the double-angle identities: sin 2x = 2 sin x cos x and cos 2x = 2 cos² x - 1. Substitute sin 2x and cos 2x into the equations: From sin x + 2 sin x cos x = 1, sin x(1 + 2 cos x) = 1. (Equation 1) From cos x + (2 cos² x - 1) = 0, 2 cos² x + cos x - 1 = 0. Solving this quadratic equation gives cos x = 1/2 or cos x = -1. For cos x = 1/2, substitute into Equation 1: sin x(1 + 2 × 1/2) = 1, sin x(1 + 1) = 1, 2 sin x = 1, so sin x = 1/2. Thus, x = 30°. Answer: a) 30°.