(1) 2 cos²x = 1 + sin x 2 - 2 sin²x = 1 + sin x 2 sin²2 + sin x - 1 = 0 D = 1 + 8 = 3² sin x = (1 + 3) / 4 = 1 sin x = (1 - 3) / 4 = - 1/2 (2) 2 sin²x - 5 sin x cos x + 5 cos²x = 1 2 sin²x - 5 sin x cos x + 5 cos²x = sin²x + cos²x sin²x - 5 sin x cos x + 4 cos²x = 0 | /cos²x tg²x - 5 tg x + 4 = 0 (tg x - 4) (tg x - 1) = 0 tg x = 4, tg x = 1 (3) sin x + cos x + sin 3x = 0 sin x + cos x + sin (2x + x) = 0 sin x + cos x + sin 2x cos x + sin x cos 2x = 0 sin x (1 + cos 2x) + cos x (1 + sin 2x) = 0 sin x (1 + cos²x - sin²x) + cos x (1 + sin 2x) = 0 sin x (2 cos²x) + cos x (1 + sin 2x) = 0 cos x (2 sin x cos x + 1 + sin 2x) = 0 cos x (2 sin 2x + 1) = 0 cos x = 0 x = ± pi / 2 + 2 pi k, k in Z 2 sin 2x + 1 = 0 sin 2x = - 1/2 2x = - pi / 6 + (-1)^m 2 pi m, m in Z x = - pi / 12 + (-1)^m pi m (5) √3 cos x - sin x = 1 | /2 (√3/2) cos x - (1/2) sin x = 1/2 cos (pi/6) cos x - sin (pi/6) sin x = 1/2 cos (pi/6 + x) = cos (pi/3) pi/6 + x = ± pi/3 + 2 pi k, k in Z x = ± pi/3 - pi/6 + 2 pi k (4) √3 cos x - sin x = 0 √3 cos x = sin x tg x = √3 x = pi / 3 + pi k, k in Z
