1) 2sin2x+3sin^2x=0 2) 5cos2x-2cosx-3=0 3) 21sin2x+2sin^2x+8=0 4) 9cosx-5sinx-5=0

1) 4 sin x cos x + 3 sin ² x = 0.

sin x (4 cos x + 3 sin x) = 0.

a) sin x = 0.

x = pi k.

b) tg x = - 4/3.

x = arctg - 4/3 + pi k.

2) 5 cos ² x - 5 sin ² x - 2 cos x - 3 = 0.

5 cos ² x - 5 (1 - cos ² x) - 2 cos x - 3 = 0.

10 cos ² x - 2 cos x - 8 = 0.

cos x = (1 ± 9) / 10.

a) cos x = 0.

x = pi/2 + pi k.

b) cos x = - 4/5.

x = ± arccos - 4/5 + pi k.

3) 42 sin x cos x + 2 sin ² x + 8 (cos ² x + sin ² x) = 0.

10 tg ² x + 42 tg x + 8 = 0.

tg x = ( - 21 ± 19) / 10.

a) tg x = - 1/5.

x = arctg - 1/5 + pi k.

b) tg x = - 4.

x = arctg - 4 + pi k.

4) Применим универсальную тригонометрическую подстановку:

t = tg x/2.

sin x = 2 t / (1 + t ²); cos x = (1 - t ²) / (1 + t ²).

тогда:

9 (1 - t ²) / (1 + t ²) - 5 2 t / (1 + t ²) - 5 = 0.

14 t ² + 10 t - 4 = 0.

t = tg x/2 = ( - 5 ± 9) / 14.

a) tg x/2 = - 1.

x/2 = - pi/4 + pi k.

x = - pi/2 + 2 pi k.

b) tg x/2 = 2/7.

x/2 = arctg 2/7 + pi k.

x = 2 arctg 2/7 + 2 pi k.