3sin²x + 1/2 sin2x = 2

3 sin (x) ^2 + (1/2) sin (2x) = 2

2 - 3 sin (x) ^2 = (1/2) sin (2x)

2 - 3 sin (x) ^2 = sin (x) cos (x)

[ 2 - 3 sin (x) ^2 ]^2 = sin (x) ^2 [ 1 - sin (x) ^2 ]

Обозначим sin (x) ^2 = t

[ 2 - 3t ]^2 = t (1 - t)

4 - 12 t + 9 t^2 = t - t^2

10 t^2 - 13 t + 4 = 0

D = 9

t = (13 (+/-) 3) / 20

t1 = 4/5

t2 = 1/2

sin (x) ^2 = t1

sin (x) ^2 = 4/5

sin (x) = (+/-) 2/sqr (5)

x = (+/-) arcsin (2/sqr (5)) + пк

sin (x) ^2 = t2

sin (x) ^2 = 1/2

sin (x) = (+/-) sqr (2) / 2

x = п/4 + пk/2