Question: solve 2sin^2(x)+3cos(x)=3.
Thius is actually a quadratic equation, in disguise. The equation is,
2sin^2(x) + 3cos(x) = 3
2(1 - cos^2(x)) + 3cos(x) = 3
now let u = cos(x). The equation becomes,
2(1 - u^2) + 3u = 0
2 - 2u^2 + 3u =0
2u^2 - 3u - 2= 0
(2u + 1)( u - 2) = 0
u = -1/2, u = 2
cos(x) = -1/2, cos(x) = 2
ignore the cos(x) = 2 solution. It is a solution to the quadratic equation, but not to the trig equation, since |cos(x)| <=1.
cos(x) = -1/2
x = 2.pi/3 + 2n.pi