Question

How do you find the general solutions for `2 cos^2 x = 3 sin x`?

Answer 1

Use `sin^2 x + cos^2 x = 1`

Explanation:

`2 cos^2 x = 3 sin x`
`2 (1 - sin^2 x) = 3 sin x`
`2 - 2 sin^2 x = 3 sin x`
`2 sin^2 x + 3 sin x - 2 = 0`
`(2 sin x - 1)(sin x + 2) = 0`
`sin x = -2` (no solutions) or `sin x = 1/2 \Rightarrow x = pi/6 + 2kpi=pi/6(1+12k), k in ZZ` or `x=(5pi)/6+2kpi=pi/6(5+12k), k in ZZ`
`therefore x=pi/6(1+12k)` or `x=pi/6(5+12k), k in ZZ`