Swap cos²x for sin²x

$2{\cos\left(x\right)}^{2}+\sin\left(x\right) = 1$ has two functions, cosine squared and sine. ${\sin\left(x\right)}^{2}+{\cos\left(x\right)}^{2} = 1$ lets you trade one for the other: cos²x = 1 − sin²x.

Substitute and expand: $2-2{\sin\left(x\right)}^{2}+\sin\left(x\right) = 1$, so $2{\sin\left(x\right)}^{2}-\sin\left(x\right)-1 = 0$, a quadratic in sin x: $\left(2\sin\left(x\right)+1\right)\left(\sin\left(x\right)-1\right) = 0$. sin x = −1/2 gives 210° and 330°; sin x = 1 gives 90°.