All the angles θ with 0° ≤ θ ≤ 180° for which sin2θ = cosθ are:

Write sin2θ = 2·sinθ·cosθ and move everything to one side: cosθ(2sinθ − 1) = 0. Do NOT divide by cosθ — that is exactly how the root θ = 90° gets lost. The two factors give cosθ = 0, i.e. θ = 90°, and sinθ = 1/2, i.e. θ = 30° or 150°. All three check: sin60° = cos30° = 0.866, sin180° = cos90° = 0, and sin300° = cos150° = −0.866. Cancelling cosθ leaves the incomplete pair 30° and 150°; keeping only the first-quadrant root gives 30° alone.