Half Angle: Sine

This identity lets you find the sine of half an angle using only the cosine of the full angle, which is especially useful when the half-angle is not given directly. It comes from the double-angle relation for cosine, since rearranging cos(2x) = 1 - 2sin^2(x) and setting x = θ/2 gives sin^2(θ/2) = (1 - cosθ)/2. Because taking a square root loses sign information, the ± symbol is required: the correct sign depends on which quadrant θ/2 lies in.

Use this when you know an angle’s cosine but need the sine of half that angle, or when simplifying expressions in trigonometric proofs and integration. It also appears in geometry and physics problems where an angle is halved and only the original angle’s cosine is available.

• Forgetting the ± sign and assuming the result is always positive
• Using the identity with cos(θ/2) instead of cosθ
• Choosing the sign without checking which quadrant θ/2 is in