AHL 3.10 (HL)—Compound-angle identities

Compound-angle identities rewrite a difficult angle.

The addition and subtraction identities express sin(A±B) and cos(A±B) using known values of A and B, so an unfamiliar angle can be decomposed into familiar ones.

Example

sin 75°=sin(45°+30°)=sin45°cos30°+cos45°sin30°=(√6+√2)/4.

Choose a decomposition that gives exact known angles, expand once, then simplify and check the sign from the quadrant.

An identity is true for every allowed angle; a numerical equation may have only selected solutions.

Identity set: $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$ and $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$. Dividing the sine expansion by the cosine expansion gives $\tan(A\pm B)=(\tan A\pm\tan B)/(1\mp\tan A\tan B)$ when defined; setting $A=B=\theta$ gives $\tan2\theta=2\tan\theta/(1-\tan^2\theta)$. Denominators and quadrant signs remain part of the answer.