Compound Angles (HL only)

\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)

\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)

\(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)

For the tangent form, both individual tangents and the final denominator must be defined. Choose signs carefully: cosine uses the opposite sign between the products.

Example

\(\begin{aligned}\sin75^\circ&=\sin(45^\circ+30^\circ)\\&=\dfrac{\sqrt2}{2}\dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\dfrac12\\&=\dfrac{\sqrt6+\sqrt2}{4}.\end{aligned}\)

Setting \(B=A\) derives the sine and cosine double-angle identities and \(\tan2A=\dfrac{2\tan A}{1-\tan^2A}\). Other multiple-angle relations can be derived by repeated use of compound-angle identities.