Angle Addition Formulae

The angle addition (compound-angle) formulae tell you the sine and cosine of a sum or difference of two angles in terms of the sines and cosines of the parts:

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

\cos(A + B) = \cos A\cos B - \sin A\sin B

The difference versions just flip the middle sign:

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

\cos(A - B) = \cos A\cos B + \sin A\sin B

Notice the pattern: \sin mixes \sin\cos with a matching sign, while \cos keeps each function with itself and flips the sign.

Why they are useful

Their real power is finding exact values of brand-new angles by splitting them into two angles you already know. For example 75^\circ = 45^\circ + 30^\circ, so

\sin 75^\circ = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ = \tfrac{\sqrt2}{2}\cdot\tfrac{\sqrt3}{2} + \tfrac{\sqrt2}{2}\cdot\tfrac12 = \tfrac{\sqrt6 + \sqrt2}{4}.

For any two angles A and B:

\sin(A + B) = \sin A\cos B + \cos A\sin B;
\cos(A + B) = \cos A\cos B - \sin A\sin B;
\sin(A - B) = \sin A\cos B - \cos A\sin B;
\cos(A - B) = \cos A\cos B + \sin A\sin B.

Splitting an angle as a sum or difference of special angles lets you build exact values for new angles such as 15^\circ (45^\circ - 30^\circ) and 75^\circ (45^\circ + 30^\circ).

Adding angles at a vertex

Placing two angles A and B next to each other at a single vertex gives the combined angle A + B — the geometric idea the formulae capture.