Compound-angle Formulae

A compound angle is a sum or difference such as $A+B$. In general, $\sin(A+B)$ is not $\sin A+\sin B$: with $A=B=30^\circ$, the first is $\sqrt3/2$ but the second is 1.

The addition formula comes from splitting a triangle into two right triangles. In the diagram, $p,q$ are side lengths and $h$ is the perpendicular height. The top angle is $A+B$.

The base pieces have lengths $q\sin A$ and $p\sin B$. The height is both $p\cos B$ and $q\cos A$. Add the two areas, each half its base times height.

The whole triangle also has area $\tfrac12pq\sin(A+B)$, taking side $q$ as the base and $p\sin(A+B)$ as its perpendicular height. Divide both expressions by the same positive $pq/2$. The included angle, between sides $p$ and $q$, is $A+B$.

This construction explains the formula for acute $A,B$. The identity also holds for other angles, using their signed sine and cosine values.

Replace $B$ by $-B$. Cosine keeps its value but sine changes sign, giving the subtraction formula.

Complementary angles add to $90^\circ$: their sine and cosine values swap. Thus $\cos(A+B)=\sin((90^\circ-A)-B)$. Substituting into the subtraction formula gives:

MF19 supplies these formulae. In its paired-sign notation, $\pm$ means choose either the upper or lower sign throughout; $\mp$ uses the opposite sign. Cosine of a sum therefore has a minus between the products.

For tangent, divide the sine expansion by the cosine expansion. Dividing numerator and denominator by $\cos A\cos B$ gives:

Use these quotient forms only when $\tan A$ and $\tan B$ exist and the final denominator is non-zero. For example, they cannot evaluate $\tan(90^\circ+30^\circ)$ directly because $\tan90^\circ$ is undefined. Evaluate the combined angle or use sine divided by cosine instead.