Trigo Formulae

The following formulae will be useful when integrating Trigonometric functions. Taken from the MF15 formula sheet for JC.

Addition Formulae

$\begin{aligned} \sin(A\pm B)&\equiv\sin A\cos B\pm\cos A\sin B\\ \cos(A\pm B)&\equiv\cos A\cos B\mp\sin A\sin B\\ \tan(A\pm B)&\equiv\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\\ \end{aligned}$

Double Angle Formulae

$\begin{aligned} \sin 2A &\equiv 2\sin A\cos A\\ \cos 2A\equiv\cos^2 A-\sin^2 A&\equiv 2\cos^2 A-1\equiv 1-2\sin^2 A\\ \tan 2A&\equiv\frac{2\tan A}{1-\tan^2 A} \end{aligned}$

Remark: The second identity is useful for integrating $\sin^2 x$ and $\cos^2 x$ .

Factor Formulae

$\begin{aligned} \sin P+\sin Q&\equiv 2\sin\frac 12(P+Q)\cos\frac 12(P-Q)\\ \sin P-\sin Q&\equiv 2\cos\frac 12(P+Q)\sin\frac 12(P-Q)\\ \cos P+\cos Q&\equiv 2\cos\frac 12(P+Q)\cos\frac 12(P-Q)\\ \cos P-\cos Q&\equiv -2\sin\frac 12(P+Q)\sin\frac 12(P-Q) \end{aligned}$