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}

Remark: The factor formulae are useful for integrating \sin nx\cos mx, \sin nx\sin mx, etc.


About mathtuition88
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