Notes
compound angle formulae

Compound Angle Formulae

The compound angle formulae are identities satisfied by the trigonometric functions. They state that for all angles $x$ and $y$ then:

\begin{aligned} \sin(x + y) &= \sin(x) \cos(y) + \cos(x) \sin(y) \\ \cos(x + y) &= \cos(x) \cos(y) - \sin(x) \sin(y) \\ \tan(x + y) &= \frac{\tan(x) + \tan(y)}{1 - \tan(x) \tan(y)} \end{aligned}

Double Angle Formulae

Applying the above with $x = y$ yields:

\begin{aligned} \sin(2 x) &= 2 \sin(x) \cos(x) \\ \cos(2 x) &= \cos^2(x) - \sin^2(x) = 2 \cos^2(x) - 1 = 1 - 2 \sin^2(x) \\ \tan(2 x) &= \frac{2 \tan(x)}{1 - \tan^2(x)} \end{aligned}

Half-Angle Formulae

Setting $t = \tan\left(\frac{1}{2} x\right)$ then the double angle formulae can be rewritten as:

\begin{aligned} \sin(x) &= \frac{2 t}{1 + t^2} \\ \cos(x) &= \frac{1 - t^2}{1 + t^2} \\ \tan(x) &= \frac{2 t}{1 - t^2} \end{aligned}

category: trigonometry

Created on November 10, 2025 21:52:04 by Andrew Stacey?