notes/education/math/MATH1060 (trig)/Double and Half Angle Identities.md

To solve for a double or half angle identity:

1. Draw a triangle
2. Choose an identity to use
3. Substitute into formula

Double Angle Identities

Sine:

 \sin(2\theta) = 2\sin\theta\cos\theta 

Cosine:

\begin{matrix}
\cos(2\theta) = \cos^2\theta - \sin^2\theta\\
= 1 - 2sin^2\theta\\
= 2cos^2\theta - 1\\
\end{matrix}

Tan:

 \tan(2\theta) = \dfrac{2\tan\theta}{1-\tan^2\theta}

Half Angle Identities

Whether the output is positive or negative depends on what quadrant the output is in. Sine:

 \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos\theta}{2}} 

Cosine:

 \cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos\theta}{2}} 

Tangent:

\begin{matrix}
\tan(\dfrac{\theta}{2}) = \pm\sqrt{\dfrac{1-\cos\theta}{1 + \cos\theta}}\\
= \dfrac{\sin\theta}{1 + \cos\theta}\\
= \dfrac{1 - cos\theta}{\sin\theta}
\end{matrix}