# Double and Half Angle Formulas (Summary)

## Double Angle

Note that for cosine, of the three, first is most commonly used.

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

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

\(\cos(2\theta) = 2\cos^2\theta - 1\)

\(\cos(2\theta) = 1 - 2 \sin^2\theta\)

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

## Half Angle

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

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

\(\tan(\frac{\alpha}{2}) = \pm \sqrt{\frac{1\space-\space\cos\alpha}{1\space+\space\cos\alpha}}\)

\(\pm\) is determined by quadrant of \(\frac{\alpha}{2}\)

## New Identities

(Paul’s Math Notes call these Half Angle Formulas, alternate form)

\(\sin^2\theta = \frac{1\space-\space\cos(2\theta)}{2}\)

\(\cos^2\theta = \frac{1\space+\space\cos(2\theta)}{2}\)

\(\tan^2\theta = \frac{1\space-\space\cos(2\theta)}{1\space+\space\cos(2\theta)}\)