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