Step By Step Calculus » 5.5 - Double and Half Angle Formulae

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Synopsis
Double Angle Formulae By a simple substitution in the addition formulae for sine, cosine and tangent, we get
\displaystyle \textbf{(Double Angle Formulae)}\qquad\left\{ \begin{array}{l} \sin(2 \alpha) = 2 \sin(\alpha) \cos( \alpha)\\ \cos(2 \alpha) = \cos^{2}(\alpha) - \sin^{2}(\alpha) \\ \tan(2\alpha)=\frac{2\tan(\alpha)}{1-\tan^2(\alpha)} \end{array}\right.
\displaystyle \textbf{(Double Angle Formulae)}\qquad\left\{ \begin{array}{l} \sin(2 \alpha) = 2 \sin(\alpha) \cos( \alpha)\\ \cos(2 \alpha) = \cos^{2}(\alpha) - \sin^{2}(\alpha) \\ \tan(2\alpha)=\frac{2\tan(\alpha)}{1-\tan^2(\alpha)} \end{array}\right.
Sin/Cos Parabolic Identities Using the Sin/Cos Circle Identity in the double angle formulae for cosine, we get
\displaystyle \textbf{(Sin/Cos Parabolic Identities) }\qquad\left\{ \begin{array}{l} \cos(2 t) = 1 - 2 \sin^{2}(t) \ \text{ (Down)}\\ \cos(2 t) = 2 \cos^{2}(t ) - 1 \ \text{ (Up)} \end{array}\right.
\displaystyle \textbf{(Sin/Cos Parabolic Identities) }\qquad\left\{ \begin{array}{l} \cos(2 t) = 1 - 2 \sin^{2}(t) \ \text{ (Down)}\\ \cos(2 t) = 2 \cos^{2}(t ) - 1 \ \text{ (Up)} \end{array}\right.
Half Angle Formulae Rearranging the Sin/Cos parabolic identities and replacing tt by \frac{\theta}{2}\frac{\theta}{2}, we have
\displaystyle \textbf{(Half Angle Formulae) }\qquad\left\{ \begin{array}{l} \cos^2( \frac{\theta}{2}) = \frac{1 +\cos(\theta)}{2} \ \\ \sin^2( \frac{\theta}{2}) = \frac{1 -\cos(\theta)}{2} \ \end{array}\right.
\displaystyle \textbf{(Half Angle Formulae) }\qquad\left\{ \begin{array}{l} \cos^2( \frac{\theta}{2}) = \frac{1 +\cos(\theta)}{2} \ \\ \sin^2( \frac{\theta}{2}) = \frac{1 -\cos(\theta)}{2} \ \end{array}\right.
Triple Angle Formulae The triple angle formulae for sine and cosine are as follows:
\displaystyle \textbf{(Triple Angle Formulae) }\qquad\left\{\begin{matrix}\cos \left(3\theta \right)=4\cos^{3} \left(\theta \right)-3\cos \left(\theta \right)\\\sin \left(3\theta \right)=3\sin \left(\theta \right)-4\sin^{3} \left(\theta \right)\end{matrix}\right.
\displaystyle \textbf{(Triple Angle Formulae) }\qquad\left\{\begin{matrix}\cos \left(3\theta \right)=4\cos^{3} \left(\theta \right)-3\cos \left(\theta \right)\\\sin \left(3\theta \right)=3\sin \left(\theta \right)-4\sin^{3} \left(\theta \right)\end{matrix}\right.