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education/math/MATH1060 (trig)/Double and Half Angle Identities.md
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# Double Angle Identities # Double Angle Identities
Sine:
$$ \sin(2\theta) = 2\sin\theta\cos\theta $$ $$ \sin(2\theta) = 2\sin\theta\cos\theta $$
Cosine:
$$ $$
\begin{matrix} \begin{matrix}
\cos(2\theta) = \cos^2\theta - \sin^2\theta\\ \cos(2\theta) = \cos^2\theta - \sin^2\theta\\
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= 2cos^2\theta - 1\\ = 2cos^2\theta - 1\\
\end{matrix} \end{matrix}
$$ $$
Tan:
$$ \tan(2\theta) = \dfrac{2\tan\theta}{1-\tan^2\theta}$$ $$ \tan(2\theta) = \dfrac{2\tan\theta}{1-\tan^2\theta}$$
## 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:
$$