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# Trigonometric Equations Using Half Angle Formulas

## Simplifying all six trigonometric functions with half a given angle.

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Practice Trigonometric Equations Using Half Angle Formulas
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Angle Identities

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#### Double Angle Identity

double angle identity relates the a trigonometric function of two times an argument to a set of trigonometric functions, each containing the original argument.

cos(α+α)=cosαcosαsinαsinαcos2α=cos2αsin2α\begin{align*}\cos (\alpha + \alpha ) = \cos \alpha \cos \alpha - \sin \alpha \sin \alpha \\ \cos 2 \alpha = \cos^2 \alpha - \sin^2 \alpha\end{align*}

There are ways in which you can manipulate the double angle identity which causes there to be three ways you can present the double angle identity.

cos2α=cos2αsin2αcos2α=2cos2α1cos2α=12sin2α\begin{align*}\cos 2 \alpha= \cos^2 \alpha - \sin^2 \alpha \\ \cos 2 \alpha = 2 \cos^2 \alpha -1 \\ \cos 2 \alpha = 1 - 2 \sin^2 \alpha\end{align*}

Hint: This double angle identity comes in handy when you are trying to solve proofs

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Half Angle Identity

half angle identity relates the a trigonometric function of one half of an argument to a set of trigonometric functions, each containing the original argument.

sinα2=1cosα2\begin{align*}\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*} if α2\begin{align*}\frac{\alpha}{2}\end{align*} is located in either the first or second quadrant.

sinα2=1cosα2\begin{align*}\sin \frac{\alpha}{2} = - \sqrt{\frac{1 - \cos \alpha}{2}}\end{align*} if α2\begin{align*}\frac{\alpha}{2}\end{align*} is located in the third or fourth quadrant.

cosα2=1+cosα2\begin{align*}\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*} if α2\begin{align*}\frac{\alpha}{2}\end{align*} is located in either the first or fourth quadrant.

cosα2=1+cosα2\begin{align*}\cos \frac{\alpha}{2} = - \sqrt{\frac{1 + \cos \alpha}{2}}\end{align*} if α2\begin{align*}\frac{\alpha}{2}\end{align*} is located in either the second or fourth quadrant.

Tip: This all depends how the values of cosine and sine are negative dependig which quadrant cosine or sine lies on.