# Double Angle and Half Angle Formulas

## Recovering the Double Angle Formulas

Using the sum formula and difference formulas for Sine and Cosine we can observe the following identities:

${\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}$
${\displaystyle \cos(2\theta )=\cos ^{2}(\theta )-\sin ^{2}(\theta )}$

Using the Pythagorean Identities for trigonometric functions we can also see that:

${\displaystyle \cos(2\theta )=1-2\sin ^{2}(\theta )}$
${\displaystyle \cos(2\theta )=2\cos ^{2}(\theta )-1}$

## Half Angle Formulas

Using the last two double angle formulas we can now solve for the half angle formulas:

${\displaystyle \sin(\theta )={\sqrt {\frac {1-\cos(2\theta )}{2}}}}$
${\displaystyle \cos(\theta )={\sqrt {\frac {1+\cos(2\theta )}{2}}}}$
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