# Double Angle Identities

## Key Questions

As below.

#### Explanation:

Following table gives the double angle identities which can be used while solving the equations.

You can also have $\sin 2 \theta , \cos 2 \theta$ expressed in terms of $\tan \theta$ as under.

$\sin 2 \theta = \frac{2 \tan \theta}{1 + {\tan}^{2} \theta}$

$\cos 2 \theta = \frac{1 - {\tan}^{2} \theta}{1 + {\tan}^{2} \theta}$

• You would need an expression to work with.

For example:
Given $\sin \alpha = \frac{3}{5}$ and $\cos \alpha = - \frac{4}{5}$, you could find $\sin 2 \alpha$ by using the double angle identity
$\sin 2 \alpha = 2 \sin \alpha \cos \alpha$.

$\sin 2 \alpha = 2 \left(\frac{3}{5}\right) \left(- \frac{4}{5}\right) = - \frac{24}{25}$.

You could find $\cos 2 \alpha$ by using any of:
$\cos 2 \alpha = {\cos}^{2} \alpha - {\sin}^{2} \alpha$
$\cos 2 \alpha = 1 - 2 {\sin}^{2} \alpha$
$\cos 2 \alpha = 2 {\cos}^{2} \alpha - 1$

In any case, you get $\cos \alpha = \frac{7}{25}$.

• Double Angle Identities

$\sin 2 \theta = 2 \sin \theta \cos \theta$

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

$\tan 2 \theta = \frac{2 \tan \theta}{1 - {\tan}^{2} \theta}$

I hope that this was helpful.