# Question #28924

Jan 19, 2018

$\sin \left(2 x\right) = - \frac{24}{25}$
$\cos \left(2 x\right) = \frac{7}{25}$
$\tan \left(2 x\right) = - \frac{24}{7}$

#### Explanation:

Since ${90}^{\circ} < x < {180}^{\circ}$ and $\cos x = - \frac{4}{5}$ we know that $\sin x = \frac{3}{5}$. (I drew a triangle to find that.)

$\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right) = 2 \left(\frac{3}{5}\right) \left(- \frac{4}{5}\right) = - \frac{24}{25}$.
$\cos \left(2 x\right) = 2 {\cos}^{2} \left(x\right) - 1 = 2 {\left(- \frac{4}{5}\right)}^{2} - 1 = 2 \left(\frac{16}{25}\right) - 1$
$\cos \left(2 x\right) = \frac{7}{25}$

Since $\tan \left(2 x\right) = \sin \frac{2 x}{\cos} \left(2 x\right)$ we know:

$\tan \left(2 x\right) = \frac{- \left(\frac{24}{25}\right)}{\frac{7}{25}} = - \frac{24}{25} \cdot \frac{25}{7} = - \frac{24}{7}$.