# If \sin x = \frac { 4} { 5], what is sin 2x?

May 21, 2018

i got $\frac{24}{25}$

#### Explanation:

We have $\sin \left(2 x\right) = 2 \sin \left(x\right) \cos \left(x\right)$ and $\cos \left(x\right) = \sqrt{1 - {\sin}^{2} \left(x\right)}$
so we get $\sin \left(2 x\right) = \frac{8}{5} \cdot \sqrt{1 - \frac{16}{25}} = \frac{8}{5} \cdot \frac{3}{5} = \frac{24}{25}$

May 22, 2018

$\sin 2 x = \pm \frac{24}{25}$

#### Explanation:

$\sin x = \frac{4}{5}$. First, find cos x
${\cos}^{2} x = 1 - {\sin}^{2} x = 1 - \frac{16}{25} = \frac{9}{25}$
$\cos x = \pm \frac{3}{5}$
Since $\sin x = \frac{4}{5}$ --> x could be in Quadrant 1 or Quadrant 2, therefor, cos x could be positive or negative.
$\sin \left(2 x\right) = 2 \sin x . \cos x = 2 \left(\frac{4}{5}\right) \left(\pm \frac{3}{5}\right) = \pm \frac{24}{25}$
If x lies in Q. 1 --> 2x lies in Q. 2 --> sin 2x is positive
If x lies in Q. 2 --> 2x lies in Q. 3 --> sin 2x is negative.