# Question #ea459

Jun 27, 2017

$\sin \left(\frac{x}{2}\right) = \frac{3 \sqrt{10}}{10}$
$\sin \left(\frac{x}{2}\right) = - \frac{\sqrt{10}}{10}$

#### Explanation:

$\sin x = - \frac{3}{5}$. First, find cos x.
${\cos}^{2} x = 1 - {\sin}^{2} x = 1 - \frac{9}{25} = \frac{16}{25}$ --> $\cos x = \pm \frac{4}{5}$
To Find sin (x/2), use trig identity:
$2 {\sin}^{2} x = 1 - \cos 2 x$.
In this case: $2 {\sin}^{2} \left(\frac{x}{2}\right) = 1 - \cos x = 1 \pm \frac{4}{5}$
a. $2 {\sin}^{2} \left(\frac{x}{2}\right) = 1 - \left(- \frac{4}{5}\right) = \frac{9}{5}$
${\sin}^{2} \left(\frac{x}{2}\right) = \frac{9}{10}$ --> $\sin \left(\frac{x}{2}\right) = \pm \frac{3}{\sqrt{10}} = \pm \frac{3 \sqrt{10}}{10}$
sin x < 0, cos x < 0, x is in Quadrant 3, $\frac{x}{2}$ is in quadrant 2, then,
$\sin \left(\frac{x}{2}\right)$ is positive.
$\sin \left(\frac{x}{2}\right) = \frac{3 \sqrt{10}}{10}$
b. $2 {\sin}^{2} \left(\frac{x}{2}\right) = 1 - \frac{4}{5} = \frac{1}{5}$
${\sin}^{2} \left(\frac{x}{2}\right) = \frac{1}{10}$ --> $\sin \left(\frac{x}{2}\right) = \pm \frac{1}{\sqrt{10}} = \pm \frac{\sqrt{10}}{10}$.
x is in Quadrant 4, $\frac{x}{2}$ is also in quadrant 4, $\sin \left(\frac{x}{2}\right)$ is negative.
$\sin \left(\frac{x}{2}\right) = - \frac{\sqrt{10}}{10}$
Check by calculator.
$\sin x = - \frac{3}{5}$ --> $x = - {36}^{\circ} 87$ and $x = {216}^{\circ} 87$
a. $x = - {36}^{\circ} 87$ --> $\frac{x}{2} = - {18}^{\circ} 46$ --> $\sin \left(\frac{x}{2}\right) = - 0.316$ = $= - \frac{\sqrt{10}}{10}$. OK
b. $x = 216.87$ --> $\frac{x}{2} = 108.46$ --> $\sin \left(\frac{x}{2}\right) = 0.948 = \left(3 \frac{\sqrt{10}}{10}\right)$. OK