# Solve the equation in x for exact solutions over the interval [0, 2pi) for sin . x/2=sqrt2-sinx/2 ?

## sin$\frac{x}{2} = \sqrt{2} -$sin$\frac{x}{2}$

Feb 9, 2018

$\textcolor{b l u e}{{90}^{\circ} , {270}^{\circ}}$

#### Explanation:

Using identity:

color(red)bb(sin^2(theta/2)=1/2(1-costheta)

color(red)bb(=> sin(theta/2)=sqrt(1/2(1-costheta))

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

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

Substituting:

$\sqrt{\frac{1}{2} \left(1 - \cos x\right)} = \frac{\sqrt{2}}{2}$

Squaring:

$\frac{1}{2} \left(1 - \cos x\right) = \frac{2}{4}$

$\frac{1}{2} - \frac{1}{2} \cos x = \frac{1}{2}$

$\cos x = 0$

$x = \arccos \left(\cos x\right) = \arccos \left(0\right) = \textcolor{b l u e}{{90}^{\circ} , {270}^{\circ}}$

Feb 10, 2018

pi/2; (3pi)/2

#### Explanation:

$2 \sin \left(\frac{x}{2}\right) = \sqrt{2}$
$\sin \left(\frac{x}{2}\right) = \frac{\sqrt{2}}{2}$
Trig table and unit circle give 2 solutions -->

a. $\frac{x}{2} = \frac{\pi}{4}$, --> $x = \frac{\pi}{2}$.

b. $\frac{x}{2} = \frac{3 \pi}{4}$ --> $x = \frac{3 \pi}{2}$