# Find all real numbers to satisfy equation? 2sin(x/2)-1=0

Apr 8, 2018

$x = \frac{\pi}{3} \pm 4 \pi n , \frac{5 \pi}{3} \pm 4 \pi n$ where n ∈ Z

#### Explanation:

$2 \sin \left(\frac{x}{2}\right) - 1 = 0$

Let $\frac{x}{2}$ be $u$:
$2 \sin \left(u\right) - 1 = 0$
$\sin \left(u\right) = \frac{1}{2}$

$u = \frac{\pi}{6} \pm 2 \pi n , \frac{5 \pi}{6} \pm 2 \pi n$

Now replace $u$ with $\frac{x}{2}$ and solve for $x$ by multiplying by $2$:

$\frac{x}{2} = \frac{\pi}{6} \pm 2 \pi n , \frac{5 \pi}{6} \pm 2 \pi n$

$x = \frac{\pi}{3} \pm 4 \pi n , \frac{5 \pi}{3} \pm 4 \pi n$ where n ∈ Z

graph{2sin(x/2)-1 [-3.04, 16.96, -5.08, 4.92]}