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

Apr 8, 2018

$x = \frac{1}{6} \pm 2 n , \frac{5}{6} \pm 2 n$ where n ∈ Z

#### Explanation:

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

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

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

Now replace $u$ with $\pi x$ and solve for $x$ by dividing by $\pi$:

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

$x = \frac{1}{6} \pm 2 n , \frac{5}{6} \pm 2 n$ where n ∈ Z

graph{sin(pix)-1/2 [-10, 10, -5, 5]}