How do you find all the solutions between 0 and 2π for 4sin^2x-3 = 0?

Aug 17, 2017

$x \in \left\{\frac{\pi}{3} , \frac{2 \pi}{3} , \frac{5 \pi}{3} , \frac{7 \pi}{3}\right\}$

Explanation:

If $4 {\sin}^{2} x - 3 = 0$
then
$\textcolor{w h i t e}{\text{XXX}} {\sin}^{2} x = \frac{3}{4}$
and
$\textcolor{w h i t e}{\text{XXX}} \sin x = \pm \frac{\sqrt{3}}{2}$

This is one of the standard reference angles $= \frac{\pi}{3}$

Between 0 and $2 \pi$, the possibilities for this reference angle are
$\textcolor{w h i t e}{\text{XXX}} \frac{\pi}{3} , \frac{2 \pi}{3} , \frac{5 \pi}{3} , \frac{7 \pi}{3}$