# How do you solve 11=3csc^2x+7 for 0<=x<=2pi?

Sep 4, 2016

$x = \frac{\pi}{3} , \frac{2 \pi}{3} , \frac{4 \pi}{3} , \frac{5 \pi}{3}$

#### Explanation:

We have: $11 = 3 {\csc}^{2} \left(x\right) + 7$; $0 \le q x \le q 2 \pi$

Let's subtract $7$ from both sides of the equation:

$\implies 3 {\csc}^{2} \left(x\right) = 4$

Then, we can divide both sides by $3$:

$\implies {\csc}^{2} \left(x\right) = \frac{4}{3}$

To simplify the left-hand side we can take the square root of both sides:

$\implies \csc \left(x\right) = \pm \sqrt{\frac{4}{3}}$

$\implies \csc \left(x\right) = \pm \frac{2 \sqrt{3}}{3}$

$\implies x = \frac{\pi}{3} , \frac{2 \pi}{3} , \frac{4 \pi}{3} , \frac{5 \pi}{3}$