Question

square root 3csc -2=0 solve the equation

Answer 1

Solve the equation $\sqrt{3\csc x} - 2 = 0$.

Solution.

$$\begin{aligned} \sqrt{3\csc(x)} - 2 &= 0 \Rightarrow \sqrt{3\csc(x)} = 2 \Rightarrow 3\csc(x) = 4 \Rightarrow \csc(x) = \frac{4}{3} \Rightarrow \sin x = \frac{3}{4} \Rightarrow \\ &\Rightarrow \left[ \begin{array}{l} x = \arcsin\left(\frac{3}{4}\right) + 2\pi k, \, k \in \mathbb{Z} \\ x = \pi - \arcsin\left(\frac{3}{4}\right) + 2\pi k, \, k \in \mathbb{Z} \end{array} \right. \end{aligned}$$

Answer.

$$U_{k \in \mathbb{Z}} \left\{ \arcsin\left(\frac{3}{4}\right) + 2\pi k, \, \pi - \arcsin\left(\frac{3}{4}\right) + 2\pi k \right\}.$$