Find all solutions of the following equation on the interval $[ 0,2 \pi )$.
$$\csc ^ { 2 } ( x ) - 2 = 0$$
A) $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$
B) $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$
C) $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$
D) $\frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$
E) $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$

Question

Quizplus · Accepted Answer

First, solve the equation $\csc^2(x) - 2 = 0$ to get $\csc^2(x) = 2$, which simplifies to $\sin^2(x) = \frac{1}{2}$. This means $\sin(x) = \pm\frac{\sqrt{2}}{2}$, leading to solutions at $x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$ within the given interval.

Find All Solutions of the Following Equation on the Interval [0,2π)[ 0,2 \pi )[0,2π)

Find All Solutions of the Following Equation on the Interval $[ 0,2 \pi )$