17. -/1 points details my notes larpcalclim4hs 5.3.056. use a graphing utility to approximate (to three…

Explanation:

Step1: Rewrite the equation

Given $\csc^{2}(x)-8 = 0$, we can rewrite it as $\csc^{2}(x)=8$, then $\csc(x)=\pm\sqrt{8}=\pm2\sqrt{2}$. Since $\csc(x)=\frac{1}{\sin(x)}$, we have $\sin(x)=\pm\frac{1}{2\sqrt{2}}=\pm\frac{\sqrt{2}}{4}$.

Step2: Find solutions for $\sin(x)=\frac{\sqrt{2}}{4}$

Using the inverse - sine function, $x_1=\sin^{-1}(\frac{\sqrt{2}}{4})\approx0.362$ and $x_2=\pi - \sin^{-1}(\frac{\sqrt{2}}{4})\approx2.780$ in the interval $[0,2\pi)$.

Step3: Find solutions for $\sin(x)=-\frac{\sqrt{2}}{4}$

Using the inverse - sine function, $x_3=\pi+\sin^{-1}(\frac{\sqrt{2}}{4})\approx3.503$ and $x_4 = 2\pi-\sin^{-1}(\frac{\sqrt{2}}{4})\approx5.921$ in the interval $[0,2\pi)$.

Answer:

$0.362,2.780,3.503,5.921$