evaluate the following expression. your answer must be in exact form: for example, type pi/6 for π/6 or dne…

Explanation:

Step1: Find coterminal angle

First, find a coterminal angle of $\frac{44\pi}{7}$ that lies in $[-\frac{\pi}{2},\frac{\pi}{2}]$. We know that $\frac{44\pi}{7}=6\pi+\frac{2\pi}{7}$. Since $\sin(x + 2k\pi)=\sin(x)$ for integer $k$, $\sin(\frac{44\pi}{7})=\sin(6\pi+\frac{2\pi}{7})=\sin(\frac{2\pi}{7})$.

Step2: Apply arcsin - sin property

The function $y = \arcsin(u)$ has a domain of $[- 1,1]$ and range of $[-\frac{\pi}{2},\frac{\pi}{2}]$. For $y=\arcsin(\sin(x))$, when $x\in[-\frac{\pi}{2},\frac{\pi}{2}]$, $\arcsin(\sin(x))=x$. Since $\frac{2\pi}{7}\in[-\frac{\pi}{2},\frac{\pi}{2}]$, $\arcsin(\sin(\frac{44\pi}{7}))=\arcsin(\sin(\frac{2\pi}{7}))=\frac{2\pi}{7}$.

Answer:

$\frac{2\pi}{7}$