Science:Math Exam Resources/Courses/MATH100/December 2014/Question 03 (f)/Solution 1

The function $\arcsin$ has domain $[-1,1]$ and range $[-\pi /2,\pi /2]$, so our answer must lie in this range.

• We know that $\sin (x)=\sin (x-2\pi )$; so $\sin (9\pi /11)=\sin (31\pi /11)$.
• Also we know that $\sin (x)=\sin (\pi -x)$ and so, $\sin (9\pi /11)=\sin (2\pi /11)$.

Since $-\pi /2\le 2\pi /11\le \pi /2$, we conclude that

$\begin{array}{rl}\arcsin (\sin (\frac{31\pi }{11}))& =\arcsin (\sin (\frac{2\pi }{11}))\\ & =\frac{2\pi }{11}\end{array}$