008A Sample Final A, Question 19

Question: Compute ${\displaystyle \arcsin -{\frac {\sqrt {3}}{2}}}$. Provide your answer in radians.

Foundations:
1) What is the range of ${\displaystyle \arcsin }$?
2) Sine of what angle in the range of ${\displaystyle \arcsin }$ yields ${\displaystyle -{\frac {\sqrt {3}}{2}}}$? Answer:
1) The range of arcsin is ${\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]}$.
2) ${\displaystyle \sin(-{\frac {\pi }{3}})=-{\frac {\sqrt {3}}{2}}}$.

Solution:

Step 1:
Since the range of ${\displaystyle \arcsin }$ is ${\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]}$, the answer must be in that range.
${\displaystyle \arcsin -{\frac {\sqrt {3}}{2}}=-{\frac {\pi }{3}}}$