005 Sample Final A, Question 13

Question Give the exact value of the following if its defined, otherwise, write undefined.
$(a)\sin ^{-1}(2)\qquad \qquad (b)\sin \left({\frac {-32\pi }{3}}\right)\qquad \qquad (c)\sec \left({\frac {-17\pi }{6}}\right)$

Foundations:
1) What is the domain of $\sin ^{-1}?$
2) What are the reference angles for ${\frac {-32\pi }{3}}$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {-17\pi }{6}}} ?
Answers:
1) The domain is $[-1,1].$
2) The reference angle for ${\frac {-32\pi }{3}}$ is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {4\pi }{3}}} , and the reference angle for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {-17\pi }{6}}} is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {7\pi }{6}}}

Step 1:
For (a), we want an angle $\theta$ such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin(\theta )=2} . Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -1\leq \sin(\theta )\leq 1} , it is impossible
for $\sin(\theta )=2$. So, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin ^{-1}(2)} is undefined.

Step 2:
For (b), we need to find the reference angle for ${\frac {-32\pi }{3}}$. If we add multiples of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\pi } to this angle, we get the
reference angle Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {4\pi }{3}}} . So, $\sin \left({\frac {-32\pi }{3}}\right)=\sin \left({\frac {4\pi }{3}}\right)={\frac {-{\sqrt {3}}}{2}}$.

Step 3:
For (c), we need to find the reference angle for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {-17\pi }{6}}} . If we add multiples of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\pi } to this angle, we get the
reference angle ${\frac {7\pi }{6}}$. Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \cos \left({\frac {7\pi }{6}}\right)={\frac {-{\sqrt {3}}}{2}}} , we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sec \left({\frac {-17\pi }{6}}\right)=\sec \left({\frac {7\pi }{6}}\right)={\frac {2}{-{\sqrt {3}}}}={\frac {-2{\sqrt {3}}}{3}}} .

Final Answer:
a) undefined
b) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {-{\sqrt {3}}}{2}}}
c)${\frac {-2{\sqrt {3}}}{3}}$