005 Sample Final A, Question 13

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

Step 1:
For (a), we want an angle $\theta$ such that $\sin(\theta )=2$. Since $-1\leq \sin(\theta )\leq 1$, it is impossible
for $\sin(\theta )=2$. So, $\sin ^{-1}(2)$ is undefined.

Step 2:
For (b), we need to find the reference angle for ${\frac {-32\pi }{3}}$. If we add multiples of $2\pi$ to this angle, we get the
reference angle ${\frac {4\pi }{3}}$. 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 ${\frac {-17\pi }{6}}$. If we add multiples of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} to this angle, we get the
reference angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7\pi}{6}} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\left(\frac{7\pi}{6}\right)=\frac{-\sqrt{3}}{2}} , we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-\sqrt{3}}{2}}
c)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-2\sqrt{3}}{3}}