005 Sample Final A, Question 13

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Question Give the exact value of the following if its defined, otherwise, write undefined.


Foundations:
1) What is the domain of
2) What are the reference angles for 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
2) The reference angle for 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 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 . 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 . 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, .
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 . 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)