Difference between revisions of "005 Sample Final A, Question 13"

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''' Question '''  Give the exact value of the following if its defined, otherwise, write undefined. <br>
 
''' Question '''  Give the exact value of the following if its defined, otherwise, write undefined. <br>
 
<math>(a) \sin^{-1}(2) \qquad \qquad (b) \sin\left(\frac{-32\pi}{3}\right) \qquad \qquad (c)\sec\left(\frac{-17\pi}{6}\right)</math>
 
<math>(a) \sin^{-1}(2) \qquad \qquad (b) \sin\left(\frac{-32\pi}{3}\right) \qquad \qquad (c)\sec\left(\frac{-17\pi}{6}\right)</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Foundations:
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|-
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|1) What is the domain of <math>\sin^{-1}?</math>
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|-
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|2) What are the reference angles for <math>\frac{-32\pi}{3}</math> and <math>\frac{-17\pi}{6}</math>?
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|Answers:
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|1) The domain is <math>[-1, 1].</math>
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|2) The reference angle for <math>\frac{-32\pi}{3}</math> is <math>\frac{4\pi}{3}</math>, and the reference angle for <math>\frac{-17\pi}{6}</math> is <math>\frac{7\pi}{6}</math>
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|}
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Latest revision as of 20:58, 21 May 2015

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


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