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

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(Created page with "''' 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}\ri...")
 
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! Final Answers
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! Step 1:
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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| For (a), we want an angle <math>\theta</math> such that <math>\sin(\theta)=2</math>. Since <math>-1\leq \sin (\theta)\leq 1</math>, it is impossible
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|for <math>\sin(\theta)=2</math>. So, <math>\sin^{-1}(2)</math> is undefined.
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! Step 2:
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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| For (b), we need to find the reference angle for <math>\frac{-32\pi}{3}</math>. If we add multiples of <math>2\pi</math> to this angle, we get the
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|reference angle <math>\frac{4\pi}{3}</math>. So, <math>\sin\left(\frac{-32\pi}{3}\right)=\sin\left(\frac{4\pi}{3}\right)=\frac{-\sqrt{3}}{2}</math>.
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! Step 3:
 
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|e) True.
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| For (c), we need to find the reference angle for <math>\frac{-17\pi}{6}</math>. If we add multiples of <math>2\pi</math> to this angle, we get the
 
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|f) False.  
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|reference angle <math>\frac{7\pi}{6}</math>. Since <math>\cos\left(\frac{7\pi}{6}\right)=\frac{-\sqrt{3}}{2}</math>, we have
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|<math>\sec\left(\frac{-17\pi}{6}\right)=\sec\left(\frac{7\pi}{6}\right)=\frac{2}{-\sqrt{3}}=\frac{-2\sqrt{3}}{3}</math>.
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! Final Answer:
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|a) undefined
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|b) <math>\frac{-\sqrt{3}}{2}</math>
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|c)<math>\frac{-2\sqrt{3}}{3}</math>
 
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Revision as of 15:59, 19 May 2015

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


Step 1:
For (a), we want an angle such that . Since , it is impossible
for . So, is undefined.
Step 2:
For (b), we need to find the reference angle for . 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{4\pi}{3}} . So, 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 \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 (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{-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}}
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