Difference between revisions of "005 Sample Final A, Question 12"

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(Created page with "''' Question ''' Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions. {| class="mw-collapsib...")
 
 
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''' Question '''  Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.
 
''' Question '''  Given that <math>\sec(\theta) = -2</math> and <math>\tan(\theta) > 0 </math>, find the exact values of the remaining trig functions.
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Foundations
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|1) Which quadrant is <math>\theta</math> in?
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|-
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|2) Which trig functions are positive in this quadrant?
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|-
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|3) What are the side lengths of the triangle associated to <math>\theta?</math>
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|-
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|Answers:
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|-
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|1) <math>\theta</math> is in the third quadrant. We know it is in the second or third quadrant since <math>\cos</math> is negative. Since \<math>\tan</math> is positive <math>\theta</math> is in the third quadrant.
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|-
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|2) <math>\tan</math> and <math>\cot</math> are both positive in this quadrant. All other trig functions are negative.
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|-
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|3) The side lengths are 2, 1, and <math>\sqrt{3}.</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 1:
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|-
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| Since <math>\sec(\theta)=-2</math>, we have <math>\cos(\theta)=\frac{1}{\sec(\theta)}=\frac{-1}{2}</math>.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2:
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|-
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| We look for solutions to <math>\theta</math> on the unit circle. The two angles on the unit circle with <math>\cos(\theta)=\frac{-1}{2}</math> are <math>\theta=\frac{2\pi}{3}</math> and <math>\theta=\frac{4\pi}{3}</math>.
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|-
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|But, <math>\tan\left(\frac{2\pi}{3}\right)=-\sqrt{3}</math>. Since <math>\tan(\theta)>0</math>. we must have <math>\theta=\frac{4\pi}{3}</math>.
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|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answers
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! Step 3:
 
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|-
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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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| The remaining values of the trig functions are
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|-
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|<math>\sin(\theta)=\sin\left(\frac{4\pi}{3}\right)=\frac{-\sqrt{3}}{2}</math>,
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|-
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|  <math>\tan(\theta)=\tan\left(\frac{4\pi}{3}\right)=\sqrt{3}</math>
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|-
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|<math>\csc(\theta)=\csc\left(\frac{4\pi}{3}\right)=\frac{-2\sqrt{3}}{3}</math> and
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|-
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|<math>\cot(\theta)=\cot\left(\frac{4\pi}{3}\right)=\frac{\sqrt{3}}{3}</math>
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
 
|-
 
|-
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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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| <math>\sin(\theta)==\frac{-\sqrt{3}}{2}</math>
 
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|-
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|c) False. <math>y = x^2</math> does not have an inverse.
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|<math>\cos(\theta)=\frac{-1}{2}</math>
 
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|-
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|<math>\tan(\theta)=\sqrt{3}</math>
 
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|-
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|e) True.
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|<math>\csc(\theta)=\frac{-2\sqrt{3}}{3}</math>
 
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|-
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|f) False.
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|<math>\cot(\theta)=\frac{\sqrt{3}}{3}</math>
 
|}
 
|}

Latest revision as of 19:52, 21 May 2015

Question Given that 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(\theta) = -2} and 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 \tan(\theta) > 0 } , find the exact values of the remaining trig functions.

Foundations
1) Which quadrant is 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 \theta} in?
2) Which trig functions are positive in this quadrant?
3) What are the side lengths of the triangle associated to 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 \theta?}
Answers:
1) 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 \theta} is in the third quadrant. We know it is in the second or third quadrant 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} is negative. 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 \tan} is positive 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 \theta} is in the third quadrant.
2) 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 \tan} and 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 \cot} are both positive in this quadrant. All other trig functions are negative.
3) The side lengths are 2, 1, and 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 \sqrt{3}.}
Step 1:
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 \sec(\theta)=-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 \cos(\theta)=\frac{1}{\sec(\theta)}=\frac{-1}{2}} .
Step 2:
We look for solutions to 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 \theta} on the unit circle. The two angles on the unit circle with 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(\theta)=\frac{-1}{2}} are 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 \theta=\frac{2\pi}{3}} and 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 \theta=\frac{4\pi}{3}} .
But, 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 \tan\left(\frac{2\pi}{3}\right)=-\sqrt{3}} . 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 \tan(\theta)>0} . we must 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 \theta=\frac{4\pi}{3}} .
Step 3:
The remaining values of the trig functions are
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(\theta)=\sin\left(\frac{4\pi}{3}\right)=\frac{-\sqrt{3}}{2}} ,
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 \tan(\theta)=\tan\left(\frac{4\pi}{3}\right)=\sqrt{3}}
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 \csc(\theta)=\csc\left(\frac{4\pi}{3}\right)=\frac{-2\sqrt{3}}{3}} and
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 \cot(\theta)=\cot\left(\frac{4\pi}{3}\right)=\frac{\sqrt{3}}{3}}


Final Answer:
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(\theta)==\frac{-\sqrt{3}}{2}}
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(\theta)=\frac{-1}{2}}
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 \csc(\theta)=\frac{-2\sqrt{3}}{3}}
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 \cot(\theta)=\frac{\sqrt{3}}{3}}
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