Difference between revisions of "005 Sample Final A, Question 11"

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(Created page with "''' Question ''' Solve the following equation in the interval <math> [0, 2\pi)</math> <br> <center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center> {|...")
 
 
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''' Question '''  Solve the following equation in the interval <math> [0, 2\pi)</math> <br>
 
''' Question '''  Solve the following equation in the interval <math> [0, 2\pi)</math> <br>
 
<center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center>
 
<center><math> \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)</math></center>
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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) Which trigonometric identities are useful in this problem?
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|-
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|Answer:
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|1) <math>\sin^2(\theta)=1-\cos^2(\theta)</math> and
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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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| We need to get rid of the <math>\sin^2(\theta)</math> term. Since <math>\sin^2(\theta)=1-\cos^2(\theta)</math>, the equation becomes
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|-
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|<math>(1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) </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 2:
 
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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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| If we simplify and move all the terms to the right hand side, we have <math>0=2\cos^2(\theta)+\cos(\theta)</math>.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 3:
 
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|-
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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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| Now, factoring, we have <math>0=\cos(\theta)(2\cos(\theta)+1)</math>. Thus, either <math>\cos(\theta)=0</math> or <math>2\cos(\theta)+1=0</math>.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 4:
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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| The solutions to <math>\cos(\theta)=0</math> in <math> [0, 2\pi)</math> are <math>\theta=\frac{\pi}{2}</math> or <math>\theta=\frac{3\pi}{2}</math>.
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 5:
 
|-
 
|-
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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| The solutions to <math>2\cos(\theta)+1=0</math> are angles that satisfy <math>\cos(\theta)=\frac{-1}{2}</math>. In <math> [0, 2\pi)</math>, the
 
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|-
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|e) True.
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| solutions are <math>\theta=\frac{2\pi}{3}</math> or <math>\theta=\frac{4\pi}{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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|-
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|f) False.  
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| The solutions are <math>\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}</math>.
 
|}
 
|}

Latest revision as of 20:43, 21 May 2015

Question Solve the following equation in the interval

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^2(\theta) - \cos^2(\theta)=1+\cos(\theta)}
Foundations:
1) Which trigonometric identities are useful in this problem?
Answer:
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 \sin^2(\theta)=1-\cos^2(\theta)} and


Step 1:
We need to get rid of the 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^2(\theta)} term. 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 \sin^2(\theta)=1-\cos^2(\theta)} , the equation becomes
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 (1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) } .
Step 2:
If we simplify and move all the terms to the right hand side, 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 0=2\cos^2(\theta)+\cos(\theta)} .
Step 3:
Now, factoring, 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 0=\cos(\theta)(2\cos(\theta)+1)} . Thus, either 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)=0} or 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\cos(\theta)+1=0} .
Step 4:
The 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 \cos(\theta)=0} in 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 [0, 2\pi)} 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{\pi}{2}} or 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{3\pi}{2}} .
Step 5:
The 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 2\cos(\theta)+1=0} are angles that satisfy 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}} . In 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 [0, 2\pi)} , the
solutions 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}} or 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}} .
Final Answer:
The solutions 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 \frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}} .
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