MathAdmin: Created page with "''' Question ''' Solve the following equation in the interval $[0, 2\pi)$
$\sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)$ {| c..."

2015-06-01T01:25:55Z

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> {| c..."

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''' 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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!Foundations:
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|1) Which trigonometric identities are useful in this problem?
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|Answer:
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|1) <math>\sin^2(\theta)=1-\cos^2(\theta)</math> and
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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
|-
|<math>(1-\cos^2(\theta))-\cos^2(\theta)=1+\cos(\theta) </math>.
|}

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! Step 2:
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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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
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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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! Step 4:
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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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 5:
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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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| solutions are <math>\theta=\frac{2\pi}{3}</math> or <math>\theta=\frac{4\pi}{3}</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
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| The solutions are <math>\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}</math>.
|}

[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

005 Sample Final A, Question 11 - Revision history

MathAdmin: Created page with "''' Question ''' Solve the following equation in the interval [0, 2\pi) \sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta) {| c..."

MathAdmin: Created page with "''' Question ''' Solve the following equation in the interval $[0, 2\pi)$
$\sin^2(\theta) - \cos^2(\theta)=1+\cos(\theta)$ {| c..."