MathAdmin: Created page with "''' Question ''' Find an equivalent algebraic expression for the following, $\cos(\tan^{-1}(x))$ {| class="mw-collapsible mw-collapsed" style = "..."

2015-06-01T01:29:44Z

Created page with "''' Question ''' Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center> {| class="mw-collapsible mw-collapsed" style = "..."

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''' Question ''' Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations
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|1) <math>\tan^{-1}(x)</math> can be thought of as <math>\tan^{-1}\left(\frac{x}{1}\right),</math> and this now refers to an angle in a triangle. What are the side lengths of this triangle?
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|Answers:
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|1) The side lengths are 1, x, and <math>\sqrt{1 + x^2}.</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
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|First, let <math>\theta=\tan^{-1}(x)</math>. Then, <math>\tan(\theta)=x</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
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| Now, we draw the right triangle corresponding to <math>\theta</math>. Two of the side lengths are 1 and x and the hypotenuse has length <math>\sqrt{x^2+1}</math>.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
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| Since <math>\cos(\theta)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}</math>, <math>\cos(\tan^{-1}(x))=\cos(\theta)=\frac{1}{\sqrt{x^2+1}}</math>.
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|
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
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| <math>\frac{1}{\sqrt{x^2+1}}</math>
|}

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

005 Sample Final A, Question 15 - Revision history

MathAdmin: Created page with "''' Question ''' Find an equivalent algebraic expression for the following, \cos(\tan^{-1}(x)) {| class="mw-collapsible mw-collapsed" style = "..."

MathAdmin: Created page with "''' Question ''' Find an equivalent algebraic expression for the following, $\cos(\tan^{-1}(x))$ {| class="mw-collapsible mw-collapsed" style = "..."