Difference between revisions of "005 Sample Final A, Question 15"
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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(\tan^{-1}(x))}
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''' Question ''' Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center> | ''' Question ''' Find an equivalent algebraic expression for the following, <center><math> \cos(\tan^{-1}(x))</math></center> | ||
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| + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | ! Foundations | ||
| + | |- | ||
| + | |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> | ||
| + | |} | ||
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
Latest revision as of 20:13, 21 May 2015
Question Find an equivalent algebraic expression for the following,
| Foundations |
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| 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 \tan^{-1}(x)} can be thought of as 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^{-1}\left(\frac{x}{1}\right),} and this now refers to an angle in a triangle. What are the side lengths of this triangle? |
| Answers: |
| 1) The side lengths are 1, x, 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{1 + x^2}.} |
| Step 1: |
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| First, let 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=\tan^{-1}(x)} . Then, 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)=x} . |
| Step 2: |
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| Now, we draw the right triangle corresponding 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} . Two of the side lengths are 1 and x and the hypotenuse has length 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{x^2+1}} . |
| Step 3: |
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| 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(\theta)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}} , 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(\tan^{-1}(x))=\cos(\theta)=\frac{1}{\sqrt{x^2+1}}} . |
| Final Answer: |
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| 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{1}{\sqrt{x^2+1}}} |