MathAdmin: Created page with "''' Question ''' Given that $\sec(\theta) = -2$ and $\tan(\theta) > 0$ , find the exact values of the remaining trig functions. {| class="mw-collapsibl..."

2015-06-01T01:26:32Z

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-collapsibl..."

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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.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations
|-
|1) Which quadrant is <math>\theta</math> in?
|-
|2) Which trig functions are positive in this quadrant?
|-
|3) What are the side lengths of the triangle associated to <math>\theta?</math>
|-
|Answers:
|-
|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.
|-
|2) <math>\tan</math> and <math>\cot</math> are both positive in this quadrant. All other trig functions are negative.
|-
|3) The side lengths are 2, 1, and <math>\sqrt{3}.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
|-
| Since <math>\sec(\theta)=-2</math>, we have <math>\cos(\theta)=\frac{1}{\sec(\theta)}=\frac{-1}{2}</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
|-
| 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>.
|-
|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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
|-
| The remaining values of the trig functions are
|-
|<math>\sin(\theta)=\sin\left(\frac{4\pi}{3}\right)=\frac{-\sqrt{3}}{2}</math>,
|-
| <math>\tan(\theta)=\tan\left(\frac{4\pi}{3}\right)=\sqrt{3}</math>
|-
|<math>\csc(\theta)=\csc\left(\frac{4\pi}{3}\right)=\frac{-2\sqrt{3}}{3}</math> and
|-
|<math>\cot(\theta)=\cot\left(\frac{4\pi}{3}\right)=\frac{\sqrt{3}}{3}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
|-
| <math>\sin(\theta)==\frac{-\sqrt{3}}{2}</math>
|-
|<math>\cos(\theta)=\frac{-1}{2}</math>
|-
|<math>\tan(\theta)=\sqrt{3}</math>
|-
|<math>\csc(\theta)=\frac{-2\sqrt{3}}{3}</math>
|-
|<math>\cot(\theta)=\frac{\sqrt{3}}{3}</math>
|}

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

005 Sample Final A, Question 12 - Revision history

MathAdmin: Created page with "''' Question ''' Given that \sec(\theta) = -2 and \tan(\theta) > 0 , find the exact values of the remaining trig functions. {| class="mw-collapsibl..."

MathAdmin: Created page with "''' Question ''' Given that $\sec(\theta) = -2$ and $\tan(\theta) > 0$ , find the exact values of the remaining trig functions. {| class="mw-collapsibl..."