005 Sample Final A, Question 12

Question Given that $\sec(\theta )=-2$ and $\tan(\theta )>0$, find the exact values of the remaining trig functions.

Foundations
1) Which quadrant is $\theta$ in?
2) Which trig functions are positive in this quadrant?
3) What are the side lengths of the triangle associated to $\theta ?$
Answers:
1) $\theta$ is in the third quadrant. We know it is in the second or third quadrant since $\cos$ is negative. Since \$\tan$ is positive $\theta$ is in the third quadrant.
2) $\tan$ and $\cot$ are both positive in this quadrant. All other trig functions are negative.
3) The side lengths are 2, 1, and ${\sqrt {3}}.$

Step 1:
Since $\sec(\theta )=-2$, we have $\cos(\theta )={\frac {1}{\sec(\theta )}}={\frac {-1}{2}}$.

Step 2:
We look for solutions to $\theta$ on the unit circle. The two angles on the unit circle with $\cos(\theta )={\frac {-1}{2}}$ are $\theta ={\frac {2\pi }{3}}$ and $\theta ={\frac {4\pi }{3}}$.
But, $\tan \left({\frac {2\pi }{3}}\right)=-{\sqrt {3}}$. Since $\tan(\theta )>0$. we must have $\theta ={\frac {4\pi }{3}}$.

Step 3:
The remaining values of the trig functions are
$\sin(\theta )=\sin \left({\frac {4\pi }{3}}\right)={\frac {-{\sqrt {3}}}{2}}$,
$\tan(\theta )=\tan \left({\frac {4\pi }{3}}\right)={\sqrt {3}}$
$\csc(\theta )=\csc \left({\frac {4\pi }{3}}\right)={\frac {-2{\sqrt {3}}}{3}}$ and
$\cot(\theta )=\cot \left({\frac {4\pi }{3}}\right)={\frac {\sqrt {3}}{3}}$

Final Answer:
$\sin(\theta )=={\frac {-{\sqrt {3}}}{2}}$
$\cos(\theta )={\frac {-1}{2}}$
$\tan(\theta )={\sqrt {3}}$
$\csc(\theta )={\frac {-2{\sqrt {3}}}{3}}$
$\cot(\theta )={\frac {\sqrt {3}}{3}}$