Back to the class
Quiz 12
1. What quadrant does $\theta$ lie in if $\sec(\theta) \lt 0$ and $\tan(\theta) \gt 0$?
Solution: Since $\sec(\theta)$ is negative, $\theta$ must lie in quadrant $II$ or quadrant $III$. Since $\tan(\theta)$ is positive, $\theta$ must lie in quadrant $I$ or quadrant $III$. Therefore the only possibility is that $\theta$ lies in quadrant $III$.
2. Use the unit circle to find the exact value of $\cos(135^{\circ})$.
Solution: Since $135^{\circ}$ lies in quadrant $II$, its reference angle is $180^{\circ}-135^{\circ}=45^{\circ}$. So $\cos(135^{\circ})$ is the same as $\cos(45^{\circ})$, but is negative. Therefore by the unit circle,
$$\cos(135^{\circ})=-\cos(45^{\circ})=-\dfrac{\sqrt{2}}{2}.$$
3. Find $\sin(\theta)$ when $\cos(\theta)=0.314$ and $\tan(\theta)<0$.
Solution: Since $\cos(\theta)=0.314$, taking the inverse cosine yields
$$\theta=\cos^{-1}(0.314)=71.7^{\circ}.$$
4. Convert to degrees: $\dfrac{\pi}{11}$ radians.
Solution: Since $360^{\circ}=\pi \mathrm{radians}$, we divide to get the conversion factor $\dfrac{360^{\circ}}{\pi \mathrm{radians}}=1$ to see
$$\dfrac{\pi}{11} \mathrm{radians} = \left( \dfrac{\pi}{11} \mathrm{radians} \right) \left( \dfrac{360^{\circ}}{\pi \mathrm{radians}} \right)=\left( \dfrac{360}{11} \right)^{\circ}=32.727^{\circ}.$$
5. Find $\theta$, in radians, such that $\sin(\theta)=0.6189$.
Solution: Using radian mode in a calculator...
$$\theta = \sin^{-1} \left( 0.6189 \right) = 0.667341\ldots$$
