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Quiz 11
1. On a test flight, during the landing of the space shuttle, the ship was 346 feet above the end of the landing strip. If it came in at a constant angle of $6.4^{\circ}$ above the landing strip, how far from the end of the landing strip did it first touch the ground?
Solution: First we draw the situation:
We seek the length of the bottom leg of the triangle. Call it $L$. Then we have
$$\tan(6.4^{\circ})=\dfrac{346}{L},$$
so we solve for $L$ to get
$$L = \dfrac{346}{\tan(6.4^{\circ})}=3080.$$
2. Headlights from a car are set so the beam drops 6 inches for every 25 feet forward. What is the angle between the beam and the road?
Solution: We diagram this in the following image:
We need to convert inches to feet or feet to inches. Recall that $1 \mathrm{foot} = 12 \mathrm{inches}$, so we will convert to foot using the conversion factor $\dfrac{1 \mathrm{foot}}{12 \mathrm{inches}}=1$ to see
$$6 \mathrm{inches} = (6 \mathrm{inches}) \left( \dfrac{1 \mathrm{foot}}{12 \mathrm{inches}} \right) = 0.5 \mathrm{feet}.$$
We seek the angle $\theta$. We will use the tangent function to write
$$\tan(\theta)=\dfrac{0.5}{25}=0.02.$$
Taking $\tan^{-1}$ gives us
$$\theta=\tan^{-1}(0.02)=1.146^{\circ}.$$
3. What is the sign of $\sin(97^{\circ})$?
Solution: It is positive, because $97^{\circ}$ lies in quadrant II.
4. What quadrant does $\theta$ lie in if $\sin(\theta)$ is positive and $\tan(\theta)$ is negative?
Solution: If $\sin(\theta)$ is positive, then $\theta$ lies in quadrant $I$ or quadrant $II$. If $\tan(\theta)$ is negative, then $\theta$ lies in quadrant $II$ or $IV$. Therefore the only possibility is that $\theta$ lies in quadrant $II$.
5. What quadrant does $\theta$ lie in if $\cos(\theta)=0.342$ and $\csc(\theta)=-1.064$?
Solution: The value of $\cos(\theta)$ tells us that $\cos(\theta)$ is positive and so $\theta$ must lie in quadrant $I$ or quadrant $IV$. The value of $\csc(\theta)$ tells us that $\csc(\theta)$ is negative and so $\theta$ must lie in quadrant $III$ or quadrant $IV$. Therefore the only possibility is tat $\theta$ lies in quadrant $IV$.
