Inverse Trigonometric Functions Exercises

Subsection 3.1 Inverse Trigonometric Functions Exercises

Suppose that $\sin \alpha = \frac{\sqrt{3}}{2}\text{.}$ Use Desmos to help you find possible value(s) of $\alpha\text{,}$ where $0 \leq \alpha \leq \frac{\pi}{2}\text{.}$
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Let $\cos \theta = .75\text{.}$
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1. Find $\theta$ if $\theta$ is a positive angle in the first quadrant.
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2. Find $\theta$ if $\theta$ is a positive angle in the fourth quadrant.
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Suppose that $\triangle DEF$ is a right triangle, with right angle at $E\text{.}$ Let $DE=5$ and $EF=3\text{.}$ Determine $DF\text{,}$ and then use the side lengths to compute $\sin \angle D\text{,}$ $\cos \angle D\text{,}$ and $\tan \angle D\text{.}$
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Suppose that $\triangle KLM$ is a right triangle, with right angle at $L\text{.}$ Let $KM=8\text{,}$ and let $\angle MKL = .25$ radians. Find the lengths $KL$ and $LM\text{.}$
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A circle has its center at $C= (3, 5)\text{,}$ and just touches the $y$-axis.
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1. Draw a diagram of the circle in the $xy$-plane.
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2. Write the equation of the circle.
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3. Find the circumference and area of the circle.
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4. Find the length of the arc of the circle cut out by an angle of $\frac{\pi}{6}$ radians.
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5. Find the area of the sector of the circle cut out by an angle of $\frac{\pi}{6}$ radians.
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6. Find the coordinates of point $A\text{,}$ if point $A$ is in the 2 o’clock position on the circle.
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A circle has its center at $V=(6, 0)\text{,}$ and contains the point $(0,0)\text{.}$
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1. Draw a diagram of the circle in the $xy$-plane.
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2. Write the equation of the circle.
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3. Find the circumference and area of the circle.
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4. Find the length of an arc of the circle cut out by an angle of $\frac{5\pi}{6}\text{.}$
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5. Find the coordinates of the point $B\text{,}$ if ray $\overrightarrow{VB}$ makes an angle of $\frac{\pi}{4}$ with the $x$-axis (there are multiple answers).
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6. What is the measure of $\angle FVP\text{,}$ where $F=(9, 3\sqrt{3})$ and $P=(12,0)\text{?}$
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The owners of a small concert venue (capacity 1050) have determined that the number of people who attend is a function of the price of the tickets. If tickets are sold for $3, the venue will sell out. On the other hand, at $38/ticket, no one will buy tickets.
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1. Assuming that the number of tickets sold, $S(t)\text{,}$ is a linear function of the price, $t\text{,}$ write an equation for $S(t)\text{.}$
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2. Write an equation for the income the theater will generate, $g(t)\text{,}$ as a function of the ticket price, $t\text{.}$
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3. What is the maximum amount of money that the theater can earn? What price should they charge, and how many people will come at this price?
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The power produced by a windmill (in watts) is given by the equation

\begin{equation*} p(v)=k\cdot v^3 \end{equation*}

where $v$ is the wind speed in mph.

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1. If a wind speed of 20 mph generates 80 watts of power, write an equation for $p(v)\text{.}$
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2. Using your equation from Item 8.a, how much power will a wind speed of 10 mph generate?
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3. Using your equation from Item 8.a, what is the wind speed when the windmill is generating 100 watts of power?
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For each angle $\theta$ below, find $\sin \theta\text{,}$ $\cos \theta\text{,}$ $\tan \theta\text{,}$ $\sec \theta\text{,}$ $\csc \theta$ and $\cot \theta\text{.}$
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1. $\displaystyle \theta=-\pi$
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2. $\displaystyle \theta=\frac{20\pi}{3}$
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3. $\displaystyle \theta=\frac{7 \pi}{2}$
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