The Unit Circle and Trigonometric Functions Exercises

Subsection 2.3 The Unit Circle and Trigonometric Functions Exercises

For each triangle below, find all missing sides and angles.
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1. Figure 2.3.14.
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2. Figure 2.3.15.
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Refer to the function \(Q(x)=\frac{(x-4)(x+1)}{(x+2)(x+6)}\text{.}\)
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1. What is the domain of \(Q\text{?}\)
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2. Graph \(Q(x)\) so that the important features are visible, and label them.
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3. Describe the interval(s) on which \(Q\) is increasing.
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4. Find the average rate of change of \(Q\) on the interval \(-1 \leq x \leq 1\text{.}\)
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5. Find the average rate of change of \(Q\) on the interval \(1 \leq x \leq 10\text{.}\)
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6. Find an interval on which the average rate of change of \(P\) is 0.
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Refer to Table 2.3.16 and Table 2.3.17.
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\(x\) \(a(x)\)

\(-1\) 3

0 6

1 9

2 12

Table 2.3.16. Table of values for \(a(x)\)
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\(x\) \(b(x)\)

-1 3

0 6

1 12

2 24

Table 2.3.17. Table of values for \(b(x)\)
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1. Decide which type of function (linear, quadratic, exponential) best fits the data for \(a(x)\text{.}\) Explain your reasoning.
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2. Write a function equation for \(a(x)\text{.}\)
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3. Solve the equation \(a(x)=12\text{.}\)
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4. Using the function equation you wrote for \(a(x)\text{,}\) solve the inequality \(a(x) \geq 12\text{.}\)
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5. Using the function equation you wrote for \(a(x)\text{,}\) solve \(a(x)=48\text{.}\)
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6. Find the inverse function \(a^{-1}\text{.}\)
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7. Decide which type of function (linear, quadratic, exponential) best fits the data for \(b(x)\text{.}\) Explain your reasoning.
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8. Write a function equation for \(b(x)\text{.}\)
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9. Solve the equation \(b(x)=12\text{.}\)
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10. Using the function equation you wrote for \(a(x)\text{,}\) solve \(b(x) < 12\text{.}\)
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11. Using the function equation you wrote for \(b(x)\text{,}\) solve \(b(x)=48\text{.}\)
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12. Find the inverse function \(b^{-1}\text{.}\)
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Solve the following equations.
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1. \(\displaystyle 2^{x+1} \cdot 3^{x+2} = 23328\)
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2. \(\displaystyle \sqrt{x+1} \sqrt{x-39}=21\)
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3. \(\displaystyle \frac{1}{2}\ln (t+3) +\frac{1}{2} \ln (t-4) = \ln (12)\)
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Let \(a(x)=3^x\text{,}\) \(b(x)=3^{-x}\text{,}\) \(c(x)=\frac{1}{x^3}\text{.}\) Write a function equation for each function below, using the exponent rules to simplify the equation and write without negative exponents.
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1. \(\displaystyle f(x)=(c \circ b)(x)\)
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2. \(\displaystyle k(x)=c(x) \cdot a(x)\)
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3. \(\displaystyle g(x)=(c(x))^{-2}\)
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4. \(\displaystyle h(x)=2 (b(x))^2\)
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5. \(\displaystyle j(x)=a(x)\cdot b(x)\)
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Suppose that \(\sin \alpha = \frac{1}{2}\text{.}\) Use your work in the “Important Angles" section to determine possible value(s) of \(\alpha\text{,}\) where \(0 \leq \alpha < 2\pi\text{.}\)
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Suppose that \(\cos \beta = \frac{\sqrt{2}}{2}\text{.}\) Use your work in the “Important Angles" section to determine possible value(s) of \(\beta\text{,}\) where \(0 \leq \beta < 2\pi\text{.}\)
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A carousel has a diameter of 44 feet. The ride turns counterclockwise.
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1. Draw a diagram of the carousel in the \(xy\)-plane with the center at the origin.
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2. What is the distance traveled by a rider along the outer rim of the carousel as she travels through an angle of \(\frac{\pi}{3}\text{?}\) If the rider began at the point \((22,0)\text{,}\) what are the rider’s coordinates after traveling through an angle of \(\frac{\pi}{3}\text{?}\)
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A Ferris wheel has a diameter of 30 feet, and is mounted so that the bottom of the wheel is 4 feet off the ground. The ride turns counterclockwise as seen by the operator.
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1. Draw a diagram of the Ferris wheel in the \(xy\)-plane.
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2. Write an equation to describe the points on the Ferris wheel.
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3. Find the distance traveled by someone riding through one complete turn of the wheel.
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4. Find the distance traveled by someone ridlng from the 7 o’clock position to the 11 o’clock position.
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5. The wheel makes 1 turn every 25 seconds. A rider boards the wheel and travels a distance of 50 feet. How long did this take?
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6. What are the coordinates of a rider when she is in the 11 o’clock position on the wheel?
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7. What is the height above ground of a rider when her position is at an angle of \(\frac{\pi}{4}\) with the horizontal?
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Solve each equation.
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1. \(\displaystyle \frac{5}{n^2}-\frac{2n-4}{5n}=\frac{n-6}{n^2}\)
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2. \(\displaystyle 3r+3-\frac{5}{r}=\frac{1}{r}\)
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3. \(\displaystyle \frac{n-4}{n}=\frac{n+5}{n^2-4n}=\frac{2n^2-9n-5}{n^2-4n}\)
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Let \(m(x)=\sqrt[5]{x}\text{,}\) \(n(x)=x^2\text{,}\) and \(p(x)=\frac{1}{x^4}\text{.}\)
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1. Write \(m(n(x))\) as a function of \(x\) with a single exponent.
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2. Write \(m(n(x)p(x))\) as a function of \(x\) with a single exponent.
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3. Solve \(n(p(x))=128\)
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4. Solve \(m(x) \leq 1024\)
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The graph of \(f(x)\) is shown below. Let \(g(x)=f(2(x+1))-3\text{.}\)

Figure 2.3.18.
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1. Describe the transformations that would be required to transform \(f(x)\) in to \(g(x)\text{.}\)
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2. On the same set of axes, sketch the graph of \(g(x)\text{.}\)
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3. What are the domain and range of \(f(x)\text{.}\)
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4. What are the domain and range of \(g(x)\text{.}\)
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\(x\)	\(a(x)\)
\(-1\)	3
0	6
1	9
2	12

\(x\)	\(b(x)\)
-1	3
0	6
1	12
2	24

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