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MAT 151: College Algebra with Trigonometry

Subsection 3.2 Solving Trigonometric Equations Exercises

  1. Let \(\sin \theta = -.15\text{.}\)
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    1. Find \(\theta\) if \(\theta\) is a negative angle in the fourth quadrant.
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    2. Find \(\theta\) if \(\theta\) is a positive angle in the fourth quadrant.
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    3. Find \(\theta\) if \(\theta\) is a positive angle in the third quadrant.
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  2. Solve each equation on the interval \([0, 2\pi)\text{.}\) Check your answers in Desmos.
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    1. \(\displaystyle 1+\cos\theta=\frac{1}{2}\)
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    2. \(\displaystyle 3\sec\theta=3\sqrt{2}\)
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    3. \(\displaystyle 1=-\sin(-\theta)\)
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    4. \(\displaystyle -2\sin(\theta+\frac{3\pi}{2})=\sqrt{3}\)
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    5. \(\displaystyle -6\cos^2\theta+3=-2\cos^2\theta\)
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    6. \(\displaystyle 0=-3\csc^2\theta+4\)
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  3. Let \(g(x) = 3\sin x -2\text{.}\)
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    1. Graph the function on the domain \(-2\pi \leq x \leq 2\pi\text{.}\) Be sure to label the midline, amplitude, and period.
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    2. Graph the function \(s(x)=\sin x\) on the same graph as in Item 3.a. Then explain how \(s(x)\) is transformed to create the graph of \(g(x)\text{.}\)
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    3. Again referring to the graph in Item 3.a, write out the intervals on the domain \(-2\pi \leq x \leq 2\pi\) on which \(g(x)\) is decreasing.
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    4. List all solutions to \(3\sin x -2 = 1\) on the interval \(-2\pi \leq x \leq 2\pi\text{.}\) Give your answer in exact form.
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  4. Refer to the graph of \(R(x)\) in Figure 3.2.6.
    A graph of an upward-opening parabola on a coordinate grid. The graph has a minimum near y = −10 at approximately x = 10. The curve crosses the x-axis near x = 0 and again near x = 20, and the y-axis is shown with labeled tick marks.
    Figure 3.2.6.
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    1. Solve the inequality \(R(x) \geq 0\text{.}\)
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    2. Write a function equation for \(R(x)\text{.}\)
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    3. What is the range of the function \(R(x)\) on the domain \(2 \leq x \leq 18\text{?}\)
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    4. What is the average rate of change of \(R(x)\) on the domain \(2 \leq x \leq 8\text{?}\)
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    5. Give an example of a domain on which the average rate of change of \(R(x)\) is 0.
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    6. Write a function equation for a function that has a graph that is shifted 3 units left and 4 units down from the graph of \(R(x)\text{.}\)
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  5. For each graph given, (a) identify the function family (linear, quadratic, exponential, linear absolute value, reciprocal, square root, cubic), and (b) identify how the basic function has been transformed to produce the graph given (c) write an equation of the graph.
    A graph of a V-shaped function on a coordinate grid. The graph has a vertex at (−2, 0). To the right of the vertex, the graph increases linearly, passing through the point (0,1). To the left of the vertex, the graph decreases linearly.
    A graph of a decreasing curved function on a coordinate grid, shown only for negative x-values. The curve decreases from left to right and ends at the point (−3, −1).
    Figure 3.2.7.
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  6. Simplify each expression. Write your answers without negative exponents.
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    1. \(\displaystyle (5x^3y^{-3}z)^{-2}\)
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    2. \(\displaystyle \sqrt[3]{8x^{-6}z^{3}}\)
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    3. \(\displaystyle \frac{x^2-3x+1}{\sqrt{x}}\)
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  7. Refer to the function \(g(x)=\frac{3x+5}{x-2}\text{.}\)
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    1. What are the domain and range of \(g(x)\text{?}\)
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    2. Graph \(g(x)\) labeling all important features.
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    3. Find a function equation for the inverse function, \(g^{-1}\text{.}\)
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    4. What are the domain and range of \(g^{-1}(x)\text{?}\)
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