5.6.2: Homework

Math 373: Trigonometry for Calculus · LibreTexts

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    Reading Questions

    1. How is the graph of the cosecant function related to the graph of the sine function?
    2. Where do the vertical asymptotes of the secant function occur?
    3. What is the natural period of the cotangent function?
    4. What is the domain of the cosecant function?
    5. What is the range of the secant function?
    6. Which of the reciprocal trigonometric functions are odd? Which are even?
    7. What is the recommended first step when graphing a transformed cosecant or secant function?
    8. Why do the cosecant, secant, and cotangent functions not have an amplitude?
    9. The zeros of the sine function correspond to what features on the graph of the cosecant function?
    10. The local maximums on the graph of the sine function correspond to what features on the graph of the cosecant function?

    Homework

    Concept Check

    1. Describe the general shape of the graph of the cosecant function.

    2. What is the relationship between the sine function and the cosecant function? How does this relationship affect their graphs?

    3. Describe the general shape of the graph of the secant function.

    4. What is the relationship between the cosine function and the secant function? How does this relationship affect their graphs?

    5. Describe the general shape of the graph of the cotangent function.

    6. What is the relationship between the tangent function and the cotangent function? How does this relationship affect their graphs?

    Basic Skills

    1. Determine the period, phase shift, and midline of the function \( y = 3 \sec(2x - \pi) + 1 \).

    2. Determine the period, phase shift, and midline of the function \( y = 2 \csc(x + \frac{\pi}{4}) - 3 \).

    3. Determine the period, phase shift, and midline of the function \( y = -\cot\left(\frac{1}{2}x - \frac{\pi}{3}\right) + 4 \).

    4. Determine the period, phase shift, and midline of the function \( y = 4 \sec\left(\frac{1}{3}x + \frac{\pi}{6}\right) - 2 \).

    5. Determine the period, phase shift, and midline of the function \( y = -2 \csc(3x - \pi) + 1 \).

    6. Determine the period, phase shift, and midline of the function \( y = 3 \cot\left(4x + \frac{\pi}{2}\right) - 5 \).

    7. Graph \( y = \sec(x - \frac{\pi}{2}) + 2 \).

    8. Graph \( y = 2 \csc\left(\frac{\pi}{2}x\right) - 1 \).

    9. Graph \( y = -\cot\left(x + \frac{\pi}{4}\right) + 3 \).

    10. Graph \( y = 3 \sec(2x) - 4 \).

    11. Graph \( y = -\csc\left(\frac{1}{2}x - \frac{\pi}{3}\right) + 2 \).

    12. Graph \( y = \cot\left(\frac{3}{2}x + \frac{\pi}{6}\right) - 1 \).

    13. Graph \( y = -2 \sec\left(\frac{1}{3}x - \frac{\pi}{4}\right) + 5 \).

    14. Graph \( y = 4 \csc(3x) - 3 \).

    15. Graph \( y = \cot(2x) + 4 \).

    16. Graph \( y = -\sec\left(\frac{2}{3}x + \frac{\pi}{2}\right) - 2 \).

    17. Graph \( y = \csc\left(\frac{\pi}{3}x - \frac{\pi}{6}\right) + 1 \).

    18. Graph \( y = -3 \cot\left(\frac{1}{2}x + \frac{\pi}{3}\right) + 2 \).

    For the following exercises, assume that the related sinusoidal function has an amplitude of \( 1 \).

    1. Given the graph of a secant function that has a period of \( \pi \), a phase shift of \( \frac{\pi}{4} \) to the right, and a vertical shift up by 2 units, determine the equation of the graph.

    2. Given the graph of a cosecant function that has a period of \( 2\pi \), a phase shift of \( \frac{\pi}{6} \) to the left, and a vertical shift down by 3 units, determine the equation of the graph.

    3. Given the graph of a reflected secant function that has a period of \( \frac{\pi}{2} \), a phase shift of \( \frac{\pi}{2} \) to the left, and a vertical shift down by 4 units, determine the equation of the graph.

    4. Given the graph of a reflected cosecant function that has a period of \( 4\pi \), a phase shift of \( \frac{\pi}{4} \) to the right, and a vertical shift up by 5 units, determine the equation of the graph.

    5. Given the graph of a secant function with a period of \( 2\pi \), a vertical shift up by 1 unit, and where the related sinusoidal function has an amplitude of 3, determine the equation of the graph.

    6. Given the graph of a cosecant function with a period of \( \pi \), a vertical shift down by 4 units, and where the related sinusoidal function has an amplitude of 2, determine the equation of the graph.

    7. Given the graph of a reflected secant function with a period of \( \frac{\pi}{3} \), a phase shift of \( \frac{\pi}{2} \) to the left, a vertical shift down by 3 units, and where the related sinusoidal function has an amplitude of 1.2, determine the equation of the graph.

    8. Given the graph of a reflected cosecant function with a period of \( \frac{\pi}{4} \), a phase shift of \( \frac{\pi}{6} \) to the right, a vertical shift up by 2 units, and where the related sinusoidal function has an amplitude of 2, determine the equation of the graph.

    For the following exercises, find the equation of the function in terms of cosecant, secant, or cotangent.

    1.  
      desmos-graph (1).png
    2.  
      desmos-graph (2).png
    3.  
      desmos-graph (3).png

    Synthesis Questions

    1. Given the function \( y = 3 \csc(2x - \frac{\pi}{2}) + 4 \), find all asymptotes within the interval \( [0, 2\pi] \). Explain the process of identifying these asymptotes.

    2. Given the function \( y = 2 \sec\left(\frac{1}{2}x - \frac{\pi}{3}\right) - 1 \), find all asymptotes within the interval \( [-2\pi, 2\pi] \). Describe the method used to determine these asymptotes.

    3. For the function \( y = -3 \csc(4x + \frac{\pi}{4}) + 2 \), determine all asymptotes within the interval \( [0, \pi] \). Explain your reasoning and the steps taken to find these asymptotes.

    4. Given the function \( y = \sec(3x - \frac{\pi}{6}) - 2 \), find the equations of all vertical asymptotes within the interval \( [-\pi, \pi] \). Describe the procedure used to identify these asymptotes.

    5. For the function \( y = -\csc\left(\frac{3}{2}x + \frac{\pi}{3}\right) + 5 \), find the equations of all vertical asymptotes within the interval \( [-\pi, \pi] \). Detail the steps and logic used to locate these asymptotes.

    6. Consider the function \( y = 4 \sec\left(2x - \frac{\pi}{2}\right) + 1 \). Identify all vertical asymptotes within the interval \( [0, 2\pi] \) and explain how you determined their locations.

    This page titled 5.6.2: Homework was last modified on Fri, 04 Jul 2025 01:18:08 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.