Questions on Trigonometric Functions

What's On This Page

This page contains sample problems on trigonometric functions. They are for Self-assessment and Review.

Each problem (or group of problems) has an "answer button" ANSWER which you can click to look at an answer. Some solutions have a "further explanation button" Explain which you can click to see a more complete, detailed solution.

What to Do

To gain the most benefit from these problems,

Work the problems on your own. Write down your solutions BEFORE looking at any answers.

Use a graphing calculator as appropriate.

If you have difficulties with this material, please contact your instructor. (See Getting Help in Stage 1.) It will be very difficult to succeed in Calculus without understanding trigonometry.

1. Use the definitions of the six trigonometric functions and the Pythagorean Identity given in the Field Guide Lesson to show that:

  1. 1 + tan2(x) = sec2(x)
  2. 1 + cot2(x) = csc2(x)


2. Draw graphs of each of the functions. Give the period and any vertical asymptotes. Comment on the amplitudes of these functions.

  1. cot (x)
  2. sec (x)
  3. csc (x)


3. Give the amplituden and period for each of the following functions. Sketch their graphs. Your graphing calculator is helpful for checking here.

  1. g(x) = 3cos 2x
  2. f(x) = -2.7sin 2(x - /4)
  3. g(x) = sin ((1/2)(x + /3)
  4. h(x) = 1.5cos (x - 1)


4. The tidal variation in Desolation Sound on the west coast of Canada is roughly 4 meters. That is, the difference between water depth at high tide and at low tide is 4 meters, with successive high tides occurring 12.5 hours apart. Suppose that at Refuge Cove in Desolation Sound, the depth of water in meters is given by

D(t) = D0 + Acos B(t - t0)
where t is measured in hours from midnight on June 1, 1996.

  1. What does D0 mean in the context of the problem?
  2. What is the value of A?
  3. What is the value of B?
  4. Give the physical meaning of t0 for this problem.


(Adapted by Judy de Szoeke from CALCULUS, (The Harvard Consortium Text), Hughes-Hallett, Gleason, et al.))

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