Solve Trigonometric Equations: Practice Problems
Test your understanding of trigonometric equations with these 10 problems. Each question requires finding all solutions, applying properties of trigonometric functions, identities, and the unit circle. Select the correct answer for each problem. Check your understanding with the answer key.
Tip: Use the interactive unit circle to visualize and locate solutions once you find the reference angle.
Problem Set
Problem 1
Solve for all solutions:
\[ 2 \cos x + 1 = 0 \]
- a) \( \dfrac{\pi}{3} + 2n\pi, \dfrac{5\pi}{3} + 2n\pi \)
- b) \( -\dfrac{1}{2} \)
- c) No solutions
- d) \( \dfrac{2\pi}{3} + 2n\pi, \dfrac{4\pi}{3} + 2n\pi \)
- e) \( \dfrac{\pi}{2} + n\pi \)
Problem 2
Solve for all solutions:
\[ 3 \sec^2 x - 4 = 0 \]
- a) \( \sqrt{3}, -\sqrt{3} \)
- b) \( \dfrac{\pi}{3} + 2n\pi, \dfrac{5\pi}{3} + 2n\pi \)
- c) \( \dfrac{\pi}{6} + 2n\pi, \dfrac{11\pi}{6} + 2n\pi , \dfrac{7\pi}{6} + 2n\pi, \dfrac{5\pi}{6} + 2n\pi\)
- d) \( \dfrac{\pi}{3} + n\pi, \dfrac{5\pi}{3} + n\pi \)
- e) \( \dfrac{\pi}{6} + n\pi, \dfrac{11\pi}{6} + n\pi \)
Problem 3
Solve for all solutions:
\[ (3 \cos x + 7)(-2 \sin x - 1) = 0 \]
- a) No solutions
- b) \( \dfrac{7\pi}{6} + 2n\pi, \dfrac{11\pi}{6} + 2n\pi \)
- c) \( \dfrac{\pi}{3} + 2n\pi, \dfrac{2\pi}{3} + 2n\pi \)
- d) \( \dfrac{7\pi}{6} + n\pi, \dfrac{11\pi}{6} + n\pi \)
- e) \( -\dfrac{7}{3}, -\dfrac{1}{2} \)
Problem 4
Solve for all solutions:
\[ (6 \tan^2 x - 2)(2 \tan^2 x - 6) = 0 \]
- a) \( \dfrac{\pi}{6} + n\pi, \dfrac{5\pi}{6} + 2n\pi, \dfrac{\pi}{3} + n\pi, \dfrac{2\pi}{3} + n\pi \)
- b) No solutions
- c) \( \dfrac{\pi}{6}, \dfrac{5\pi}{6} \)
- d) \( -\sqrt{3}, \sqrt{3} \)
- e) \( \dfrac{\pi}{3} + n\pi, \dfrac{2\pi}{3} + n\pi \)
Problem 5
Find all solutions in the interval \( [0, 2\pi) \):
\[ -2 \sec^2 x + 4 = -2\sec x \]
- a) \( \dfrac{\pi}{3}, \dfrac{5\pi}{3}, \pi \)
- b) No solutions
- c) \( \pi \)
- d) \( -1, 2 \)
- e) \( \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \pi \)
Problem 6
Find all solutions in the interval \( [0, 2\pi) \):
\[ 2\sin (x) \cos (-x) = 2 \sin (-x) \sin (x) \]
- a) \( 0, \pi \)
- b) \( 0, \pi, \dfrac{3\pi}{4}, \dfrac{7\pi}{4} \)
- c) \( \dfrac{3\pi}{4}, \dfrac{7\pi}{4} \)
- d) \( 0, \dfrac{\pi}{2} \)
- e) \( \dfrac{\pi}{6}, \dfrac{4\pi}{3} \)
Problem 7
Find all solutions in the interval \( [0, 2\pi) \):
\[ \sin 2x = -\sin (-x) \]
- a) No solutions
- b) \( 0 \)
- c) \( 0, \dfrac{\pi}{3}, \pi, \dfrac{5\pi}{3} \)
- d) \( 0, \pi \)
- e) \( \dfrac{\pi}{3}, \pi \)
Problem 8
Which of these equations has no solution?
- a) \( \sin(100x) = 0.1 \)
- b) \( \tan(x) = 10000 \)
- c) \( 2\sin x = -3 \)
- d) \( \cos^2 x - \dfrac{1}{4} = 0 \)
- e) \( \csc x = -2000 \)
Problem 9
Which of these equations has no solution?
- a) \( 5\sin(x) = 4.9 \)
- b) \( 4\cot(x) = 400 \)
- c) \( 3\tan x = 0 \)
- d) \( \sec x = -\dfrac{1}{2} \)
- e) \( \cos x = -0.001 \)
Problem 10
Find all solutions in the interval \( [0, 2\pi) \):
\[ \sin^2 x + \sin x = 6 \]
- a) \( 2, -3 \)
- b) \( 0 \)
- c) \( 0, \dfrac{\pi}{3}, \dfrac{\pi}{2} \)
- d) \( 2 \)
- e) No solutions
Further Resources