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Mathematics LibreTexts

6.4E: Solving Trigonometric Equations (Exercises)

  • Page ID
    13926
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    Section 6.4 Exercises

    Give all answers in radians unless otherwise indicated.

    Find all solutions on the interval \(0\le \theta <2\pi\).

    1. \(2\sin \left(\theta \right)=-\sqrt{2}\)

    2. \(2\sin \left(\theta \right)=\sqrt{3}\)

    3. \(2\cos \left(\theta \right)=1\)

    4. \(2\cos \left(\theta \right)=-\sqrt{2}\)

    5. \(\sin \left(\theta \right)=1\)

    6. \(\sin \left(\theta \right)=0\)

    7. \(\cos \left(\theta \right)=0\)

    8. \(\cos \left(\theta \right)=-1\)

    Find all solutions.

    9. \(2\cos \left(\theta \right)=\sqrt{2}\)

    10. \(2\cos \left(\theta \right)=-1\)

    11. \(2\sin \left(\theta \right)=-1\)

    12. \(2\sin \left(\theta \right)=-\sqrt{3}\)

    Find all solutions.

    13. \(2\sin \left(3\theta \right)=1\)

    14. \(2\sin \left(2\theta \right)=\sqrt{3}\)

    15. \(2\sin \left(3\theta \right)=-\sqrt{2}\)

    16. \(2\sin \left(3\theta \right)=-1\)

    17. \(2\cos \left(2\theta \right)=1\)

    18. \(2\cos \left(2\theta \right)=\sqrt{3}\)

    19. \(2\cos \left(3\theta \right)=-\sqrt{2}\)

    20. \(2\cos \left(2\theta \right)=-1\)

    21. \(\cos \left(\frac{\pi }{4} \theta \right)=-1\)

    22. \(\sin \left(\frac{\pi }{3} \theta \right)=-1\)

    23. \(2\sin \left(\pi \theta \right)=1.\)

    24. \(2\cos \left(\frac{\pi }{5} \theta \right)=\sqrt{3}\)

    Find all solutions on the interval \(0\le x<2\pi\).

    25. \(\sin \left(x\right)=0.27\)

    26. \(\sin \left(x\right)= 0.48\)

    27. \(\sin \left(x\right)= -0.58\)

    28. \(\sin \left(x\right)=-0.34\)

    29. \(\cos \left(x\right)=-0.55\)

    30. \(\sin \left(x\right)= 0.28\)

    31. \(\cos \left(x\right)= 0.71\)

    32. \(\cos \left(x\right)=-0.07\)

    Find the first two positive solutions.

    33. \(7\sin \left(6x\right)=2\)

    34. \(7\sin \left(5x\right)= 6\)

    35. \(5\cos \left(3x\right)=-3\)

    36. \(3\cos \left(4x\right)=2\)

    37. \(3\sin \left(\frac{\pi }{4} x\right)=2\)

    38. \(7\sin \left(\frac{\pi }{5} x\right)=6\)

    39. \(5\cos \left(\frac{\pi }{3} x\right)=1\)

    40. \(3\cos \left(\frac{\pi }{2} x\right)=-2\)

    Answer

    1. \(\dfrac{5\pi}{4}\), \(\dfrac{7\pi}{4}\)

    3. \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\)

    5. \(\dfrac{\pi}{2}\)

    7. \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\)

    9. \(\dfrac{\pi}{4} + 2 \pi k\), \(\dfrac{7\pi}{4} + 2 \pi k\), where \(k\) is an integer

    11. \(\dfrac{7\pi}{6} + 2 \pi k\), \(\dfrac{11\pi}{6} + 2 \pi k\), where \(k\) is an integer

    13. \(\dfrac{\pi}{18} + \dfrac{2 \pi}{3} k\), \(\dfrac{5\pi}{18} + \dfrac{2 \pi}{3} k\), where \(k\) is an integer

    15. \(\dfrac{5\pi}{12} + \dfrac{2 \pi}{3} k\), \(\dfrac{7\pi}{12} + \dfrac{2 \pi}{3} k\), where \(k\) is an integer

    17. \(\dfrac{\pi}{6} + \pi k\), \(\dfrac{5\pi}{6} + \pi k\), where \(k\) is an integer

    19. \(\dfrac{\pi}{4} + \dfrac{2 \pi}{3} k\), \(\dfrac{5\pi}{12} + \dfrac{2 \pi}{3} k\), where \(k\) is an integer

    21. \(4 + 8k\), where \(k\) is an integer

    23. \(\dfrac{1}{6} + 2k\), \(\dfrac{5}{6} + 2k\), where \(k\) is an integer

    25. 0.2734, 2.8682

    27. 3.7603, 5.6645

    29. 2.1532, 4.1300

    31. 0.7813, 5.5019

    33. 0.04829, 0.47531

    35. 0.7381, 1.3563

    37. 0.9291, 3.0709

    39. 1.3077, 4.6923