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Mathematics LibreTexts

7.1E: Solving Trigonometric Equations with Identities (Exercises)

  • Page ID
    13934
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    Section 7.1 Exercises

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

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

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

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

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

    Find all solutions.

    5. \(2\sin \left(\frac{\pi }{4} x\right)= 1\)

    6. \(2\sin \left(\frac{\pi }{3} x\right)=\sqrt{ 2}\)

    7. \(2\cos \left(2t\right)=-\sqrt{3}\)

    8. \(2\cos \left(3t\right)=-1\)

    9. \(3\cos \left(\frac{\pi }{5} x\right)=2\)

    10. \(8\cos \left(\frac{\pi }{2} x\right)=6\)

    11. \(7\sin \left(3t\right)=-2 12. 4\sin \left(4t\right)=1\)

    Find all solutions on the interval \([0, 2\pi )\).

    13. \(10\sin \left(x\right)\cos \left(x\right)=6\cos \left(x\right)\)

    14. \(-3\sin \left(t\right)=15\cos \left(t\right)\sin \left(t\right)\)

    15. \(\csc \left(2x\right)-9=0\)

    16. \(\sec \left(2\theta \right)=3\)

    17. \(\sec \left(x\right)\sin \left(x\right)-2\sin \left(x\right)= 0\)

    18. \(\tan \left(x\right)\sin \left(x\right)-\sin \left(x\right)=0\)

    19. \(\sin ^{2} x=\frac{1}{4}\)

    20. \(\cos ^{2} \theta =\frac{1}{2}\)

    21. \(\sec ^{2} x=7\)

    22. \(\csc ^{2} t=3\)

    23. \(2\sin ^{2} w+3\sin w+1=0\)

    24. \(8\sin ^{2} x+6\sin \left(x\right)+1=0\)

    25. \(2\cos ^{2} t+\cos \left(t\right)=1\)

    26. \(8\cos ^{2} \left(\theta \right)=3-2\cos \left(\theta \right)\)

    27. \(4\cos ^{2} (x)-4=15\cos \left(x\right)\)

    28. \(9\sin \left(w\right)-2=4\sin ^{2} (w)\)

    29. \(12\sin ^{2} \left(t\right)+\cos \left(t\right)-6=0\)

    30. \(6\cos ^{2} \left(x\right)+7\sin \left(x\right)-8=0\)

    31. \(\cos ^{2} \phi =-6\sin \phi\)

    32. \(\sin ^{2} t=\cos t\)

    33. \(\tan ^{3} \left(x\right)=3\tan \left(x\right)\)

    34. \(\cos ^{3} \left(t\right)=-\cos \left(t\right)\)

    35. \(\tan ^{5} \left(x\right)=\tan \left(x\right)\)

    36. \(\tan ^{5} \left(x\right)-9\tan \left(x\right)=0\)

    37. \(4\sin \left(x\right)\cos \left(x\right)+2\sin \left(x\right)-2\cos \left(x\right)-1=0\)

    38. \(2\sin \left(x\right)\cos \left(x\right)-\sin \left(x\right)+2\cos \left(x\right)-1=0\)

    39. \(\tan \left(x\right)-3\sin \left(x\right)= 0\)

    40. \(3\cos \left(x\right)=\cot \left(x\right)\)

    41. \(2\tan ^{2} \left(t\right)=3\sec \left(t\right)\)

    42. \(1-2\tan \left(w\right)=\tan ^{2} \left(w\right)\)

    Answer

    1. \(\dfrac{7\pi}{6}\), \(\dfrac{11\pi}{6}\)

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

    5. \(\dfrac{2}{3} + 8k\), and \(\dfrac{10}{3} + 8k\), where \(k\) is an integer

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

    9. \(0.1339 + 10k\) and \(8.6614 + 10k\), where \(k\) is an integer

    11. \(1.1438 + \dfrac{2\pi}{3} k\) and \(1.9978 + \dfrac{2\pi}{3} k\), where \(k\) is an integer

    13. \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\), 0.644, 2.498

    15. 0.056, 1.515, 3.197, 4.647

    17. 0, \(\pi\), \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\)

    19. \(\dfrac{\pi}{6}\), \(\dfrac{5\pi}{6}\), \(\dfrac{7\pi}{6}\), \(\dfrac{11\pi}{6}\)

    21. 1.183, 1.958, 4.325, 5.100

    23. \(\dfrac{3\pi}{2}\), \(\dfrac{7\pi}{6}\), \(\dfrac{11\pi}{6}\)

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

    27. 1.823, 4.460

    29. 2.301, 3.983, 0.723, 5.560

    31. 3.305, 6.120

    33. 0, \(\dfrac{\pi}{3}\), \(\dfrac{2\pi}{3}\), \(\pi\), \(\dfrac{4\pi}{3}\), \(\dfrac{5\pi}{3}\)

    35. 0, \(\dfrac{\pi}{4}\), \(\dfrac{3\pi}{4}\), \(\pi\), \(\dfrac{5\pi}{4}\), \(\dfrac{7\pi}{4}\)

    37. \(\dfrac{\pi}{6}\), \(\dfrac{2\pi}{3}\), \(\dfrac{5\pi}{6}\), \(\dfrac{4\pi}{3}\)

    39. 0, \(\pi\), 1.231, 5.052

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