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Mathematics LibreTexts

6.4E: Solving Trigonometric Equations (Exercises)

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    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. \[33. 7\sin \left(6x\right)=2~ 34. 7\sin \left(5x\right)=\; 6 35. 5\cos \left(3x\right)=-3 36. 3\cos \left(4x\right)=237. 3\sin \left(\frac{\pi }{4} x\right)=2~ 38. 7\sin \left(\frac{\pi }{5} x\right)=6 39. 5\cos \left(\frac{\pi }{3} x\right)=1 40. 3\cos \left(\frac{\pi }{2} x\right)=-2\]

    Section 6.5 Modeling with Trigonometric Functions 451