
# 6.4E: Solving Trigonometric Equations (Exercises)

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

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