7.4E: Solving Trigonometric Equations (Exercises)

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. $\cos \left(\dfrac{\pi }{4} \theta \right)=-1$

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

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

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

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. $3\sin \left(\dfrac{\pi }{4} x\right)=2$

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

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

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

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