7.1E: Solving Trigonometric Equations with Identities (Exercises)

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. $2\sin \left(\dfrac{\pi }{4} x\right)= 1$

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

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

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

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

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

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. $\sin ^{2} x=\dfrac{1}{4}$

20. $\cos ^{2} \theta =\dfrac{1}{2}$

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. $\dfrac{\pi}{3}$, $\dfrac{5\pi}{3}$