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# 7.1E: Solving Trigonometric Equations with Identities (Exercises)

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