6.4E: 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
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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