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
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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. \(\dfrac{\pi}{3}\), \(\dfrac{5\pi}{3}\)