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Mathematics LibreTexts

7.1E: Solving Trigonometric Equations with Identities (Exercises)

  • Page ID
    13934
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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 \([0,\; 2\pi )\). \[13. 10\sin \left(x\right)\cos \left(x\right)=6\cos \left(x\right) 14. -3\sin \left(t\right)=15\cos \left(t\right)\sin \left(t\right)\] \[15. \csc \left(2x\right)-9=0 16. \sec \left(2\theta \right)=3\] \[17. \sec \left(x\right)\sin \left(x\right)-2\sin \left(x\right)=\; 0 18. \tan \left(x\right)\sin \left(x\right)-\sin \left(x\right)=0\] \[19. \sin ^{2} x=\frac{1}{4} 20. \cos ^{2} \theta =\frac{1}{2}\] \[21. \sec ^{2} x=7 22. \csc ^{2} t=3\] \[23. 2\sin ^{2} w+3\sin w+1=0 24. 8\sin ^{2} x+6\sin \left(x\right)+1=0\] \[25. 2\cos ^{2} t+\cos \left(t\right)=1 26. 8\cos ^{2} \left(\theta \right)=3-2\cos \left(\theta \right)\] \[27. 4\cos ^{2} (x)-4=15\cos \left(x\right) 28. 9\sin \left(w\right)-2=4\sin ^{2} (w)\] \[29. 12\sin ^{2} \left(t\right)+\cos \left(t\right)-6=0 30. 6\cos ^{2} \left(x\right)+7\sin \left(x\right)-8=0\] \[31. \cos ^{2} \phi =-6\sin \phi 32. \sin ^{2} t=\cos t\] \[33. \tan ^{3} \left(x\right)=3\tan \left(x\right) 34. \cos ^{3} \left(t\right)=-\cos \left(t\right)\] \[35. \tan ^{5} \left(x\right)=\tan \left(x\right) 36. \tan ^{5} \left(x\right)-9\tan \left(x\right)=0\] \[37. 4\sin \left(x\right)\cos \left(x\right)+2\sin \left(x\right)-2\cos \left(x\right)-1=0\] \[38. 2\sin \left(x\right)\cos \left(x\right)-\sin \left(x\right)+2\cos \left(x\right)-1=0\] \[39. \tan \left(x\right)-3\sin \left(x\right)=\; 0 40. 3\cos \left(x\right)=\cot \left(x\right)\] \[41. 2\tan ^{2} \left(t\right)=3\sec \left(t\right) 42. 1-2\tan \left(w\right)=\tan ^{2} \left(w\right)\] Section 7.2 Addition and Subtraction Identities 467