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3.7: Exercises - Double Angle, Half-Angle, and Power Reductions

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Chapter 3 Practice:

1. If \(\sin \left(x\right)=\dfrac{1}{8}\) and \(x\) is in quadrant I, then find exact values for (without solving for \(x\)):

a. \(\sin \left(2x\right)\) b. \(\cos \left(2x\right)\) c. \(\tan \left(2x\right)\)

2. If \(\cos \left(x\right)=\dfrac{2}{3}\) and \(x\) is in quadrant I, then find exact values for (without solving for \(x\)):

Simplify each expression.

3. \(\cos ^{2} \left(28{}^\circ \right)-\sin ^{2} (28{}^\circ )\)

4. \(2\cos ^{2} \left(37{}^\circ \right)-1\)

5. \(1-2\sin ^{2} (17{}^\circ )\)

6. \(\cos ^{2} \left(37{}^\circ \right)-\sin ^{2} (37{}^\circ )\)

7. \(\cos ^{2} \left(9x\right)-\sin ^{2} (9x)\)

8. \(\cos ^{2} \left(6x\right)-\sin ^{2} (6x)\)

9. \(4\sin \left(8x\right){\rm cos}(8x)\)

10. \(6\sin \left(5x\right){\rm cos}(5x)\)

Solve for all solutions on the interval \([0, 2\pi )\).

11. \(6\sin \left(2t\right)+9\sin \left(t\right)=0\)

12. \(2\sin \left(2t\right)+3\cos \left(t\right)=0\)

13. \(9\cos \left(2\theta \right)=9\cos ^{2} \left(\theta \right)-4\)

14. \(8\cos \left(2\alpha \right)=8\cos ^{2} \left(\alpha \right)-1\)

15. \(\sin \left(2t\right)=\cos \left(t\right)\)

16. \(\cos \left(2t\right)=\sin \left(t\right)\)

17. \(\cos \left(6x\right)-\cos \left(3x\right)=0\)

18. \(\sin \left(4x\right)-\sin \left(2x\right)=0\)

Use a double angle, half angle, or power reduction formula to rewrite without exponents.

19. \(\cos ^{2} (5x)\)

20. \(\cos ^{2} (6x)\)

21. \(\sin ^{4} (8x)\)

22. \(\sin ^{4} \left(3x\right)\)

23. \(\cos ^{2} x\sin ^{4} x\)

24. \(\cos ^{4} x\sin ^{2} x\)

25. If \(\csc \left(x\right)=7\) and \(90{}^\circ <x<180{}^\circ\), then find exact values for (without solving for \(x\)):

a. \(\sin \left(\dfrac{x}{2} \right)\) b. \(\cos \left(\dfrac{x}{2} \right)\) c. \(\tan \left(\dfrac{x}{2} \right)\)

26. If \(\sec \left(x\right)=4\) and \(270{}^\circ <x<360{}^\circ\), then find exact values for (without solving for \(x\)):

Prove the identity.

27. \(\left(\sin t-\cos t\right)^{2} =1-\sin \left(2t\right)\)

28. \(\left(\sin ^{2} x-1\right)^{2} =\cos \left(2x\right)+\sin ^{4} x\)

29. \(\sin \left(2x\right)=\dfrac{2\tan \left(x\right)}{1+\tan ^{2} \left(x\right)}\)

30. \(\tan \left(2x\right)=\dfrac{2\sin \left(x\right)\cos \left(x\right)}{2\cos ^{2} \left(x\right)-1}\)

31. \(\cot \left(x\right)-\tan \left(x\right)=2\cot \left(2x\right)\)

32. \(\dfrac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)} =\tan \left(\theta \right)\)

33. \(\cos \left(2\alpha \right)=\dfrac{1-\tan ^{2} \left(\alpha \right)}{1+\tan ^{2} \left(\alpha \right)}\)

34. \(\dfrac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos \left(t\right)} =\dfrac{2\cos \left(t\right)}{2\sin \left(t\right)-1}\)

35. \(\sin \left(3x\right)=3\sin \left(x\right)\cos ^{2} \left(x\right)-\sin ^{3} (x)\)

36. \(\cos \left(3x\right)=\cos ^{3} (x)-3\sin ^{2} (x)\cos \left(x\right)\)

1. a. \(\dfrac{3\sqrt{7}}{32}\) b. \(\dfrac{31}{32}\) c. \(\dfrac{3\sqrt{7}}{31}\)

3. \(\cos(56^{\circ})\)

5. \(\cos(34^{\circ})\)

7. \(\cos(18x)\)

9. \(2\sin(16x)\)

11. 0, \(\pi\), 2.4189,3.8643

13. 0.7297, 2.4119, 3.8713, 5.5535

15. \(\dfrac{\pi}{6}\), \(\dfrac{\pi}{2}\), \(\dfrac{5\pi}{6}\), \(\dfrac{3\pi}{2}\)

17. a. \(\dfrac{2\pi}{9}\), \(\dfrac{4\pi}{9}\), \(\dfrac{8\pi}{9}\), \(\dfrac{10\pi}{9}\), \(\dfrac{14\pi}{9}\), \(\dfrac{16\pi}{9}\), 0, \(\dfrac{2\pi}{3}\), \(\dfrac{4\pi}{3}\)

19. \(\dfrac{1 + \cos(10x)}{2}\)

21. \(\dfrac{3}{8} - \dfrac{1}{2} \cos(16x) + \dfrac{1}{8} \cos(32x)\)

23. \(\dfrac{1}{16} - \dfrac{1}{16} \cos(2x) + \dfrac{1}{16} \cos(4x) - \dfrac{1}{16} \cos(2x) \cos(4x)\)

25. a. \(\sqrt{\dfrac{1}{2}+\dfrac{2 + \sqrt{7}}{7}}\) b. \(\sqrt{\dfrac{1}{2}-\dfrac{2 + \sqrt{7}}{7}}\) c. \(\dfrac{1}{7 - 4\sqrt{3}}\)

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Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

(This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)

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