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# 7.3E: Double Angle Identities (Exercises)

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Section 7.3 Exercises

1. If $$\sin \left(x\right)=\frac{1}{8}$$ 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 $$\cos \left(x\right)=\frac{2}{3}$$ 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. $$\tan \left(2x\right)$$

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. $$\sin \left(\frac{x}{2} \right)$$
b. $$\cos \left(\frac{x}{2} \right)$$
c. $$\tan \left(\frac{x}{2} \right)$$

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

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

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

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

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

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

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

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

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. $$\dfrac{1}{7 - 4\sqrt{3}}$$