7.3E: Double Angle Identities (Exercises)
Section 7.3 Exercises
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 \(x\)):
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. \(\sin \left(\dfrac{x}{2} \right)\)
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 \(x\)):
a. \(\sin \left(\dfrac{x}{2} \right)\)
b. \(\cos \left(\dfrac{x}{2} \right)\)
c. \(\tan \left(\dfrac{x}{2} \right)\)
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. \(\sin \left(2x\right)=\dfrac{2\tan \left(x\right)}{1+\tan ^{2} \left(x\right)}\)
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. \(\cos \left(3x\right)=\cos ^{3} (x)-3\sin ^{2} (x)\cos \left(x\right)\)
- Answer
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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. \(\dfrac{1}{7 - 4\sqrt{3}}\)