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Exercises 20.4, page 553

1. $\cos 57 ° = 0.544639035$ $\cos 57 ° = 0.544639035$
3. $\begin{array}{l} \cos 15 ° = \cos \frac{1}{2} (30 °) = \sqrt{\frac{1 + \cos 30 °}{2}} = \sqrt{\frac{1.8660}{2}} \\ = 0.9659 \end{array}$ $\begin{array}{l} \cos 15 ° = \cos \frac{1}{2} (30 °) = \sqrt{\frac{1 + \cos 30 °}{2}} = \sqrt{\frac{1.8660}{2}} \\ = 0.9659 \end{array}$
5.

$\begin{array}{l} \sin 105 ° = \sin \frac{1}{2} (210 °) = \sqrt{\frac{1 - \cos 210 °}{2}} \\ = \sqrt{\frac{1.8660}{2}} = 0.9659 \end{array}$ $\begin{array}{l} \sin 105 ° = \sin \frac{1}{2} (210 °) = \sqrt{\frac{1 - \cos 210 °}{2}} \\ = \sqrt{\frac{1.8660}{2}} = 0.9659 \end{array}$
7. $\begin{array}{l} \cos \frac{3 π}{8} & = \cos \frac{1}{2} (\frac{3 π}{4}) = \sqrt{\frac{1 + \cos \frac{3 π}{4}}{2}} \\ = \sqrt{\frac{0.29289}{2}} = 0.3827 \end{array}$ $\begin{array}{l} \cos \frac{3 π}{8} & = \cos \frac{1}{2} (\frac{3 π}{4}) = \sqrt{\frac{1 + \cos \frac{3 π}{4}}{2}} \\ = \sqrt{\frac{0.29289}{2}} = 0.3827 \end{array}$
9. $\sin 118 ° = 0.8829476$ $\sin 118 ° = 0.8829476$
11. $\cos 48 ° = 0.6691$ $\cos 48 ° = 0.6691$
13. $\pm \sin 4 x$ $\pm \sin 4 x$
15. $\pm \cos 3 x$ $\pm \cos 3 x$
17. $\pm 2 \sqrt{2} \sin 5 θ$ $\pm 2 \sqrt{2} \sin 5 θ$
19. 1
21. $\frac{1}{26} \sqrt{26}$ $\frac{1}{26} \sqrt{26}$
23. 0.1414
25. $\pm \sqrt{\frac{2 \sec α}{\sec α - 1}}$ $\pm \sqrt{\frac{2 \sec α}{\sec α - 1}}$
27. $\tan \frac{1}{2} α = \frac{1 - \cos α}{\sin α} = \frac{\sin α}{1 + \cos α}$ $\tan \frac{1}{2} α = \frac{1 - \cos α}{\sin α} = \frac{\sin α}{1 + \cos α}$
29. $\frac{1 - \cos α}{2 \sin \frac{1}{2} α} = \frac{1 - \cos α}{2 \sqrt{\frac{1}{2} (1 - \cos α)}} = \sqrt{\frac{1 - \cos α}{2}}$ $\frac{1 - \cos α}{2 \sin \frac{1}{2} α} = \frac{1 - \cos α}{2 \sqrt{\frac{1}{2} (1 - \cos α)}} = \sqrt{\frac{1 - \cos α}{2}}$
31. $2 \sqrt{2 \frac{(1 + \cos x) (1 + \cos x)}{2 (2) (1 + \cos x)}} = \frac{2 (1 + \cos x)}{2} ∙ \sqrt{\frac{2}{1 + \cos x}}$ $2 \sqrt{2 \frac{(1 + \cos x) (1 + \cos x)}{2 (2) (1 + \cos x)}} = \frac{2 (1 + \cos x)}{2} ∙ \sqrt{\frac{2}{1 + \cos x}}$
33.
35.
37. $\pm \frac{24}{7}$ $\pm \frac{24}{7}$
39. $\sqrt{3 - 2 \sqrt{2}}$ $\sqrt{3 - 2 \sqrt{2}}$
41. $\frac{1}{5} \sqrt{5}$ $\frac{1}{5} \sqrt{5}$
43. $4 \sin \frac{θ}{2}$ $4 \sin \frac{θ}{2}$
45.

$\begin{array}{l} \sin ω t = \pm \sqrt{\frac{1 - \cos 2 ω t}{2}}, \\ \sin^{2} ω t = \frac{1}{2} (1 - \cos 2 ω t) \end{array}$ $\begin{array}{l} \sin ω t = \pm \sqrt{\frac{1 - \cos 2 ω t}{2}}, \\ \sin^{2} ω t = \frac{1}{2} (1 - \cos 2 ω t) \end{array}$
47. $\frac{\sqrt{\frac{1 - \cos (A + ϕ)}{2}}}{\sqrt{\frac{1 - \cos A}{2}}} = \sqrt{\frac{1 - \cos (A + ϕ)}{1 - \cos A}}$ $\frac{\sqrt{\frac{1 - \cos (A + ϕ)}{2}}}{\sqrt{\frac{1 - \cos A}{2}}} = \sqrt{\frac{1 - \cos (A + ϕ)}{1 - \cos A}}$

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Table of Contents for Exercises 20.4, page 553

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Table of Contents for
Exercises 20.4, page 553