Exercises 20.2, page 546

1. $\frac{16}{65}$ $\frac{16}{65}$
3.

$\begin{array}{l} \sin 105 ° = \sin 60 ° \cos 45 ° + \cos 60 ° \sin 45 ° \\ = \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} + \frac{1}{2} \frac{\sqrt{2}}{2} = 0.9659 \end{array}$ $\begin{array}{l} \sin 105 ° = \sin 60 ° \cos 45 ° + \cos 60 ° \sin 45 ° \\ = \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} + \frac{1}{2} \frac{\sqrt{2}}{2} = 0.9659 \end{array}$
5.

$\begin{array}{l} \cos 15 ° = \cos (60 ° - 45 °) = \cos 60 ° \cos 45 ° + \sin 60 ° \sin 45 ° \\ = (\frac{1}{2}) (\frac{1}{2} \sqrt{2}) + (\frac{1}{2} \sqrt{3}) (\frac{1}{2} \sqrt{2}) \\ = \frac{1}{4} \sqrt{2} + \frac{1}{4} \sqrt{6} = \frac{1}{4} (\sqrt{2} + \sqrt{6}) = 0.9659 \end{array}$ $\begin{array}{l} \cos 15 ° = \cos (60 ° - 45 °) = \cos 60 ° \cos 45 ° + \sin 60 ° \sin 45 ° \\ = (\frac{1}{2}) (\frac{1}{2} \sqrt{2}) + (\frac{1}{2} \sqrt{3}) (\frac{1}{2} \sqrt{2}) \\ = \frac{1}{4} \sqrt{2} + \frac{1}{4} \sqrt{6} = \frac{1}{4} (\sqrt{2} + \sqrt{6}) = 0.9659 \end{array}$
7. $- \frac{33}{65}$ $- \frac{33}{65}$
9. $- \frac{56}{65}$ $- \frac{56}{65}$
11. $\sin 3 x$ $\sin 3 x$
13. $\cos 4 x$ $\cos 4 x$
15. $\cos x$ $\cos x$
17. $\tan x$ $\tan x$
19. 0
21. 1
23. 0
25.

$\begin{array}{l} (\sin x \cos y + \cos x \sin y) (\sin x \cos y - \cos x \sin y) \\ = \sin^{2} x \cos^{2} y - \cos^{2} x \sin^{2} y \\ = \sin^{2} x (1 - \sin^{2} y) - (1 - \sin^{2} x) \sin^{2} y = \sin^{2} x - \sin^{2} y \end{array}$ $\begin{array}{l} (\sin x \cos y + \cos x \sin y) (\sin x \cos y - \cos x \sin y) \\ = \sin^{2} x \cos^{2} y - \cos^{2} x \sin^{2} y \\ = \sin^{2} x (1 - \sin^{2} y) - (1 - \sin^{2} x) \sin^{2} y = \sin^{2} x - \sin^{2} y \end{array}$
27.

$\begin{array}{l} (\cos α \cos β - \sin α \sin β) + (\cos α \cos β + \sin α \sin β) \\ = 2 \cos α \cos β \end{array}$ $\begin{array}{l} (\cos α \cos β - \sin α \sin β) + (\cos α \cos β + \sin α \sin β) \\ = 2 \cos α \cos β \end{array}$
29.
31.
33, 35, 37, 39. Use the indicated method.
41. $\frac{1}{2}$ $\frac{1}{2}$
43.

$\begin{array}{l} \frac{\sin (x + x)}{\sin x} = \frac{\sin x \cos x + \cos x \sin x}{\sin x} \\ = \frac{2 \sin x \cos x}{\sin x} \end{array}$ $\begin{array}{l} \frac{\sin (x + x)}{\sin x} = \frac{\sin x \cos x + \cos x \sin x}{\sin x} \\ = \frac{2 \sin x \cos x}{\sin x} \end{array}$
45. Using Eq. (20.9) , $\sin 75 ° = \sin (30 ° + 45 °) .$ $\sin 75 ° = \sin (30 ° + 45 °) .$ Using Eq. (20.11) , $\sin 75 ° = \sin (135 ° - 60 °) .$ $\sin 75 ° = \sin (135 ° - 60 °) .$ (Other angles are also possible.)
47.

$\begin{array}{l} I (\cos θ \cos 30 ° - \sin θ \sin 30 °) \\ + I (\cos θ \cos 150 ° - \sin θ \sin 150 °) \\ + I (\cos θ \cos 270 ° - \sin θ \sin 270 °) \\ = I [\frac{1}{2} \sqrt{3} \cos θ - \frac{1}{2} \sin θ - \frac{1}{2} \sqrt{3} \cos θ - \frac{1}{2} \sin θ \\ + 0 - (- \sin θ)] = 0 \end{array}$ $\begin{array}{l} I (\cos θ \cos 30 ° - \sin θ \sin 30 °) \\ + I (\cos θ \cos 150 ° - \sin θ \sin 150 °) \\ + I (\cos θ \cos 270 ° - \sin θ \sin 270 °) \\ = I [\frac{1}{2} \sqrt{3} \cos θ - \frac{1}{2} \sin θ - \frac{1}{2} \sqrt{3} \cos θ - \frac{1}{2} \sin θ \\ + 0 - (- \sin θ)] = 0 \end{array}$
49.
51. $d_{0} \sin (ω t + α) = d_{0} (\sin ω t \cos α + \cos ω t \sin α)$ $d_{0} \sin (ω t + α) = d_{0} (\sin ω t \cos α + \cos ω t \sin α)$
53.

$\begin{array}{l} \tan α (R + \cos β) = \sin β, R = \frac{\sin β - \tan α \cos β}{\tan α} \\ = \frac{\sin β \cos α - \cos β \sin α}{\cos α \tan α} \end{array}$ $\begin{array}{l} \tan α (R + \cos β) = \sin β, R = \frac{\sin β - \tan α \cos β}{\tan α} \\ = \frac{\sin β \cos α - \cos β \sin α}{\cos α \tan α} \end{array}$

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Table of Contents for Exercises 20.2, page 546

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Table of Contents for
Exercises 20.2, page 546