Chapter 20

1. $\begin{array}{rcl} \sec θ - \frac{\tan θ}{\csc θ} & = & \frac{1}{\cos θ} - \frac{\frac{\sin θ}{\cos θ}}{\frac{1}{\sin θ}} = \frac{1}{\cos θ} - \frac{\sin^{2} θ}{\cos θ} \\ = & \frac{1 - \sin^{2} θ}{\cos θ} = \frac{\cos^{2} θ}{\cos θ} = \cos θ \\ \cos θ & = & \cos θ \end{array}$
2. $\begin{array}{l} \sin 2 x + \sin x = 0 \\ 2 \sin x \cos x + \sin x = 0 \\ \sin x (2 \cos x + 1) = 0 \\ \sin x = 0 \cos x = - \frac{1}{2} \\ x = 0, π, \frac{2 π}{3}, \frac{4 π}{3} \end{array}$
3. $\begin{array}{rcl} θ & = & \sin^{- 1} x \\ \cos θ & = & \cos (\sin^{- 1} x) \\ = & \sqrt{1 - x^{2}} \end{array}$
4. $i = 8.00 e^{- 20 t} (1.73 \cos 10.0 t - \sin 10.0 t)$ Because $8.00 e^{- 20 t}$ will not equal zero, $i = 0$ if $1.73 \cos 10.0 t - \sin 10.0 t = 0$

$\begin{array}{rcl} 1.73 & = & \tan 10.0 t \\ 10.0 t & = & \tan^{- 1} 1.73 \\ t & = & 0.105 s \end{array}$
5. $\begin{array}{rcl} \frac{\tan α + \tan β}{\tan α - \tan β} & = & \frac{\frac{\sin α}{\cos α} + \frac{\sin β}{\cos β}}{\frac{\sin α}{\cos α} - \frac{\sin β}{\cos β}} = \frac{\sin α \cos β + \sin β \cos α}{\sin α \cos β - \sin α \cos β} \\ = & \frac{\sin (α + β)}{\sin (α - β)} \end{array}$
6. $\begin{array}{l} \cot^{2} x - \cos^{2} x = \frac{\cos^{2} x}{\sin^{2} x} - \cos \end{array}$

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Basic Technical Mathematics, 11th Edition by Allyn J. Washington, Richard Evans