Chapter 6.1, Problem 50E

(a)

To determine

Towrite:The equation of $\alpha$ as a function of $\beta$ .

Explanation of Solution

The equation for the law of sine is given by,

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

The expression for $\alpha$ is calculated as,

  $\begin{array}{c}\frac{9}{\sin \alpha }=\frac{18}{\sin \beta }\\ \sin \alpha =\frac{1}{2}\sin \beta \\ \alpha ={\sin }^{-1}\left(\frac{\sin \beta }{2}\right)\end{array}$

Therefore, the expression for $\alpha$ is $\alpha ={\sin }^{-1}\left(\frac{\sin \beta }{2}\right)$ .

(b)

To determine

Tograph:The function $\alpha$ and write its domain.

Explanation of Solution

The expression for $\alpha$ is,

  $\alpha ={\sin }^{-1}\left(\frac{\sin \beta }{2}\right)$

The graph of the function is shown below.

  

Figure (1)

The domain of the function includes any real number.

The range of function is,

  $R=\left[-30,30\right]$

(c)

To determine

Towrite:The equation for $c$

Explanation of Solution

Apply the law of sine to get the expression for $c$ .

  $\frac{c}{\sin \gamma }=\frac{18}{\sin \beta }$

Also,

  $\begin{array}{c}\sin \gamma =\sin \left(\alpha +\beta \right)\\ =\sin \alpha \cos \beta +\cos \alpha \sin \beta \\ =\frac{\sin \beta \cos \beta }{2}+\frac{1}{2}\sqrt{4-{\sin }^2\beta }\end{array}$

Thus,

  $\begin{array}{c}c=\frac{18\sin \gamma }{\sin \beta }\\ =\frac{18\frac{\sin \beta \cos \beta }{2}+\frac{1}{2}\sqrt{4-{\sin }^2\beta }}{\sin \beta }\\ =\frac{9\sin \beta \cos \beta +9\sqrt{4+{\sin }^2\beta }}{\sin \beta }\end{array}$

Therefore, the expression for $c$ is $c=\frac{9\sin \beta \cos \beta +9\sqrt{4+{\sin }^2\beta }}{\sin \beta }$ .

(d)

To determine

To graph:The function $c$ and write its domain.

Explanation of Solution

The expression for $c$ is,

  $c=\frac{9\sin \beta \cos \beta +9\sqrt{4+{\sin }^2\beta }}{\sin \beta }$

The graph of the function is shown below.

  

Figure (2)

The domain of the function includes any real number.

The range of function is,

  $R=\left[-12.176,21.492\right]$

(e)

To determine

To fill:The table.

Explanation of Solution

The expression for $c$ is,

  $c=\frac{9\sin \beta \cos \beta +9\sqrt{4+{\sin }^2\beta }}{\sin \beta }$

The expression for $\alpha$ is,

  $\alpha ={\sin }^{-1}\left(\frac{\sin \beta }{2}\right)$

The complete filled table is shown in figure below.

$\beta$	$0.4$	$0.8$	$1.2$	$1.6$	$2.0$	$2.4$	$2.8$
$\alpha$	$0.195$	$0.37$	$0.48$	$0.52$	$0.471$	$0.344$	$0.17$
$c$	$20.88$	$21.3$	$18.97$	$15.33$	$12.62$	$12.46$	$14.9$

Table (1)

Therefore, the complete filled table is shown in Table (1)