Chapter 6.1, Problem 54E

(a)

To determine

To Find: The angle $\alpha$ as the function of $\beta$ .

Answer to Problem 54E

The required expression is $\alpha =\arcsin \frac{\sin \beta }{2}$ .

Explanation of Solution

Given:

The given figure is shown in Figure 1

  

Figure 1

Calculation:

Consider the relation by the sine angle property as,

  $\begin{array}{l}\frac{\sin \alpha }{9}=\frac{\sin \beta }{18}\\ \sin \alpha =\frac{\sin \beta }{2}\\ \alpha =\arcsin \frac{\sin \beta }{2}\end{array}$

(b)

To determine

To Find: The angle of the function in part (a) and determine its domain and the range.

Answer to Problem 54E

The graph is shown in Figure 2 and the range lies between $0$ and $\frac{\pi }{6}$ as the angle can only be positive.

Explanation of Solution

Consider the expression is,

  $\alpha =\arcsin \frac{\sin \beta }{2}$

The required diagram for the given expression is shown in Figure 2

  

Figure 2

From the above graph, the domain is between 0 and $\pi$ , the range lies between $0$ and $\frac{\pi }{6}$ as the angle can only be positive.

(c)

To determine

To Find: The value of c as the function of $\beta$ .

Answer to Problem 54E

The required equation is $\frac{18\sin \pi -\arcsin \frac{\sin \beta }{2}-\beta }{\sin \beta }$ .

Explanation of Solution

Consider the law of sine to set up equation with the variables as,

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

Then,

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

Consider the value of the angle $\gamma$ is obtained as,

  $\begin{array}{c}\gamma =\pi -\alpha -\beta \\ =\pi -\arcsin \frac{\sin \beta }{2}-\beta \end{array}$

Then,

  $c=\frac{18\sin \pi -\arcsin \frac{\sin \beta }{2}-\beta }{\sin \beta }$

(d)

To determine

To Find: The graph of the function in the part (c) and determine its domain and the range.

Answer to Problem 54E

The domain is between 0 and $\pi$ , the range is between 9 and 27 and the graph for the function is shown in Figure 3

Explanation of Solution

Consider the function is,

  $\frac{18\sin \pi -\arcsin \frac{\sin \beta }{2}-\beta }{\sin \beta }$

The graph for the above function is shown in Figure 3

  

Figure 3

The domain is between 0 and $\pi$ , the range is between 9 and 27.

(e)

To determine

To Find: The completed form of the given table.

Answer to Problem 54E

The completed table is shown in Table 1

Explanation of Solution

The complete given table is shown in Table 1

Table 1

$\beta$	0.4	0.8	1.2	1.6	2.0	2.4	2.8
$\alpha$	0.20	0.37	0.48	0.52	0.47	0.34	0.17
$c$	25.95	23.07	19.19	15.33	12.29	10.31	9.27