Chapter 14.3, Problem 31PPS

(a)

To determine

To Find: The value of the sine angle $\angle BAC$_.

Answer to Problem 31PPS

The value of the angle as $\sin \angle BAC=0.9928$ .

Explanation of Solution

Given:

The other swatch has sides 8 inches, $8\sqrt{3}$ inches, and 16 inches long. The pieces are placed with the sides of eight inches against each other, as shown in the figure, to form triangle ABC.

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Mark the values and draw the diagram as shown in Figure 1

  

Figure 1

Consider the angle relationship as,

  $\begin{array}{c}\sin \angle BAC=\sin \left(\angle BAX+\angle XAC\right)\\ =\sin \angle BAX\cos \angle XAC+\sin \angle XAC\cos \angle BAX\end{array}$

Then, solve for the angle as,

  $\begin{array}{c}\sin \angle BAC=\frac{8\sqrt{3}}{16}\left(\frac{8}{10}\right)+\frac{8}{16}\left(\frac{6}{10}\right)\\ =\frac{4\sqrt{3}+3}{10}\\ =0.9928\end{array}$

(b)

To determine

To Find: The value of the cosine angle $\angle BAC$_.

Answer to Problem 31PPS

The value of the angle as $\cos \angle BAC=-0.1196$ .

Explanation of Solution

Calculation:

Consider the angle relationship as,

  $\begin{array}{c}\cos \angle BAC=\cos \left(\angle BAX+\angle XAC\right)\\ =\cos \angle BAX\cos \angle XAC-\sin \angle XAC\sin \angle BAX\end{array}$

Then, solve for the angle as,

  $\begin{array}{c}\cos \angle BAC=\frac{8}{16}\left(\frac{8}{10}\right)-\frac{8\sqrt{3}}{16}\left(\frac{6}{10}\right)\\ =\frac{4-3\sqrt{3}}{10}\\ =-0.1196\end{array}$

(c)

To determine

To Find: The value of the angle $\angle BAC$_.

Answer to Problem 31PPS

The value of the angle is $96.9°$ .

Explanation of Solution

Calculation:

Consider the angle relationship as,

  $\cos \angle BAC=-0.1196$

Then, solve for the angle as,

  $\begin{array}{c}\angle BAC={\cos }^{-1}\left(-0.1196\right)\\ =96.9°\end{array}$

(d)

To determine

To Find: Whether the new triangle formed form the two triangles also the right triangle.

Answer to Problem 31PPS

The triangle is not formed.

Explanation of Solution

Calculation:

Since, the triangle formed form the two triangles is not the right triangle so the triangle is not formed