Chapter 6.5, Problem 3E

To determine

To calculate: The value of side $x$ or angle $\theta$ in the given figure,

Answer to Problem 3E

The value of $x\approx 318.8$ .

Explanation of Solution

Given information:

The figure,

Formula used:

Sum of interior angles of any triangle is ${180}^{\circ }$ . So, in any $\text{ΔABC}$ , $\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }$ .

According to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

Calculation:

Consider the given figure,

Recall that the sum of interior angles of any triangle is ${180}^{\circ }$ . So, in any $\text{ΔABC}$ , $\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }$ .

In the given figure, $\angle \text{A = 98}{\text{.4}}^{\circ }$ and $\angle \text{B}={24.6}^{\circ }$ .

So, $\angle \text{C}$ will be calculated as,

  $\begin{array}{c}\angle \text{A}+\angle \text{B}+\angle \text{C}={180}^{\circ }\\ {\left(98.4+24.6\right)}^{\circ }+\angle \text{C}={180}^{\circ }\\ \angle \text{C}={180}^{\circ }-{\left(123\right)}^{\circ }\\ \angle \text{C}={57}^{\circ }\end{array}$

Now, to find the length of side $x$ , apply law of sines.

Recall that according to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

So, the value of x will be calculated as,

  $\begin{array}{c}\frac{\text{sin A}}{376}=\frac{\text{sin C}}{x}\\ \frac{\sin 98.4}{376}=\frac{\sin 57}{x}\\ x=\frac{376\sin 57}{\sin 98.4}\\ x\approx 318.8\end{array}$

Thus, the value of $x\approx 318.8$ .