Chapter 5.5, Problem 11E

Solution

To find: The value of angle $A$ and $C$ and side $a$ in triangle $ABC$ .

The value of angles $A$ and $C$ are $72.8°$ and $37.2°$ , and side $a$ is $14.2$ .

Given information: In triangle $ABC$ , the measure of angle $B$ is $70°$ and the measure of sides $b$ and $c$ are $14$ and $9$ .

Formula Used: The formula for law of sine is as follows.

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

Calculation:

Draw the diagram of the triangle.

  

Apply law of sine to determine the value of angle $C$ .

  $\frac{\sin B}{b}=\frac{\sin C}{c}$

Substitute $14$ for $b$ , $9$ for $c$ , and $70°$ for $B$ .

  $\begin{array}{c}\frac{\sin 70°}{14}=\frac{\sin C}{9}\\ \sin C=\frac{\sin 70°\times 9}{14}\end{array}$

Use calculator to determine the value of the expression.

  $\begin{array}{c}\sin C=0.6040\\ C={\sin }^{-1}(0.6040)\\ =37.2°\end{array}$

Calculate the value of angle $A$ using triangle.

  $\begin{array}{c}A+70°+37.2°=180°\\ A=180°-70°-37.2°\\ =72.8°\end{array}$

Apply law of sine to determine the value of side $a$ .

  $\frac{\sin A}{a}=\frac{\sin B}{b}$

Substitute $14$ for $b$ , $72.8°$ for $A$ , and $70°$ for $B$ .

  $\begin{array}{c}\frac{\sin 72.8°}{a}=\frac{\sin 70°}{14}\\ a=\frac{\sin 72.8°\times 14}{\sin 70°}\end{array}$

Use calculator to determine the value of the expression.

  $a=14.2$

Therefore, the value of angles $A$ and $C$ are $72.8°$ and $37.2°$ , and side $a$ is $14.2$ .