Chapter 5.6, Problem 8E

Solution

To find: The measure of angles $B$ and $C$ , and side $a$ in the triangle.

The measures are $B={37.9}^{\circ }$ , $C={60.1}^{\circ }$ , and $a=35.4$ .

-Given information: In the triangle, angle and two sides of triangle are $A={82}^{\circ }$ , $b=22$ , and $c=31$ .

Formula used:

The law of cosines to find the angles and sides are given by,

  $a^2=b^2+c^2-2bc\cos A$  ...... (1)$b^2=c^2+a^2-2ca\cos B$  ....... (2)

Calculation:

Substitute ${82}^{\circ }$ for $A$ , $22$ for $b$ , and $31$ for $c$ in equation (1).

  $\begin{array}{c}{a}^2={\left(22\right)}^2+{\left(31\right)}^2-2\cdot 22\cdot 31\cdot \cos {82}^{\circ }\\ a^2=484+961-2\cdot 22\cdot 31\cdot 0.1391\\ a^2=1255.25\\ a=35.4\end{array}$

Substitute $22$ for $b$ , and $31$ for $c$ , and $35.4$ for $a$ in equation (2).

  $\begin{array}{c}{\left(22\right)}^2={\left(31\right)}^2+{\left(35.4\right)}^2-2\cdot 31\cdot 35.4\cdot \cos B\\ 484=961+1253.16-2194.8\cdot \cos B\\ \cos B=\frac{1730.16}{2194.8}\\ B={37.9}^{\circ }\end{array}$

The sum of interior angles in the triangle is ${180}^{\circ }$ . Calculate the measure of angle $C$ .

  $\begin{array}{c}A+B+C={180}^{\circ }\\ C={180}^{\circ }-({37.9}^{\circ }+{82}^{\circ })\\ C={60.1}^{\circ }\end{array}$

Therefore, the measures are $B={37.9}^{\circ }$ , $C={60.1}^{\circ }$ , and $a=35.4$ .