Chapter EP, Problem 13.5.7EP

To determine

To find: The given triangle can be solved by beginning with the law of Sines or law of Cosines. And hence find the solution of the given triangle

Answer to Problem 13.5.7EP

The solution of the triangle is $b\approx 18.7,\angle B\approx 33°,m\angle C\approx 35°$ .

Explanation of Solution

Given:

The given information of the triangle is:

  $a=32,c=20,A=112°$

As two sides and any angle is given so the triangle will be solved by beginning with the law of Sines.

Use the law of Sines to find the missing side length is given as:

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

The solution of the triangle is calculated as:

  $\begin{array}{c}\frac{\sin 112°}{32}=\frac{\sin C}{20}\\ \sin C=\frac{20\left(0.927\right)}{32}\\ \sin C=0.579\\ C=35°\end{array}$

The measure of the third angle is given as:

  $\begin{array}{c}m\angle B\approx 180°-\left(112°+35°\right)\\ m\angle B\approx 33°\end{array}$

Use the law of Sines to find the missing angle measure.

Use the law of Cosines to find the missing side measure.

  $b^2=c^2+a^2-2ac\cos B$

  $\begin{array}{c}{b}^2={32}^2+{20}^2-2\left(32\right)\left(20\right)\cos 33°\\ b^2=1424-1073.49\\ b^2=350.51\\ b=18.7\end{array}$

Therefore, $b\approx 18.7,\angle B\approx 33°,m\angle C\approx 35°$ .