Chapter EP, Problem 13.5.8EP

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 13.5.8EP

The solution of the triangle is $a\approx 3.8,\angle B\approx 82°,m\angle C\approx 70°$ .

Explanation of Solution

Given:

The given information of the triangle is:

  $b=8,c=7,A=28°$

As two sides and their included angle is given so the triangle will be solved by beginning with the law of Cosines.

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

  $a^2=b^2+c^2-2bc\cos A$

The solution of the triangle is calculated as:

  $\begin{array}{c}{a}^2=8^2+7^2-2\left(8\right)\left(7\right)\cos 28°\left[\because b=8,c=7\text{ \&}\text{angle}A=28°\right]\\ a^2\approx 113-98.89\\ a^2\approx 14.11\\ a\approx 3.8\end{array}$

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

  $\begin{array}{c}\frac{\sin 28°}{3.8}\approx \frac{\sin B}{8}\left[\because \frac{\sin A}{a}=\frac{\sin B}{b}\right]\\ \sin B\approx \frac{8\left(0.47\right)}{3.8}\\ \sin B\approx 0.99\\ B\approx 82°\end{array}$

The measure of the third angle is given as:

  $\begin{array}{c}m\angle C\approx 180°-\left(28°+82°\right)\\ m\angle C\approx 70°\end{array}$

Therefore, $a\approx 3.8,B\approx 82°,m\angle C\approx 70°$ .