Chapter EP, Problem 13.5.1EP

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 13.5.1EP

The solution of the triangle is $a\approx 36.9,B\approx 58°,m\angle C\approx 71°$ .

Explanation of Solution

Given:

The given triangle is:

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:

  $\begin{array}{c}{a}^2={40}^2+{45}^2-2\left(40\right)\left(45\right)\cos 51°\left[\because b=40,c=45\text{ \&}\text{angle}A=51°\right]\\ a^2\approx 3625-2265.55\\ a^2\approx 1359.45\\ a\approx 36.9\end{array}$

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

  $\begin{array}{c}\frac{\sin 51°}{36.9}\approx \frac{\sin B}{40}\left[\because \frac{\sin A}{a}=\frac{\sin B}{b}\right]\\ \sin B\approx \frac{40\left(0.78\right)}{36.9}\\ \sin B\approx 0.85\\ B\approx 58°\end{array}$

The measure of the third angle is given as:

  $\begin{array}{c}m\angle C\approx 180°-\left(51°+58°\right)\\ m\angle C\approx 71°\end{array}$

Therefore, $a\approx 36.9,B\approx 58°,m\angle C\approx 71°$ .