Question

In Problems 3–5, use the given information to determine the three remaining parts of each triangle.

Short answer

The missing values are$a=18$,$B\approx 48.09°$, and$C=79.9°$

Step-by-step solution

Step 1. Given information

The given triangle is

Step 2. Use the Rule of Cosines 

• By the rule of cosines, $a^2=b^2+c^2-2bc\cos A$
• Substitute the values $b=17,c=19,A=52°$ to find the value of a.

$\begin{array}{rcl}{a}^2& =& {17}^2+{19}^2-2\left(17\right)\left(19\right)\cos \left(52\right)\\ & \approx & 289+361-397.72\\ & \approx & 324.28\\ a& \approx & 18\end{array}$

Step 3. Use the rule of Sines

• According to the rule of sines, $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$.
• Substitute $b=17,a=18,A=52°$ into the formula and find the value of angle B.

$\begin{array}{rcl}\frac{\sin \left(52\right)}{18}& =& \frac{\sin B}{17}\\ \sin B& =& 17\times \frac{0.788}{18}\\ B& \approx & 48.09\end{array}$

Step 4. Use Angle sum Property

The sum of angles of a triangle is $180°$.

So,$\begin{array}{rcl}C& =& 180°-52°-48.09°\\ & =& 79.9°\end{array}$