Chapter 5.7, Problem 19E

To determine

To find: all the solutions of the triangle and round to the nearest tenth. In case no solutions exist, write none.

Answer to Problem 19E

First solution: $B=71.1°,C=50.9°,c=23.8$

Second solution: $B=129.9°,C=7.1°,c=3.8$

Explanation of Solution

Given information:

  $A=58°,a=26,b=29$

Calculation:

Since the angle given is acute i.e. $A<90°$ , case I will be considered. Evaluate $b\sin A$ and compare with $a$ as shown below.

  $\begin{array}{c}b\sin A=29\sin 58°\\ =24.6\end{array}$

Since $b\sin A<a$ , so there are two solutions for the triangle.

Use laws of sine to evaluate $B$ as follows:

  $\begin{array}{c}\frac{a}{\sin A}=\frac{b}{\sin B}\\ \frac{26}{\sin 58°}=\frac{29}{\sin B}\left[\text{Substitute the values}\right]\\ \sin B=\frac{29\left(\sin 58°\right)}{26}\\ \sin B=0.9459\\ B={\sin }^{-1}\left(0.9459\right)\left[\text{Taking sin inverse each side}\right]\\ B=71.1°\end{array}$

The third angle $\left(C\right)$ of this triangle can be evaluated as follows:

  $\begin{array}{c}\angle C=180°-\left(\angle A+\angle B\right)\\ =180°-\left(58°+71.1°\right)\left[\text{Put the values}\right]\\ =50.9°\end{array}$

Use laws of sine to evaluate $c$ for first triangle as follows:

  $\begin{array}{c}\frac{a}{\sin A}=\frac{c}{\sin C}\\ \frac{26}{\sin 58°}=\frac{c}{\sin 50.9°}\left[\text{Substitute the values}\right]\\ c=\frac{26\left(\sin 50.9°\right)}{\sin 58°}\\ c=23.8\end{array}$

Thus first triangle will have $B=71.1°,C=50.9°\text{ and }c=23.8$ .

The value of $B$ for second triangle will be as evaluated as follows:

  $\begin{array}{c}\angle B=180°-\left(B\right)\\ =180°-\left(50.9°\right)\left[\text{Put the values}\right]\\ =129.1°\end{array}$

The third angle $\left(C\right)$ of the triangle can be evaluated as follows:

  $\begin{array}{c}\angle C=180°-\left(\angle A+\angle B\right)\\ =180°-\left(58°+129.1°\right)\left[\text{Put the values}\right]\\ =7.1°\end{array}$

Use laws of sine to evaluate $c$ for second triangle as follows:

  $\begin{array}{c}\frac{a}{\sin A}=\frac{c}{\sin C}\\ \frac{26}{\sin 58°}=\frac{c}{\sin 7.1°}\left[\text{Substitute the values}\right]\\ c=\frac{26\left(\sin 7.1°\right)}{\sin 58°}\\ c=3.8\end{array}$

Thus second triangle will have $B=7129.9°,C=7.1°\text{ and }c=3.8$ .