Chapter 5.7, Problem 21E

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 21E

First solution: $b=13.42,A=78.28°,\text{ and }B=31.72°$

Second solution: $b=3.63,A=101.82°,andB=8.18°$

Explanation of Solution

Given information:

  $C=70°,c=24,a=25$

Calculation:

Since the angle given is acute i.e. $C<90°$ , case I will be considered. In order to check the number of possible solutions, first evaluate $a\sin C$ as shown below.

  $\begin{array}{c}a\sin C=25\sin 70°\\ =23.49\end{array}$

The value of $c>a\sin C$ , so the number of possible solution in this case will be two.

Use law of sine to evaluate $A$ as follows:

  $\begin{array}{c}\frac{c}{\sin C}=\frac{a}{\sin A}\\ \sin A=\frac{a\sin C}{c}\left[\text{Put the value of }c\text{ and }a\sin C\right]\\ =\frac{23.49}{24}\\ =0.9788\\ A={\sin }^{-1}\left(0.9788\right)\left[{\text{Taking sin}}^{-1}\text{ on each side}\right]\\ =78.18°\end{array}$

The other possible measurement of $A$ must be less than $180°$ and have same sine value. For $\alpha <90°$ , the value of $\sin \left(\alpha \right)$ is $\sin \left(180°-\alpha \right)$ . Thus angle A will be evaluated as follows:

  $\begin{array}{c}\sin A=\sin \left(180°-A\right)\\ A=180°-A\\ =180°-78.18°\\ =101.82°\end{array}$

The other measurement of A is $101.82°$ .

Let’s find the measurements of possible number of solution.

Solution I: First possible solution

  $a=25,c=24,C=70°\text{ and }A=78.28°$

The angle B will be evaluated as follows:

  $\begin{array}{c}A+B+C=180°\\ B=180°-\left(A+C\right)\\ =180°-\left(78.28°+70°\right)\\ =31.72°\end{array}$

Now evaluate the side b for the triangle by using the laws of sine as shown below.

  $\begin{array}{c}\frac{a}{\sin A}=\frac{b}{\sin B}\\ \frac{25}{\sin 78.82°}=\frac{b}{\sin 31.72°}\\ b=\frac{25\left(\sin 31.72°\right)}{\sin 78.82°}\\ =13.42\end{array}$

Thus the measurements of first possible triangle are as follows: $\begin{array}{c}a=25,b=13.42,andc=24\\ A=78.28°,B=31.72°\text{ and }C=70°\end{array}$

Solution II: Second possible solution

  $a=25,c=24,C=70°\text{ and }A=101.82°$

The angle B will be evaluated as follows:

  $\begin{array}{c}A+B+C=180°\\ B=180°-\left(A+C\right)\\ =180°-\left(101.82°+70°\right)\\ =8.18°\end{array}$

Now evaluate the side b for the triangle by using the laws of sine as shown below.

  $\begin{array}{c}\frac{a}{\sin A}=\frac{b}{\sin B}\\ \frac{25}{\sin 101.82°}=\frac{b}{\sin 8.18°}\\ b=\frac{25\left(\sin 8.18°\right)}{\sin 101.82°}\\ =3.63\end{array}$

Thus the measurements of second possible triangle are as follows: $\begin{array}{c}a=25,b=3.63,andc=24\\ A=101.82°,B=8.18°\text{ and }C=70°\end{array}$