Chapter 9.5, Problem 19E

Solution

To solve: the $△ABC$ .

  $B=119°,a\approx 6.5,c\approx 8.5$

Given information:

  $A=26°,C=35°,b=13$

Definition Used:

Law of Sines

The law of sines can be written in either of the following forms for $△ABC$ with sides of length a , b and c .

  

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Explanation:

It is known that the sum of all the angles of a triangle is $180°$ .

Using this knowledge to find angle A,

  $\begin{array}{c}A+B+C=180°\\ 26°+B+35°=180°\\ B=180°-26°-35°\\ B=119°\end{array}$

Now, by the law of Sines,

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{a}{\sin 26°}=\frac{13}{\sin 119°}=\frac{c}{\sin 35°}\end{array}$

Use this to write two equations with one variable each and solve them independently to find the value of the variable.

  $\begin{array}{c}\frac{a}{\sin 26°}=\frac{13}{\sin 119°}\\ a=\frac{13\sin 26°}{\sin 119°}\\ a\approx 6.5\end{array}$

And second equation is:

  $\begin{array}{c}\frac{13}{\sin 119°}=\frac{c}{\sin 35°}\\ c=\frac{13\sin 35°}{\sin 119°}\\ c\approx 8.5\end{array}$