Chapter 9.5, Problem 25E

Solution

To solve: the $△ABC$ .

  $C=99°,a\approx 62.66,c\approx 82$

Given information:

  $A=49°,B=32°,b=44$

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,

  $\begin{array}{c}A+B+C=180°\\ 49°+32°+C=180°\\ C=180°-49°-32°\\ C=99°\end{array}$

Now, by the law of Sines,

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{a}{\sin 49°}=\frac{44}{\sin 32°}=\frac{c}{\sin 99°}\end{array}$

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

  $\begin{array}{c}\frac{a}{\sin 49°}=\frac{44}{\sin 32°}\\ a=\frac{44\sin 49°}{\sin 32°}\\ a\approx 62.66\end{array}$

And second equation is:

  $\begin{array}{c}\frac{44}{\sin 32°}=\frac{c}{\sin 99°}\\ c=\frac{44\sin 99°}{\sin 32°}\\ c\approx 82\end{array}$