Chapter 9.5, Problem 22E

Solution

To solve: the $△ABC$ .

  $C=61°,a\approx 31.84,b\approx 34.94$

Given information:

  $A=55°,B=64°,c=34$

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,

  $\begin{array}{c}A+B+C=180°\\ 55°+64°+C=180°\\ C=180°-55°-64°\\ C=61°\end{array}$

Now, by the law of Sines,

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{a}{\sin 55°}=\frac{b}{\sin 64°}=\frac{34}{\sin 61°}\end{array}$

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

  $\begin{array}{c}\frac{a}{\sin 55°}=\frac{34}{\sin 61°}\\ a=\frac{34\sin 55°}{\sin 61°}\\ a\approx 31.84\end{array}$

And second equation is:

  $\begin{array}{c}\frac{b}{\sin 64°}=\frac{34}{\sin 61°}\\ b=\frac{34\sin 64°}{\sin 61°}\\ b\approx 34.94\end{array}$