Chapter 9, Problem 21REAE

Solution

To solve: the $△ABC$ .

  $A=47°,a\approx 16.89,c\approx 22.08$

Given information:

  $B=60°,C=73°,b=20$

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 .

  

  $\begin{array}{l}\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\\ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\end{array}$

Explanation:

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

Using this knowledge to find angle A,

  $\begin{array}{c}A+B+C=180°\\ A+60°+73°=180°\\ A=180°-60°-73°\\ A=47°\end{array}$

Now, by the law of Sines,

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{a}{\sin 47°}=\frac{20}{\sin 60°}=\frac{c}{\sin 73°}\end{array}$

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

  $\begin{array}{c}\frac{a}{\sin 47°}=\frac{20}{\sin 60°}\\ a=\frac{20\sin 47°}{\sin 60°}\\ a\approx 16.89\end{array}$

Now, the second equation is:

  $\begin{array}{c}\frac{20}{\sin 60°}=\frac{c}{\sin 73°}\\ c=\frac{20\sin 73°}{\sin 60°}\\ c\approx 22.08\end{array}$