Chapter 8, Problem 16RE

To determine

To find: The remaining angle(s) and side(s) of the triangle, if it(they) exist.

Answer to Problem 16RE

The remaining measurements are sides $b=2.38$ and $a=0.54$ , and the angle $B=130°$ .

Explanation of Solution

Given:

The side $c=2$ and the angles $A=10°$ , $C=40°$ .

Calculation:

The Sine Law for a triangle $\Delta ABC$ is given by $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ , where capital letters represent the angles and the small letters represent the length of the respective sides of the triangle.

Given the angles A and C , and the side c , side a can be determined as follows:

  $\begin{array}{c}\frac{\sin A}{a}=\frac{\sin C}{c}\\ \frac{\sin 10°}{a}=\frac{\sin 40°}{2}\\ a=\frac{2\sin 10°}{\sin 40°}\\ =0.54\end{array}$

With the 2 angles known, the third angle B is determined as shown below:

  $\begin{array}{c}A+B+C=180°\\ 10°+B+40°=180°\\ B=180°-50°\\ =130°\end{array}$

Use the sine law to determine the third side b .

  $\begin{array}{c}\frac{\sin A}{a}=\frac{\sin B}{b}\\ \frac{\sin 10°}{0.54}=\frac{\sin 130°}{b}\\ b=\frac{0.54\sin 130°}{\sin 10°}\\ =2.38\end{array}$

So, the remaining measurements are sides $b=2.38$ and $a=0.54$ , and the angle $B=130°$ .