Chapter 8, Problem 18RE

To determine

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

Answer to Problem 18RE

No triangle.

Explanation of Solution

Given:

The sides $a=2$ , $c=5$ and the angle $A=60°$ .

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 sides a and c , and the angle A , angle C can be determined as follows:

  $\begin{array}{c}\frac{\sin A}{a}=\frac{\sin C}{c}\\ \frac{\sin 60°}{2}=\frac{\sin C}{5}\\ \sin C=\frac{5\sin 60°}{2}\\ =2.165\end{array}$

Since the range of sine function is $\left[-1,1\right]$ , $\sin C\ne 2.165$ .

So, no triangle exists with the given measurements.