Chapter 6, Problem 1RE

To determine

Using Law of Sines solve the triangle, find solutions that exist.

Answer to Problem 1RE

The values on solving by Law of Sines related to triangle are:

  $\angle C={72}^{\circ }$

  $b\approx 12.21,c\approx 12.36$

Explanation of Solution

Given:

Data of a triangle $A={38}^{\circ },B={70}^{\circ },a=8$

Sum of three angles in a triangle is ${180}^{\circ }$

  $\begin{array}{l}\angle A+\angle B+\angle C={180}^{\circ }\hfill \\ {38}^{\circ }+{70}^{\circ }+\angle C={180}^{\circ }\hfill \\ {108}^{\circ }+\angle C={180}^{\circ }\hfill \\ \angle C={180}^{\circ }-{108}^{\circ }\hfill \\ \angle C={72}^{\circ }\hfill \end{array}$

Apply Sine rule to find other coordinates

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}\hfill \\ \frac{8}{\sin {38}^{\circ }}=\frac{b}{\sin {70}^{\circ }}\hfill \\ \frac{8}{\sin {38}^{\circ }}\times \sin {70}^{\circ }=b\hfill \\ b\approx 12.21\hfill \end{array}$

  $\begin{array}{l}\frac{b}{\sin B}=\frac{c}{\sin C}\hfill \\ \frac{12.21}{\sin {70}^{\circ }}=\frac{c}{\sin {72}^{\circ }}\hfill \\ \frac{12.21}{\sin {70}^{\circ }}\times \sin {72}^{\circ }=c\hfill \\ c=\frac{12.21}{\sin {70}^{\circ }}\times \sin {72}^{\circ }\hfill \\ c=12.358\hfill \\ c\approx 12.36\hfill \end{array}$

Hence, the other two solutions are: $b\approx 12.21,c\approx 12.36$