Chapter 12.7, Problem 13WE

To determine

To determine the exact value of $\frac{a}{b}$ in triangle ABC without using tables or calculators.

Answer to Problem 13WE

  $\frac{a}{b}=\frac{10}{9}$

Explanation of Solution

Given information :

  $\sin A=\frac{2}{3},\cos B=\frac{4}{5}$

According to law of sines,

In any triangle ABC,

  

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Hence,

  $\frac{a}{b}=\frac{\sin A}{\sin B}$

now we shall use the identity ${\sin }^{2}x+{\cos }^{2}x=1$

hence we get,

  $\begin{array}{l}{\sin }^{2}B+{\cos }^{2}B=1\\ {\sin }^{2}B=1-\frac{16}{25}\\ \sin B=\sqrt{\frac{9}{25}}=\frac{3}{5}\end{array}$

Therefore $\frac{a}{b}=\frac{\sin A}{\sin B}=\frac{\left(\frac{2}{3}\right)}{\left(\frac{3}{5}\right)}=\frac{10}{9}$