Question

If in a $△ABC,c=2b$ and $C=B+\frac{\pi }{3}$ then the measure of $\angle A$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{3}$

$c=2b$

$c=B+\pi /3$

$\angle A=?$

$A+B+C=180$

$A+B+B+60=180$

$A+2B=120$

$\frac{9}{\sin A}=\frac{b}{\sin B}=\frac{2b}{\sin B+\pi /3}$

$\frac{b}{\sin B}=\frac{2b}{\sin (B+60)}$

$B={30}^o$

$A+60={120}^o$

$A={60}^o$

$\sin B\cos {60}^o+\cos B\sin {60}^o=2\sin B$

$\frac{\sin B}{2}+\frac{\sqrt{3}}{2}=2\sin B$

$\frac{\sqrt{3}}{2}\cos B=\frac{\sqrt{3}}{2}\sin B$

$\cot B=\sqrt{3}$