Question

If the bisector of angle A of a triangle ABC makes an angle theta with BC then sin $theta$ is equal to?

Answer 1

ADC and ADB form a triangle
Apply sine rule,
$\frac{sin\varphi }{b}=\frac{sin\frac{A}{2}}{DC}\frac{sin(180-\varphi )}{c}=\frac{sin\frac{A}{2}}{BD}\frac{sin\varphi }{c}=\frac{sin\frac{A}{2}}{BD}BD+DC=\frac{sin\left(\frac{A}{2}\right[b+c\left]\right)}{sin\left(\varphi \right)}\alpha =\frac{sin\left(\frac{A}{2}\right)[b+c]}{sin\left(\varphi \right)}sin\left(\varphi \right)=\frac{sin\left(\frac{A}{2}\right)[b+c]}{sin\left(\alpha \right)}$