Question

If the area of the triangle $ABC$ is $\Delta ,$ such that $b^2\sin 2C+c^2\sin 2B=k\Delta ,$ then the value of $k$ is

Answer 1

$b^2\sin 2C+c^2\sin 2B$
$=b^2\cdot 2\sin C\cos C+c^2\cdot 2\sin B\cos B$
$=2b\cos C\cdot b\sin C+2c\cos B\cdot c\sin B$
$=2b\cos C\cdot c\sin B+2c\cos B\cdot c\sin B$
$[\because \frac{b}{\sin B}=\frac{c}{\sin C}\Rightarrow b\sin C=c\sin B]$
$=2c\sin B(b\cos C+c\cos B)$
$=2c\sin B\cdot a$
$=4\cdot \frac{1}{2}ac\sin B$
$=4\Delta$
$\therefore k=4$