Question

Solve the following equations.
$sin3x+3sin4x+sin5x=0.$

Answer 1

$sin3x+3sin4x+sin5x=0$

$\Rightarrow (sin3x+sin5x)+3sin4x=0$

$\Rightarrow 2sin\left(\frac{3x+5x}{2}\right)cos\left(\frac{3x-5x}{2}\right)+3sin4x=0$

(by $sinC+sinD$ formula)

$\Rightarrow 2sin4xcosx+3sin4x=0$

$\Rightarrow sin4x(2cosx+3)=0$

$\Rightarrow sin4x=0$ or $2cosx+3=0$

$\Rightarrow sin4x=sin\left(x\pi \right)$ or $cosx=-3/2$

$\Rightarrow 4x=x\pi$

$\Rightarrow x=\frac{x\pi }{4}$