Question

sin(x+60)=5sinx.sin60

Answer 1

Given,

$\sin(x+60)=5\sin x.\sin60\textdegree\\\;\\\sin x.\cos60+cos x.\sin60=5\sin x.\sin60\\\;\\\sin x\frac{1}{2}+\cos x.\frac{\sqrt{3}}{2}=5\sin x.\frac{\sqrt{3}}{2}\\\;\\\sin x + \sqrt{3}\cos x=5\sqrt{3}\sin x\\\;\\\sqrt{3}\cos x =\sin x(5\sqrt{3}-1)\\\;\\\frac{\sin x}{\cos x}=\frac{\sqrt{3}}{5\sqrt{3}-1}\\\;\\\tan x=\frac{\sqrt{3}}{5\sqrt{3}-1}\\\;\\x=\tan^{-1}(\frac{\sqrt{3}}{5\sqrt{3}-1})\\\;\\x=\tan^{-1}(\frac{15+\sqrt{3}}{14})$

$Note:1.\sin(A+B)=\sin A.\cos B+\cos A.\sin B\\\;\\2.\sin(A-B)=\sin A.\cos B-\cos A.\sin B$