Question

Calcule sen3x =sen7x
Urgente!!!

Answer 1

$\mathrm{\sin{3x}=\sin{7x}\ \to\ \sin{7x}-\sin{3x}=0\ \to}\\\\ \mathrm{*\ Prostaf\'erese\ \to\ \sin{a}-\sin{b}=2\sin{\bigg(\dfrac{a-b}{2}\bigg)}\cos{\bigg(\dfrac{a+b}{2}\bigg)}}\\\\\\ \mathrm{\to\ 2\sin{\bigg(\dfrac{7x-3x}{2}\bigg)}\cos{\bigg(\dfrac{7x+3x}{2}\bigg)}=0\ \to}\\\\\\ \mathrm{\to\ 2\sin{\bigg(\dfrac{4x}{2}\bigg)\cos{\bigg(\dfrac{10x}{2}\bigg)}}=0\ \to\ 2\sin{2x}\cos{5x}=0}$

$\mathrm{\mathbf{i)}\ \sin{2x}=0\ \to\ 2x=\arcsin{0}\ \to\ 2x=\pi k\ \to\ x=\dfrac{\pi k}{2}}\\\\ \mathrm{\mathbf{ii)}\ \cos{5x}=0\ \to\ 5x=\arccos{0}\ \to\ 5x=\dfrac{\pi}{2}+\pi k\ \to\ }\\\\ \mathrm{\to\ 5x=\dfrac{\pi+2\pi k}{2}\ \to\ x=\dfrac{\pi(2k+1)}{2.5}\ \to\ x=\dfrac{\pi}{10}(2k+1)}\\\\\\ \boxed{\mathbf{S=\bigg\{x\in\mathbb{R}\ |\ x=\dfrac{\pi k}{2}\ ou\ x=\dfrac{\pi}{10}(2k+1),\ k\in\mathbb{Z}\bigg\}}}$