Question

3Sin x + 4cos x +5 sin 3x=0

Answer 1

$3\sin x+4\cos x+5\sin3x=0\\ \\ 5\sin(x+ \frac{\pi}{4} )+5\sin 3x=0\\ \\ \sin (x+\frac{\pi}{4} )+\sin 3x=0\\ \\ 2\sin \frac{x+\frac{\pi}{4} +3x}{2} \cos \frac{x+\frac{\pi}{4} -3x}{2} =0\\ \\ 2\sin(x+\frac{\pi}{8} )\cos(\frac{\pi}{8} -x)=0\\ \\ \left[\begin{array}{ccc}\sin (x+\frac{\pi}{8})=0 \\ \cos (\frac{\pi}{8} -x)=0\end{array}\right~~~\Rightarrow~~~~ \boxed{ \left[\begin{array}{ccc}x_1=-\frac{\pi}{8} + \pi k,k \in \mathbb{Z}\\ x_2=\frac{\pi}{8} -2 \pi k,k \in \mathbb{Z}\end{array}\right}$