Question

sin x + sin 3x + cos x = 0​

Answer 1

$\sin(x)+\sin(3x)+\cos(x)=0\\2\sin(2x)\cos(x)+\cos(x)=0\\\cos(x)(2\sin(2x)+1)=0\\\begin{cases}\cos(x)=0\\\sin(2x)=-\dfrac{1}{2}\end{cases}\Rightarrow\begin{cases}x=\dfrac{\pi}{2}+\pi k\\2x=-\dfrac{\pi}{6}+2\pi k\\2x=-\frac{7\pi}{6}+2\pi k\end{cases}, k \in \mathbb Z\Rightarrow\begin{cases}x=\dfrac{\pi}{2}+\pi k\\x=-\dfrac{\pi}{12}+\pi k\\x=-\frac{7\pi}{12}+\pi k\end{cases}k\in\mathbb Z$