[tex]\displaystyle\\
\text{We use formula:}\\
\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+cos(\alpha)\sin(\beta)\\\\
Resolve:\\\\
\sin(\Theta)cos(3\Theta)+cos(\Theta)sin(3\Theta)=0\\\\
\sin(\Theta+3\Theta)=0\\\\
\sin(4\Theta)=0\\\\
\text{We have two solutions:}\\\\
4\Theta=0~~\Longrightarrow~~\Theta=\frac{0}{4}=0~~\Longrightarrow~~ \boxed{\Theta_1=0+2k\pi}\\\\
4\Theta=\pi~~\Longrightarrow~~\Theta=\frac{\pi}{4}~~\Longrightarrow~~ \boxed{\Theta_2=\frac{\pi}{4}+2k\pi}\\\\[/tex]
