Question

$\sin 7\theta +\sin 4\theta +\sin \theta =0,0<\theta <\frac{\pi }{2}$.

Answer 1

$\begin{array}{c}\Rightarrow \sin 7\theta +\sin \theta +\sin 4\theta \\ \Rightarrow 2sin4\theta \cdot \cos 3\theta +\sin 4\theta =0\\ \Rightarrow \sin 4\theta \left[2\cos 3\theta +1\right]=0\\ \therefore \sin 4\theta =0\\ 4\theta =n\pi ,wheren\in I\\ \therefore \theta =\frac{n\pi }{4}\\ \cos \theta =-\frac{1}{2}\\ \Rightarrow \cos \theta =\cos \left(\frac{2\pi }{3}\right)\\ \Rightarrow \cos 3\theta =2k\pi +\frac{2\pi }{3},k\in I\\ \theta =2k\frac{\pi }{3}\pm \frac{2\pi }{9}\end{array}$