Question

Find all solution of the equation.
$\sin x+\sin \frac{\pi }{8}\sqrt{(1-\cos x{)}^2+{\sin }^{2}x}=0$ in $[\frac{5\pi }{2},\frac{7\pi }{2}]$.

Answer 1

$\begin{array}{c}\sin x+\sin \frac{\pi }{8}\sqrt{{(1-\cos x)}^2+{\sin }^{2}x}=0\\ \Rightarrow \sin x+\sin \frac{\pi }{8}\sqrt{1-2\cos x+{\cos }^{2}x+{\sin }^{2}x}=0\\ \Rightarrow \sin x+\sin \frac{\pi }{8}\sqrt{2\left(1-\cos x\right)}=0\\ \Rightarrow \sin x+2\sin \frac{\pi }{8}\sqrt{{\sin }^2\frac{x}{2}}=0\\ \Rightarrow 2\sin \frac{x}{2}\cdot \cos \frac{x}{2}+2\sin \frac{x}{2}\cdot \sin \frac{\pi }{8}=0\\ \Rightarrow 2\sin \frac{x}{2}\left[\cos \frac{x}{2}+\sin \frac{\pi }{8}\right]=0\\ \therefore \sin \frac{x}{2}=0\\ \frac{x}{2}=n\pi \\ x=2n\pi ,n\in I\\ putn=1,2,3\\ x=2\pi ,3\pi \in \left[\frac{5\pi }{2},\frac{7\pi }{2}\right]\end{array}$