Question

If ${180}^0<\theta <{270}^0$ and $\sin \theta =\frac{-4}{5}$. If $\sin \left(\frac{\theta }{2}\right)=\sqrt{\frac{1}{m}}$.Find $m$

Answer 1

${180}^{\circ }<\theta <{270}^{\circ }$

$\Rightarrow {90}^{\circ }<\frac{\theta }{2}<{135}^{\circ }$

$\sin \theta =\frac{-4}{5}$

$\cos \theta =\sqrt{1-{\sin }^2\theta }$

$=\frac{-3}{5}$ ( As $\cos \theta =0$)

$\Rightarrow 1-2{\sin }^2\left(\frac{\theta }{2}\right)=\frac{-3}{5}$

$\Rightarrow 2{\sin }^2\left(\frac{\theta }{2}\right)=\frac{8}{5}$

$\Rightarrow \sin \left(\frac{\theta }{2}\right)=\sqrt{\frac{4}{5}},\cos \left(\frac{\theta }{2}\right)=-\sqrt{\frac{1}{5}}$