Question

If ${180}^o<\theta <{270}^o$ and $\sin \theta =\frac{-4}{5}$, calculate $\sin \frac{\theta }{2}$ and $\cos \frac{\theta }{2}$

Answer 1

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

Hence $\cos \theta =-\frac{\sqrt{5^2-4^2}}{5}=\frac{-3}{5}$

As we know $\cos \theta =2{\cos }^2\left(\theta /2\right)$

$⟹\cos \left(\theta /2\right)=\pm \sqrt{\frac{1-3/5}{2}}=\frac{1}{\sqrt{5}}$

As $\theta /2$ lies in second quadrant so, $\cos \left(\theta /2\right)=\frac{1}{\sqrt{5}}$

Also $2\sin \left(\theta /2\right)\cos \left(\theta /2\right)=\sin \theta$

$⟹\frac{-4}{5}=2\sin \left(\theta /2\right).(-\frac{1}{\sqrt{5}}$

$⟹\sin \left(\theta /2\right)=\frac{2}{\sqrt{5}}$