Question

Value of $\sin \left(\frac{1}{2}{\cot }^{-1}(-\frac{3}{4})\right)$ is $\sin \theta /2$, then find the value of $\sin \theta /2$

• A. $1/\sqrt{5}$
• B. $2/\sqrt{5}$
• C. $1/\sqrt{3}$
• D. $2/\sqrt{3}$

Answer 1

The correct option is B $2/\sqrt{5}$

Here, ${\cot }^{-1}(-3/4)=\theta$ or $\cot \theta =-3/4$ $\Rightarrow$ $\theta ϵ(\pi /2,\pi )$

$\Rightarrow$ $\cos \theta =-3/5(\theta ϵ\pi /2,\pi )$ $\Rightarrow$ $1-2{\sin }^2\left(\theta /2\right)=-3/5$

$\therefore$ ${\sin }^2\left(\theta /2\right)$ or $\theta /2={\sin }^{-1}\left(2/\sqrt{5}\right)$