Question

If $\alpha$ is a root of $25{\cos }^2\theta +5\cos \theta -12=0,\frac{\pi }{2}<\alpha <\pi .$ Then the value of $\sin 2\alpha$ is equal to

• A. $\frac{24}{25}$
• B. $-\frac{24}{25}$
• C. $\frac{13}{18}$
• D. $-\frac{13}{18}$

Answer 1

The correct option is B $-\frac{24}{25}$

$25{\cos }^2\theta +5\cos \theta -12=0\Rightarrow 5\cos \theta (5\cos \theta +4)-3(5\cos \theta +4)=0\Rightarrow \cos \theta =\frac{3}{5},\cos \theta =-\frac{4}{5}$

As $\frac{\pi }{2}<\alpha <\pi ,\cos \alpha =-\frac{4}{5},\sin \alpha =\frac{3}{5}$
$\sin 2\alpha =2\sin \alpha \cos \alpha =-\frac{24}{25}$