Question

$\mathrm{If}\sin \theta =0.8\mathrm{and}cot\theta <0,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\sin 2\theta \mathrm{is}$.

• A. 0.64
• B. 0.96
• C. -0.96
• D. -0.64

Answer 1

The correct option is C -0.96

Given $\sin \theta =0.8$ and $cot\theta <0$.
This means $\theta$ lies in the second quadrant.
Now we know the identity $\sin 2\theta =2\sin \theta \cos \theta$.
Then we will calculate $\cos \theta$ from the identity ${\sin }^2\theta +{\cos }^2\theta =1\Rightarrow \cos \theta =\pm \sqrt{1-{\sin }^2\theta }$
Since $\theta$ lies in the second quadrant thus $\cos \theta <0$.

$\Rightarrow \cos \theta =-\sqrt{1-{\sin }^2\theta }\Rightarrow \cos \theta =-\sqrt{1-0.8^2}\Rightarrow \cos \theta =-0.6$
Thus, $\sin 2\theta =2\sin \theta \cos \theta \Rightarrow \sin 2\theta =-2\times 0.8\times 0.6=-0.96$
Hence, $\sin 2\theta =-0.96.$