Question

If in a $△ABC,$ 4 $\tan A+3=0$ then $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ are the roots of

• A. $10x^2-4\sqrt{10}x+3=0$
• B. $10x^2+4\sqrt{10}x+3=0$
• C. $10x^2-4\sqrt{10}x-3=0$
• D. $10x^2+4\sqrt{10}x-3=0$

Answer 1

The correct option is A $10x^2-4\sqrt{10}x+3=0$

$\tan A=\frac{-3}{4}$.
Hence
$\sin A=\frac{-3}{5}$ and $\cos A=\frac{4}{5}$.
Then
$\cos \frac{A}{2}=\sqrt{\frac{1+\cos A}{2}}$

$=\sqrt{\frac{9}{10}}$

$=\frac{3}{\sqrt{10}}$ ...(i)
And
$\sin \left(\frac{A}{2}\right)=\sqrt{\frac{1-\cos A}{2}}$

$=\frac{1}{\sqrt{10}}$.
Hence
Sum of roots $=\frac{4}{\sqrt{10}}$
And
Product of roots $=\frac{3}{10}$.
Hence the correct answer is option A.