Question

$\cos A=\frac{3}{4}\Rightarrow 32\sin \left(\frac{A}{2}\right)\sin \left(\frac{5A}{2}\right)=$

• A. 7
• B. 8
• C. 13
• D. 11

Answer 1

The correct option is A 7

Given, $\cos A=\frac{3}{4}$
To Solve for ,
$32\sin \left(\frac{A}{2}\right)\sin \left(\frac{5A}{2}\right)$
$16\times 2\sin \left(\frac{A}{2}\right)\sin \left(\frac{5A}{2}\right)$
$16[\cos (\frac{A}{2}-\frac{5A}{2})-\cos (\frac{A}{2}+\frac{5A}{2})]$
$16[\cos (-2A)-\cos \left(3A\right)]$
$16[1-2{\cos }^{2}A-4{\cos }^{3}A+3\cos A]$
Now, as we have
$\cos A=\frac{3}{4}$
So, $16[1-2{\left(\frac{3}{4}\right)}^2-4{\left(\frac{3}{4}\right)}^3+3\left(\frac{3}{4}\right)]$
$=16\times \left[\frac{7}{16}\right]$
$=7$