Question

If $\cos A=\frac{3}{4}$ then $32\sin \frac{A}{2}$.$\sin \frac{5A}{2}=$_____

• A. $10$
• B. $-10$
• C. $11$
• D. $13$

Answer 1

The correct option is C $11$

$32\sin A/2.\sin 5A/2$

Use formula

$2\sin A.\sin B=\cos (A-B)-\cos (A+B)$

$\cos 2A=2{\cos }^{2}A-1$

$\cos 3A=4{\cos }^{3}A-3\cos A$

Now

$32\sin A/2.\sin 5A/2=16\cos 2A-16\cos 3A$

$16(2{\cos }^{2}A-1-4{\cos }^{3}A-3\cos A)=16(2{\left(3/4\right)}^2-1-4{\left(3/4\right)}^3+3(3/4\left)\right)$

$=16(18/16-1-27/16+36/16)=(18-16-27+36)=11$