Question

If \( \cos A = \frac{3}{4} \), then \( 32 \sin \frac{A}{2} \sin \frac{5A}{2} = \text{?} \)

• A. 7
• B. 16
• C. 11
• D. 8

Hint:

Use the half-angle and multiple-angle identities to simplify trigonometric expressions.

Answer 1

The Correct Option is C

Step 1: Using the half-angle and multiple-angle identities.
We are given \( \cos A = \frac{3}{4} \). To find \( \sin \frac{A}{2} \), we use the half-angle identity: \[ \sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}. \] Substituting \( \cos A = \frac{3}{4} \), we get: \[ \sin \frac{A}{2} = \sqrt{\frac{1 - \frac{3}{4}}{2}} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}}. \]

Step 2: Using the identity for \( \sin \frac{5A}{2} \).
Now, we use the formula for \( \sin \frac{5A}{2} \) and the given angle, and apply the appropriate simplifications. After calculating, we find that the value of the expression is 11.

Step 3: Conclusion.
Thus, the correct answer is 11, corresponding to option (3).