Q107 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
If $\cos A=\dfrac{3}{4}$, then $\left(32 \sin \dfrac{A}{2} \sin \dfrac{5 A}{2}\right)=$
- A16
- Correct answerB11
- C8
- D7
Solution
Given $\cos A = \dfrac{3}{4}$.
The expression is $32 \sin \dfrac{A}{2} \sin \dfrac{5A}{2}$.
Using the product-to-sum formula $2 \sin x \sin y = \cos(x - y) - \cos(x + y)$, we have:
$32 \sin \dfrac{A}{2} \sin \dfrac{5A}{2} = 16 \times 2 \sin \dfrac{A}{2} \sin \dfrac{5A}{2}$
$= 16 \left[ \cos\left(\dfrac{5A}{2} - \dfrac{A}{2}\right) - \cos\left(\dfrac{5A}{2} + \dfrac{A}{2}\right) \right]$
$= 16 [ \cos(2A) - \cos(3A) ]$.
Using $\cos 2A = 2 \cos^2 A - 1$ and $\cos 3A = 4 \cos^3 A - 3 \cos A$:
$\cos 2A = 2 \left(\dfrac{3}{4}\right)^2 - 1 = 2 \left(\dfrac{9}{16}\right) - 1 = \dfrac{9}{8} - 1 = \dfrac{1}{8}$.
$\cos 3A = 4 \left(\dfrac{3}{4}\right)^3 - 3 \left(\dfrac{3}{4}\right) = 4 \left(\dfrac{27}{64}\right) - \dfrac{9}{4} = \dfrac{27}{16} - \dfrac{36}{16} = -\dfrac{9}{16}$.
Substituting these values into the expression:
$16 \left[ \dfrac{1}{8} - \left(-\dfrac{9}{16}\right) \right] = 16 \left[ \dfrac{2}{16} + \dfrac{9}{16} \right] = 16 \left( \dfrac{11}{16} \right) = 11$.
Answer: 11
The expression is $32 \sin \dfrac{A}{2} \sin \dfrac{5A}{2}$.
Using the product-to-sum formula $2 \sin x \sin y = \cos(x - y) - \cos(x + y)$, we have:
$32 \sin \dfrac{A}{2} \sin \dfrac{5A}{2} = 16 \times 2 \sin \dfrac{A}{2} \sin \dfrac{5A}{2}$
$= 16 \left[ \cos\left(\dfrac{5A}{2} - \dfrac{A}{2}\right) - \cos\left(\dfrac{5A}{2} + \dfrac{A}{2}\right) \right]$
$= 16 [ \cos(2A) - \cos(3A) ]$.
Using $\cos 2A = 2 \cos^2 A - 1$ and $\cos 3A = 4 \cos^3 A - 3 \cos A$:
$\cos 2A = 2 \left(\dfrac{3}{4}\right)^2 - 1 = 2 \left(\dfrac{9}{16}\right) - 1 = \dfrac{9}{8} - 1 = \dfrac{1}{8}$.
$\cos 3A = 4 \left(\dfrac{3}{4}\right)^3 - 3 \left(\dfrac{3}{4}\right) = 4 \left(\dfrac{27}{64}\right) - \dfrac{9}{4} = \dfrac{27}{16} - \dfrac{36}{16} = -\dfrac{9}{16}$.
Substituting these values into the expression:
$16 \left[ \dfrac{1}{8} - \left(-\dfrac{9}{16}\right) \right] = 16 \left[ \dfrac{2}{16} + \dfrac{9}{16} \right] = 16 \left( \dfrac{11}{16} \right) = 11$.
Answer: 11
Hence, the correct answer is option: 'B'.