First slide

Multiple and sub- multiple Angles

Question

If $\cos A = \frac{3}{4}$ , then the value of $\sin \frac{A}{2} \sin \frac{5 A}{2}$ , is

Moderate

A

$\frac{1}{32}$
B

$\frac{11}{8}$
C

$\frac{11}{32}$
D

$\frac{11}{16}$

Solution

We have,
$\begin{array}{l} \sin \frac{A}{2} \sin \frac{5 A}{2} \\ = \frac{1}{2} (2 \sin \frac{A}{2} \sin \frac{5 A}{2}) \\ = \frac{1}{2} (\cos 2 A - \cos 3 A) \\ = \frac{1}{2} \{2 \cos^{2} A - 1 - 4 \cos^{3} A + 3 \cos A\} \\ = \frac{1}{2} \{2 \times \frac{9}{16} - 1 - 4 \times \frac{27}{64} + 3 \times \frac{3}{4}\} = \frac{11}{32} \end{array}$

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