Q81 / 160 · TS EAMCET 2011 · Mathematics

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

  • A
    7
  • B
    8
  • C
    13
  • Correct answer
    D
    11

Solution

$\begin{aligned} & 16\left[2 \sin \frac{A}{2} \sin \left(\frac{5 A}{2}\right)\right]=16[\cos 2 A-\cos 3 A] \\ & =16\left[2 \cos ^2 A-1-\left(4 \cos ^3 A-3 \cos A\right)\right] \\ & =16\left[2\left(\frac{3}{4}\right)^2-1-\left(4 \times \frac{27}{64}-3 \times \frac{3}{4}\right)\right] \\ & =16\left[\frac{1}{8}-\left(\frac{27}{16}-\frac{9}{4}\right)\right] \\ & =16\left[\frac{2-27+36}{16}\right] \\ & =11\end{aligned}$

Hence, the correct answer is option: 'D'.

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