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If $a = \sin \frac{\pi }{{18}}\sin \frac{{5\pi }}{{18}}\sin \frac{{7\pi }}{{18}}$ and $x$ is the solution of the equatioin $y = 2\left[ x \right] + 2$ and $y = 3\left[ {x - 2} \right] ,$ where $\left[ x \right]$ denotes the integral part of $x,$ then $a$ is equal to :-
$\left[ x \right]$
$\frac{1}{{\left[ x \right]}}$
$2\left[ x \right]$
${\left[ x \right]^2}$
Solution
$a= \sin \frac{\pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18} $
$=\sin 10^{\circ} \sin 50^{\circ} \sin 70^{\circ} $
$=\frac{1}{2}\left[2 \sin 70^{\circ} \sin 10^{\circ}\right] \sin 50^{\circ}$
$=\frac{1}{2}\left[\cos 60^{\circ}-\cos 80^{\circ}\right] \sin 50^{\circ}$
$=\frac{1}{4} \sin 50^{\circ}-\frac{1}{4}\left(2 \cos 80^{\circ} \sin 50^{\circ}\right)$
$=\frac{1}{4} \sin 50^{\circ}-\frac{1}{4}\left(\sin 130^{\circ}-\sin 30^{\circ}\right)$
$=\frac{1}{4} \sin 50^{\circ}-\frac{1}{4} \sin 50^{\circ}+\frac{1}{4} \cdot \frac{1}{2}=\frac{1}{8}$
$y=2[x]+2 \text { and } y=3[x-2]$
$\Rightarrow 2[\mathrm{x}]+2=3[\mathrm{x}-2]$
$=3[\mathrm{x}]+3[-2] $
$=3[\mathrm{x}]-6 $
$\Rightarrow [\mathrm{x}]=8 $
$\therefore \mathrm{a}=\frac{1}{[\mathrm{x}]}$
