If $sin \frac{π}{2 n} + cos \frac{π}{2 n} = \frac{\sqrt{n}}{2}$ , where $n \in Z$ , then the number of values of n between $4$ and $8$ for which the equation is true is?

1

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2

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3

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More than

3

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Solution

The correct option is B $3$
$sin \frac{π}{2 n} + cos \frac{π}{2 n} = \frac{\sqrt{n}}{2} n \in Z$
Squaring both sides
$\begin{matrix} {sin}^{2} \frac{π}{2 n} + {cos}^{2} \frac{π}{2 n} + 2 sin \frac{π}{2 n} cos \frac{π}{2 n} = \frac{n}{4} \Rightarrow 1 + sin \frac{π}{n} = \frac{n}{4} \Rightarrow sin \frac{π}{n} = \frac{n}{4} - 1 sin \frac{π}{n} \in [- 1, 1] \end{matrix}$
$∴ - 1 ⩽ \frac{n}{4} - 1 ⩽ 1$
$\frac{n}{4} - 1 ⩾ - 1 \Rightarrow \frac{n}{4} ⩾ 0$
$\Rightarrow n ⩾ 0$
$\frac{n}{4} - 1 ⩽ 1 \Rightarrow \frac{n}{4} - 2 ⩽ 0$
$\Rightarrow \frac{n}{4} ⩽ 2 \Rightarrow n ⩽ 8$
$∴$ between $4 & 8$ excluding 4 and 8
we have three integers.
so Answer : option $(c)$ .

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