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  • Let n be a fixed positive integer such that <img alt="\sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2}," src="https://entrancecorner.oncodecogs.com/gif.latex?%5Csin%20%5C

Let n be a fixed positive integer such that \sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\frac{\sqrt{n}}{2},  then

Option: 1

n=4


Option: 2

n=5


Option: 3

n=6


Option: 4

None of these 


Answers (1)

best_answer

\begin{aligned} \\& \sin \frac{\pi}{2 n}+\cos \frac{\pi}{2 n}=\sqrt{2} \sin \left(\frac{\pi}{4}+\frac{\pi}{2 n}\right) \\ \\\Rightarrow& \frac{\sqrt{n}}{2}=\sqrt{2} \sin \left(\frac{\pi}{4}+\frac{\pi}{2 n}\right) \\ & \end{aligned}

\\\text{So for} n>1, \frac{\sqrt{n}}{2 \sqrt{2}}=\sin \left(\frac{\pi}{4}+\frac{\pi}{2 n}\right)>\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}} or n>4 \\\\\text{Since}, \sin \left(\frac{\pi}{4}+\frac{\pi}{2 n}\right)<1 \text{ for all } n>2,we get

So that 4<n<8. By actual verification we find that only n=6 satisfies the given relation.

Posted by

Divya Prakash Singh

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