First slide

Trigonometric equations

Question

$L e t n b e a p o s i t i v e i n t e g e r s u c h t h a t \sin \frac{π}{2 n} + \cos \frac{π}{2 n} = \frac{\sqrt{n}}{2}, t h e n n =$

Moderate

A

n=6
B

n=123..8
C

n=5
D

n=4

Solution

$W e h a v e \sin \frac{π}{2 n} + \cos \frac{π}{2 n} = \frac{\sqrt{n}}{2} S i n c e - \sqrt{2} \leq \sin \frac{π}{2 n} + \cos \frac{π}{2 n} \leq \sqrt{2}$
$\Rightarrow - \sqrt{2} \leq \frac{\sqrt{n}}{2} \leq \sqrt{2} \Rightarrow - 2 \sqrt{2} \leq \sqrt{n} \leq 2 \sqrt{2} \Rightarrow 0 \leq n \leq 8 W e c a n s e e t h a t n = 6 s a t i s f i e s t h e g i v e n e q u a t i o n$

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