First slide

Trigonometric equations

Question

If $n â N$ and $\sin â¡ (\frac{Ï}{2 n}) + \cos â¡ (\frac{Ï}{2 n}) = \frac{\sqrt{n}}{2}$ then a possible value of $n$ is

Moderate

A

4
B

6
C

8
D

12

Solution

$\frac{n}{4} = \sin^{2} â¡ (\frac{Ï}{2 n}) + \cos^{2} â¡ (\frac{Ï}{2 n}) + \sin â¡ (\frac{2 Ï}{2 n})$
$â \sin â¡ (\frac{Ï}{n}) = \frac{n â 4}{4}$
For $n â N, \sin â¡ (\frac{Ï}{n}) > 0$
$â´ \frac{n â 4}{4} > 0$ or $n > 4$
For $n > 4, \frac{Ï}{n} < \frac{Ï}{2} â \sin â¡ (\frac{Ï}{n}) < 1$
$â 0 < \frac{1}{4} (n â 4) < 1 â 4 < n < 8$
Thus, a possible value of $n = 6$
In fact, for $n = 6$
$\sin â¡ (\frac{Ï}{12}) + \cos â¡ (\frac{Ï}{12})$
$= \sqrt{2} \sin â¡ (\frac{Ï}{4} + \frac{Ï}{12}) = \sqrt{2} \sin â¡ (\frac{Ï}{3})$
$= \sqrt{2} (\frac{\sqrt{3}}{2}) = \frac{\sqrt{6}}{4} = \frac{\sqrt{n}}{4}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App