Question

If $f_n\left(\theta \right)=\frac{\cos \frac{\theta }{2}+\cos 2\theta +\cos \frac{7\theta }{2}+\cdots +\cos \frac{(3n-2)\theta }{2}}{\sin \frac{\theta }{2}+\sin 2\theta +\sin \frac{7\theta }{2}+\cdots +\sin \frac{(3n-2)\theta }{2}},$ then which among the following is (are) CORRECT?

• A. $f_1\left(\frac{\pi }{2}\right)=1$
• B. $f_3\left(\frac{3\pi }{16}\right)=\sqrt{2}+1$
• C. $f_3\left(\frac{3\pi }{16}\right)=\sqrt{2}-1$
• D. $f_5\left(\frac{\pi }{28}\right)=\sqrt{2}+1$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $f_1\left(\frac{\pi }{2}\right)=1$
C $f_3\left(\frac{3\pi }{16}\right)=\sqrt{2}-1$
D $f_5\left(\frac{\pi }{28}\right)=\sqrt{2}+1$

$\cos \frac{\theta }{2}+\cos 2\theta +\cos \frac{7\theta }{2}+\cdots +\cos \frac{(3n-2)\theta }{2}=\cos \frac{\theta }{2}+\cos (\frac{\theta }{2}+\frac{3\theta }{2})+\cos (\frac{\theta }{2}+2\left(\frac{3\theta }{2}\right))+\cdots +\cos (\frac{\theta }{2}+(n-1\left)\left(\frac{3\theta }{2}\right)\right)$
$=\frac{\sin n\left(\frac{3\theta }{4}\right)}{\sin \left(\frac{3\theta }{4}\right)}\times \left[\cos \left(\frac{\theta +(n-1)\frac{3\theta }{2}}{2}\right)\right]$

Similarly, $\sin \frac{\theta }{2}+\sin 2\theta +\sin \frac{7\theta }{2}+\cdots +\sin \frac{(3n-2)}{2}\theta =\frac{\sin n\left(\frac{3\theta }{4}\right)}{\sin \left(\frac{3\theta }{4}\right)}\times \left[\sin \left(\frac{(3n-1)\theta }{4}\right)\right]$
$\therefore f_n\left(\theta \right)=\cot \frac{(3n-1)\theta }{4}$
$f_1\left(\frac{\pi }{2}\right)=\cot \frac{\pi }{4}=1$
$f_3\left(\frac{3\pi }{16}\right)=\cot 2\left(\frac{3\pi }{16}\right)=\sqrt{2}-1$
$f_5\left(\frac{\pi }{28}\right)=\cot \left(\frac{14}{4}\left(\frac{3\pi }{16}\right)\right)=\cot \frac{\pi }{8}=\sqrt{2}+1$