Question

The amplitude $f \sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$ is

(A) $\frac{\pi}{5}$

(B) $\frac{2 \pi}{5}$

(C) $\frac{\pi}{10}$

(D) $\frac{\pi}{15}$

Answer 1

$\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$

$=2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}+i 2 \sin ^2 \frac{\pi}{10}$

$=2 \sin \frac{\pi}{10}\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$

For amplitude :

$\begin{aligned} \tan \theta & =\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{10}}=\tan \frac{\pi}{10} \\ \Rightarrow \quad \theta & =\frac{\pi}{10}\end{aligned}$

So, The correct option will be (C).