Question

Statement I The amplitude of $\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$ is $\frac{\pi}{10}$.
Statement II If complex number lies on the first quadrant, then $\arg (z)=\theta$.

• A. Statement $I$ is correct
• B. Statement II is correct
• C. Both are correct
• D. Neither I nor II is correct

Answer 1

Answer: (C)

Here, $r \cos \theta=\sin \left(\frac{\pi}{5}\right)$ and $r \sin \theta=1-\cos \frac{\pi}{5}$
$\Rightarrow r \cos \theta=2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}$ and $r \sin \theta=2 \sin ^2 \frac{\pi}{10}$
$\Rightarrow \tan \theta=\tan \left(\frac{\pi}{10}\right) \Rightarrow \theta=\frac{\pi}{10}$
Statements I and II are true.