Complex Numbers 3 Question 3
3. If $\arg (z)<0$, then $\arg (-z)-\arg (z)$ equals
$(2000,2 M)$
(a) $\pi$
(b) $-\pi$
(c) $-\pi / 2$
(d) $\pi / 2$
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Answer:
Correct Answer: 3. (a)
Solution:
- Since, $\arg (z)<0 \Rightarrow \quad \arg (z)=-\theta$
$$ \begin{aligned} & \Rightarrow \quad z=r \cos (-\theta)+i \sin (-\theta) \\ & =r(\cos \theta-i \sin \theta) \\ & \text { and } \quad-z=-r[\cos \theta-i \sin \theta] \\ & =r[\cos (\pi-\theta)+i \sin (\pi-\theta)] \\ & \therefore \quad \arg (-z)=\pi-\theta \\ & =\pi-\theta-(-\theta)=\pi \end{aligned} $$
Alternate Solution
Reason $\arg (-z)-\arg z=\arg \frac{-z}{z}=\arg (-1)=\pi$ and also $\arg z-\arg (-z)=\arg \frac{z}{-z}=\arg (-1)=\pi$
