Question

If $a$ and $b$ are real, then show that the principal value of arg $a$ is $0$ or $\pi$ according to $a$ is positive or negative and that of arg $b$ is $\frac{\pi }{2}$ or $-\frac{\pi }{2}$ according to $b$ is positive or negative.

Answer 1

Let $a=r\cos \alpha$ and $0=r\sin \alpha$ ..... (i)
$\Rightarrow a^2+0^2$

$=r^2({\cos }^2\alpha +{\sin }^2\alpha )$

$=r^2\left(1\right)$

$\Rightarrow a^2=r^2$
$\therefore r=|a|$
then $a=|a|\cos \alpha$ {from (i)}
$\therefore \cos \theta =\pm 1$
Then $\cos \alpha =1$ or $\cos \alpha =-1$

According as $a$ is positive or negative and $\sin \alpha =0$. Hence $\alpha =0$ or $\pi$ according as a is positive or negative
Again, let

$0=r_1\cos \beta$ and $b=r_1\sin \beta$ ....... (ii)
$\Rightarrow 0^2+b^2=r_1^2$

$=r_1^2({\cos }^2\beta +{\sin }^2\beta )$

$=r_1^2\left(1\right)$

$=r_1^2$

$\Rightarrow b^2=r_1^2$
$\therefore r_1=|b|$
from (ii) $b=|b|\sin \beta$
$\therefore \sin \beta =\pm 1$
Then $\sin \beta =1$ or $\sin \beta =-1$

According as $b$ is positive or negative and $\cos \beta =0$.

Hence, $\frac{\pi }{2}$ or $-\frac{\pi }{2}$ according as $b$ is positive or negative.