$arg(r_+) = 0$ and $arg(r_-) = pi$

$r_+$ = positive real number

$r_-$ = negative real number

I am confused $arg(r_+) = 0$ and $arg(r_-) = pi$

for $arg(r-) = pi$ :

for example, $arg(-1)$ can be represented as $-1 + 0i $

$cos(arg(-1)) = frac{-1}{1}
= -1$; this equals to $pi$

$sin(arg(-1)) = frac{0}{1}
= 0$ this equals to $0$ or $pi$

$arctan({arg(-1)}) = 0$

so we care about real number is $(-1)$ so its cosine and is $pi$, correct?

for next one is where I have confusing of $arg(r_+) = 0$

for example $arg(1) = 0$

$cos(arg(1)) = frac{1}{1}
= 1$ ,
$sin(arg(1)) = frac{0}{1}
= 0$ ,
$arctan({arg(1)}) = 0 $, so the answer is zero. So we chose value of $arctan$?

now let’s do example of $arg(lvert2+irvert)$ and $r = sqrt{5}$ , so automatically is zero? This is where I am confused. I hope from the start to now is correctly shown so far?