Mathematics Asked on January 5, 2022

$r_+$ = 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?

You seem to confuse $arg(|2+i|)$ and $arg(2+i)$. The first is indeed the argument of a positive real number and is $0$.

For every nonzero $z$, $arg(|z|)=arg(|re^{itheta}|)=arg(r)=arg(re^{i0})=arg(e^{i0})=0$, while $arg(z)=arg(re^{itheta})=arg(e^{itheta})=theta$.

Answered by user65203 on January 5, 2022

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