Determining arg of a complex number in Euler form

Question

I am trying to solve $arg(3e^{10i})$ and the solution should be $10-4\pi=-2,5664$. However, I have my problems understanding and getting to this solution.

I would assume that you could rewrite $$arg(3e^{10i})=arg (3(cos(10)+isin(10))$$ and since we want to find $\theta$ from this solution it would equal $arccos cos \theta)=\theta=10$. However, since we know that every rotation of radian $2\pi$ would lead to the same point, I would divide $10:2\pi=1.5915$ and $10-2\pi$ since it is only possible to go one full round (1.5915).

This ansatz and reasoning seems wrong. Can anyone help to explain it to me or show me a detailed solution path?

Answer 1

They want you to determine $arg(z)$ in the interval $(-\pi,\pi]$. Since $arg(z)=Arg(z)+2\pi k$ for $k\in\mathbb Z$ where $Arg(z)\in (-\pi,\pi]$, we have $Arg(z)=10-2\pi k$. Which value of $k$ will give you $Arg(z)$ in the correct interval?

Answer 2

The argument is simply $10.$ If you don't want arguments less than $0$ or more than $2π,$ then this is expressible as $10-2π.$ If you allow negative arguments then the least such in magnitude is $10-4π.$