I want to solve the following equation: $$e^{i\cdot m2\pi}=1$$ Where $i$ is the imaginary unit and $m$ is some constant that I want to determine. According to Euler's formula this equation is equal to: $$e^{i\cdot m2\pi}=\cos(2\pi m)+i\cdot \sin(2\pi m)=1$$ Furthermore, this link says that the $x$ in $e^{ix}$ can be obtained by: $$\tan^{-1}\bigg(\frac{\sin(x)}{\cos(x)}\bigg)=x$$ In my case, $x=2\pi m$ which must be $0$ to give $e^{i\cdot m2\pi}=1$. When plotting the following equation with $m$ as the variable: $$\tan^{-1}\bigg(\frac{\sin(2\pi m)}{\cos(2\pi m)}\bigg)$$ I get solutions for $0$ when $m=\pm1, 1.5, 2, 2.5,..$ However, this link says at Equation 7.3.15 that $m$ can only be integers $\pm 1, 2, 3, 4$
Why is my method not working?
From your equation $$ \cos(2 \pi m) + \mathrm{i} \sin(2\pi m) = 1 + \mathrm{i} 0 $$ we have $$ \cos(2\pi m) = 1 \quad \text{ and } \quad \sin(2\pi m) = 0 \text{.} $$ Cosine is only $1$ at integer multiples of $2\pi$, so when $m$ is an integer, and sine is zero at multiples of $\pi$, so when $m$ is a half integer. Notice that the incorrect set of solutions is one of these and the correct set is their intersection. The intersection of the set of $m$s giving the correct real part and the set of $m$s giving the correct imaginary part is the set of $m$s giving $1$ and is the integers.