Angle units in complex planes $$ z = re^{i\theta} $$
I have seen that, when specifying a complex number, most people would rather use radians as the units for $\theta.$ Is it incorrect to use degrees? Why is there a preference for radians?
 A: You can use degrees… However, if you do so, then for a flat angle of $180°$ you don’t have the famous relation $$e^{i\pi}=-1$$ which holds if you use radians.
A: The formula $z=re^{i\theta}$ is based on the observation that
$$\begin{align}
e^{i\theta}
= \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!} 
&= \sum_{n=0\\n\text{ even}}^{\infty} \frac{(i\theta)^n}{n!}
+ \sum_{n=0\\n\text{ odd}}^{\infty} \frac{(i\theta)^n}{n!}
\\
&= \sum_{k=0}^{\infty} \frac{(-1)^k\theta^{2k}}{(2k)!}
+ i \sum_{k=0}^{\infty} \frac{(-1)^k\theta^{2k+1}}{(2k+1)!}
= \cos\theta + i\sin\theta,
\end{align}$$
which is only valid if $\theta$ is in radians.
From this we get
$$
z=x+iy=r\cos\theta+ir\sin\theta=r(\cos\theta+i\sin\theta)=re^{i\theta}.
$$
Radians is the natural unit of angle in mathematics. You should learn and use them almost everywhere in mathematics.
A: *

*The angle $73^\circ=1.274\,$rad is a dimensionless quantity: there's no dimension (in the sense of dimensional analysis) involved in measuring $\frac{73}{360}$ of a full turn/circle (a turn, like a cycle, is dimensionless), while the length dimensions get cancelled out when dividing arc length by radius. It's a misconception that unit and dimension are synonyms.


*The input of the natural (radian) trigonometric function $\sinθ$ is not actually an angle, but a number corresponding to some angle. E.g., $\theta$ might be $2.51,$ which corresponds to—but doesn't equal—$2.51\,\mathrm{rad}=144^\circ.$


*$$z=r(\cos\theta+i\sin\theta)$$ is the polar form of a complex number. By Euler's formula, $$z=re^{iθ},\tag1$$ while by the Taylor series for sine and cosine, $$z=r\left(\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\ldots\right)+i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\ldots\right)\right)$$$$=r\left(1+(i\theta)+\frac{(i\theta)^2}{2!}-\frac{(i\theta)^3}{3!}+\frac{(i\theta)^4}{4!}+\frac{(i\theta)^5}{5!}+\ldots\right).\tag2$$
Complex Analysis (specifically, applications involving $\exp()$ and $e^{()}$) has been developed based on the natural circular measure radian and the natural (radian) trigonometric functions. For example, the various derivations of formulae $(1)$ and $(2)$ all involve the natural trigonometric functions.
Electing to work in ‘degrees’ is fine if we avoid $\exp()$ and $e^{()}$ and stick to purely geometric analyses/perspectives. In this case, $e^{iθ}$ is not a useful shorthand for $(\cos\theta+i\sin\theta)$ due to potential for confusion; stick to the $\mathrm{cis}\,\theta$ notation instead.
