Angle units in complex planes

Question

$$ z = re^{i\theta} $$

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?

Answer 1

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.

Answer 2

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.

Answer 3

1. 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.

2. 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—$2.51\,\mathrm{rad}=144^\circ.$

3. $$z=r(\cos\theta+i\sin\theta)$$ 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.