(3) Alan Cooper’s answer to How do I find the period of $e^{ix}$ without using trigonometry? – Quora

Standard

This question has been around for a while and has some decent answers. But I want to suggest a simpler and more intuitive version. (And it will be easier to follow if I replace the variable $x$ by $t$ so as not to confuse it with the real part of the complex function value.)

First, to define $f(z)=e^z$ without using trigonometry or ever mentioning trig functions, we can use either the power series or the complex differential equation $f’=f$ with $f(0)=1$. And either way we get $\frac{d}{dt}e^{it}=ie^{it}$.

Now multiplication by $i$ just rotates the complex plane by a right angle, so the curve in the plane given parametrically by $(x(t),y(t))$ with $x(t)+iy(t)=e^{it}$ has a tangential velocity vector which is always perpendicular to its position vector and equal in magnitude.

Since it starts at $t=0$ at $(x,y)=(1,0)$ the curve is just the unit circle centred at the origin.

And since its velocity vector is always of length 1, if we think of the parameter $t$ as representing time, then the point moves with speed 1 and so the time taken to complete a circuit, ie the period of $e^{it}$, is just the same as the circumference of the unit circle (commonly denoted by $2\pi$ ).