As an example, the function $\cos(\theta)$ represents the ratio of, relative to the angle $\theta$, the adjacent-side and hypotenuse of a right triangle. Strictly speaking, $\theta$ is measured in any units we want - it's the interpretation/definition of the $\cos(\theta)$ function relative to our chosen "units" for $\theta$ that matters. That's why it's just a matter of economics when choosing between radians, degrees, or anything else.
Of course, when finding arcs on a circle, it's convenient to define the angle $\theta$ in terms of the ratio between the radius of a circle and its circumference, $1/2\pi$ (e.g. when you've covered an angle $\pi$ in these units along the circumference of a circle, you've traveled $\pi/(2\pi)=1/2$ of the circumference). That way, calculating arc-lengths becomes simple multiplication by the angle $\theta$ (this obviously isn't true for degrees!).
However, I don't know how to interepret the power (Taylor/Maclaurin) series for trig functions like $\cos(\theta)$ and $\sin(\theta)$.
Why must we use radians in the above series representation? Why don't we use for the "units" of $\theta$, for example, the fraction of the entire circle that it covers (e.g. $1/4$ instead of $\pi/2$, $1$ instead of $2\pi$, etc.)? That would seem more natural and "unitless".