In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, as:
$$x(t)=a\theta-asin(\theta)$$ $$y(t)=a-acos(\theta)$$
He then simplified the parametric by giving the circle that creates the cycloid a radius of $a=1$ and unit speed. The professor explained that by giving the circle unit speed, we can just replace the angle, which we had previously used as the parameter, with time. So the parametrization simplifies to: $$x(t)=t-sin(t)$$ $$y(t)=1-cos(t)$$ However, I don't understand the intuition that allowed him to immediately know that replacing $\theta$ with $t$ sets the parent circle at unit speed. Is there a way to think about the substitution that can give me some sort of intuition on why this is true?
(edited to reflect actual conceptual issue)