Why do we have different units for same physical quantity?

A function $f\left(\theta \right)$ is defined as:

$f\left(\theta \right)=1-\theta +\frac{{\theta }^2}{2!}-\frac{{\theta }^2}{3!}+\frac{{\theta }^4}{4!}-\dots \dots$

Why is it necessary for $f\left(\theta \right)$ to be a dimensionless quantity?