Constant Function is of Exponential Order Zero

Theorem
Let $f_C :\R \to \mathbb{F}: t \mapsto C$ be a constant function, where $\mathbb F \in \left \{ {\R, \C }\right\}$.

Then $\cos t$ is of exponential order $0$.

Proof
If $C = 0$, the theorem holds trivially.

Let $C \ne 0$.