Cosine is of Exponential Order Zero

Theorem
Let $\cos t$ be the cosine of $t$, where $t \in \R$.

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

Proof 1
The result follows from the definition of exponential order with $M = 1$, $K = 2$, and $a = 0$.

Proof 2
The result follows from Boundedness of Sine and Cosine and Bounded Function is of Exponential Order Zero.