Cosine is of Exponential Order Zero

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Theorem

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


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


Proof 1

\(\ds \forall t \ge 1: \, \) \(\ds \size {\cos t}\) \(\le\) \(\ds 1\) Real Cosine Function is Bounded
\(\ds \leadsto \ \ \) \(\ds \size {\cos t}\) \(<\) \(\ds 2\)
\(\ds \) \(=\) \(\ds 2 e^{0 t}\) Exponential of Zero

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

$\blacksquare$


Proof 2

The result follows from Real Cosine Function is Bounded and Bounded Function is of Exponential Order Zero.

$\blacksquare$


Sources