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
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Functions of Exponential Order