Sine is of Exponential Order Zero

Theorem

Let $\sin t$ be the sine of $t$, where $t \in \R$.

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

Proof 1

\(\ds \size {\sin t}\)

\(\le\)

\(\ds 1\)

Real Sine Function is Bounded

\(\ds \leadsto \ \ \)

\(\ds \size {\sin t}\)

\(<\)

\(\ds 2\)

\(\ds \)

\(=\)

\(\ds 2 e^{0 t}\)

Exponential of Zero

$\blacksquare$

Proof 2

The result follows from Real Sine 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