Laplace Transform of Sine/Proof 2

Theorem

Let $\sin$ denote the real sine function.

Let $\laptrans f$ denote the Laplace transform of a real function $f$.

Then:

$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$

where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$.

Proof

\(\ds \laptrans {e^{i a t} }\)

\(=\)

\(\ds \frac 1 {s - i a}\)

Laplace Transform of Exponential

\(\ds \)

\(=\)

\(\ds \frac {s + i a} {s^2 + a^2}\)

multiplying top and bottom by $s + i a$

Also:

\(\ds \laptrans {e^{i a t} }\)

\(=\)

\(\ds \laptrans {\cos a t + i \sin a t}\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds \laptrans {\cos a t} + i \laptrans {\sin a t}\)

Linear Combination of Laplace Transforms

So:

\(\ds \laptrans {\sin a t}\)

\(=\)

\(\ds \map \Im {\laptrans {e^{i a t} } }\)

\(\ds \)

\(=\)

\(\ds \map \Im {\frac {s + i a} {s^2 + a^2} }\)

\(\ds \)

\(=\)

\(\ds \frac a {s^2 + a^2}\)

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $2$