Laplace Transform of Sine/Proof 3

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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

\(\ds \laptrans {\sin a t}\) \(=\) \(\ds \laptrans {\frac {e^{i a t} - e^{-i a t} } {2 i} }\) Euler's Sine Identity
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\laptrans {e^{i a t} } - \laptrans {e^{-i a t} } }\) Linear Combination of Laplace Transforms
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\frac 1 {s - i a} - \frac 1 {s + i a} }\) Laplace Transform of Exponential
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\frac {s + i a - s + i a} {s^2 + a^2} }\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\frac {2 i a} {s^2 + a^2} }\) simplifying
\(\ds \) \(=\) \(\ds \frac a {s^2 + a^2}\) simplifying

$\blacksquare$