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 |
$\blacksquare$