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== Theorem ==
{{:Laplace Transform of Sine}}
== Proof ==
<onlyinclude>
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\sin {a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} \sin {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \sin {a t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \intlimits {\frac {e^{-s t} \paren {-s \sin a t - a \cos a t} } {\paren {-s}^2 + a^2} } 0 L
| c = [[Primitive of Exponential of a x by Sine of b x|Primitive of $e^{a x} \sin b x$]]
}}
{{eqn | r = \lim_{L \mathop \to \infty} \paren {\dfrac {e^{-s L} \paren {-s \sin a L - a \cos a L} } {s^2 + a^2} - \dfrac {e^{-s \times 0} \paren {-s \, \map \sin {0 \times a} - a \, \map \cos {0 \times a} } } {s^2 + a^2} }
| c =
}}
{{eqn | r = \lim_{L \mathop \to \infty} \paren {\dfrac {s \sin 0 + a \cos 0} {s^2 + a^2} - \dfrac {e^{-s L} \paren {s \sin a L + a \cos a L} } {s^2 + a^2} }
| c = [[Exponential of Zero]] and rearranging
}}
{{eqn | r = \dfrac {s \sin 0 + a \cos 0} {s^2 + a^2} - 0
| c = [[Exponential Tends to Zero and Infinity|Exponential Tends to Zero]]
}}
{{eqn | r = \frac a {s^2 + a^2}
| c = [[Sine of Zero is Zero]], [[Cosine of Zero is One]]
}}
{{end-eqn}}
{{qed}}
</onlyinclude>
== Sources ==
* {{BookReference|Theory and Problems of Laplace Transforms|1965|Murray R. Spiegel|prev = Laplace Transform of Exponential/Real Argument/Proof 1|next = Laplace Transform of Cosine/Proof 1}}: Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $2 \ \text{(a)}$
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