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 $\map \Re s > a$.

Proof

 $\displaystyle \laptrans {e^{i a t} }$ $=$ $\displaystyle \frac 1 {s - i a}$ $\quad$ Laplace Transform of Exponential $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {s + i a} {s^2 + a^2}$ $\quad$ multiplying top and bottom by $s + i a$ $\quad$

Also:

 $\displaystyle \laptrans {e^{i a t} }$ $=$ $\displaystyle \laptrans {\cos a t + i \sin a t}$ $\quad$ Euler's Formula $\quad$ $\displaystyle$ $=$ $\displaystyle \laptrans {\cos a t} + i \laptrans {\sin a t}$ $\quad$ Linear Combination of Laplace Transforms $\quad$

So:

 $\displaystyle \laptrans {\sin a t}$ $=$ $\displaystyle \map \Im {\laptrans {e^{i a t} } }$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \map \Im {\frac {s + i a} {s^2 + a^2} }$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac a {s^2 + a^2}$ $\quad$ $\quad$

$\blacksquare$