# Laplace Transform of Cosine/Proof 2

## Theorem

Let $\cos$ be the real cosine function.

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

Then:

$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$

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

## Proof

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

Also:

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

So:

 $\displaystyle \laptrans {\cos a t}$ $=$ $\displaystyle \map \Re {\laptrans {e^{i a t} } }$ $\displaystyle$ $=$ $\displaystyle \map \Re {\frac {s + i a} {s^2 + a^2} }$ $\displaystyle$ $=$ $\displaystyle \frac s {s^2 + a^2}$

$\blacksquare$