- #1

dRic2

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## Homework Statement

I have to find the L-transform of ##f(x) = cos(\omega t + \phi)##

## Homework Equations

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## The Attempt at a Solution

The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: $$Lf(s) = \frac {s cos(\phi) - \omega sin(\phi)} {s^2 + \omega ^2}$$.

But can I try this other way ?

##cos ( \omega t + \phi ) = cos \left[ \omega \left( t - \left( - \frac { \phi} { \omega} \right) \right) \right]## and now I can use the t-shift relation to get: $$ Lf(s) = e^{- \left( - \frac {\phi} {\omega} \right) s} L(cos(\omega t)) = e^{ \frac {\phi} {\omega} s} \frac {s} {s^2 + \omega ^2}$$

I don't know if there is a way to simplify my last result or if it is wrong...