mathbb L {cos omega t color(red)(-) i sin omega t } = mathbb L {e^(-i omega t) }
= int_0^(oo) e^(-(i omega +s)t) \ dt
= - (1)/(i omega + s) [ e^(-(i omega +s)t)]_0^(oo)
= (1)/(i omega + s) = (s - i omega)/(s^2 + omega ^2)
implies mathbb L {cos omega t } = \mathcal (Re) (mathbb L {cos omega t color(red)(-) i sin omega t } )= (s )/(s^2 + omega ^2)
AND
implies mathbb L {sin omega t } = -\mathcal (Im) (mathbb L {cos omega t color(red)(-) i sin omega t } )= (omega )/(s^2 + omega ^2)
Repeat for: omega = 4