Question

Find the Laplace transform of cos⁡ωt u(t).

(a) \(\frac{s}{s^2+ω^2}\)

(b) \(\frac{s}{s^2-ω^2}\)

(c) \(\frac{ω}{s^2+ω^2}\)

(d) \(\frac{ω}{s^2-ω^2}\)

I have been asked this question in an interview.

Origin of the question is The Laplace Transform in section Laplace Transform and System Design of Signals and Systems

Answer 1

The correct answer is (a) \(\frac{s}{s^2+ω^2}\)

For explanation I would say: Laplace transform, L{x(t)} = X(s) = \(\int_{-∞}^∞ x(t) e^{-st} \,dt\)

X(s) = L{cos⁡ωt u(t)} = \(L[\frac{e^{jωt} + e^{-jωt}}{2} \,u(t)] = \frac{1}{2} L[e^{jωt} \,u(t)] + \frac{1}{2} L[e^{jωt} \,u(t)]\)

X(s) = \(\frac{1}{2} (\frac{1}{s-jω}) + \frac{1}{2} (\frac{1}{s+jω}) = \frac{s}{s^2+ω^2}\).