Question

Let us assume x (t) = A cos(ωt + φ), then the Laplace transform of x (t) is?

(a) X(S)=A(Scos Ø-ω sinØ)/(S^2-ω^2)

(b) X(S)=A(Scos Ø+ω sinØ)/(S^2+ω^2)

(c) X(S)=A(Scos Ø+ω sinØ)/(S^2-ω^2)

(d) X(S)=A(Scos Ø-ω sinØ)/(S^2+ω^2)

This question was addressed to me in an online quiz.

I would like to ask this question from Transfer Function topic in portion Application of the Laplace Transform in Circuit Analysis of Network Theory

Answer 1

The correct choice is (d) X(S)=A(Scos Ø-ω sinØ)/(S^2+ω^2)

Easiest explanation: We use the transfer function to relate the study state response to the excitation source. And we had assumed that x (t) = A cos(ωt + φ). On expanding and taking the laplace transform we get X(s) = AcosØs/(s^2+ω^2)-AsinØs/(s^2+ω^2) = A(Scos Ø-ω sinØ)/(S^2+ω^2).