Problem 1 (9pt) In this question, we will find the Laplace transform of cos wt in three different ways. (a) Follow the three steps to find the Laplace transform of cos wt from scratch. Show Cletat = L[cos wt] + iL[sin wt], (1) from efwt = cos wt + i sinwt and the linearity property. Show Let] = +i- 82 + w? (2) using Lleat] = for any complex number a. Using that fact that (1) and (2) are identical, show L[cos wt) : %3D s2 + w? (b) Let f(t) = sin wt. We have F(s): (3) s2 + w?" Show cos wt = (4) Find the Laplace transform of (4) Using L[f'(t)) = sF(s) f(0) and (3). (c) Let g(t) = cos wt. Show g"(t) = -w? cos wt. (5) Take the Laplace transform of (5) to show sG(s) s = -w*G(s). - 8 (6) Solve (6) for G(s).