Question

5. In this problem we will derive some trigonometric identities important for deriving Fourier series. Let wo = 7, where T is a given constant. Assume k, n, m > 0 are integers. 27 (a) Show that T/2 T when k = 0 ejkwot dt =. J-T/2 O when k >1 (b) Use the above result, ej(n-m)wot = ejnwote-jmwot, ej(n+m)wot = ejnwotejmwot, and Euler's identi- ties to deduce (T/2 when n = m, n = 0, m = 0 i. 1112, cos(nwot) cos(mwot)dt = {0 when n #m T when n= m = 0 (T/2 when n=m,n # 0,m # 0 ii. S-41 sin(nwot) sin(mwot)dt = {0 when n #m To when n = m = 0) iii. S172, sin(nwot) cos(mwot)dt = 0 for all m and n. (c) Note that a coswt + b sin wt = real part of ((a – jb)ejwt). Use polar form a – jb = ce-jo, where c=va+ and 0 = tan-- () to show that a cos wt + b sin wt = c cos(wt – 0).

Answer 1

12 eskwot 1 dt = T2+ I ET , when kao 112 Ak wat gokwot IKT - OKT kwo. kw. when kai De5+B) te ba-B), ja jb te ja E-3B -(uz AtiSA)(Los BigsinB) + (Los Atisin A) Los B-gsinB) = (LA LOBB-sin Asin B) + g (sing ARBA CASA sinB) + (A cosB + sinA sinB) tj(sin A cosB LOB A sin B) = 2/63A wsB + j sin a cosB) & p; GAB) _ e 8 (4.B) =-2 sin A ello + 2 j us Asinß (nWot) was from wat dit 112 = Refe in+m) wot jn-m) wot . (nem) wit shm) wot when nam , n7o,mto so + 1/2 = -2 oto=0 when nm (nam, onde when when n=mzo + PT wie (nw.t) sinsenrot) det seleslo wot-in wot) en tm) wot) at

12 (nm) wot at htm) wot ( 12 T -0 = 7/2 , when nam to lo-o-o a when nam (1 T r / 2 Tao, when namo اوملا ۳ بارے - ادما sin (n wot) as (m wot) dt = Im dt =0 a coblut)+bsin (wt) = Rella-; b) e swt = he iceio e swt] و عة = (5 - 4 دم) نے = Re [ e oscut-o) + ģc sin(wt-o - clos (wt-o)