Question

please include the graph

1. Expand 7T if 0 <<< f(x) = 1 if <<, in a half-range: (a) Sine series. (b) Cosine series.

Answer 1

Page No. Date f(x) = if ocz </ if I ca sa this graph + TY (a hulfrary, sirs seris this F(X) = & bn sin (nat n=1 bn = 2 L I f(x) sin Sin (ntx) dx this L = And T би 2 ES f) sin f(x) sin nx dx TT 1/2 2 f(x) Sinnadet +(fsw sinnxo ㅠ AT 2 f Sinnxdy T 112 dll 2 ㅠ conx h 2 [-costint costyn] ㅠn

Page No. Date This 1 n = 2k con nt = + con/ ciskt 2 + k=zk! coskt = k = 2k't -] + Coskiy n = 2K+ ) = IC فيلم and the 2 thy f(x) = (2 (į (w cos TX 2 cos nt)) Sininx) n=1 Tin cast = n=1 Tn " Sinny nar Ž 2 ( (-1)", sin nx thy fix) 2 1 135 mp + -v **") & reg. half rany Suns half saxy corire rew fia t. 3 Cortes seres 16 at an cos ntr L 2 no ao = 2 f from da t 2 t f fel ata 2 (TT-TT) T on Go i na 8

Page No. Dato And an= 2 dll f(x) cos (ntx du 2 f(x) cosnx dy 2 f conx dy Ħ # ½ 2 (Sumber Day ㅠ es - finarz 10- Sin inox.) 2 ATT Note n= 2k Sin T ni/ 2 n = (@k+1) I sin Kak+1) 17) n=4k-1 - + the Sin (nt n = 0,24 Sin ant 2 2n-2 El a in = 1,3,5 the flx) = Š 2 (-sin(atz )) + 2 Th + E-21-12 2 2 (2001)-2 tent)*) 4 Sin n=1 BT (2n-1) 2(-1)? sin (2n-1)x) (2n-1) IT nei registris fix) ü 놀 2