Question

there are 4 questions in 1 here

Find the Fourier Coefficients an for the periodic function f(x) So for – 4 < x < 0 f(x+8) = f(x) for 0 < x < 4 { Find the Fourier Coefficients bn for the periodic function f(x) = X for – 3 < x < 0 O for 0 < x <3 f(x+6) = f(x) Determine the half range sine series of f(x) = 1 - x 0 < x < TT, f(x + 25) = f(x) Find the first 2 harmonics of the Fourier series for the values given in the following table: t 0° 30° 60° 90° 120° 150° 180° 210° 240° 270º 300° 330° f(t) 3.14 0.52 1.05 1.57 2.09 2.62 3.14 0.52 1.05 1.57 2.09 2.62

Answer 1

Q. 19 Ansit Given ful 42nto f(x + 8) = fu + on 24 fince fee + C) = f f is perlodie with period 8 T=2f 8=21 - 1=4 fe so Fourier Coefficients an given by an = f fw cos nin du 15 l an 4 fe) Cosmit -4 4 feu contex)dx+ fe. corn grable 4 4

q= 1 Focoshlxide + fx.com Hotel 4 & & I Г 4 = [o+ (r. Singaps) EN 4 1. Sin (2x) dx 4 9. Bialas –o) Binay - o) -4 simba) 4 4 4 Į singth) du hlo 4 Sinet &n=1? feinander -) hI Coghi - - Coson 4 (osh + coso coso

sq = -4 22 1- Cosni 4 h2o Cosmin =(-uh Ane Q. 1 6 Answeron Ginen. foo - -32nLo futb) fry on 23 ou f(x+b) = f'lu =) Periool of function is 6. T=28 EJ 6 cztett 3 So 'bn is Fourier Coefficients for given by s for sin trip) don bn = 1

I foco. Sinchith) du will 3 3 f(x) sincetto dx+ fx) Sirhinde 3 O that sinceta de 3 k-sinfnte) de to. Sinhita) 31 3 using productrule Integration. X.-osan) 3 nil Ho - Cash In den 3 n ! 10 te wet + 3 coratin) on 3 ht hT -3 - 9 Cashll + 3 hill Simona HT 3 ocor)com

=) by a I 3 -g Cosnt +9 hit Co- 4272 o sinozo & sinta -o &n=13 bn Eighth ht Ans - 9.1 © Ansot Ginen, f(x) =1-X OLXZT fu+ a11)=fa. Recall :-) Half range of Fouries. fine serices is given by

Whe flu bn sin (hix ) hal where 2. br fo) Sinha) da since period of function of (le f (x +297) = f(n)) of اعلان الموت So m2 1 w ginahata) die (1) sinhxs du "repetamu ya 2 I 1-1 X-Casha ²) Cosat TI) (03.HTT +(16) Caso Cosnach

(-1) CORAT + - Cosnada no ou Coso=1 (IT (sinine) 고 - 2 ХО oo Sin Ost + -) €)") Vnals sinif Hence fu Ź 2 (1+ (1 -)Edu) Sinha Aug Q. 1 @ Ansi) Recalling Harmonic fourtv series is given by of tu