Question

mechanical engineering analysis help, get from problem to solution, pls show all work, thanks.

Problem 2. Find the Fourier series approximation of the following periodic function f(x), where the first two leading cosine and sine functions must be included. f(x) Angle sum formulas for sine / cosine functions sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B TT cos(A + B) = cos A cos B – sin A sin B cos(A - B) = cos A cos B + sin A sin B TC 0 TT TC Answer: f(x) = 2 + *sin 2x – 1 cos 4x + cos 6x - cos 8x + ...

Answer 1

We see We The graph is given. the function is f(1) = In cosul find the fonnier expansion f(u) = It cosul in (-T, T) First we find whether it is even fr ou odd. f(-x) IT Cos(-4) 1 = In cosal = f(1) . f(a) is an even function, Fonnier expansion for even function is f(x) = a + E ancosna [bn = o 70 where a. - S SC) one 6 an = f(u) cosna alu + Sam (7) 71 f(n) Cosnu du Also In cosul - T cosa ocul 12 - Losu 7 kn la S. It cosul du T cosnan + sa cond * [sinn] {* - *(-1)} a [sinn } 1 7

sin cosul cos nada TY 2.T con cosna dut fe con los nx de rcos (1+) x + cos I--)] & Sconce anix cumhu] aix] Sin (tn)x + sin (1-na sin (1+r)n + sin(1-nu In Itn sin ((+1) I (1+n) + sin (1-2) (in) 19}-{o-{sin (com)* + dim (in) DW] We know, sin (1 nn) - cos na 2 cos / con + cos / cos na + + Itn In los ni 2 + Itn In 2. (e) η η 이기플 ("*++ 2 Cos na (1-2) This value of an valid for all values of n except = 1. T2 a = 2 cosa da + Swin du 기 71 dn + cosan 3 [ S* (** 022) 어리

2 дана", - [ sia). 7 то - (n-1) - к The founier series S () - + { an cos na ою со М. І - 2 ts ч tos • tоз ми 2 + се)37. Cоn 33 t - 3) (-1) Le 2 + ч (-8) 2 (-15) to) 2 л. соч+ ton 51 cөз 5 + (-) te) 37 0 6 x + 4 се ат сөзах + ч Сочи со 3х (-63) 4 (35) - а + 3 Со 2х + O - че Чи + O + чCo 6 и 15 25 Ч Со 8 к 63 - 2 + - twо зах че 4х + 1С 6 - 98 15 35 C3