Question

(1 pt) Prove this orthogonality relation: sin(ma)sin(na)dx O if m n T = r if m n where m and n are integers. For full credit you must explicitly show the steps to evalu ate the integral, and may not use a computer or integral table. Hint: We encountered some integrals like this while discussing complex exponentials

Answer 1

Ans Page No. Date Sincmr sincnrydde т If(T) dx=2 f(x)da so I=2 / sin/mx] sin (nade fasinimog Sihtne9d27 if my n i by Toignonnotory we know that COSCA-B2-COscat B) = 2 Sin Asin B So asin ma simnx= cos(mx-nx) - Cos(mx+nx) = Cos Cm-nDye - Cos (m+mo se - © But this value in equation of = cos(m-n) x_ Cos (m + 2 x dze

Page No. Date ł I= cos(m-nix dx - (cos(m+n2xd ze (Cos O = sino cos (m-nized ae = cm-no Sinem-n2 sin (mtno Coscm+nox dx = cmtn 1 = sin (man 20 1 Sincmtn min - (man) o put value of x I- sina (m-no-sinco)(mon cm_ nu - imam simlocm+m?)-sin comum (minal Sin (0) = 0 put this value in above equation - I = sinn (m-n -0 (m-n) - Tsinr sinnomtno (min)

Page No Date Sincm-ngh -Sin (men in Tmn men) I I = I xort xo 6 m and n are integars, so Sincm-non=0 because Sina=0 Sincm+min to Sinan zo Simsa 20 (man) cm+no I=0-0 Simon zo I=0 So if m £n then I = 0 if m= n / then I = 2 sin masin nada

Page No. Dale asinmxsinna dax man, put it 2 simma sinn [asin mx]dx 0 COS20 [- 2 asinho = 1- Cosao Sesin m x = (1 - Cosa cm x ]] put this value in above aquation II-Cosa cm xolde nt I = (dx - Cosa (mx) dx I Sin2cm x)" om 2 m

I= ln - Sinamn - sinomn am am so I = TT-o-0 7 I = n where man This shows that if m=n , I =* I = sinmrsinarda men I= 0 Thank you
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