2sin cos dx. 2sin sin dx

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Rudolf Haynes
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1 Fou S & Fou To Th udtl d tht y podc ucto (.. o tht pt t ptcul tvl) c b pd t u o d co wv o dt pltud d qucy. Th c b glzd to tgl (Th Fou To), whch h wd gg pplcto o th oluto o dtl quto whch occu th Phycl cc. (.g. pctl ly o oud d lght, coducto o ht, dcto o wv... ). I c, th Fou to o t vyg gl wll yld t hoc cott,.. th quc whch cop t. A Dct Fou To (DFT). (whch opt o gl pltud whch pld t d t) c b ctly pltd coput pog v Ft Fou To (FFT). Fou S Cod ucto whch podc t tvl ( ) ( ) Hc: co co d Joph Fou D th t : ot th ollowg tgootc dtt: I I ( ) co b How do w d out wht th coct d b? co A co B co A B co A B Aco B ( A B) ( A B) A B co( A B) co( A B) Hc, tg, d co co d ( ) ( ) co co d ( ) ( ) I I I = 4 I co d co d 4 I 4 W tly d to pov o tgl ult volvg d co W c pt th ly o oth cobto o d co t I, tg, d I I co d ( ) ( ) d I = ( ) ( ) I co co I I, tg, d I I d ( ) ( ) co co d I = ( ) ( ) I I co 4 I co d d 4 I co 4 4 co d 4 I d co d 4 I 4 co d co co Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE

2 ( ) co b Rtug to ou Fou S... co co d d co d Hc: ( ) co d co d co co d b co d ( )co d ( )co d ( ) d d co d b d ( )co d b b ( ) d ( ) d d co d b d ( )co d ( ) d Epl: Fou S o qu wv Wth th tvl ( ) ( ) d () d ( ) d obvou lly o yty! ( )co d co d co d b ( ) d b d d co co b b co co co v v b 4 odd odd 4 ( ).. plc wth odd tg + Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE

3 W c td th d dvlopd o th Fou S to d Fou To ( ) co ( ) ( ) ( ) ( ) d ( ) d ( ) d d d Hc: ( ) ( ) ( ), D th copl d-mov Tho co d ( ) ( ) d.. d ( ) ( ) d d D th Fou To o () ( ) ( ) d ( ) ( ) d.. th Iv Fou To Copl Fou S ( ) ( ) d I th lt w c tu th t u to tgl Fo ly o ptl wvo w c oct wth th wvub Fo t gl ( ) ( t) d ( t) ( ) d Agul qucy T Pod Fou To o top ht ucto ( ) ( ) ( ) ( ) d ( ) d ( ) ( ) ( ) ( ) ( ) c c( ) Th ult hghly uul Optc. Etlly, th -ld Fuho dcto ptt th Fou To o th ptu ucto A dcto gtg llutd by l wll ult ptt o bght l gully ptd. Th l pcg dd by th pto o th th lt whch cop th gtg. Th t ptt wll lo hv pltud vlop (gv by c ucto) whch ult o th t wdth o th lt. I lt uoly llutd by th l, w c odl t ptu ucto by top ht. Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE 3

4 Fou To o Gu ucto ( ) ( ) d d ( ) ( ) d ( ) d ( ) p d ( ) p d ( ) p d ( ) p d z, dz d z ( ) p dz ( ) So th Fou To o Gu ucto lo Gu ucto. ot th tdd dvto o th Gu d t Fou To vly ltd. Th popty udp th Uctty Pcpl Qutu Mchc, wh th wvucto o qutt uch otu ltd v Fou To to th wvucto o poto. * d Th olzto th Gu ucto c b ud pobblty dtbuto., ( ) E[ ] ( ) d [ ] E d V E E [ ] [ ] [ ] V[ ] d V[ ] d V[ ] V [ ] V[ ] td[ ] V[ ] ( ) Sc tgd odd ( ) d d 4 d So th tdd dvto o Gu ucto : Th tdd dvto o th Fou To o th Gu (whch lo Gu) Th : I Qutu Mchc, th uctt poto d otu ltd to th tdd dvto o th pobblty dty ucto octd wth th wvucto. (Actully th odulu qu o th wvucto th Bo Itptto). ow o ptcl o wvvcto, t otu p : h p p p p Th wvucto ot lwy Gu, o th Uctty Pcpl ctully qulty p d-bogl lto *Stctly pg w ought to po cotou tgl h, but Rly, Hobo, Bc; Mthtcl Mthod o Phyc d Egg, pp436, w hll b cvl d u th hgh qucy ocllto cud by th gy t glgbl! Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE 4

5 Suy o Fou To tho - ( ) ( ) d - - ( ) ( ) d Dto o th Fou To d Iv Fou To ot th Fou To pcl c o gl cl o Itgl To. Aoth, whch popul gl pocg, th plc To. t F( ) ( t) dt ( ) ( ) Opto otto o d th Fou To o th ucto () ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) d d ( ) ( ) d Fou Sht Tho. Th th lgothc b o y ptch htg gl pocg ct Sclg tho Th tho volvg dvtv povd powul tool o olvg dtl quto. Applyg Fou To wll tu dtl quto to lgbc o. Oc olvd, pplcto o th Iv Fou To wll yld th oluto. ( ) g( ) ( ) g( ) ( ) g( ) ( ) g( ) ( ) g ( ) d ( ) g ( ) d ( ) d ( ) d Covoluto Tho. Th vy uul gl pocg..g. Pul Copo Rd pplcto. ( ) g( ) ( z) g( z) dz ( ) g( ) Pvl lto.. th ud th Pow Spctu th th tgl o gl pow. Th ut b tu c gy covd! Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE 5

6 Th Dct Fou to d pctl ly I gl (t) pld t t, w c d dgtl lt, whch c b ud to tct uul tu, whl php hbtg oth y w y th lt output, whch cob ubqut wghtd pl o th gl. A Dgtl Fou To (DFT) lt b wth lt wght w, ( )( ) Th lt h qucy po (.. th output v qucy du to pl o pu uod) ( ) t t y ( ) w, y ( ) ( )( ) ( ) Th c b pld by codg th u o gotc pogo ( ) ( ) ( ) ( ) ( ) y ( ) y ( ) ( ) ( ) ( ) ( ) y ( ) y ( ) ( ) ( ) ( ) ( ) ( ) ( ) Mu po wh: ( ) ' ( ) ' t th output o th lt b copod to dd g o quc, wth u po t th quc. Th lt qucy po bguou (.. pt) vy qucy tvl o, o th u vlu c b t ot th pl t. So o th DFT, d: ( ) w ( ) ( ) y( ) ( ) Th ltt tlly dcpto o th DFT. ( ) ( ) ( ) ( ) Th clld th yqut Cto DFT Epl gl u o th uod 3 ( t) ( t ),,.5.8,.6,.5, 4, 6 Fo th Pow Spctu t y to d o th copot quc d th pctv gtud. ot th Pow Spctu p d to b hp, o pl qud. I th pl t cd, th pctl p wll bod, o to hp th, you hv to pl o log Mthtc topc hdout: Clculu Fou S & Fou To D Adw Fch. PAGE 6

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,