Section P.1 Notes Page 1 Section P.1 Precalculus and Trigonometry Review

Transcription

1 Secion P Noe Pge Secion P Preclculu nd Trigonomer Review ALGEBRA AND PRECALCULUS Eponen Lw: Emple: 8 Emple: Emple: Emple: b b Emple: 9 EXAMPLE: Simplif: nd wrie wi poiive eponen Fir I will flip e frcion o mke e ouide number poiive: Now rie evering o e power of In doing o, we mulipl e eponen: 8 8 We will mulipl cro e op nd boom We will need o dd eponen: 0 8 Now ubrc eponen: 0 8 Finll we cn wrie wi poiive eponen: 0 8 EXAMPLE: Divide: 9 For i problem i will be be o divide evering on op b e boom o we cn brek up e frcion: 9 Now reduce ec frcion b ubrcing eponen

2 Fcoring nd Solving Wi Rionl Eponen Secion P Noe Pge EXAMPLE: Solve: 0 7 Look for e mlle frcion, nd i w ou fcor ou Here we fcor ou Fcoring i e me diviion We re rell dividing ec monomil b Here i w i look like: 7 Wen ou divide ou ubrc e eponen Once ou ubrc ou will ge Don forge o wrie e number ou fcored ou, o now ou ve: 0 Ti i no full fcored, o we fcor one more ime: 0 re done Se ec fcor equl o zero o ge: = 0, /, nd -/ EXAMPLE: Fcor compleel: ( ) ( ) Here e common fcor i ou will ge: ( ) Ti cn be implified o: ( ) Now we You cn onl pull ou fcor wi e lowe power If ou fcor i ou Now epnd e epreion inide e brcke Simplif We re no done becue we cn fcor one more ime ( )( ) Ti i our finl nwer EXAMPLE: Solve: 8 Te eie w o olve i i o fir cro mulipl o cler e frcion: 8( ) Now epnd bo ide: 8 Now e i equl o zero: 8 0 Now fcor ( )( ) 0 Se bo equl o zero = Ti i our onl nwer

3 EXAMPLE: Le f ( ) Find e difference quoien uing f ( ) f ( ) Secion P Noe Pge We will do i e me w bove Fir we will find f ( ) f ( ) ( ) W i f ( ) ( )( ) f ( ) ( ) ( )? If ou re inking ou re wrong Ti i cull ( )( ) wic i FOIL I i f ( ) Now we ve implified i muc poible, we will pu i ino e difference quoien formul ( ) Now we will diribue e minu ino f () Now cncel nd implif Now we cn fcor ou n from e op ( ) L ing i we cn cncel e from op nd boom Ti i our nwer Grp of bic funcion Trnformion nd Grp Skece Suppoe = f() i e originl funcion Y = f() + k move f() k uni up Y= f() k move f() k uni down Y = f( - ) move f() uni o e rig Y= f( + ) move f() uni o e lef Y = -f() flip e grp over orizonl i Y = f(-) flip e grp over vericl i

4 EXAMPLE: Skec b uing rnformion Find e inercep Secion P Noe Pge Now le pu i ll ogeer Uull ou will be uing more n one rnformion Ti problem doe pu e l ree emple ogeer We r wi our bolue vlue grp e origin We will move i one plce o e lef, en up one uni nd en flip i upide down: -in: (-, 0) nd (0, 0) -in: (0, 0) EXAMPLE: Skec b uing rnformion Find e inercep In order o ue e rnformion rule e mu come fir nd ere mu be one in fron of In our problem bove we need o fir pu e fir nd en we will fcor ou negive: Here we pu e erm fir ( ) Here we fcored ou - Now we re red o grp Since we lred fcored ou e negive We need o move e grp plce o e rig nd en up uni Tere i negive inide e funcion, o we need o ue rule of e rnformion wic we will flip e grp over e vericl i -in: none -in: (0, ) TRIGONOMETRY You wn o mke ure ou know e following ble of vlue Tble of rigonomeric vlue (degree) (rdin) in co n undefined

5 Secion P Noe Pge If ere re rigonomeric vlue of ngle re more n 90 degree, en ou will need reference ngle Reference Angle n ngle beween 0 nd 90 i formed b e erminl ide of n ngle nd e -i Te reference ngle i lbeled below I i indiced b e double curved line Noice no mer were e ngle i drwn i i meured from e -i Under ec drwing i ell ou ow o find e reference ngle: If en If en If 70 0 en Ref ngle = 80 Ref ngle = 80 Ref ngle = 0 If en If en If en Ref ngle = Ref ngle = Ref ngle = Sign vlue of ine, coine, nd ngen in ec qudrn in co n in co n in co n in co n Depending on wic qudrn ou re in e ine, coine, nd ngen funcion will be eier poiive or negive You will need i for uing reference ngle o find rigonomeric vlue An e w o remember e ign cr i e pre All Suden Tke Clculu Te fir leer of ec word in e pre ell ou w i poiive in ec qudrn, ring in qud nd going counerclockwie ALL Men ll of em re poiive in e fir qudrn S Men ine i e onl one poiive in qud T Men ngen i e onl one poiive in qud C Men coine i e onl one poiive in qud

6 How o find e rigonomeric vlue for n ngle: Secion P Noe Pge ) Find e reference ngle ) Appl e rig funcion o e reference ngle ) Appl e pproprie ign EXAMPLE: Find e ec vlue of indice e reference ngle We will follow e ree ep from bove co uing reference ngle Drw e ngle in ndrd poiion nd ) Fir we will drw i ngle in ndrd poiion Te reference ngle i indiced b e double curved line We found e reference ngle b king 80 ) We need o ppl e rig funcion o our reference ngle, o we will do co ) We need o ppl e pproprie ign Ti i were we will ue e ign cr from e l pge Ti ngle i in e econd qudrn, o coine need o be negive ere So now we cn wrie our nwer: co EXAMPLE: Find e reference ngle of in nd find i EXACT vlue 80 We wn o cnge our ngle ino degree: 0 So now our problem become: in( 0 ) Uing e even nd odd properie, in( 0 ) in 0 We ve cnged e problem, o now we will look poiive 0 degree We cn rewrie e problem in(00 0 ) wic become in(00 ) Ti i in e four qudrn, o our reference ngle i , or We will now look in 0 We cn pu in e vlue off our ble, nd we will ge negive, o we need o cnge e ign of our nwer, o in In e four qudrn, ecn i

7 Grp of Sine nd Coine Secion P Noe Pge 7 EXAMPLE: Grp over one period uing rnformion: in Ti will ve e ke poin in Te onl difference i e mpliude i, o e ige nd lowe poin on e grp will be nd 0 EXAMPLE: Grp over one period uing rnformion: co Ti one will ve n mpliude of nd lo e negive will flip over our grp I will ill ve e me ke poin co 0 Li of Trigonomeric Ideniie in n co co co in cc in ec co co n in co in co co in ec n n ec cc co co cc Proving Trigonomeric Ideni Aloug we will no pecificll be doing ee pe of problem in Clculu, ee problem re good review of our rig ideniie

8 EXAMPLE: Ebli e ideni: co in in co ec Secion P Noe Pge 8 You wn o ow one ide of e equion equl e oer ide In ee problem ou re NOT llowed o do operion like dding or ubrcing ing from one ide o e oer Tink of ec ide independen Once gin we wn o fir ge ingle frcion o we need common denominor co co in in ec in co co in Now mulipl nd wrie ingle frcion co ( in ) co ( in ) ec We will epnd e numeror co in in ec co ( in ) We will ue e ideni co in in co ( in ) in co ( in ) (in ) co ( in ) ec ec ec Simplif e numeror Fcor e numeror We cn cncel e in from e op nd boom co ec We will ue e ideni ec co ec ec Bo ide re e me, o we re done Solving Trigonomeric Equion EXAMPLE: Solve e equion: co co 0 on [ 0, ) We cn fcor i one: ( co )(co ) 0 Now e ec pr equl o zero We ge co 0 nd co 0 Solving e fir equion we will ge co We look on e uni circle nd look for n vlue re Ti will ppen nd Solving e econd equion ou will ge co Te ngle give n vlue of negive one i Terefore our nwer re,, nd

9 Secion P Noe Pge 9 EXAMPLE: Solve e equion: in co in on [ 0, ) We need o e i equl o zero: in co in 0 Now fcor ou common fcor of in : in (co ) 0 Now e ec fcor equl o zero We ve in 0 Looking our uni circle we ee i 0 nd Te oer equion give u co Tking e qure roo we ge co Ti men co Since i number i lrger n one, i will no give u n oluion becue of e domin of e coine So our nwer for re 0 nd EXAMPLE: Solve e equion: in co 0 on [ 0, ) Since ere re bo ine nd coine we need o ue n ideni o ge ll e erm o ve e me rig vlue Since I noice ere i lred ine in e problem I wn o ue co in So now e problem i: in ( in ) 0 Simplifing we ge: in 0 Wen we olve i we ge in o 7 in Ten vlue off e uni circle re:,,,