Amath Notes

iiiiiigane.gs t sn otoot.iiiiiiiiieimanumapara completelyabove completelybelow µg4 discrimina iiiiiiii.it a.iteEiiiiLIiii eiiiii.fii yii in iminimumcurvecoeffof.es Ifunsusti Iii mm iiii.it iiiiiiiiiiiman t.amAlwayspositive awaynegative Inmaypgpq.ae qq.q.gguyyqqyyneergy 2 I ieii a T.IEii.tiiiiiiii Iiiiiiiiiiiiiiii.am 1iigone hansetofactomedfomii.is iEsasonefo.yuseauaaraticeguationorfaconse man chapters cnn.mn fisiisiiiiitÉ msn.miatiiiieauation.pauiitinmngam.n.pl P ii It.itiiiiiismT.EE iiiiii iiiiiii.li iiiiiii.isi aiiiiiiiiiii qiyascp.fisrationalnumber IiiiEii an IiiliIIIIIIIIII iii Iii am iiIiiiisii.IE iiiiiii improperfractiondegreeofnumeratordegreeofdenominatorfactorne.uaexpression toesan pe.g.cito ipropertractionsdistinctlinearfactorsdenominator IEiiiiiiiiiiIiiii si Iiiiiiiiiiiman iiiiiiiii.ie 91,1me Bytactortheoremxoisafa.to E.ie tstEes.iiiiiiiit IEEEEEE iieie.IE na IiIEiiiiiiiiii Im me IIIFII IiiiI.fiiitEisa 159 pergmagggimiiggg.gg Perpendatarbisectorofline midpoint m.IE iei iiiiiiet.ie iiiii iiiiiii iiIE iiiIi.ism iiiiiiiiii.mnsiiiiiEi IittipppL jjgjppn.nameI Iii.liiiiiiiIiit.aeiiiiiiiiii iff.fiIiIIIftp.i eemt fia i mean nineties.itiii.an itiiiÉ

isopposite isequal anfous.aeequal opposites 0I sinthetasinalphacosthetacosalphatanthetatanalphaI0 cagnisdifferent lceil Qinist manaa 3rdint_alpha 1theta2ndpropto IswarrowThetalsteqTrigofunctionanagrams 4h Qhandno tano 11sigma lefttright4leftrightleftright alphatheta quadrantsinceeysin80circ0985sinleft180circ80circrightsin100circ0985anleft180circ80circrightsin260circ0985sinleft360circ80circrightsin280circ0985 leftright A unthetathetaGalphaG andifforget squatne value e.g leftsin45circright2frac12 theta30circ45circ60circfracpi6fracpi4fracpi3nthetafracsqrt12frac12frac1sqrt2fracsqrt22fracsqrt32 sthetafracsqrt32frac1sqrt2fracsqrt22fracsqrt12frac12 anthetafrac1sqrt3fracsqrt331sqrt3 sandquadrantance thetain4thsinthetacostheta taQc QmidsQileftlanglerightrangle 4thctano T3d4h 2nd 15t 3rdquadrant one.sunaQalpha1 alpha 4thquadrantsince frac1sqrt2fracsqrt21 rangle rationale morethan tenthanalphaJtheta thetaalphadtheta obtuse morethan lessthan36reflexGraphysinx tanleftfracpi2rightisundefined onesperiod anperiodan ytanxywedgeycosxywedgesnoraexpressemothefollowinginternofa ywedge period noamplitudewithoutacaimatorandvalueofthefollowing sina.io.amplitude amplitude Istanad tan40circ sqrt1a2a 11Cfracb3snathquadrant 2pifrac53pifracpi3alphafracpi3cosfracpi3frac12cosfrac5pi3frac12 tanfrac11pi6frac11611snathcos130circrightarrowalpha50circ2ndquadrant sqrt1a2 x2pimesonatasquadrant 0fracpi2 x2pi x2piprod 0fracpi20fracpi2pifrac3pi2 pianadian 2pifrac11pi6fracpi6alphafracpi6tanfracpi6fracsqrt33tanfrac11pi6fracsqrt33 frac3pi21 1 1 210circ180circ30circalpha30circsin30circfrac12sin210circfrac12 periodangsinywedge 0leqxleq2piperiodanycosx ytanxamplitude ywedgeywedge amplitudebeginmatrix 140circa150circsqrtsqrt12a2endmatrix 1 periodpinoamplitude1 1 x2pimidlinel0 x2pi x2pipi 00fracpi2 tfrac3pi2 pi1 fracpi2frac3pi2 11l2cosleft90circthetarightsintheta3tanleft90circthetarightfrac1tanthetasinleft900thetarightcostheta alsodetermineshapeofgraph yasin xcyacosbxc Refertodasskicttsno.tl forexample and90circA270circfindwithoutcalculator a11icotA yatandxcmidline ycperiodfracpib giventhatsinAfrac45cnn.ec90circA frac1sinlefta0circArightfrac1cosAfrac1frac35frac53 leftiirightecleft180circAright midlineisycaverageoftwopoints amplitudeaapositive maxmin.ca period lcolleft180circArightquadrant frac45 frac1tanAfrac1frac43frac34Qsinleft900left1800Arightright 5alphalceilalphafrac35cosAfrac35alphafrac43tanAfrac43 Qsin1left900Aright4 frac1sinleftleft90circArightrightfrac1sinleft90circArightTheta 1 sina.aesina ccalcosatancastanasafe cosleft90circleft90circArightrightfrac1cosleftArightfrac1cosAfrac1leftfrac35rightfrac53ecdotgcdotcosecleftx50circright1350circx360circfrac1sinleftx50circright135leftsinx50circright074074 c0a1 Itasexplainwhysinx i.tha.nosolution Fromgraphimaximumvaneotsin x1 sinx11nanosaution bexplainwnsthesimultaneomeauation.in casconavensation sinx0 x00180circ360circ again wedge 1e.g25in2xsqrt30 180circx180circTrigeequations 0sinssqrt3sinsfracsqrt321leano 180360leftrightxQQiiwhenquadrantes geofxQangles aQbasedonanadrantanarange a 1 y0cosx0 x90circ210circ1conatrangeand sign1letx50circQ2Qininlandquadrant 00x360050circC 50circ310circ 2Qns.dorathauadrant y0902701 180circx180circ360circ2x360circ360circtheta360circsinthetafracsqrt32sinalphafracsqrt32 backslashsimultaneousequations x0cosx0havenosautio.to 00leqxleq36003 candonegative anginaandanquadrant anox180circ50circQ310circsinalpha074074alphasin1left014074right47794circetheta47794circo x50circ41794circx97794circ978circ Q 1 alphasin1leftfracsqrt32rightaalpha60circ alphalambda annotrelevantasQimmlandquadrant4 theta Q180circ47794circx50circ180circ47194circx182206circ1822circ or theta180circ60circ240circ Q600alphadtheta3rd4thsqrt100alphapartial Q180circ QGalphatheta360circ60circ300circtheta180circ60circ120circ4h3rd propto YQ3rd4th 2pi60circx30circ2pi120circx60circ62x240circx120circ2x300circx150circ 1117111backslash Pleftx1right10descendingpowers left1xright10ascendingpower cowenaegree left3kxright20cdotcoeffcircfx4x5316Findk Trleftbeginmatrix20vendmatrixrightleft320rrightleftkrxrright midpointofacparallelogram Dfracnrleftnrrightbeginmatrix 95endmatrixwithoutaccurate 95left95right9times8backslash xox4times3times2times1 trapezium midpointofBDFindthegeneraltermofleftx2frac1xright12furore c1 AB xinATr1leftbeginmatrix12rendmatrixrightleftx2right12rleftfrac1xrightrleftbeginmatrix12rendmatrixrightleftx242rrightleft1rightleftfrac1xrightrleftbeginmatrix12rendmatrixrightleft1rightleftx242rrrightleftbeginmatrix12rendmatrixrightleft1rightrleftx243rrightandtermcontainingx9 one 7beta1 aas betweenparamelsideadduptoso ABclapairofequal sidesopposite sideequalandparallel oppositeeequal adjacentasaddupto180 diagonalsneareachothernotperpendicular asax diagonalsareperpendina mentorix paranedontcaniate separately axoxxon savan adjacentaadduptonocappiementary diagonalsanperpendicularbisectors beginmatrixleftbeginmatrix204endmatrixrightleft3204rightleftk4rightfrac316leftbeginmatrix205endmatrixrightleft3205rightleftk5rightendmatrixfrac15k416k5frac316frac1516kfrac31639times8times7x leftbeginmatrix125endmatrixrightleft1right5leftx9right792x9243r93r15newnoen.es r54times3times2times1 must 18times7126 ysmranglelangle y1 pi y1 y1ay1hasnosolution tanthetafracsintheta101theta sinleftthetarightsintheta 10.6RFormulae asinthetapmbcosthetaRsinleftthetaalpharightacosthetapmbsinthetaRcosleftthetampalpharightab0Rsqrta2b2alphatan1fracba express7on08inQoneformRinleftthetaalpharightR0so.asa form1let7cos0 8sinthetaRcosleftthetaalpharightRcosthetacosalphaRsinthetasinalpha 8Rsinalphaleft2right 7Rcosalphaleft1rightleft2rightdiv111tanalphafrac87alpha48814circmaxofsin xcosx1minotsin x10x1therefore maxotsinleftthetaalpharight1mn.tnleftthetaalpharight1 left1right2left2right2R2cos2alphaR2sin2alpha72R2leftcos2alphasin2alpharight72fracRsinRcosαalphafrac87 1therefore7costheta8sintheta106coslefttheta488circright Rsqrt728210630amencesinagreatestleastvamentslstheta8sinthetacorrespondingvaluesoftheta 0circleqthetaleq360circmaxuameofio.es 0coslefttheta488circrightis106max theta48814circ360circcresathetaleq360circ letalefttheta488circright1theta48814circ0circtheta488circ rightarrowsin2Afrac12left1cos2Aright cos0circcos360circ1commanana cos2Afrac12leftcos2A1right tan1frac2tanfracA21tan2fracA2 minvalueof 10630coslefttheta488circrightis106letcoslefttheta48814circright1theta48814circ180circQ2288circcos180circ1 linearlaw Ydownarrow practicenow yfracax2bgraphoffrac 1yagainstx2drawn inepanatmonguaasa.ae c1is constant ymxcleft0right1leftfrac45rightthetaprinciplevalues 90circleqsin1xleq90circ 0circleqcos1xleq180circ 90circleqtan1xleq90circ yinterceptgradient costhetafrac45 variableIfsassandexappyiogan.inenaavameotaanasanvaneoymena.es 1 leftiright2x2y12x2frac1y2XY xfracy2Subxfracy2intoyfrac13xfrac53Yfrac13leftfracY2rightfrac53 yfracy6frac53Yleft1frac16rightfrac53Yfrac53divl