Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse...
Transcript of Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse...
InverseTrigonometricfunctions
Andthe“HelenofMathematics”
UBCMath102
2000DecemberQ9RelatedRates
Thecycloid.
•
SolutionoftheExamproblem
• SeAppendix(Documentcamerawork)
Doesthiscurvehaveanx-yequation?
• Isitpossibleto“eliminatetheangletheta”andwritethisasasingleequationforxandy?
• Yes,andthisisonemotivationforinversetrigfunctions.
InverseTrigonometricfunctions
UBCMath102
InverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
UBCMath102
“theanglewhosesineisx”
MeaningofInverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
• Theabovetriangleisconstructedsothatsin(θ)=x,whichmeansthatθ=arcsin(x)
UBCMath102
“theanglewhosesineisx”
(1)ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x))leadsto:(A)(B)(C)(D)(E)
UBCMath102
(1)ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x))leadsto:(A)(B)(C)(D)(E)
UBCMath102
ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x)):
θ=arcsin(x)andtan(θ)=opp/adj=soy=tan(arcsin(x))=
UBCMath102
Backtocycloid• à
• Drawarighttrianglewiththisrelationsatisfied.Wecanusethattofindallothertrigquantities
• Usetheinversetrigfn:• Plugintotheequationforxandsimplify
θ1-y
Backtocycloid
• Simplifybyusingthistriangle• Sin(θ)=opp/hypot
θ1-y
Inversefunctions
UBCMath102
Inversefunctionsonrestricteddomains
UBCMath102
Onrestricteddomains
UBCMath102
f(g(x))=xandg(f(x))=x
DomainsofInverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
• Becausesin(x)isperiodic(repeatsitself),wehavetorestrictthedomaintodefineaninversefunction.
UBCMath102
“theanglewhosesineisx”
Domainofsin(x)
UBCMath102
Restrictingthedomain
UBCMath102
Domainofarcsin(x)
UBCMath102
(2)DomainsThefunctionssin(x)andarcsin(x)areinversefunctionsonthefollowingdomains:(A)–π≤x≤πand-1≤x≤1(B)–π/2≤x≤π/2and-1≤x≤1(C)-1≤x≤1and–π/2≤x≤π/2(D)-1≤x≤1and–π≤x≤π(E)–π/2≤x≤π/2and–π/2≤x≤π/2
UBCMath102
(2)DomainsThefunctionssin(x)andarcsin(x)areinversefunctionsonthefollowingdomains:(A)–π≤x≤πand-1≤x≤1(B)–π/2≤x≤π/2and-1≤x≤1(C)-1≤x≤1and–π/2≤x≤π/2(D)-1≤x≤1and–π≤x≤π(E)–π/2≤x≤π/2and–π/2≤x≤π/2
UBCMath102
Ontherestricteddomains,arcsin(sin(x))=x
–π/2≤x≤π/2and-1≤x≤1
•
Similarly,cosineandarccosine
• Domains
Symmetryabouty=x
• Domains
Whatisthederivativeofarccos(x)?
• (A)– arcsin(x)
• (B)arccos(x)
• (C)
• (D)-
• (E)
Whatisthederivativeofarccos(x)?
• (A)– arcsin(x)
• (B)arccos(x)
• (C)
• (D)-
• (E)
Derivativeofarccos(x)• Rewriteintermsoffamiliarfunction
• NowuseimplicitdifferentiationThissideisjust=1
yx
1Atriangleinwhichx=cos(y).
Wecanuseittoexpresssin(y)in
termsofx.
Tan(x)andarctan(x)
Figureoutthedomainsofeachofthesefunctions
UBCMath102
DerivativesofTan(x)andarctan(x)(Showthisusingquotientruleon)Rewriteasx=tan(y),useimplicitdifferentiation
UBCMath102
Mostimportantderivativesofinversetrigfunctions
UBCMath102
Practiceofthisandimplicitdiff.
• Findanequationfordy/dxforthecurve
• (WSNov27Q6)
• Simplifiesto:
Cycloid
Thecycloid.
• Alivingdemoofthecycloid…
CycloidonDesmos:1.Prepareforanimation
• Startbydefiningaparameterthatwillgetanimatedlater
• Giveitarangeof0<a<6π
2.Addacircle
• Inputtheequationofacircleofradius1centeredattheorigin.
• Addapointontherimofthecircleusingsineandcosineofaasthexandycoordinates.
• Makeyourpointgoaroundthecirclebyanimatinga
Itshouldlooklike:
Animatingawillmakethepointmovearoundthecircle.OK,nowturnoffthatpoint,sowecanmoveon..
Makeyourcirclemove
• Changetheequationofthecirclesothatitscenterisat(a,1)
• Whathappenswhenyouanimatea?
Clicktoanimate
Itshouldlooklike:
• Youshouldseeacircleslidingupthexaxis
Clicktoanimate
Addthecoordinatesofthecycloid
• Addthis:
• Seewhathappenswhenyouanimatea
Itshouldlooklike:
• Apointstuckontherimoftherollingcircle
Addthecycloidpath
• Thiswilladdthecurve,notjustthepointalreadyonyourgraph.
Addthecycloidpath
•
Sowhocares?
• What’sspecialaboutthisfunnycurve?
Famousproblem:
“FindthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.”
“Brachistochrone”
“FindthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.”JohannBernoulli1667–1748Posedtheproblem(1696)https://en.wikipedia.org/
Solutionisthecycloid!
ItisthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.Straightline=shortestdistanceCycloid=shortesttime!!
Demonstration
• Marbleraceonalinearandcycloidalpath.
Ahistoryofquarrels
GillesdeRoberval(1628)PierredeFermat
ReneDescartes
https://en.wikipedia.org/wiki/Gilles_de_Robervalhttps://en.wikipedia.org/wiki/Ren%C3%A9_Descartes
Iknowthesecret,butI’llnevertellyouthesolutionto
thisexamproblem.
ThemostridiculousgibberishI’veeverseen
Isolvedittoo!
Tautochrone
• Findacurvesuchthatslidingdownthattothelowestpointdoesnotdependonthestartingpoint
• Solution:acycloid,time=
• Samecurveasthebrachistochrone!
• SolutionbyChristiaanHuygens1659
https://en.wikipedia.org/wiki/Tautochrone_curvehttps://en.wikipedia.org/wiki/Christiaan_Huygens
JohanBernoulli’ssolution:usesSnell’slaw
α1
α2
v1
v2
Supposetherearemanylayers
• Lightpassingthroughlayerswithdecreasingdensity:speedincreases,andlightraybendsaccordingtoSnell’sLaw
α1
α2
α3
α4
v1
v2
v3
v4
Snell’slawformanylayers
α1
α2
α3
α4
v1
v2
v3
v4
Idea
• Findasolutionbyimmitatingthebehaviouroflight
• Usetwothings:(1) Velocityofparticlechangesduetogravity(2) Wecanrelatesin(α)totheslopeofthe
tangentlineofthecurve(seepreviouslecture)
(1)velocityatagivenheight
• Ballstartswithvelocityv=0fromsomeheight
• Afterfallingverticaldistancey,ithastradeditspotentialenergy=mgyforkineticenergy:
½mv2=mgyàv=(2gy)1/2y
Relatesin(α)tody/dx
• (fromlastlecture)
α Δy
Δx
α
y
(2)sin(α)
Weshowedlastweekthatwecanrelatethistotheslopeofthetangentline:sin(α)=opp/hyp
Δyα
Δx
Putthesefactstogether
Whatitmeans:Conclusion:Thedifferentialequationbelowdescribesthepathofleasttime:Wecanshowthatthecycloidsatisfiesthisdifferentialequationusingeitherimplicitdifferentiationortrigidentities!(CalculationshownintheAppendix:documentcamerawork)
•
JohanBernoulli’sCollectedwork
• “Suprainvidiam”(Aboveenvy)• https://www.maa.org/press/periodicals/convergence/mathematical-treasure-collected-works-of-johann-bernoulli
Cycloid:HelenofGeometry
• Beautifulpropertiesbutcausedterriblequarrelsbetweenitsmathematician“lovers”in17thcentury.
TheHelenofGeometry,JohnMartindoi:10.4169/074683410X475083
HelenofTroy
ThefacethatlaunchedathousandshipsLithographbyWalterCranehttps://en.wikipedia.org/wiki/Helen_of_Troy
HelenofMathematics
Theshapethatlaunchedathousandquips.
Fillinacourseevaluation
• https://eval.ctlt.ubc.ca/science
• Yourconstructivefeedbackandindicationofwhichaspectsofthiscourseworkedforyouwouldbegreatlyappreciated.
ProblemsolvingsessionWeranoutoftime,butshowedtheproblemthatwewillsolvenexttime.
Fromrecentresearch
UBCMath102
Seenextslideforenlargeddiagram
Enlargeddiagram
UBCMath102
Otherpracticequestions:
• SeeourQuestionChallengewiki
http://wiki.ubc.ca/Course:MATH102/Question_Challenge