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InverseTrigonometricfunctions
Andthe“HelenofMathematics”
UBCMath102
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2000DecemberQ9RelatedRates
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Thecycloid.
•
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SolutionoftheExamproblem
• SeAppendix(Documentcamerawork)
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Doesthiscurvehaveanx-yequation?
• Isitpossibleto“eliminatetheangletheta”andwritethisasasingleequationforxandy?
• Yes,andthisisonemotivationforinversetrigfunctions.
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InverseTrigonometricfunctions
UBCMath102
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InverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
UBCMath102
“theanglewhosesineisx”
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MeaningofInverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
• Theabovetriangleisconstructedsothatsin(θ)=x,whichmeansthatθ=arcsin(x)
UBCMath102
“theanglewhosesineisx”
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(1)ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x))leadsto:(A)(B)(C)(D)(E)
UBCMath102
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(1)ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x))leadsto:(A)(B)(C)(D)(E)
UBCMath102
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ManipulatingInversetrigfunctions
Simplifyingtheexpressiony=tan(arcsin(x)):
θ=arcsin(x)andtan(θ)=opp/adj=soy=tan(arcsin(x))=
UBCMath102
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Backtocycloid• à
• Drawarighttrianglewiththisrelationsatisfied.Wecanusethattofindallothertrigquantities
• Usetheinversetrigfn:• Plugintotheequationforxandsimplify
θ1-y
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Backtocycloid
• Simplifybyusingthistriangle• Sin(θ)=opp/hypot
θ1-y
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Inversefunctions
UBCMath102
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Inversefunctionsonrestricteddomains
UBCMath102
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Onrestricteddomains
UBCMath102
f(g(x))=xandg(f(x))=x
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DomainsofInverseTrigFunctions
• Letf(x)=sin(x)thenf-1(x)=arcsin(x)
• Becausesin(x)isperiodic(repeatsitself),wehavetorestrictthedomaintodefineaninversefunction.
UBCMath102
“theanglewhosesineisx”
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Domainofsin(x)
UBCMath102
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Restrictingthedomain
UBCMath102
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Domainofarcsin(x)
UBCMath102
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(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
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(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
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Ontherestricteddomains,arcsin(sin(x))=x
–π/2≤x≤π/2and-1≤x≤1
•
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Similarly,cosineandarccosine
• Domains
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Symmetryabouty=x
• Domains
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Whatisthederivativeofarccos(x)?
• (A)– arcsin(x)
• (B)arccos(x)
• (C)
• (D)-
• (E)
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Whatisthederivativeofarccos(x)?
• (A)– arcsin(x)
• (B)arccos(x)
• (C)
• (D)-
• (E)
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Derivativeofarccos(x)• Rewriteintermsoffamiliarfunction
• NowuseimplicitdifferentiationThissideisjust=1
yx
1Atriangleinwhichx=cos(y).
Wecanuseittoexpresssin(y)in
termsofx.
![Page 29: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/29.jpg)
Tan(x)andarctan(x)
Figureoutthedomainsofeachofthesefunctions
UBCMath102
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DerivativesofTan(x)andarctan(x)(Showthisusingquotientruleon)Rewriteasx=tan(y),useimplicitdifferentiation
UBCMath102
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Mostimportantderivativesofinversetrigfunctions
UBCMath102
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Practiceofthisandimplicitdiff.
• Findanequationfordy/dxforthecurve
• (WSNov27Q6)
• Simplifiesto:
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Cycloid
![Page 34: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/34.jpg)
Thecycloid.
• Alivingdemoofthecycloid…
![Page 35: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/35.jpg)
CycloidonDesmos:1.Prepareforanimation
• Startbydefiningaparameterthatwillgetanimatedlater
• Giveitarangeof0<a<6π
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2.Addacircle
• Inputtheequationofacircleofradius1centeredattheorigin.
• Addapointontherimofthecircleusingsineandcosineofaasthexandycoordinates.
• Makeyourpointgoaroundthecirclebyanimatinga
![Page 37: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/37.jpg)
Itshouldlooklike:
Animatingawillmakethepointmovearoundthecircle.OK,nowturnoffthatpoint,sowecanmoveon..
![Page 38: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/38.jpg)
Makeyourcirclemove
• Changetheequationofthecirclesothatitscenterisat(a,1)
• Whathappenswhenyouanimatea?
Clicktoanimate
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Itshouldlooklike:
• Youshouldseeacircleslidingupthexaxis
Clicktoanimate
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Addthecoordinatesofthecycloid
• Addthis:
• Seewhathappenswhenyouanimatea
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Itshouldlooklike:
• Apointstuckontherimoftherollingcircle
![Page 42: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/42.jpg)
Addthecycloidpath
• Thiswilladdthecurve,notjustthepointalreadyonyourgraph.
![Page 43: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/43.jpg)
Addthecycloidpath
•
![Page 44: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/44.jpg)
Sowhocares?
• What’sspecialaboutthisfunnycurve?
![Page 45: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/45.jpg)
Famousproblem:
“FindthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.”
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“Brachistochrone”
“FindthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.”JohannBernoulli1667–1748Posedtheproblem(1696)https://en.wikipedia.org/
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Solutionisthecycloid!
ItisthepathofleasttimebetweenpointsAandBforaparticlemovingunderforceofgravity.Straightline=shortestdistanceCycloid=shortesttime!!
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Demonstration
• Marbleraceonalinearandcycloidalpath.
![Page 49: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/49.jpg)
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!
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Tautochrone
• Findacurvesuchthatslidingdownthattothelowestpointdoesnotdependonthestartingpoint
• Solution:acycloid,time=
• Samecurveasthebrachistochrone!
• SolutionbyChristiaanHuygens1659
https://en.wikipedia.org/wiki/Tautochrone_curvehttps://en.wikipedia.org/wiki/Christiaan_Huygens
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JohanBernoulli’ssolution:usesSnell’slaw
α1
α2
v1
v2
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Supposetherearemanylayers
• Lightpassingthroughlayerswithdecreasingdensity:speedincreases,andlightraybendsaccordingtoSnell’sLaw
α1
α2
α3
α4
v1
v2
v3
v4
![Page 53: Inverse Trigonometric functionskeshet/M102/Lect2017_13.1.pdf · 2017. 11. 29. · Inverse Trigonometric functions And the “Helen of Mathematics” UBC Math 102 ... Inverse Trig](https://reader035.vdocument.in/reader035/viewer/2022062318/5fe728af799f2500a549f5f9/html5/thumbnails/53.jpg)
Snell’slawformanylayers
α1
α2
α3
α4
v1
v2
v3
v4
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Idea
• Findasolutionbyimmitatingthebehaviouroflight
• Usetwothings:(1) Velocityofparticlechangesduetogravity(2) Wecanrelatesin(α)totheslopeofthe
tangentlineofthecurve(seepreviouslecture)
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(1)velocityatagivenheight
• Ballstartswithvelocityv=0fromsomeheight
• Afterfallingverticaldistancey,ithastradeditspotentialenergy=mgyforkineticenergy:
½mv2=mgyàv=(2gy)1/2y
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Relatesin(α)tody/dx
• (fromlastlecture)
α Δy
Δx
α
y
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(2)sin(α)
Weshowedlastweekthatwecanrelatethistotheslopeofthetangentline:sin(α)=opp/hyp
Δyα
Δx
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Putthesefactstogether
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Whatitmeans:Conclusion:Thedifferentialequationbelowdescribesthepathofleasttime:Wecanshowthatthecycloidsatisfiesthisdifferentialequationusingeitherimplicitdifferentiationortrigidentities!(CalculationshownintheAppendix:documentcamerawork)
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•
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JohanBernoulli’sCollectedwork
• “Suprainvidiam”(Aboveenvy)• https://www.maa.org/press/periodicals/convergence/mathematical-treasure-collected-works-of-johann-bernoulli
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Cycloid:HelenofGeometry
• Beautifulpropertiesbutcausedterriblequarrelsbetweenitsmathematician“lovers”in17thcentury.
TheHelenofGeometry,JohnMartindoi:10.4169/074683410X475083
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HelenofTroy
ThefacethatlaunchedathousandshipsLithographbyWalterCranehttps://en.wikipedia.org/wiki/Helen_of_Troy
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HelenofMathematics
Theshapethatlaunchedathousandquips.
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Fillinacourseevaluation
• https://eval.ctlt.ubc.ca/science
• Yourconstructivefeedbackandindicationofwhichaspectsofthiscourseworkedforyouwouldbegreatlyappreciated.
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ProblemsolvingsessionWeranoutoftime,butshowedtheproblemthatwewillsolvenexttime.
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Fromrecentresearch
UBCMath102
Seenextslideforenlargeddiagram
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Enlargeddiagram
UBCMath102
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Otherpracticequestions:
• SeeourQuestionChallengewiki
http://wiki.ubc.ca/Course:MATH102/Question_Challenge