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