ERTH430:
FluidDynamics
inEarthSystems
TheSineandCosineFunctions Dr.DaveDempsey
andtheirDerivatives
Dept.ofEarth
&ClimateSciences,SFSU
I.Functions
Afunctionisarulethatassociateseachmemberofonesetof
values(theindependentvariable)toavalueinanothersetofvalues(the
dependentvariable).
II.SineandCosineFunctions
Forthesetwotrigonometricfunctions,wecanask,whatisthe
independentvariable,andwhatistherulethatassociatesvaluesofthe
independentvariabletovaluesofthedependentvariable?
First,somecontext.Setupatwo-dimensionalrectangular
coordinatesystem.Pickapointanywhere,anddrawalinefromthe
originofthecoordinatesystemtoyourpoint.Callthecoordinatesofthis
pointxandy.Callthedistancefromtheorigintothispointd.(Seethe
diagrambelow.)
Thex-axisandthelinefromtheorigintothepointintersectand
formanangle.Callthisangle,measuredcounterclockwisefromthexaxis,θ.
[Note:Althoughwedefineunitsformeasuringangles(namely,
degreesorradians),anglesaredimensionlessquantities.Whowould
havethoughtit:adimensionlessquantitythathasunits!]
So,forthesineandcosinefunctions,whatistheindependent
variable?Answer:theangle,θ.
Ÿ Forthesinefunction,whatistherule
d
y
thatassociatesθwiththevalueofthe
dependentvariable(thatis,thesineofθ,or
θ
sin(θ))?Itisasfollows:Givenavalueofθas
x
definedabove,dividethey-coordinateofthe
pointbythedistanceofthepointfromthe
origin,d.Mathematically:sin(θ)=y/d.
Thisshouldmakeitclearthatsin(θ)isdimensionless(becauseitis
definedastheratiooftwoquantitieswiththesamedimensions).
Forthecosinefunction,theruleis“givenavalueofθ,dividethexcoordinateofthepointbythedistanceofthepointfromtheorigin,d.
Mathematically:cos(θ)=x/d.
TheSineandCosineFunctionsandtheirDerivatives
Youmighthavelearnedaboutthesineandcosinefunctionsin
termsoftheratioofthelengthsoftwosidesofarighttriangle.The
problemwithdefiningsineandcosinefunctionsinthiswayisthatit
worksonlyforpointsintheupper-rightquadrantofthecoordinate
system.Inanyoftheotherthreequadrants,eitherthex-coordinateor
they-coordinate,orboth,isnegative,whereasthelengthsofthesidesof
atrianglearepositivequantities(becausetheyarelengthsordistances,
notcoordinates).
Considerthepointinthediagramat
right.Inthiscase,y>0butx<0,sosin(θ)=
y/dispositive(and<1)butcos(θ)=x/dis
negative(and|x/d|<1).(Becausedisa
distance,notacoordinateintherectangular
coordinatesystem,itisalwayspositive.)
Forthepointinthediagramatright,y
andxareboth<0,sobothsin(θ)andcos(θ)
arenegative(andhavemagnitude<1).
Forapointinthelower-righthand
quadrant(notshown),sin(θ)wouldbe
negativebutcos(θ)wouldbepositive.
y
Ÿ d
θ
x
θ
x
y
Ÿ
d
Forthecaseswherethepointliesdirectlyoneitherthex-oryaxis,sothatthey-coordinateorthex-coordinateofthepointiszero,the
rulestillappliesinaverystraightforwardway.
Forexample,whenthepointlieson
Ÿ
y=d
they-axiswithy>0,asinthediagramat
right,sothatθ=90°=π/2radians,the
θ
x-coordinateofthepointiszeroandthe
x=0
y-coordinateequalsthedistanceofthe
pointfromtheorigin,d.Hence,
sin(θ)=y/d=1,andcos(θ)=x/d=0.
θ
Whenthepointliesonthex-axiswith
d
x<0,sothatθ=180°=πradians(asinthe
y=0 Ÿ
diagramatright),thex-coordinateisequal
x=−d
inmagnitudetothedistanced(butunliked,
isnegative)andthey-coordinateiszero.
Hence,sin(θ)=y/d=0,and
cos(θ)=x/d= −d/d=−1.
2
TheSineandCosineFunctionsandtheirDerivatives
Similarly,forthepointonthex-axiswithx>0(notshown),so
thaty=0,x=d,andθ=0,thensin(θ)=y/d=0and
cos(θ)=x/d=d/d=1.
Andfinally,forapointlyingonthey-axiswithy<0,sothaty=−d,
x=0,andθ=270°=3π/2radians, thensin(θ)=y/d=−d/d=−1and
cos(θ)=x/d=0.
Foranglesgreaterthan360° (2πradians),thefunctionvalues
repeatthemselves,sothesineandcosinefunctionsareperiodic.
Forangleslessthanzero(thatis,anglesmeasuredclockwisefrom
thex-axis),thefunctionvaluesalsorepeatthemselves.
Ifweplotthevaluesofthesinefunctionvs.theindependent
variable,θ,thenwegetthefollowing:
sin(θ)
θ
-360°-270°-180°-90°0°90°180°270°360°
Noteagainthatthemagnitudeofthesinefunctionneverexceeds
1.Itsmaximumvalueis1anditsminimumvalueis−1.Itiszeroat0°
(0radians)andatintervalsof180°(πradians).Thesinefunctionforms
apatternthatrepeatsitselfwithaperiodof360°(2πradians),sothat
sin(θ)=sin(θ+2π)forallθ.
3
TheSineandCosineFunctionsandtheirDerivatives
Theangle,θ,iscalledthephaseoftheperiodicpattern.It
determinesthepartofthepatternyouarein(apeak,atrough,or
somewhereinbetween).
Aplotofthecosinefunctionvs.θlookslikethis:
cos(θ)
θ
-360°-270°-180°-90°0°90°180°270°360°
Notethatlikethesinefunction,themaximumvalueofthecosine
functionis1anditsminimumvalueis−1,sothemagnitudeofthecosine
functionneverexceeds1.Unlikethesinefunction,thecosinefunction
hasavalueof1at0°(0radians)and−1at180°(πradians),andiszero
at90° (π/2radians)andat270° (3π/2radians). Likethesinefunction,
itformsapatternthatrepeatsitselfwithaperiodof360°(2πradians),
sothatcos(θ)=cos(θ+2π)forallθ.
Theplotofcos(θ)vs.θisthesameastheplotofthesinefunction
shiftedtotheleft(phaseshifted)by90°(π/2radians).Asaresult,we
canwritecos(θ)=sin(θ+π/2).Thatis,thesineandcosinefunctionsare
“outofphase”by90°(π/2radians).
III.DerivativesoftheSineandCosineFunctions
Youlearnedinfirst-semestercalculusthatthederivativesofthe
sineandcosinefunctionswithrespecttotheirindependentvariableare:
d sin (θ )
= cos (θ )
dθ
d cos (θ )
= −sin (θ )
dθ
4
TheSineandCosineFunctionsandtheirDerivatives
(Youmighthavelearnedthiswithadifferentnotation,wherethe
symbol“x”representstheindependentvariableratherthanθ,butthis
justchangesthenameassignedtotheindependentvariable,andthe
namecanbeanythingyouwant.Changingthenamedoesn’tchangethe
factthattheindependentvariableisanangleandisdimensionless,nor
doesitchangethefactthatthesineandcosinefunctionsaredefinedas
ratiosofapositioncoordinate(xory)toadistance(d)andhenceare
alsodimensionless.)
Howdoweknowwhatthesederivativesare?Thatis,howdowe
knowthat d sin (θ ) dθ = cos (θ ) ,forexample?Toanswerthat,weneedto
applythedefinitionofaderivative,atrigonometricidentity,andan
approximation.
First, d sin (θ ) dθ isjustashorthandnotationforthederivativeof
thesinefunctionwithrespecttoitsindependentvariable:
d sin (θ )
sin (θ + Δθ ) − sin (θ )
≡ lim
Δθ →0
dθ
(θ + Δθ ) − θ
= lim
Δθ →0
sin (θ + Δθ ) − sin (θ )
Δθ
Second,ausefultrigonometricidentity(whichwewon’ttryto
provehere)is:
sin ( a + b) = sin ( a) cos ( b) + cos ( a) sin ( b) Applythisidentityto sin (θ + Δθ ) inthedefinitionof d sin (θ ) dθ above:
⎡⎣sin (θ ) cos ( Δθ ) + cos (θ ) sin ( Δθ )⎤⎦ − sin (θ )
d sin (θ )
= lim
Δθ →0
dθ
Δθ
Reorganizethetermsinthenumeratorandbreakintotwoterms:
sin (θ ) cos ( Δθ ) − sin (θ )
cos (θ ) sin ( Δθ )
+ lim
= lim
Δθ →0
Δθ →0
Δθ
Δθ
5
TheSineandCosineFunctionsandtheirDerivatives
Factorsin(θ)outofthenumeratoroftheleft-handterm,and
recognizethatsin(θ)andcos(θ)don’tvaryaswetakethelimitas
Δθ → 0 (becauseθandΔθareindependentofeachother), sowecan
pullsin(θ)andcos(θ)outofthelimit(becausetheyareeffectively
constantsasfarasthelimit-takingoperationisconcerned):
sin (θ ) (cos ( Δθ ) −1)
d sin (θ )
sin ( Δθ )
= lim
+ cos (θ ) × lim
Δθ →0
Δθ →0
dθ
Δθ
Δθ
cos
Δ
θ
−1
( ( ) ) + cos (θ ) × lim sin (Δθ )
= sin (θ ) × lim
Δθ →0
Δθ →0
Δθ
Δθ
Finally,weinvokeanapproximation.Itispossibletoshowthatif
θissmall,thensin(θ)≅θ(consistentwiththefactthatsin(0)=0).You
canseewhythisapproximationmightbetrueontheplotofsin(θ)vs.θ
onpage3.Forvaluesofθclosetozero,theplotisalmostlinear,withan
interceptof0andaslopeofabout45°.Thesearecharacteristicsofa
plotofthefunctionf(x)=xvs.x,thatis,aplotofavariableagainstitself.
Hence,forvaluesofθclosetozero,thevaluesofsin(θ)areverynearly
equaltoθ,andtheygetcloserandclosertoθasθgetssmaller.Thatis,
theapproximationgetsbetterandbetterthesmallerθgets(and
becomesexactinthelimitasθgoesto0).(Itispossibletoshowthis
moreformallybyusingaTaylorseriesexpansionofthesinefunction.)
Similarly,itispossibletoshowthatforsmallvaluesofθ,
cos(θ)≅1–θ2(consistentwiththefactthatcos(0)=1).Youcansee
whythisapproximationmightbetrueontheplotofcos(θ)vs.θonpage
4.Forvaluesofθclosetozero,theplothasashapeverymuchlikean
invertedparabolawithinterceptof1,whichiswhatcos(θ)≅1–θ2is.
Hence,forvaluesofθclosetozero,thevaluesofcose(θ)arevarynearly
equalto1–θ2,andtheygetcloserandcloserto1–θ2asθgetssmaller.
(ItispossibletoshowthismoreformallybyusingaTaylorseries
expansionofthesinefunction.)
6
TheSineandCosineFunctionsandtheirDerivatives
Applyingtheseapproximationstosin(Δθ)andcos(Δθ)asΔθ
becomesverysmallintheexpressionforthederivativeabove,weget:
(
2
)
1− ( Δθ ) −1
d sin (θ )
Δθ
= sin (θ ) × lim
+ cos (θ ) × lim
Δθ →0
Δθ →0 Δθ
dθ
Δθ
= sin (θ ) × lim
− ( Δθ )
2
+ cos (θ ) × lim 1
Δθ →0
Δθ
= sin (θ ) × lim (−Δθ ) + cos (θ )
Δθ →0
Δθ →0
= sin (θ ) × 0 + cos (θ )
= cos (θ )
Wecandothesamesortofthingtoshowthat
d cos (θ )
dθ
= −sin (θ ) .
IV.DerivativesofCompositeFunctionsInvolvingSineandCosine
Functions
Inmanyapplications,theindependentvariableofthesineand
cosinefunctions,theangleθ,itselfdependsonsomethingelse,suchas
timeorlocation(position),orboth. Inthosecaseswewouldwrite
θ=θ(t)orθ=θ(x)orθ=θ(x,t),wheretrepresentstimeandx
representspositioninspacealongacoordinateaxis.Thevalueofthe
sineandcosinefunctionsdependsonlyindirectlyontand/orx,orboth,
andwecouldask,howmuchdoesthesineorcosinefunctionvaryper
unitchangeintorx?Sincethesineandcosinefunctionsdependonθ
directlyandontorxonlyindirectly(throughthedependenceofθont
and/orx),wehavetoanswerthequestionbyapplyingthechainrule.
Thechainrulecanbewrittenlikethis:
df (g(x)) df ( g) dg ( x )
=
×
dx
dg
dx
wheref(g)isafunctionof(thatis,dependsupon)theindependent
variableg,andg(x)itselfisafunctionthatdependsuponthe
independentvariablex.Thechainrulesaysthatthechangeinthe
functionfperunitchangeintheindependentvariablexdependsupon
howmuchgchangesperunitchangeinxandalsohowmuchfchanges
perunitchangeing.Thisshouldmakegood,intuitivesense!
7
TheSineandCosineFunctionsandtheirDerivatives
Ifthesineorcosinefunctiondependsupontheangle,θ,and θ
dependsupontorxorboth,thenwecanapplythechainruletoanswer
thequestion:howmuchdoesthesineorcosinefunctionvaryin
responsetoaunitchangeintorx?
d sin (θ (t )) d sin (θ ) dθ (t )
=
×
dt
dθ
dt dθ (t )
= cos (θ ) ×
dt
orinthecasewhereθ=θ(x,t):
∂sin (θ ( x, t )) d sin (θ ) ∂θ ( x, t )
=
×
∂t
dθ
∂t ∂θ ( x, t )
= cos (θ ) ×
∂t
andsimilarlyforaderivativewithrespecttox,andderivativesofthe
cosinefunction.
Acommonrelationshipbetweenθandtmightbe: θ (t ) = 2π t T ,
whereTistheperiodoftheperiodicsineorcosinefunction.(Thas
dimensionsoftime,justastdoes,sot/Tisdimensionless,andhenceso
isθ,whichinthiscaseisexpressedinradians.Theversionbasedon
unitsofdegreeswouldbe θ (t ) = 360° × t T .Eitherway,astvariesfrom
0toT,theanglevariesfrom0to2πor360°(onefullcycleofthe
periodicsineorcosinefunction).
dθ (t ) d ( 2π t T ) 2π dt 2π
2π
=
=
=
×1 =
So,wecouldwrite:
dt
dt
T dt T
T
andso:
d sin (θ (t )) d sin (θ ) dθ (t )
=
×
dt
dθ
dt
d ( 2π t T )
= cos (θ ) ×
dt
= cos (θ ) × ( 2π T )
= ( 2π T ) × cos ( 2π t T )
8
TheSineandCosineFunctionsandtheirDerivatives
Acommonrelationshipbetweenθandbothtandxmightbe:
⎛x t ⎞
2π
θ (x, t) =
( x − ct ) = 2π ⎜⎝ − ⎟⎠ L
L T
whereListhewavelengthofspatiallyperiodicvariationandc=L/Tis
thephasespeed,thespeedatwhichtheperiodicpatterntravelsparallel
tothex-axis).Thesineandcosinefunctionsinthiscasevary
periodicallyinbothspatialpositionandintime.
(Notethatforthephase,θ,toremainconstantastprogresses,
thenxmustalsovary—thatis,thepositionatwhichthephase(the
particularlocationrelativetotheperiodicpattern)remainsconstant
varieswithrespecttotime,anditmustvaryattheratec.) ⎛x t ⎞
2π
When θ (x, t) =
( x − ct ) = 2π ⎜⎝ − ⎟⎠ ,then:
L
L T
∂sin (θ ( x, t )) d sin (θ ) ∂θ ( x, t )
=
×
∂t
dθ
∂t
⎛ 2π
⎞
∂ ⎜ ( x − ct ) ⎟
⎝ L
⎠
= cos (θ ) ×
∂t
2π ∂ ( x − ct )
= cos (θ ( x, t )) ×
L
∂t
⎛ 2π
⎞ 2π ⎛ ∂t ⎞
= cos ⎜ ( x − ct ) ⎟ ×
⎜ −c ⎟
⎝ L
⎠ L ⎝ ∂t ⎠
⎛ 2π
⎞
2π
= −c
cos ⎜ ( x − ct ) ⎟
⎝ L
⎠
L
⎛ 2π
⎞
2π
=−
cos ⎜ ( x − ct ) ⎟
⎝ L
⎠
T
∂sin (θ ( x, t )) ∂cos (θ ( x, t ))
Applyingthechainruletodetermine
,
,
∂x
∂x
∂cos (θ ( x, t ))
and
isanalogous.
∂t
9