gonometrT:The Wave Function
(e.9.the interaction
of soundwaves
It is possible
to modelthe behaviour
of wavesin real-lifesituations
the
Consider
the resultof combining
or thetideswheretwo bodiesof watermeet)usingtrigonometry.
=
=
graph
y
resultant
would
look
wavesrepresented
by the functions/ sinx" and cosxo.To findwhatthe
plot
on the axesbelow.
like,complete
the tableof values(accurate
to 1 d.p.)and
0o
450
90'
135'
180"
2250
270'
315.
3600
sinx o
cosxo
sinxo + cosxo
Max=
Min=
Maxwhenx=
Minwhenx=
y=
Theaboveshowsthataddingthegraphsof y = sinxoand)/ = cosxo produces
the cosinegraphshifted
45"to the right(i.e.y = cos(x-a)o),andstretched
vertically
by a factorofroughly1.4(i.e.y= kcosx").
Asdrawinga graphis slow,thiscanalsobeshownalgebraically,
asfollows.
Example1: Writesinxo+ cosxoin theformkcos(X- a)o,where0 < cl < 360.
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betweenwaves;to expressthe answerin
to includethe difference
canbe expanded
Thistechnique
otherforms;andto includedouble(or higher)angles,but only when the angles of both the sin and
to a wavefunction,but 2cos2x+5sin3x
bechanged
cos term are the same (i.e.2coslx+5sin2xcan
couldnot).
Example2: Writesinx -{3cosx in the formkcos(X' a), where0 3 u < 2n
Example3: Writel2cosxo- 5sinx' in theformksin(x- a)o,where0 < cl < 360
Exampfe4: Write2sin20- cos20in the formksin(2O+ o), where0 < a < 2n
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questions
In most cases,
likethesewillbesplitintotwo parG,witha) beinga "write
in theform
y = k cos(x- a)" followedby b) asking"henceor otherwise
so|ve...........".
Usethe wave function from
paft a) to solve the equation!
Example5:
a) WriteZcosxo- sinxoin theform kas(x- o)' where0 < a < 360
b) Hencesolve2cosxo- sinxo = -1 where0 < x< 360
Lookbackat the graphyoudrewof sinxo+ cosxo.Themaximum
valueof the graphis r/2at the point
-{Z
=
=
wherex 45o,andthe minimum
valueis
at the pointwherex 225. Compare
theseto the
maximumandminimumofy=cosxo,i.e.amaximumoflwhereX=0oor360oandaminimum
wherex = 180o.
Sincesinxo+ cosxo= {2 cos(x- 45)",we canseethatthe maximum
andminimum
valueschange
from+ltotk
Themaximum
valueoccurswherer/2cos(x- 45)' - r/2,i.e.cos(x- 45)o= 1. Similarly,
the minimum
-{2,
-1
=
valueoccurswhere!2 cos(x- 45;"
i.e.cos(x- 45)o
Forasinx+ bcosx= {cos(x-g):
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Wecanusethisinformation
to solveproblems
or sketchgraphsof y = a sinx+ b cosx.
Example6:
a) Write{3sinx+cosxin theform kcos(x-o), where0 < a<2n
b) Hencesketchandfullyannotatethegraph
of y= {3sinx+ cosx for0 < x<2n
PastPaperExample:Statethe maximum
valueof the functionf(x) = 5 - 13,ot( ,-i).
\
2)
The WaveFunction:TooicChecklist
kcos(x- a)
kcos(x+cr)
ksin(xt o)
Maximaand minima
Solving
equations
Basic Exercise
16C,p 304,O 1 - 5
Exam Exercise16D,p 304,Q 2
Basic Exercise16E,p 305,O 1
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Basic
Exam
Basic
Exam
Basic
Exam
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Exercise
16E.o 305.O 2. 3
Exercise
16E,p 306,Q 4, 5
Exercise16G,p 307,Q L
Exercise
16G,p 308,O 3, 4, 5,7
Exercise
16H,p 309,O 1, 2
Exercise
16H.o 310.O 3. 4
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