Higher Mathematics Unit 3.4
The Wave Function
Contents
1.
1.
2.
3.
4.
5.
6.
7.
Preliminaries/Revision.
Sketching Graphs.
Waves on the Graphical Calculator.
Writing y  a cos x   b sin x  as a single sine or cosine.
Equations/Problems.
Further Considerations.
References/Resources.
Preliminaries/Revision of S4 Work
(i)
(ii)
(iii)
(iv)
(v)
Pupils must be familiar with degreee/radian measure.
Pupils must be familiar with basic sine/cosine graphs, both in degrees and radians.
Pupils must be able to use reference triangles to obtain exact values of ratios of simple
angles.
Pupils must be able to obtain exact values of ratios using CAST diagram and related
1
3
angles. e.g. tan 210   tan 30  
;
cos 210    cos 30   
. And so on.
2
3
Pupil must be familiar with graphs of sines/cosines of multiple angles, i.e. graphs of
functions such as f ( x)  3 cos 2 x  .
Abler pupils should be proficient in these, but weaker pupils might need some revision.
2.
Sketching Graphs
Given sufficient practice, most candidates should become highly proficient here.
The approach is best illustrated by example:
Sketch the graph of y  10 sin( 6 x  30) 
(0  x  60).
This is best done in the following steps:
(i)
(ii)
Write down maximum and minimum values. (10 and –10 here.)
Draw a set of axes on which to draw the graph:
y
10
O
15
30
45
-10
(iii)
Draw “basic” sine/cosine graph. (sine here):
60
x
(iv)
Now find maximum/minimum turning points and the intersection points with the axes.
When doing this, make use of the basic sine graph, above.
Also, as each point is obtained, mark it on the diagram that has been drawn in (ii), above.
For a maximum, 6x  30  90
x  20
Hence the max T.P. has coords (20,10).
For a minimum, 6x  30  270
x  50
Hence the min T.P. has coords (50,-10).
On the x axis, 6 x  30  0, 180, 360
x  5, 35, 65
Hence the points on x-axis are (5,0), (35,0).
[Note that (65,0) is outside the range here.]
Now obtain the y-intercept: On y axis, x  0  y  10 sin( 30)   5 .
Hence the point of intersection with the y axis is (0,-5).
(v)
Now, having been plotting the points as they have been found, “join the dots” to obtain a
fully annotated graph.
Notes
(a)
Consider at first only a single cycle of the graph. Once comprehension has been achieved,
go on to consider several cycles.
(b)
Do initial examples in degree measure. Once comprehension has been achieved, do
examples in radian measure also.
(c)
Graphs must be annotated, as indicated, above.
(d)
It is important to consider periods of graphs. For example, in radians, we have:
sin x  period 2
2
sin 2 x  period

2
2
sin 3 x  period
3
2
sin x  period
2

(e)
3.
Do examples that include vertical translations, such as y  100  50 sin( 2t  60)  , for
0  t  180.
The approach is to sketch the graph of y  50 sin( 2t  60)  first, then apply the vertical
translation.
Waves on the Graphic Calculator
The graphic calculator is an ideal resource for introducing the Wave Function.
Get the class to obtain the graphs of functions such as:
y  3 sin x   4 cos x  ,
y  5 cos x   12 sin x 
Help will be given so that the appropriate scale is chosen.
The pupils should be encouraged to make conjectures about the answers to the following
questions:
(i)
Max/min values?
(ii)
Period of graph?
(iii) Is the shape of the graph familiar?
The conclusion is that we should try to write functions such as the above ones as single sine or
cosine functions.
Don’t spend excessive time on this: the purpose of this activity is to provide a justification for
proceeding as we do in section 4, below.
4.
Writing y  a cos x   b sin x  as a single sine or cosine.
The approach is best illustrated by example:
Write f ( x)  3 sin x   4 cos x  in the form k sin( x   )  , where k  0 and 0    90.
The best method is to equate coefficients, as follows:
3 sin x   4 cos x   k sin( x   ) 
3 sin x   4 cos x   k[sin x  cos    cos x  sin   ]
3 sin x   4 cos x   k cos   sin x   k sin   cos x 
Equate coefficients to get k cos    3 and k sin    4
These must be stated.
Square both sides and add, to get
(k cos   ) 2  (k sin   ) 2  32  4 2
k 2 cos 2    k 2 sin 2    3 2  4 2
k 2 (cos 2    sin 2   )  32  4 2
k 2  1  32  4 2
k  32  4 2
k =5
The result for k can be stated without the above work being shown.
k sin  0 4

Dividing to get
k cos  0 3
4
tan  0 
3
Now, sin    0 and cos    0 .  0    90. And   53.1
Thus 3 sin x   4 cos x   5 sin( x  53.1)  .
Further examples could be:
5 cos x   12 sin x   R cos( x   ) 
2 cos x   3 sin x   k cos( x   ) 
5 sin 10T   10 cos10T   k sin( 10T   ) 
sin 3x   cos 3x   r sin( 3x   ) 
When picking examples at this stage always choose the function so that the phase angle,   , is
acute. Obtuse angles may be considered later.
At this stage, examples in radian measure may be done. Possible examples could be:
sin x  cos x  r sin( x   )
3 cos x  sin x  k cos( x   )
3 sin 5 x  6 cos 5 x  R sin( 5 x   )
[The first two will give an exact value for the phase angle, but in the third example the radian
measure will be obtained (to 3 dec. places, say) using the calculator in “rad” mode.]
Question
Part of the graph of
y  2 sin( x 0 )  5 cos( x 0 ) is shown in the
diagram.
a) Express y  2 sin( x 0 )  5 cos( x 0 ) in the form
k sin( x 0  a 0 ) where k >0 and 0  a  360 .
b) Find the coordinates of the minimum turning point P.
Answer
 1
 2
5.
k sin x 0 cos a 0  k cos x 0 sin a 0
k cos a 0  2, k sin a 0  5



3
4
5
k = 29 , (5.4...)
a = 68.20
29 sin( x  68.2) 0   29

6
x p  201.80

7
y p   29
(4)
(3)
stated explicitly
stated explicitly
Equations/Problem Solving
There are many interesting possibilities. Pupils should be good at:
(a)
Questions where there is a “lead in”. For example:
(i)
Write 5 sin x   3 cos x  in the form k sin( x   ) 
Hence solve the equation 5 sin x   3 cos x   1  2
(0  x  360).
(ii)
f ( x)  5 sin x   2 cos x 
(0  x  360) .
Express f (x) in the form R sin( x   )  , where R  0 and 0    90.
Hence write down the maximum and minimum values of f (x) , and the values of
 at which the maximum and minimum occur.
Another possibility here is:
1
and the value of x at which the maximum
10  f ( x)
occurs.
[Make sure that the denominator is always positive when choosing
an example such as this.]
Find the maximum value of
Another possibility is:
Write down the maximum and minimum value of ( f ( x)) 2 .
(b)
(c)
Harder questions with no “lead in”, e.g. questions such as:
(i)
Solve the equation 3 cos x   2 sin x   2.5
(ii)
f ( x)  cos x  sin x . Write down the maximum value of f (x ) , and the value of x,
with 0  x  2 , at which the maximum occurs.
(0  x  360).
Drawing graphs, once the wave equation has been written as a single sine or cosine is
worth considering ( with exact values).
An example of such a question is:
f ( x)  sin x  cos x (0  x  2 ) . Express f (x ) in the form R sin( x   ) with R  0
and 0   

.
2
Hence sketch the graph of f (x) , for 0  x  2 .
(d)
Solving Inequations. For example:
The depth of water in a harbour is modelled by the equation d  8  4 sin( 10T  30)  ,
(0  T  72) ,where d is the depth in metres and T is the time in hours after midnight.
Between which times is the depth of water less than 6 metres?
The method here is to draw a sketch graph of the function and solve the equation
algebraically. Then use the solutions in conjunction with the sketch graph to obtain the
required time intervals.
Example
a) Express 3 cos( x 0 )  5 sin( x 0 ) in the form k cos( x 0  a 0 ) where k>0 and 0  a  90
(4)
0
0
b) Hence solve the equation 3 cos( x )  5 sin( x ) =4 for 0  a  90 .
(3)
Answer
 1
 2





6.
3
4
5
6
7
k cos x cos a  k sin x sin a
k cos a  3, k sin a  5
stated explicitly
stated explicitly
k = 34
a = 59
34 cos( x  59) 0  4
x – 59 = any one of -46.7, 46.7, 313.3
x = 12.3
Further Considerations
(i)
This kind of question has come up:
k sin    3 and k cos    6, with k  0 and 0    90. Find the value of k and .
There are several approaches. The first (which nobody seems to think of!) is geometrical:
Using the fact that k  0 and   is acute, we can rearrange the equations and draw a
reference triangle.
6
3
Now, given that cos    and sin    , we get the following figure:
k
k
k
3

6
And the value of k and  may be got by P.T./Trig.
There are also Algebraic approaches, such as:
Divide, to get
k sin   3
sin  

 0.5  tan    0.5, etc.



6
cos 
k cos 
Or even square each equation to get k 2 sin 2    9 and k 2 cos 2    36 . Then add the two
resulting equations, take out k 2 as a common factor, use the fact that c 2  s 2  1 and the
value of k can be easily obtained.  can then be found by simple Trig.
(ii)
There might be a question where the equation of a wave must be obtained from the graph
of the wave. For example:
y
(60,10)
O 15
D
F
x
E
f ( x)  a sin( bx  c)  . Find the value of a, b, c and find the coords of D, E and F.
(iv)
Solving a wave equation without a “lead in”.
Thorough knowledge of the Addition Formulae is essential, so that the most convenient
form may be selected.
Give the pupils practice at making their selection.
For example,
3 sin x   5 cos x   1.
4 cos x   7 sin x   2.
k sin( x   )  is best.
k cos( x   )  is best.
The best approach is to write down the four different addition formulae and select the
R.H.S. that most resembles the L.H.S. of the wave equation. For example, in the first of
the two examples, above, it can be seen that the L.H.S. has the same form as the R.H.S. of
the formula sin( A  B)  sin A cos B  cos A sin B.
This is a “high level” skill, and even quite bright pupils will need practice.
7.
(v)
Phase angles that are not acute should be considered. This has never come up, but there is
always a “first time”. For phase angles that are obtuse or reflex, cause havoc early on
with weaker pupils!
(vi)
When doing a wave question the values of k cos   and k sin   must be stated. If they
are not both stated then the mark is not given.
References/Resources
1
2
“H.H.M. pp 300-314.
M.I.A. pp 250 - 260