Download Periodic Function

Topic 4
Periodic Functions & Applications II
1.
2.
3.
4.
5.
6.
7.
8.
Definition of a radian and its relationship with degrees
Definition of a periodic function, the period and amplitude
Definitions of the trigonometric functions sin, cos and tan
of any angle in degrees and radians
Graphs of y = sin x, y = cos x and y = tan x
Significance of the constants A, B, C and D on the
graphs of y = A sin(Bx + C) + D, y = A cos(Bx + C) + D
Applications of periodic functions
Solutions of simple trigonometric equations within a
specified domain
Pythagorean identity sin2x + cos2x = 1
1. Definition of a radian and
its relationship to degrees
Radians
In the equilateral triangle, each angle is 60o
r
60
r
If this chord were pushed onto the
circumference,
this radius would be pulled back onto the
other marked radius
Radians
1 radian
2 radians


3 radians 
57o18’
114o36’
171o54’
 radians = 180o
Radians
 radians = 180o
/2 radians = 90o
/3 radians = 60o
/4 radians = 45o
etc
Model
Express the following in degrees: (a) 2
3
(b) 4
5
(c)
13
6
Remember  = 180o
2 2  180



 2  60  120
3
3
4 4  180

 4  36  144
5
5
13 13  180

 13  30  390
6
6
Model
Express the following in radians: (a)
225
(b) 72
(c) 43
Remember  = 180o
225
225  180
   54   
72
72  180
   52   
43
43  180
 
43
180
5
4
2
5
Exercise
NewQ P 298
Set 9.1
Numbers 1 - 4
2. Definition of a periodic function, period and
amplitude
• Consider the function shown here.
• A function which repeats values in
this way is called a Periodic
Function
• The minimum time taken for it to
repeat is called the Period (T).
This graph has a period of 4
• The average distance between
peaks and troughs is called
Amplitude (A). This graph has an
amplitude of 3
3. Definition of the trigonometric functions sin,
cos & tan of any angle in degrees and
radians
Unit Circle
Model
Find the exact value of: (a)
cos 225 
(b)
tan 225 
(c)
sin 300 
Model
Find the exact value of: (a)
cos 225 
(b)
tan 225 
(c)
sin 300 
cos 225
  cos 45

1
2


45
Model
Find the exact value of: (a)
cos 225 
(b)
tan 225 
(c)
sin 300 
tan 225
 tan 45
 1


45
Model
Find the exact value of: (a)
cos 225 
(b)
tan 225 
(c)
sin 300 
sin 300
  sin 60

 
3
2

60
Now let’s do the same
again, using radians
Model
Find the exact value of: (a)
(b)
(c)
cos 225  5
tan 225
sin 300


4
5
3
Model
5
cos
4
5
(b) tan
4
5
sin
(c)
3
Find the exact value of: (a)
5
cos
4
  cos

1
2

4

4
Model
Find the exact value of: (a)
(b)
(c)
5
tan
4
 tan
 1

4
5
cos
4
5
tan
4
5
sin
3

4
Model
Find the exact value of: (a)
(b)
(c)
5
sin
3
  sin
 
3
2

3
5
cos
4
5
tan
4
5
sin
3

3
Exercise
NewQ P 307
Set 9.2
Numbers 1, 2, 8-11
4. Graphs of y = sin x, y = cos x and y = tan x
The general shapes of the three major trigonometric graphs
y = sin x
y = cos x
y = tan x
5. Significance of the constants A,B and D on
the graphs of…
y = A sinB(x + C) + D
y = A cosB(x + C) + D
1. Open the file y = sin(x)
y = A cos B(x + C) + D
A: adjusts the amplitude
B: determines the period (T). This is the
distance taken to complete one cycle
where T = 2/B. It therefore, also
determines the number of cycles between
0 and 2.
C: moves the curve left and right by a
distance of –C (only when B is outside the brackets)
D: shifts the curve up and down the y-axis
Graph the following curves for 0 ≤ x ≤ 2
a) y = 3sin(2x)
b) y = 2cos(½x) + 1
Exercise
NewQ P 318
Set 9.4 1 - 6
6. Applications of periodic functions
Challenge question
Assume that the time between successive
high tides is 4 hours
High tide is 4.5 m
Low tide is 0.5m
It was high tide at 12 midnight
Find the height of the tide at 4am
Assume that the time between successive high tides is 4 hours
High tide is 4.5 m
Low tide is 0.5m
It was high tide at 12 midnight
Find the height of the tide at 4am
Assume that the time between successive high tides is 4 hours
High tide is 4.5 m
Low tide is 0.5m
It was high tide at 12 midnight
y = a sin b(x+c) + d
Tide range = 4m  a = 2
Find the height of the tide at 4am
High tide = 4.5  d = 2.5
y
Period = 4
 b = 0.5
Period = 2/b
4
3
2
1
x
0











Assume that the time between successive high tides is 4 hours
High tide is 4.5 m
Low tide is 0.5m
It was high tide at 12 midnight
y = 2 sin 0.5(x+c) + 2.5
Find the height of the tide at 4am
At the moment, high tide
is at  hours
y
We need a
phase shift of 
units to the left
4
3
c=
2
1
x
0











Assume that the time between successive high tides is 4 hours
High tide is 4.5 m
Low tide is 0.5m
It was high tide at 12 midnight
y = 2 sin 0.5(x+) + 2.5
Find the height of the tide at 4am
We want the height of
the tide when t = 4
On GC, use 2nd Calc,
value
y
4
3
 h= 1.667m
2
1
x
0











Model: The graph below shows the horizontal displacement of a
pendulum from its rest position over time:
(a) Find the period and amplitude of the movement.
(b) Predict the displacement at 10 seconds.
(c) Find all the times up to 20 sec when the displacement will be 5 cm
to the right (shown as positive on the graph)
Y
8
6
4
2
0
-2
-4
-6
-8
X
1
2
3
4
5
Model: The graph below shows the horizontal displacement of a
pendulum from its rest position over time:
(a) Find the period and amplitude of the movement.
(b) Predict the displacement at 10 seconds.
(c) Find all the times up to 20 sec when the displacement will be 5 cm
to the right (shown as positive on the graph)
Period = 4.5 - 0.5
Y
= 4 sec
8
6
4
2
0
-2
-4
-6
-8
X
1
2
3
4
5
Model: The graph below shows the horizontal displacement of a
pendulum from its rest position over time:
(a) Find the period and amplitude of the movement.
(b) Predict the displacement at 10 seconds.
(c) Find all the times up to 20 sec when the displacement will be 5 cm
to the right (shown as positive on the graph)
Amplitude = 8
Y
8
6
4
2
0
-2
-4
-6
-8
X
1
2
3
4
5
Model: The graph below shows the horizontal displacement of a
pendulum from its rest position over time:
(a) Find the period and amplitude of the movement.
(b) Predict the displacement at 10 seconds.
(c) Find all the times up to 20 sec when the displacement will be 5 cm
to the right (shown as positive on the graph)
Since the period = 4 sec
Y
8
Displacement after 10 sec should be the same
as displacement after 2 sec
6
4
2
0
-2
X
1
-4
-6
-8
= 5.7cm to the left
2
3
4
5
Model: The graph below shows the horizontal displacement of a
pendulum from its rest position over time:
(a) Find the period and amplitude of the movement.
(b) Predict the displacement at 10 seconds.
(c) Find all the times up to 20 sec when the displacement will be 5 cm
to the right (shown as positive on the graph)
Displacement= 5cm
Y
8
t=
6
4
1.1
5.1, 9.1, 13.1, 17.1
3.9
7.9, 11.9, 15.9, 19.9
2
0
-2
-4
-6
-8
X
1
2
3
4
5
Exercise
NewQ P 179
Set 5.2 1,3
Model: Find the equation of the curve below.
y = a sin b(x+c)
Amplitude = 2.5
Y
2
X
0
-2
1
2
3
4
5
6
7
8
9
10
Model: Find the equation of the curve below.
y = 2.5 sin b(x+c)
Amplitude = 2.5
Period = 6
 6 = 2/b
Period = 2/b
b = /3
Y
2
X
0
-2
1
2
3
4
5
6
7
8
9
10
Model: Find the equation of the curve below.
Amplitude = 2.5
Period = 6
 6 = 2/b
Period = 2/b
y = 2.5 sin /3(x+c)
Phase shift = 4 ()
so c = -4
b = /3
Y
2
X
0
-2
1
2
3
4
5
6
7
8
9
10
Model: Find the equation of the curve below.
Amplitude = 2.5
Period = 6
 6 = 2/b
Period = 2/b
y = 2.5 sin /3(x-4)
Phase shift = 4 ()
so c = -4
b = /3
Y
2
X
0
-2
1
2
3
4
5
6
7
8
9
10
Exercise
NewQ P 183
Set 5.3 1,4
Find the equation of the curve below in
terms of the sin function and the cosine
function.