topic 4 periodic functions & applications ii 1.definition of a radian and its relationship with...
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Topic 4
Periodic Functions & Applications II
1. Definition of a radian and its relationship with degrees
2. Definition of a periodic function, the period and amplitude
3. Definitions of the trigonometric functions sin, cos and tan of any angle in degrees and radians
4. Graphs of y = sin x, y = cos x and y = tan x
5. 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
6. Applications of periodic functions
7. Solutions of simple trigonometric equations within a specified domain
8. Pythagorean identity sin2x + cos2x = 1
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RadiansIn the equilateral triangle, each angle is 60o
rr
60
If this chord were pushed onto the circumference,
this radius would be pulled back onto the other marked radius
1.1. Definition of a radian and Definition of a radian and its relationship to degreesits relationship to degrees
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Radians1 radian 57o18’
2 radians 114o36’
3 radians 171o54’
radians = 180o
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Radians
radians = 180o
/2 radians = 90o
/3 radians = 60o
/4 radians = 45o
etc
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ModelExpress the following in degrees: (a)
(b)
(c)
Remember = 180o
613
54
32
1443645
1804
5
4
1206023
1802
3
2
39030136
18013
6
13
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ModelExpress the following in radians: (a)
(b)
(c)
Remember = 180o
43
72
225
45
45
180225225
52
52
1807272
18043
1804343
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Exercise
NewQ P 294
Set 8.1
Numbers 2 – 5
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2. Definition of a periodic function, period and 2. Definition of a periodic function, period and amplitudeamplitude
• 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
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3. Definition of the trigonometric functions sin, 3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and cos & tan of any angle in degrees and
radiansradians
Unit Circle
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ModelFind the exact value of: (a)
(b)
(c)
300sin
225tan
225cos
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ModelFind the exact value of: (a)
(b)
(c)
300sin
225tan
225cos
225cos45cos
21
45
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ModelFind the exact value of: (a)
(b)
(c)
300sin
225tan
225cos
225tan45tan
1
45
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ModelFind the exact value of: (a)
(b)
(c)
300sin
225tan
225cos
300sin60sin
23
60
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Now let’s do the same Now let’s do the same again, using radiansagain, using radians
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ModelFind the exact value of: (a)
(b)
(c)
300sin
225tan
225cos4
5
3
5
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ModelFind the exact value of: (a)
(b)
(c) 3
5sin
4
5tan
4
5cos
4
5cos
4cos
21
4
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ModelFind the exact value of: (a)
(b)
(c)
4
5tan
4tan
1
3
5sin
4
5tan
4
5cos
4
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ModelFind the exact value of: (a)
(b)
(c)
3
5sin
3sin
23
3
5sin
4
5tan
4
5cos
3
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Exercise
NewQ P 307
Set 9.2
Numbers 1, 2, 8-11
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4. Graphs of y = sin x, y = cos x and y = tan x4. Graphs of y = sin x, y = cos x and y = tan x
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The general shapes of the three major trigonometric graphsThe general shapes of the three major trigonometric graphs
y = sin x
y = cos x
y = tan x
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5. Significance of the constants A,B, C and D 5. Significance of the constants A,B, C and D on the graphs of…on the graphs of…
y = A sinB(x + C) + Dy = A sinB(x + C) + D
y = A cosB(x + C) + Dy = A cosB(x + C) + D
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1. Open the file y = sin(x)
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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
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Graph the following curves for 0 ≤ x ≤ 2a) y = 3sin(2x)
b) y = 2cos(½x) + 1
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Exercise
NewQ P 318
Set 9.4 1 - 6
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6. Applications of periodic functions6. Applications of periodic functions
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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
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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
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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
y = a sin b(x+c) + d
Tide range = 4m a = 2
Period = 4
Period = 2/b
High tide = 4.5 d = 2.5
b = 0.5
x
y
0
1
2
3
4
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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
y = 2 sin 0.5(x+c) + 2.5
We need a phase shift of units to the left
At the moment, high tide is at hours
c =
x
y
0
1
2
3
4
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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
y = 2 sin 0.5(x+) + 2.5
x
y
0
1
2
3
4
We want the height of the tide when t = 4
On GC, use 2nd Calc, value
h= 1.667m
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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)
X
Y
1 2 3 4 5
-8
-6
-4
-2
2
4
6
8
0
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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)
X
Y
1 2 3 4 5
-8
-6
-4
-2
2
4
6
8
0
Period = 4.5 - 0.5
= 4 sec
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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)
X
Y
1 2 3 4 5
-8
-6
-4
-2
2
4
6
8
0
Amplitude = 8
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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)
X
Y
1 2 3 4 5
-8
-6
-4
-2
2
4
6
8
0
Since the period = 4 sec
Displacement after 10 sec should be the same as displacement after 2 sec
= 5.7cm to = 5.7cm to the leftthe left
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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)
X
Y
1 2 3 4 5
-8
-6
-4
-2
2
4
6
8
0
Displacement= 5cm
t = 1.1
3.9 7.9, 11.9, 15.9, 19.9
5.1, 9.1, 13.1, 17.1
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Exercise
NewQ P 179
Set 5.2 1,3
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Model: Find the equation of the curve below.
X
Y
1 2 3 4 5 6 7 8 9 10
-2
2
0
Amplitude = 2.5 y = a sin b(x+c)
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Model: Find the equation of the curve below.
X
Y
1 2 3 4 5 6 7 8 9 10
-2
2
0
Amplitude = 2.5 y = 2.5 sin b(x+c)
Period = 6
Period = 2/b 6 = 2/b
b = /3
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Model: Find the equation of the curve below.
X
Y
1 2 3 4 5 6 7 8 9 10
-2
2
0
Amplitude = 2.5 y = 2.5 sin /3(x+c)
Period = 6
Period = 2/b 6 = 2/b
b = /3
Phase shift = 4 ()
so c = -4
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Model: Find the equation of the curve below.
X
Y
1 2 3 4 5 6 7 8 9 10
-2
2
0
Amplitude = 2.5 y = 2.5 sin /3(x-4)
Period = 6
Period = 2/b 6 = 2/b
b = /3
Phase shift = 4 ()
so c = -4
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Exercise
NewQ P 183
Set 5.3 1,4
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Find the equation of the curve below in terms of the sin function and the cosine
function.