Topic 4. Periodic Functions & Applications II 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
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Topic 4
Periodic Functions & Applications II
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
Radians
In the equilateral triangle, each angle is 60o
r
If this chord were pushed onto the circumference,
r
this radius would be pulled back onto the other marked radius
60
1 radian 57o18’
2 radians 114o36’
Radians
3 radians 171o54’
radians = 180o
Radians
radians = 180o
/2 radians = 90o
/3 radians = 60o
/4 radians = 45o
etc
ModelExpress the following in degrees: (a) (b) (c)
Remember = 180o
ModelExpress the following in radians: (a) (b) (c)
Remember = 180o
Exercise
NewQ P 298
Set 9.1
Numbers 1 - 4
2. Definition of a periodic function, period and amplitude
3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians
Unit Circle
45
45
60
Now let’s do the same again, using radians
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
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
Exercise
NewQ P 318
Set 9.4 1 - 6
6. Applications of periodic functions
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
y = a sin b(x+c) + d
Tide range = 4m a = 2
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
High tide = 4.5 d = 2.5
Period = 4
Period = 2/b
b = 0.5
y = 2 sin 0.5(x+c) + 2.5
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
At the moment, high tide is at hours
We need a phase shift of units to the left
c =
y = 2 sin 0.5(x+) + 2.5
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
We want the height of the tide when t = 4
On GC, use 2nd Calc, value
h= 1.667m
(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)
(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
= 4 sec
(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
(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
Displacement after 10 sec should be the same as displacement after 2 sec
= 5.7cm to the left
(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
t =
1.1
5.1, 9.1, 13.1, 17.1
3.9
7.9, 11.9, 15.9, 19.9
Exercise
NewQ P 179
Set 5.2 1,3
y = a sin b(x+c)
Amplitude = 2.5
y = 2.5 sin b(x+c)
Amplitude = 2.5
Period = 6
6 = 2/b
b = /3
Period = 2/b
y = 2.5 sin /3(x+c)
Amplitude = 2.5
Phase shift = 4 ()
so c = -4
Period = 6
6 = 2/b
b = /3
Period = 2/b
y = 2.5 sin /3(x-4)
Amplitude = 2.5
Phase shift = 4 ()
so c = -4
Period = 6
6 = 2/b
b = /3
Period = 2/b
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.