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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

ModelExpress the following in degrees: (a) (b) (c)

Remember = 180o

ModelExpress the following in radians: (a) (b) (c)

Remember = 180o

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 of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

Model of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

45

Model of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

45

Model of any angle in degrees and radiansFind the exact value of: (a) (b)(c)

60

Now let’s do the same again, using radians of any angle in degrees and radians

Model of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

Model of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

Model of any angle in degrees and radiansFind the exact value of: (a) (b) (c)

Model of any angle in degrees and radiansFind the exact value of: (a) (b)(c)

4. Graphs of y = sin x, y = cos x and y = tan x of any angle in degrees and radians

The general shapes of the three major trigonometric graphs of any angle in degrees and radians

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 = of…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 of… ≤ x ≤ 2

- y = 3sin(2x)
- y = 2cos(½x) + 1

Challenge question of…

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 of… 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

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

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+ of…) + 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

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)

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

= 4 sec

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

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time: (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.

Since the period = 4 sec

Displacement after 10 sec should be the same as displacement after 2 sec

= 5.7cm to the left

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

t =

1.1

5.1, 9.1, 13.1, 17.1

3.9

7.9, 11.9, 15.9, 19.9

Model: Find the equation of the curve below. a pendulum from its rest position over time:

y = a sin b(x+c)

Amplitude = 2.5

Model: Find the equation of the curve below. a pendulum from its rest position over time:

y = 2.5 sin b(x+c)

Amplitude = 2.5

Period = 6

6 = 2/b

b = /3

Period = 2/b

Model: Find the equation of the curve below. a pendulum from its rest position over time:

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

Model: Find the equation of the curve below. a pendulum from its rest position over time:

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

Find the equation of the curve below in terms of the sin function and the cosine function.

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