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Topic 4

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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

- Definition of a radian and its relationship to degrees

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

- 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

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

- 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

- y = 3sin(2x)
- y = 2cos(½x) + 1

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.

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.