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

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

Radians1 radian 57o18’

2 radians 114o36’

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

613

54

32

1443645

1804

5

4

1206023

1802

3

2

39030136

18013

6

13

ModelExpress the following in radians: (a)

(b)

(c)

Remember = 180o

43

72

225

45

45

180225225

52

52

1807272

18043

1804343

Exercise

NewQ P 294

Set 8.1

Numbers 2 – 5

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

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

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos

225cos45cos

21

45

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos

225tan45tan

1

45

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos

300sin60sin

23

60

Now let’s do the same Now let’s do the same again, using radiansagain, using radians

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos4

5

3

5

ModelFind the exact value of: (a)

(b)

(c) 3

5sin

4

5tan

4

5cos

4

5cos

4cos

21

4

ModelFind the exact value of: (a)

(b)

(c)

4

5tan

4tan

1

3

5sin

4

5tan

4

5cos

4

ModelFind the exact value of: (a)

(b)

(c)

3

5sin

3sin

23

3

5sin

4

5tan

4

5cos

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 x4. Graphs of y = sin x, y = cos x and y = tan x

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

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

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 ≤ 2a) y = 3sin(2x)

b) y = 2cos(½x) + 1

Exercise

NewQ P 318

Set 9.4 1 - 6

6. Applications of periodic functions6. 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

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

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

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

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

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

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

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

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

Exercise

NewQ P 179

Set 5.2 1,3

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)

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

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

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

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.