Download presentation

Presentation is loading. Please wait.

Published byJordan Donahue Modified over 2 years ago

1
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 sin 2 x + cos 2 x = 1

2
Radians In the equilateral triangle, each angle is 60 o r r 60 If this chord were pushed onto the circumference, this radius would be pulled back onto the other marked radius 1.Definition of a radian and its relationship to degrees

3
Radians 1 radian 57 o 18 2 radians 114 o 36 3 radians 171 o 54 radians = 180 o

4
Radians radians = 180 o /2 radians = 90 o /3 radians = 60 o /4 radians = 45 o etc

5
Model Express the following in degrees: (a) (b) (c) Remember = 180 o

6
Model Express the following in radians: (a) (b) (c) Remember = 180 o

7
Exercise NewQ P 294 Set 8.1 Numbers 2 – 5

8
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

9

10

11
3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians 3. Definition of the trigonometric functions sin, cos & tan of any angle in degrees and radians Unit Circle

12
Model Find the exact value of: (a) (b) (c)

13
45

14
Model Find the exact value of: (a) (b) (c) 45

15
Model Find the exact value of: (a) (b) (c) 60

16
Now lets do the same again, using radians

17
Model Find the exact value of: (a) (b) (c)

18

19

20

21
Exercise NewQ P 307 Set 9.2 Numbers 1, 2, 8-11

22
4. Graphs of y = sin x, y = cos x and y = tan x

23
The general shapes of the three major trigonometric graphs y = sin x y = cos x y = tan x

24
5. Significance of the constants A,B, C and D on the graphs of… y = A sinB(x + C) + D y = A cosB(x + C) + D

25
1.Open the file y = sin(x)Open the file y = sin(x)

26
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

27
Graph the following curves for 0 x 2 a)y = 3sin(2x) b)y = 2cos(½x) + 1

28
Exercise NewQ P 318 Set

29
6. Applications of periodic functions

30
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

31
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

32
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

33
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) We need a phase shift of units to the left At the moment, high tide is at hours c =

34
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+ ) We want the height of the tide when t = 4 On GC, use 2 nd Calc, value h= 1.667m

35
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)

36
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 sec

37
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

38
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) Since the period = 4 sec Displacement after 10 sec should be the same as displacement after 2 sec = 5.7cm to the left

39
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 = , 11.9, 15.9, , 9.1, 13.1, 17.1

40
Exercise NewQ P 179 Set 5.2 1,3

41
Model: Find the equation of the curve below. Amplitude = 2.5 y = a sin b(x+c)

42
Model: Find the equation of the curve below. Amplitude = 2.5 y = 2.5 sin b(x+c) Period = 6 Period = 2 /b 6 = 2 /b b = /3

43
Model: Find the equation of the curve below. 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

44
Model: Find the equation of the curve below. 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

45
Exercise NewQ P 183 Set 5.3 1,4

46

47

48

49

50

51

52

53

54

55

56

57
Find the equation of the curve below in terms of the sin function and the cosine function.

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google