Download presentation

Presentation is loading. Please wait.

1
**Periodic Functions & Applications II**

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

2
**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

3
**Radians 1 radian 57o18’ 2 radians 114o36’ 3 radians 171o54’**

4
**Radians radians = 180o /2 radians = 90o /3 radians = 60o**

etc

5
**Model Express the following in degrees: (a) (b) (c)**

Remember = 180o

6
**Model Express the following in radians: (a) (b) (c)**

Remember = 180o

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

11
**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
**Model Find the exact value of: (a) (b) (c)**

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 let’s do the same again, using radians**

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

18
**Model Find the exact value of: (a) (b) (c)**

19
**Model Find the exact value of: (a) (b) (c)**

20
**Model Find the exact value of: (a) (b) (c)**

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
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**

y = 3sin(2x) 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 High tide = 4.5 d = 2.5 Period = 4 Period = 2/b 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) + 2.5 At the moment, high tide is at hours We need a phase shift of units to the left 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+) + 2.5 We want the height of the tide when t = 4 On GC, use 2nd 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 = 1.1 5.1, 9.1, 13.1, 17.1 3.9 7.9, 11.9, 15.9, 19.9

40
Exercise NewQ P 179 Set ,3

41
**Model: Find the equation of the curve below.**

y = a sin b(x+c) Amplitude = 2.5

42
**Model: Find the equation of the curve below.**

y = 2.5 sin b(x+c) Amplitude = 2.5 Period = 6 6 = 2/b b = /3 Period = 2/b

43
**Model: Find the equation of the curve below.**

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

44
**Model: Find the equation of the curve below.**

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

45
Exercise NewQ P 183 Set ,4

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

Similar presentations

Presentation is loading. Please wait....

OK

Right Triangle Trigonometry

Right Triangle Trigonometry

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on power grid system Ppt on 3 idiots movie download Ppt on good engineering practices Ppt on resistance temperature detector calibration Download ppt on community planning for disaster management Ppt on polynomials for class 9 download Earthquakes for kids ppt on batteries Ppt on weapons of mass destruction wow Ppt on data handling for class 8th Ppt on security features of atm machine