1. 11B Topic 4_2

2. Model Find the exact value of: (a) (b) (c) We are now familiar with the Unit Circle, but to answer these questions we will need to use the Unit Triangles as well…

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4. Model Find the exact value of: (a) (b) (c) 45 1 1 1 2

5. Model Find the exact value of: (a) (b) (c) 45 1 1 1 2

6. Model Find the exact value of: (a) (b) (c) 60 1 1 1 2

7. Now let’s do the same again, using radians Scootle: 11 Maths B folder Topic 4 (PWJXSR) Topic 4 (PWJXSR) Trig Radians

8. Model Find the exact value of: (a) (b) (c) 1 1 1 2

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12. Exercise NewQ P 307 Set 9.2 Numbers 1, 2, 8-11 For Homework, look at… Scootle: 11 Maths B folder Topic 4 (PWJXSR) Topic 4 (PWJXSR) Trig degrees Trig radians

13. For Homework, look at… Scootle: 11 Maths B folder Topic 4 (PWJXSR) Topic 4 (PWJXSR) Trigonometry: assessment

14. You should now be familiar with the general shape of the three major trignometric graphs

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

16. 5. Significance of the constants A,B and D on the graphs of… y = A sin[B(x + C)] + D y = A cos[B(x + C) ]+ D

17. 2.Open the file y = sin(x)Open the file y = sin(x) (Excel File) Scootle: 11 Maths B folder Topic 4 (PWJXSR) Topic 4 (PWJXSR) Eagle Cat 1.Open the file y = Asin[B(x+C)]+dOpen the file y = Asin[B(x+C)]+d (Autograph file)

18. 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

19. Graph the following curves for 0 ≤ x ≤ 2  a)y = 3sin(2x) b)y = 2cos(½x) + 1 c)y = sin[2(x +  )] d)y = 4cos[2(x -  /2)] – 3

20. Exercise NewQ P 318 Set 9.4 1 - 6

21. 6. Applications of periodic functions

22. Challenge Question (1) High tide is 4.5 m at midnight Low tide is 0.5m at 6am i)Find the height of the tide at 7pm? ii)Between what times will the tide be greater than or equal to 3m?

23. Use y = A cos B(x+C) + D i)Find “A” Tide range = 4.5 - 0.5 = 4  A = 2  y = 2cos B(x+C) + D iii) Find “B” Period = 12 ii) Find “D” D = 4.5 – 2 = 2.5  y = 2cos B(x+C) + 2.5 iv) Find “C” We can see from the graph that no C-value is needed High tide is 4.5 m at midnight Low tide is 0.5m at 6am i)Find the height of the tide at 7pm? ii)Between what times will the tide be greater than or equal to 3m?

24. By use of TI calculator… i)What is the tide height at 7pm? Graph using suitable windows 2 nd  Calc  option 1. Value Enter 19 Answer = 0.77m (2D.P.) ii)Tide above 3m Add y = 3 to the graph 2 nd  Calc  option 5. Intersect Follow prompts Answer = MN – 2:31am 9:29am – 2:31pm 9:29pm – MN

25. Challenge Question (2) High tide of 4.2m occurs in a harbor at 4am Tuesday and the following low tide of 0.8m occurs 6¼ hours later. If a ship entering the harbor needs a minimum depth of water of 3m, what times on Tuesday can this vessel enter?

26. 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)

27. 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

28. 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

29. 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 will be the same as displacement after 2 sec = 5.7cm to the left

30. 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 3.97.9, 11.9, 15.9, 19.9 5.1, 9.1, 13.1, 17.1

31. Exercise NewQ P 179 Set 5.2 1,3

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

33. 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

34. 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

35. 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

36. Exercise NewQ P 183 Set 5.3 1,4

37. Exercise 5.3 pg 183, No.4

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