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11B Topic 4_2

11B Topic 4_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…. 1. 1. 2. 1.

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11B Topic 4_2

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  1. 11B Topic 4_2

  2. ModelFind 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…

  3. 1 1 2 1

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

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

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

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

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

  9. ModelFind the exact value of: (a) (b) (c) 1 2 1 1

  10. ModelFind the exact value of: (a) (b) (c) 1 2 1 1

  11. ModelFind the exact value of: (a) (b)(c) 1 2 1 1

  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) • Trig degrees • Trig radians

  13. For Homework, look at… • Scootle: 11 Maths B folder • 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. Open the file y = Asin[B(x+C)]+d (Autograph file) Open the file y = sin(x) (Excel File) • Scootle: 11 Maths B folder • Topic 4 (PWJXSR) • Eagle Cat

  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 • y = 3sin(2x) • y = 2cos(½x) + 1 • y = sin[2(x + )] • 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 • Find the height of the tide at 7pm? • Between what times will the tide be greater than or equal to 3m?

  23. High tide is 4.5 m at midnight Low tide is 0.5m at 6am • Find the height of the tide at 7pm? • Between what times will the tide be greater than or equal to 3m? iii) Find “B” Period = 12 • Use y = A cos B(x+C) + D • Find “A” • Tide range = 4.5 - 0.5 = 4 • A = 2 • y = 2cos B(x+C) + D 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

  24. By use of TI calculator… • What is the tide height at 7pm? • Graph using suitable windows • 2nd Calc  option 1. Value • Enter 19 • Answer = 0.77m (2D.P.) • Tide above 3m • Add y = 3 to the graph • 2nd 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 5.1, 9.1, 13.1, 17.1 3.9 7.9, 11.9, 15.9, 19.9

  31. Exercise NewQ P 179 Set 5.2 1,3

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

  33. 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

  34. 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

  35. 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

  36. Exercise NewQ P 183 Set 5.3 1,4

  37. Exercise 5.3 pg 183, No.4

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

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