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11B Topic 4_2

This resource explores the fundamentals of trigonometry using unit circles and triangles, focusing on finding exact values for angles in both degrees and radians. Students will engage with exercises related to sine, cosine, and tangent graphs, learning to identify the significance of constants in graph transformations. Applications of periodic functions, such as modeling tides and pendulum movement, are also examined. This comprehensive guide prepares learners for assessment by reinforcing core concepts and offering challenge questions to deepen understanding.

yael-fitzgerald
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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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