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445.102 Mathematics 2

445.102 Mathematics 2. Module 4 Cyclic Functions Lecture 3 Making Waves. Transforming Trigonometric Graphs

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445.102 Mathematics 2

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  1. 445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 3 Making Waves

  2. Transforming Trigonometric Graphs Graphs of cyclic functions can be transformed the same way as polynomial, exponential and rational functions. We will present this as modelling the motion of waves under different conditions, but it is also helpful when identifying solutions of trigonometric equations.

  3. Post Lecture Exercises 1. 2. sin x and tan x are ODD, cos x is EVEN 3. cos π/4 = 1/√2 cosec π/3 = 2/√3 4. sin2x + cos2x = 1 Divide by sin2x: 1 + cos2x/sin2x = 1/sin2x => 1 + cot2x = cosec2x

  4. Post Lecture Exercises continued… a) cos x = -0.3 => x = 1.875 ± 2πn and x = (2π - 1.875) ± 2πn = 4.408 ± 2πn b) tan x = 5 => x = 1.373 ± 2πn and x = (2π + 1.373) ± 2πn = 7.657 ± 2πn c) sec x = 3 => cos x = 1/3 x = 1.231 ± 2πn and x = (2π - 1.231) ± 2πn = 5.052 ± 2πn d) csc x = -2 => sin x = -12 x = -0.524 ± 2πn and x = (2π + 0.524) ± 2πn = 6.807 ± 2πn

  5. 445.102 Lecture 4/4 • Administration • Last Lecture • Up & Down • Left & Right • Squishing & Stretching • Changing the Outline • Summary

  6. Preliminary Exercise y = x^2 y = x^2 – 5 y = (x – 4)^2 y = (x – 4)^2 +3

  7. Transforming Vertically

  8. 445.102 Lecture 4/4 • Administration • Last Lecture • Up & Down • Left & Right • Squishing & Stretching • Changing the Outline • Summary

  9. Transforming Horizontally

  10. 445.102 Lecture 4/4 • Administration • Last Lecture • Up & Down • Left & Right • Squishing & Stretching • Changing the Outline • Summary

  11. Horizontal Scaling

  12. 445.102 Lecture 4/4 • Administration • Last Lecture • Up & Down • Left & Right • Squishing & Stretching • Changing the Outline • Summary

  13. Transforming Amplitude

  14. Transforming Outlines

  15. 445.102 Lecture 4/4 • Administration • Last Lecture • Up & Down • Left & Right • Squishing & Stretching • Changing the Outline • Summary

  16. An Example .... The height of the tide below a wharf is given by the function: H(t) = -3 + 2sin 0.56t where t is the time after midnight in hours and H is the distance in metres When will the waterlevel be exactly 5 metres below the wharf?

  17. Solving Trigonometric Equations H(x) = 1 + 3sin(x/2) = 2.5

  18. Solving Trigonometric Equations sin(x/2) = (2.5 – 1)/3 = 0.5

  19. Lecture 4/3 – Summary • The graphs of cyclic functions are transformed in the same manner as graphs of other functions. • Such transformations can be seen as ways of modelling waves or cyclic phenomena which occur in our world. • They can also be used to “see” the solutions of trigonometric equations.

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