Polar Co Ordinates

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Polar Co Ordinates

  1. 1. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 1 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Plotting curves given in polar form. </li></ul><ul><li>● Investigating the shape of curves given in polar form with a graphical calculator or computer. </li></ul>
  2. 2. Theory An equation in polar form is given in the r = f( θ ) where θ is an angle measured anti-clockwise from the origin/positive x-axis and r is the distance from the origin. e.g. If we are working with r = 2 + sin θ when θ = π /2, r = 3.
  3. 3. Example I Plot the curve r = θ r 0 π /12 π /6 π /4 π /3 … 2 π θ
  4. 5. Example II Plot the curve r = 2 sin( θ ) r 0 π /12 π /6 π /4 π /3 … 2 π θ
  5. 7. Practice <ul><li>Construct tables showing value of θ between 0 and 2 π in steps of π /12. Use these tables to plot the following curves on polar paper. </li></ul><ul><ul><li>r = θ + sin (2 θ ) </li></ul></ul><ul><ul><li>r = 2 + cos ( θ ) </li></ul></ul><ul><ul><li>r = 3 sin ( θ ) </li></ul></ul><ul><ul><li>r = 1 + sin ( θ ) + cos ( θ ) </li></ul></ul><ul><ul><li>r = 1 + sin ( θ ) + cos (2 θ ) </li></ul></ul><ul><ul><li>r = 2 + sin ( θ ) + cos (2 θ ) </li></ul></ul>
  6. 8. Polar Co-ordinates Experiment to learn the ‘classic’ curve shapes:
  7. 9. Polar Co-ordinates Experiment to learn the ‘classic’ curve shapes: Ray from origin Circle, centred on the origin, radius a Circle Four-leafed clover Cardioid Lima ç on Spiral Rose curve – see investigation Lemniscate? Daisy
  8. 10. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 2 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Converting between Cartesian and Polar Co-ordinates. </li></ul>
  9. 11. To start with … Polar equations/graphs matching activity.
  10. 12. Theory The 2 π convention refers to when all angles are given as a positive number between 0 and 2 π e.g. all angles are measured anti-clockwise from the origin/positive x-axis. The π convention refers to when all angles are given as a positive or negative number between - π and + π e.g. all angles are at most half a turn either way from the origin/positive x-axis.
  11. 13. Examples 1.) Using a.) 2 π and b.) π convention, express the Cartesian point (3, -2) in polar form. 2.) Express the polar co-ordinate (2, 3 π /4) in Cartesian form.
  12. 14. Practice 1.) Using a.) 2 π and b.) π convention, express the Cartesian point (-2, -4) in polar form. 2.) Express the polar co-ordinate (3, - π /4) in Cartesian form. 3.) Find the area of the triangle form by the origin and the polar co-ordinates (2, π /4) and (4, 3 π /8). 4.) FP2&3, page 96, questions 7 and 8.
  13. 15. Homework See ‘Homework 1’ posted online.
  14. 16. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 3 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Polar co-ordinates and the use of symmetry. </li></ul>
  15. 17. r = 2 + cos θ If f( θ ) = f(- θ ) for all values of θ , the graph with polar equation r = f( θ ) is symmetrical about the line θ = 0.
  16. 18. Theory More generally if f(2 α – θ ) = f( θ ) for all values of θ , then graph with equations r = f( θ ) is symmetrical about the line θ = α .
  17. 19. Example Plot the graph r = 2 sin 2 θ for 0 ≤ θ ≤ π /2. Prove that the graph is symmetrical about the line θ = π /4.
  18. 21. Practice Further Pure Mathematics 2 and 3 Exercise 6C Questions 1, 3 and 4
  19. 22. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 4 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Determining the maximum and minimum values of curves given in Polar form. </li></ul>
  20. 23. Derivates from Core 4 Function Derivative sin (ax) a cos (ax) cos (ax) - a sin (ax) tan (ax) a sec 2 (ax) sec (ax) a sec (ax) tan (ax)
  21. 24. Example Determine the maximum and minimum values of r = 2 + cos θ .
  22. 25. Example II Determine the maximum and minimum values of r = 1 + cos 2 θ .
  23. 26. Practice Further Pure Mathematics 2 and 3 Exercise 6D Questions 2 – part ii.) of each question only
  24. 27. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 5 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Finding the equations of tangents at the pole (origin). </li></ul>
  25. 28. Theory If f( α ) = 0 but f( α ) > 0 in an interval α < θ < … or … < θ < α then the line θ = α is a tangent to the graph r = f( θ ) at the pole (origin)
  26. 29. Example Find the equations of the tangents of r = 1 + cos 3 θ at the pole using the π convention.
  27. 30. Practice Further Pure Mathematics 2 and 3 Exercise 6D Questions 2 – part iii.) of each question only
  28. 31. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 6 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Converting between Cartesian and Polar Equations. </li></ul>
  29. 32. sin θ = y/r -> y = r sin θ cos θ = x/r -> x = r cos θ x 2 + y 2 = r 2 Theory
  30. 33. Examples Convert the following equations into polar form: i.) y = x 2 ii.) (x 2 + y 2 ) 2 = 4xy
  31. 34. Examples (continued) Convert the following equations into Cartesian form: iii.) r = 2a cos θ iv.) r 2 = a 2 sin 2 θ
  32. 35. Practice Further Pure Mathematics 2 and 3 Exercise 6E Questions 1 and 2
  33. 36. Homework See ‘Homework 2’ posted online.
  34. 37. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 7 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● Review of Double Angle Formulae from Core 3: </li></ul><ul><li>e.g. cos 2 θ = … and sin 2 θ = … </li></ul><ul><li>● Finding areas using Polar co-ordinates. </li></ul>
  35. 38. Integrals from Core 4 … see next slide for more detail … Function Derivative sin (ax) - (1/a) . cos (ax) cos (ax) (1/a) . sin (ax)
  36. 40. Theory The area of the region bounded by the graph r = f ( θ ) and the radii θ = α and θ = β is given by
  37. 41. Example Find the area enclosed by the curve r = a θ for 0 < θ < 2 π .
  38. 42. Example II Find the area enclosed by the curve r = 2 + cos θ for - π < θ < π .
  39. 43. Practice Further Pure Mathematics 2 and 3 Exercise 6F Questions 1 onwards
  40. 44. Homework See ‘Homework 3’ posted online.
  41. 45. <ul><li>Further Pure Mathematics II </li></ul><ul><li>Polar Co-ordinates </li></ul><ul><li>Lesson 8 - </li></ul><ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>● End of Topic Test based on FMN OCR FP2 materials. </li></ul><ul><li>● Learning Summary. </li></ul><ul><li>● Past Exam Questions. </li></ul>

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