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Lecture 5
- 1. Further Pure Mathematics II Polar Co-ordinates - Lesson 1 - Key Learning Points/Vocabulary: ● Plotting curves given in polar form. ● Investigating the shape of curves given in polar form with a graphical calculator or computer.
- 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. Example I Plot the curve r = θ r 0 π/12 π/6 π/4 π/3 … 2π θ
- 5. Example II Plot the curve r = 2 sin(θ) r 0 π/12 π/6 π/4 π/3 … 2π θ
- 7. Practice Construct tables showing value of θ between 0 and 2π in steps of π/12. Use these tables to plot the following curves on polar paper. • r = θ + sin (2θ) • r = 2 + cos (θ) • r = 3 sin (θ) • r = 1 + sin (θ) + cos (θ) • r = 1 + sin (θ) + cos (2θ) • r = 2 + sin (θ) + cos (2θ)
- 8. Polar Co-ordinates Experiment to learn the ‘classic’ curve shapes: cos _( . ._ sin ) sin 2 (1 cos ) (1 2cos ) cos sin cos r a r a c f r a r a r a r a r a r a b r a a θ α θ θ θ θ θ θ θ θ θ = = = = = = + = + = = =
- 9. Polar Co-ordinates Experiment to learn the ‘classic’ curve shapes: cos _( . ._ sin ) sin 2 (1 cos ) (1 2cos ) cos sin cos r a r a c f r a r a r a r a r a r a b r a a θ α θ θ θ θ θ θ θ θ θ = = = = = = + = + = = = Ray from origin Circle, centred on the origin, radius a Circle Four-leafed clover Cardioid Limaçon Spiral Rose curve – see investigation Lemniscate? Daisy
- 10. Further Pure Mathematics II Polar Co-ordinates - Lesson 2 - Key Learning Points/Vocabulary: ● Converting between Cartesian and Polar Co- ordinates.
- 11. To start with … Polar equations/graphs matching activity.
- 12. Theory The 2π convention refers to when all angles are given as a positive number between 0 and 2π 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.
- 13. Examples 1.) Using a.) 2π and b.) π convention, express the Cartesian point (3, -2) in polar form. 2.) Express the polar co-ordinate (2, 3π/4) in Cartesian form.
- 14. Practice 1.) Using a.) 2π 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, 3π/8). 4.) FP2&3, page 96, questions 7 and 8.
- 15. Homework See ‘Homework 1’ posted online.
- 16. Further Pure Mathematics II Polar Co-ordinates - Lesson 3 - Key Learning Points/Vocabulary: ● Polar co-ordinates and the use of symmetry.
- 17. - 4 - 2 2 4 6 - 2 2 4 x y 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.
- 18. Theory More generally if f(2α – θ) = f(θ) for all values of θ, then graph with equations r = f(θ) is symmetrical about the line θ = α.
- 19. Example Plot the graph r = 2 sin 2θ for 0 ≤ θ ≤ π/2. Prove that the graph is symmetrical about the line θ = π/4.
- 21. Practice Further Pure Mathematics 2 and 3 Exercise 6C Questions 1, 3 and 4
- 22. Further Pure Mathematics II Polar Co-ordinates - Lesson 4 - Key Learning Points/Vocabulary: ● Determining the maximum and minimum values of curves given in Polar form.
- 23. Derivates from Core 4 Function Derivative sin (ax) a cos (ax) cos (ax) - a sin (ax) tan (ax) a sec2 (ax) sec (ax) a sec (ax) tan (ax)
- 24. - 4 - 2 2 4 6 - 2 2 4 x y Example Determine the maximum and minimum values of r = 2 + cos θ.
- 25. Example II - 4 - 2 2 4 6 -2 2 4 x yDetermine the maximum and minimum values of r = 1 + cos 2θ.
- 26. Practice Further Pure Mathematics 2 and 3 Exercise 6D Questions 2 – part ii.) of each question only
- 27. Further Pure Mathematics II Polar Co-ordinates - Lesson 5 - Key Learning Points/Vocabulary: ● Finding the equations of tangents at the pole (origin).
- 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)
- 29. - 6 - 4 - 2 2 4 6 - 4 - 3 - 2 - 1 1 2 3 4 x yExample Find the equations of the tangents of r = 1 + cos 3θ at the pole using the π convention.
- 30. Practice Further Pure Mathematics 2 and 3 Exercise 6D Questions 2 – part iii.) of each question only
- 31. Further Pure Mathematics II Polar Co-ordinates - Lesson 6 - Key Learning Points/Vocabulary: ● Converting between Cartesian and Polar Equations.
- 32. sin θ = y/r → y = r sin θ cos θ = x/r → x = r cos θ x2 + y2 = r2 Theory
- 33. Examples Convert the following equations into polar form: i.) y = x2 ii.) (x2 + y2 )2 = 4xy
- 34. Examples (continued) Convert the following equations into Cartesian form: iii.) r = 2a cos θ iv.) r2 = a2 sin 2θ
- 35. Practice Further Pure Mathematics 2 and 3 Exercise 6E Questions 1 and 2
- 36. Homework See ‘Homework 2’ posted online.
- 37. Further Pure Mathematics II Polar Co-ordinates - Lesson 7 - Key Learning Points/Vocabulary: ● Review of Double Angle Formulae from Core 3: e.g. cos2 θ = … and sin2 θ = … ● Finding areas using Polar co-ordinates.
- 38. Integrals from Core 4 Function Derivative sin (ax) - (1/a) . cos (ax) cos (ax) (1/a) . sin (ax) … see next slide for more detail …
- 39. [ ] [ ] 2 2 cos sin sin sin sin cos cos cos 1 cos2 sin 2 2 1 cos 1 cos2 2 1 sin 1 cos2 2 bb a a bb a a b b a a b b a a b b a a d b a d a b d d d d d θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = = − = − = − = = + = − ∫ ∫ ∫ ∫ ∫ ∫ ∫
- 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 θθθ β α β α dfdr 22 )]([ 2 1 2 1 ∫∫ =
- 41. Example - 4 - 2 2 4 6 - 4 - 2 2 x y Find the area enclosed by the curve r = aθ for 0 < θ < 2π.
- 42. Example II - 4 - 2 2 4 6 - 4 - 2 2 x y Find the area enclosed by the curve r = 2 + cos θ for - π < θ < π.
- 43. Practice Further Pure Mathematics 2 and 3 Exercise 6F Questions 1 onwards
- 44. Homework See ‘Homework 3’ posted online.
- 45. Further Pure Mathematics II Polar Co-ordinates - Lesson 8 - Key Learning Points/Vocabulary: ● End of Topic Test based on FMN OCR FP2 materials. ● Learning Summary. ● Past Exam Questions.