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 … 2 π θ

5. Example II Plot the curve r = 2 sin( θ ) r 0 π /12 π /6 π /4 π /3 … 2 π θ

8. Polar Co-ordinates Experiment to learn the ‘classic’ curve shapes:

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

11. To start with … Polar equations/graphs matching activity.

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.

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.

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.

15. Homework See ‘Homework 1’ posted online.

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.

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 .

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.

21. Practice Further Pure Mathematics 2 and 3 Exercise 6C Questions 1, 3 and 4

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)

24. Example Determine the maximum and minimum values of r = 2 + cos θ .

25. Example II Determine the maximum and minimum values of r = 1 + cos 2 θ .

26. Practice Further Pure Mathematics 2 and 3 Exercise 6D Questions 2 – part ii.) of each question only

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. Example Find the equations of the tangents of r = 1 + cos 3 θ 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

32. sin θ = y/r -> y = r sin θ cos θ = x/r -> x = r cos θ x 2 + y 2 = r 2 Theory

33. Examples Convert the following equations into polar form: i.) y = x 2 ii.) (x 2 + y 2 ) 2 = 4xy

34. Examples (continued) Convert the following equations into Cartesian form: iii.) r = 2a cos θ iv.) r 2 = a 2 sin 2 θ

35. Practice Further Pure Mathematics 2 and 3 Exercise 6E Questions 1 and 2

36. Homework See ‘Homework 2’ posted online.

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)

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

41. Example Find the area enclosed by the curve r = a θ for 0 < θ < 2 π .

42. Example II 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.