1. Graphing in polar coordinate

2. Reminder (1) The following polar representations represent the same point: ( r, θ ) ( r, θ + 2n π ) ; nεZ ( - r, θ + (2n+1)π ) ; nεZ

3. Examples (1) The following polar representations represent the same point ( 5, π/2 ) ( 5, π/2 + 2 π ) = ( 5, 5π/2) ( 5, π/2 - 2 π ) = ( 5, - 3π/2) ( - 5, π/2 + π ) = ( - 5, 3π/2 ) ( - 5, π/2 – π ) = ( - 5, -π/2 )

4. Reminder (2) All of the polar representations of the form (0, θ) represent the pole ( the origin): Examples ( 0, 0 ) ( 0, π ) ( 0, π/6 )

5. The Graph of θ = θ 0 For any value of r, θ is always θ 0 Any point of the form ( r, θ 0 ), where r is any real number, belongs to the graph of the curve All the following points belongs to that graph: ( 1, θ 0 ), ( 2, θ 0 ), ( 3, θ 0 ) ( -1, θ 0 ) ≡(1, θ 0 + π), ( -2, θ 0 ) ≡ (2, θ 0 + π) ( √3/5, θ 0 ) The graph is a straight line making an angle equal to θ 0 with the polar axis

6. Example θ = π/4 The graph is a straight line making an angle equal to π/4 with the polar axis

8. Reminder The equation of this straight line is y = x We can arrive at this equation, as follows: We have, θ = π/4 → tan θ = 1 → sin θ / cos θ = 1 → r sin θ / r cos θ = 1 → y / x = 1 → y = x

9. The Graph of r = cθ, where c is nonzero constant Example r = θ θr = θ( r, θ ) 00 ( 0, 0 ) π/2 (π/2, π/2 ) ππ(π, π) 3π/2 (3π/2, 3π/2 ) 2π2π2π2π(2π, 2π ) 2π+π/2 (2π+ π/2, 2π+ π/2 ) 3π3π3π3π(3π, 3π )

10. The graph is a spiral

12. The Graph of r = r 0, where are is a positive number For any value of θ, r is always r 0 Any point of the form (r 0, θ ), where θ is any angle ( or any real number ), belongs to the graph of the curve All the following points belongs to that graph: ( r 0, 0 ), ( r 0, π/2 ), ( r 0, π), ( r 0, π/6) ( - r 0, π ) ≡ ( r 0, 0 ), (- r 0, 3π/2 ), ≡ ( r 0, π/2 ) The graph is a circle centered at the origin and of radius equal to r 0

13. Example: r = 5 The graph is a circle centered at the origin and of radius equal to 5

14. r = 5

15. The graph of: r = c sinθ Or r = c cosθ Where c is a nonzero constant

16. Example r = 6 sin θ

17. θ 00(0,0) π/26(6, π/2) π0(0, π) 3π/2- 6(- 6, 3π/2)≡ (6, π/2) 2π2π0(0, 2π) ≡ (0,0)

18. r = 6sin θ

19. The graph is a circle centered at (0,3) of radius equal to 6/2 = 3

20. The graph of: r = a sinθ + b r = a cosθ + b Where, a/b = 1 or - 1 Cardioids ( Heart) ( special case of the lima ς on (

21. Example r = 5 - 5sinθ

22. The Graph of r = 5 - 5sinθ = 5(1-sin θ) θ r = 5(1-sin θ)( r, θ ) 05(5,0) π/20(0, π/2) π5(5, π) 3π/210(10, 3π/2) 2π2π5(5, 2π)

23. The Graph of r = 5 - 5sinθ

24. The graph of: r = a sinθ + b r = a cosθ + b Where a and b are nonzero. Lima ς on ( Snail)

25. Example r = 2 - 4sinθ = 2 ( 1 – 2sinθ)

26. The graph of r = 2 – 4sinθ = 2 ( 1 – 2 sinθ) r = 0 if sinθ = ½ Or θ = π/6, θ= π - π/6 = 5π/6 Thus, we must pay attention to these values, when graphing

27. The Graph of r = 2( 1 - 2sinθ( θr = θ( r, θ ) 02 ( 2, 0 ) π/60(0, π/6 ) π/2-2(-2, π/2) = (2, 3π/2) 5π/60(0, 5π/6 ) π2(2, π ) 3π/26(6, 3π/2 ) 2π2π2(2, 2π )

28. The Graph of r = 2( 1 - 2sinθ)

29. The graph of: r = a sinnθ r = a cosnθ Where a is positive and n a natural number Rose Curves

30. Example r = 4sin3θ

31. The graph of r = 4sin3θ r = 0 if sin3θ = 0 Or 3θ = 0, π, 2π, 3π, 4π, 5π,… Or θ = 0, π/3, 2π/3, π, 4π/3, 5π/3,… r = 4, which is maximum if sin3θ = 1 Or 3θ = π/2, 5π/2, 9π/2 Or θ = π/6, 5π/6, 3π/2 r = - 4, which is minimum if sin3θ = -1 Or 3θ = 3π/2, 7π/2, 11π/2 Or θ = π/2, 7π/6, 11π/6

32. The graph of r = 4sin3θ θr = 4sin3θ ( r, θ ) 00 ( 0, 0 ) π/64(4, π/6 ) π/30(0, π/3) ≡ (0,0) π/2- 4( - 4, π/2 ) ≡ ( 4, 3π/2 ) 2π/30(0, 2π/3 ) = (0,0) 5π/64(4, 5π/6 ) π0(0, π ) = (0,0)

33. The graph of r = 4sin3θ Values of v grater than v will not result in new points It will just trace the curve again and again θr = 4sin3θ( r, θ ) π0(0, π ) = (0,0) 7π/6- 4(- 4, 7π/6 ) ≡ ( 4, π/6 ) 4π/30(0, 4π/3) = (0,0) 3π/24( 4, 3π/2 ) 5π/30(0, 5π/3 ) ≡ (0,0) 11π/6- 4 (- 4, 11π/6) ≡ (4, 5π/6)

34. r = 4 sin3θ

36. The graph of: r = a sinnθ Or r = a cosnθ A rose curve consists of: n leaves, if n is odd 2n leaves, if n is even Remark

37. Example r = 4sin2θ

38. The graph of r = 4sin2θ r = 0 if sin2θ = 0 Or 2θ = 0, π, 2π, 3π, 4π, 5π, 6π,… Or θ = 0, π/2, π, 3π/2, 2π, 5π/2, 3π … r = 4, which is maximum if sin2θ = 1 Or 2θ = π/2, 5π/2, 9π/2, 13π/2,… Or θ = π/4, 5π/4, 9π/4, 13π/4,… r = - 4, which is minimum if sin2θ = -1 Or 2θ = 3π/2, 7π/2, 11π/2,.. Or θ = 3π/4, 7π/4, 11π/4,..

39. The graph of r = 4sin2θ θr = 4sin2θ ( r, θ ) 00 ( 0, 0 ) π/44(4, π/4 ) π/20(0, π/2) 3π/4- 4( - 4, 3π/4 ) ≡ ( 4, 7π/4 ) π0(0, π ) 5π/44(4, 5π/4 ) 3π/20(0, π ) = (0,0)

40. The graph of r = 4sin2θ Values of θ grater than 2π will not result in new points It will just trace the curve again and again θr = 4sin2θ ( r, θ ) 7π/4- 4(- 4, 7π/4 ) = (4, 3π/4 ) 2π2π0(0, 2π ) 9π/44(4, 9π/4) 5π/20( 0, 5π/2 ) 11π/4- 4(- 4, 11π/4 ) ≡ (4, 7π/4 ) 3π3π0 (0, 3π)

41. The graph of r = 4sin2θ

42. Homework: Graph each of the following Curves r = π/2 r = 2θ θ = π/4 r = 3sinθ r = 3+3sinθ r = 3sin4θ

43. r = 2cosθ r = 4 + 4cosθ r = 4 - 4cosθ r = 3 - 6cosθ r = 3 +6cosθ r = 5cos2θ r = 5cos3θ