1. Graphing r = θ on the Polar Coordinate Plane

2. Graphing r = θ on the Polar Coordinate Plane
To sketch the graph of r = θ, we should find some ordered pairs that satisfy the function. Since this function involves polar coordinates, we are looking for pairs of the form (r, θ).

3. Graphing r = θ on the Polar Coordinate Plane
To sketch the graph of r = θ, we should find some ordered pairs that satisfy the function. Since this function involves polar coordinates, we are looking for pairs of the form (r, θ).
Let’s construct a table so that we can find ordered pairs on the graph of r = θ.

4. Graphing r = θ on the Polar Coordinate Plane
To construct a table of values for r = θ, let’s choose values for θ and find the corresponding values of r, and then form ordered pairs (r, θ).

5. Graphing r = θ on the Polar Coordinate Plane
To construct a table of values for r = θ, let’s choose values for θ and find the corresponding values of r, and then form ordered pairs (r, θ).
We can then plot these ordered pairs on a polar coordinate plane to obtain a graph of the function.

6. Graphing r = θ on the Polar Coordinate Plane
Let’s start by choosing θ = 0.

7. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
Let’s start by choosing θ = 0.

8. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
Let’s start by choosing θ = 0.
If θ = 0, then since r = θ we see that r also equals 0.

9. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
Let’s start by choosing θ = 0.
If θ = 0, then since r = θ we see that r also equals 0.

10. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
Let’s start by choosing θ = 0.
If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0).

11. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 )
Let’s start by choosing θ = 0.
If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0).

12. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 )
Let’s start by choosing θ = 0.
If θ = 0, then since r = θ we see that r also equals 0. This gives us the ordered pair (0, 0).
Let’s plot this ordered pair on our polar coordinate plane.

13. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 )
Now let’s choose another value for θ.

14. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4
Now let’s choose another value for θ.
If θ = , π 4

15. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 0.8. π 4 π 4

16. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 0.8. π 4 π 4

17. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair π 4 π 4 π 4 ( )
0.8,

18. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair π 4 π 4 π 4 ( )
0.8,

19. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, π 4 π 4 π 4 ( )
0.8,

20. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, and connect the point we already graphed with this new point. π 4 π 4 π 4 ( )
0.8,

21. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.

22. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , π 2

23. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 1.6. π 2 π 2

24. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 1.6. π 2 π 2

25. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair π 2 π 2 π 2 ( )
1.6,

26. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair π 2 π 2 π 2 ( )
1.6,

27. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, π 2 π 2 π 2 ( )
1.6,

28. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far. π 2 π 2 π 2 ( )
1.6,

29. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.

30. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , 3π 4

31. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 2.4. 3π 4 3π 4

32. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately 2.4. 3π 4 3π 4

33. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair 3π 4 3π 4 3π 4 ( )
2.4,

34. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair 3π 4 3π 4 3π 4 ( )
2.4,

35. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, 3π 4 3π 4 3π 4 ( )
2.4,

36. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
Now let’s choose another value for θ.
If θ = , then since r = θ we see that r also equals , which is approximately This gives us the ordered pair
Let’s plot this ordered pair on our polar coordinate plane, and connect it with what we’ve graphed so far. 3π 4 3π 4 3π 4 ( )
2.4,

37. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
We can continue to choose values for θ,

38. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
We can continue to choose values for θ, find the corresponding values for r,

39. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
We can continue to choose values for θ, find the corresponding values for r,
plot the ordered pair (r, θ) on our polar coordinate plane,

40. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,
We can continue to choose values for θ, find the corresponding values for r,
plot the ordered pair (r, θ) on our polar coordinate plane, and connect our points to obtain a graph of r = θ.

41. Graphing r = θ on the Polar Coordinate Planer
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

42. Graphing r = θ on the Polar Coordinate Planeπ r
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

43. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 ) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

44. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

45. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

46. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

47. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4,

48. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, (

49. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, (

50. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, (

51. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, (

52. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( ( ) 3π 2
4.7,

53. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( ( ) 3π 2
4.7,

54. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( 3π 2 ( )
4.7,

55. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 7π 4
≈ 5.5 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( 3π 2 ( )
4.7,

56. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 7π 4
≈ 5.5 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( 3π 2 ( )
4.7, ) 7π 4
5.5, (

57. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 7π 4
≈ 5.5 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( ( ) ) 7π 4
5.5, ( 3π 2
4.7,

58. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 7π 4
≈ 5.5 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( ( ) ) 7π 4
5.5, ( 3π 2
4.7,
Finally, we can continue drawing our graph along this same trajectory.

59. Graphing r = θ on the Polar Coordinate Planeπ r
π ≈ 3.1
(r, θ )
(0, 0 )
(3.1, π) π 4 π 2 3π 4 5π 4 3π 2 7π 4 π 4
≈ 0.8 π 2
≈ 1.6 3π 4
≈ 2.4 5π 4
≈ 3.9 3π 2
≈ 4.7 7π 4
≈ 5.5 π 4 ( )
0.8, π 2 ( )
1.6, 3π 4 ( )
2.4, ) 5π 4
3.9, ( ( ) ) 7π 4
5.5, ( 3π 2
4.7,
Finally, we can continue drawing our graph along this same trajectory.
The graph of r = θ is called the Archimedean Spiral.