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Lecture 5(polar coordinates)
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.
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.
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.
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.
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.
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.
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.
