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.

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.

Example I

Plot the curve r = θ

r 0 π/12 π/6 π/4 π/3 … 2π

θ

Example II

Plot the curve r = 2 sin(θ)

r 0 π/12 π/6 π/4 π/3 … 2π

θ

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)

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

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

Ray from origin

Circle, centred on the origin, radius a

Circle

Four-leafed clover

Cardioid

Limaçon

Spiral

Rose curve – see investigation

Lemniscate? Daisy

Further Pure Mathematics II

Polar Co-ordinates

- Lesson 2 -

Key Learning Points/Vocabulary:

● Converting between Cartesian and Polar Co-ordinates.

To start with …

Polar equations/graphs matching activity.

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.

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.

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.

Homework

See ‘Homework 1’ posted online.

Further Pure Mathematics II

Polar Co-ordinates

- Lesson 3 -

Key Learning Points/Vocabulary:

● Polar co-ordinates and the use of symmetry.

−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.

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.

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.

Practice

Further Pure Mathematics 2 and 3

Exercise 6C

Questions 1, 3 and 4

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.

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)

−4 −2 2 4 6

−2

2

4

x

y

Example

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

Example II

−4 −2 2 4 6

−2

2

4

x

yDetermine the maximum and minimum values of r = 1 + cos 2θ.

Practice

Further Pure Mathematics 2 and 3

Exercise 6D

Questions 2 – part ii.) of each question only

Further Pure Mathematics II

Polar Co-ordinates

- Lesson 5 -

Key Learning Points/Vocabulary:

● Finding the equations of tangents at the pole (origin).

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)

−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.

Practice

Further Pure Mathematics 2 and 3

Exercise 6D

Questions 2 – part iii.) of each question only

Further Pure Mathematics II

Polar Co-ordinates

- Lesson 6 -

Key Learning Points/Vocabulary:

● Converting between Cartesian and Polar Equations.

sin θ = y/r

→ y = r sin θ

cos θ = x/r→ x = r cos θ

x2 + y2 = r2

Theory

Examples

Convert the following equations into polar form:

i.) y = x2

ii.) (x2 + y2)2 = 4xy

Examples (continued)

Convert the following equations into Cartesian form:

iii.) r = 2a cos θ

iv.) r2 = a2 sin 2θ

Practice

Further Pure Mathematics 2 and 3

Exercise 6E

Questions 1 and 2

Homework

See ‘Homework 2’ posted online.

Further Pure Mathematics II

Polar Co-ordinates

- Lesson 7 -

Key Learning Points/Vocabulary:

● Review of Double Angle Formulae from Core 3:

e.g. cos2θ = … and sin2 θ = …

● Finding areas using Polar co-ordinates.

Integrals from Core 4

Function Derivative

sin (ax) - (1/a) . cos (ax)

cos (ax) (1/a) . sin (ax)

… see next slide for more detail …

2

2

cos sin sin sin

sin cos cos cos

1cos 2 sin 2

2

1cos 1 cos 2

21

sin 1 cos 22

bb

a a

bb

a a

bb

aa

b b

a a

b b

a a

d b a

d a b

d

d d

d d

Theory

The area of the region bounded by the graph r = f(θ) and the radii θ = α and θ = β is given by

dfdr 22 )]([2

1

2

1

Example

−4 −2 2 4 6

−4

−2

2

x

yFind the area enclosed by the curve r = aθ for 0 < θ < 2π.

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.

Practice

Further Pure Mathematics 2 and 3

Exercise 6F

Questions 1 onwards

Homework

See ‘Homework 3’ posted online.

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.