polar co ordinates
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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.
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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.
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Example I
Plot the curve r = θ
r 0 π/12 π/6 π/4 π/3 … 2π
θ
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Example II
Plot the curve r = 2 sin(θ)
r 0 π/12 π/6 π/4 π/3 … 2π
θ
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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)
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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
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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
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Further Pure Mathematics II
Polar Co-ordinates
- Lesson 2 -
Key Learning Points/Vocabulary:
● Converting between Cartesian and Polar Co-ordinates.
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To start with …
Polar equations/graphs matching activity.
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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.
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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.
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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.
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Homework
See ‘Homework 1’ posted online.
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Further Pure Mathematics II
Polar Co-ordinates
- Lesson 3 -
Key Learning Points/Vocabulary:
● Polar co-ordinates and the use of symmetry.
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−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.
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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.
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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.
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Practice
Further Pure Mathematics 2 and 3
Exercise 6C
Questions 1, 3 and 4
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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.
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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)
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−4 −2 2 4 6
−2
2
4
x
y
Example
Determine the maximum and minimum values of r = 2 + cos θ.
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Example II
−4 −2 2 4 6
−2
2
4
x
yDetermine the maximum and minimum values of r = 1 + cos 2θ.
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Practice
Further Pure Mathematics 2 and 3
Exercise 6D
Questions 2 – part ii.) of each question only
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Further Pure Mathematics II
Polar Co-ordinates
- Lesson 5 -
Key Learning Points/Vocabulary:
● Finding the equations of tangents at the pole (origin).
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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)
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−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.
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Practice
Further Pure Mathematics 2 and 3
Exercise 6D
Questions 2 – part iii.) of each question only
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Further Pure Mathematics II
Polar Co-ordinates
- Lesson 6 -
Key Learning Points/Vocabulary:
● Converting between Cartesian and Polar Equations.
![Page 32: Polar Co Ordinates](https://reader033.vdocuments.mx/reader033/viewer/2022051515/54c481b94a7959cc7c8b45e8/html5/thumbnails/32.jpg)
sin θ = y/r
→ y = r sin θ
cos θ = x/r→ x = r cos θ
x2 + y2 = r2
Theory
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Examples
Convert the following equations into polar form:
i.) y = x2
ii.) (x2 + y2)2 = 4xy
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Examples (continued)
Convert the following equations into Cartesian form:
iii.) r = 2a cos θ
iv.) r2 = a2 sin 2θ
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Practice
Further Pure Mathematics 2 and 3
Exercise 6E
Questions 1 and 2
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Homework
See ‘Homework 2’ posted online.
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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.
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Integrals from Core 4
Function Derivative
sin (ax) - (1/a) . cos (ax)
cos (ax) (1/a) . sin (ax)
… see next slide for more detail …
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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
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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
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Example
−4 −2 2 4 6
−4
−2
2
x
yFind the area enclosed by the curve r = aθ for 0 < θ < 2π.
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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.
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Practice
Further Pure Mathematics 2 and 3
Exercise 6F
Questions 1 onwards
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Homework
See ‘Homework 3’ posted online.
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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.