FP2 Chapter 7 – Polar Coordinates
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What the devil are polar coordinates?
You’ve actually encountered polar coordinates already via complex numbers.
Recall that you could define complex numbers either in Cartesian form, or in ‘polar form’ using the distance and angle from the origin.
Cartesian Mike
Polar Mike
initial line
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Converting to/from polar coordinates
You should already know how to do this from Chapter 3.
But a reminder:
For 1st and 4th quadrants – use a diagram for others
Cartesian | Polar |
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Quickfire Questions:
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Sometimes it is possible to switch from a Cartesian equation to a polar one.
Find Cartesian equations for the following:
Note that the polar form is much more elegant!
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Why polar form sometimes kicks Cartesian’s butt
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Think why
Find polar equations for the following:
We know how to simplify expressions like this from C3.
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Converting to polar is easier, but the harder part is often finding how to simplify the expression. Know your double angle formulae!
Exercise 7B
Sketching Curves of Polar Equations
How would you sketch each of the following?
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Sketching using tables of values
Sketching using tables of values
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This is a cardioid, and this animation shows how it would be emerge in practice!
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Sketch ?
More sketchies
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Sketch ?
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(Known as a ‘polar rose’)
Sketch ?
Just for fun… (not in the syllabus)
Just for fun… (not in the syllabus)
And it can get rather pretty…
(This is a spiral combined with a polar rose)
Just for fun… (not in the syllabus)
Egg vs Dimple
Case 1
Case 2
Case 3
(We will see why we get the ‘egg’ vs ‘dimple’ later.)
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Egg vs Dimple
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Sketch ?
Sketch ?
Exercise 7C
Summary
Bro Helping Hand:
You can prove these by converting equation to Cartesian.
Bro Exam Tip:
I lifted each of these forms directly out of the Edexcel specification.
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Summary
(special name: cardioid)
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Summary
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Integration
Bro Exam Note: These last two sections: integration and tangents/normal, are what your exam questions will probably be based on.
When integrating normal Cartesian 2D areas, we know we’re summing a bunch of infinitely thin rectangles:
Area of each rectangle:
Adding them all for area:
Similarly in C4, we could get a volume of revolution by summing the volumes of infinitely thin cylinders:
Volume of each cylinder
Adding them all for total volume:
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Integration
Area of each sector:
Adding them all for total area:
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Example
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Another Example
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Test Your Understanding
FP2 June 2009 Q4
Bro Exam Note: As far as I can tell, the sketch is always given to you in exams. However, it does explicitly say in the specification that you are required to know how to sketch certain polar equation forms.
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Intersecting Areas
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B
(Use your silver calculators to check!)
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Test Your Understanding
FP2 June 2010 Q5
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Exercise 7D
Tangents and Normals
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Example
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Test Your Understanding
FP2 June 2012 Q2
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Example
(a) point?
(a) tangent?
Example
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Proof of dimple vs egg
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Exercise 7E
Summary
Official Edexcel Specification
Sketching either by using tables of values or converting to a Cartesian equation.
Let’s remind ourselves of those sketches for different equation forms…
Summary
Bro Helping Hand:
You can prove these by converting equation to Cartesian.
Bro Exam Tip:
I lifted each of these forms directly out of the Edexcel specification.
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Summary
(special name: cardioid)
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Summary
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