FP2 Chapter 3 – Further Complex Numbers
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OVERVIEW
This chapter is divided up into two main sections.
You are pretty much guaranteed one exam question on each of the two halves:
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Im[z]
Re[z]
RECAP :: Modulus-Argument Form
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modulus
argument
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RECAP :: Modulus-Argument Form
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Exponential Form
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Exponential Form
You need to be able to convert to and from exponential form.
| Mod-arg form | Exp Form |
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Notice this is not a principal argument.
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To get Cartesian form, put in modulus-argument form first.
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A Final Example
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Exercise 3A
Multiplying and Dividing Complex Numbers
FP1 recap:
i.e. If you multiply two complex numbers, you multiply the moduli and add the arguments, and if you divide them, you divide the moduli and subtract the arguments.
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Examples
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Test Your Understanding
FP2 June 2013 Q2
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Exercise 3B
De Moivre’s Theorem
We saw that:
They so went there...
FP2 June 2013 Q4
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Secret Alternative Way
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De Moivre’s Theorem for Exponential Form
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Examples
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Exercise 3C
Applications of de Moivre #1: Trig identities
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Applications of de Moivre #1: Trig identities
Test Your Understanding
FP2 June 2011 Q7
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Notice how the powers descend by 2 each time.
Starting point?
Using identities below.
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Bro Speed Tip: Remember that terms in such an expansion oscillate between positive and negative.
Again using identities.
Starting point?
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Test Your Understanding
Starting point?
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Exercise 3D
Applications of de Moivre #2: Roots
Plot these roots on an Argand diagram.
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Applications of de Moivre #2: Roots
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Test Your Understanding
FP2 June 2012 Q3
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Exercise 3E
Chapter 3 Part II – Locus of Points
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Loci
You have already encountered loci in FP1 as a set of points which satisfy some restriction.
For example, the definition of a parabola:
A set of points equidistant from a line (the directrix) and a point (the focus).
We often want to find an equation which defines the set of points or the opposite.
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Click to Brosketch >
On quick reminder…
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Find the Cartesian equation of this locus.
Find modulus and square both sides.
Group by real/imaginary.
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Click to Brosketch >
Note that both sides were squared.
Equation ?
Test Your Understanding
Argand Diagram ?
Equation ?
Equation ?
Argand Diagram ?
Test Your Understanding
FP2 June 2012 Q8a
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Click to Brosketch >
This bit is important. The locus is referred to as a ‘half line’.
Equation ?
Click to Brosketch >
Equation ?
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How could we find the centre of the circle?
Angle at centre will be double. Resulting triangle will need to be isosceles as radius of circle is constant.
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Another Example
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Another Example
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Exercise 3F
Regions
How would you describe each of the following in words? Therefore draw each of the regions on an Argand diagram.
An intersection of the three other regions.
(I couldn’t be bothered to draw this. Sorry)
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Test Your Understanding
P6 June 2003 Q4(i)(b)
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Transformations
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| Enlargement followed by translation. |
STEP 4: Substitute using original equation.
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More Difficult Example
STEP 4: Substitute using original equation.
STEP 5: Subsequent manipulation to put in form we recognise the loci of.
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Sketch ?
Test Your Understanding
FP2 June 2009 Q6
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STEP 4: Substitute using original equation.
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A slightly different one
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Sketch ?
Exercise 3H
Summary
Euler’s relation and de Moivre’s theorem/applications
(this is a screenshot from the official specification)
Loci/Transformations
You’ve been asked to prove these before in exams.