Further Complex Numbers – Edexcel FP2

3. Further Complex Numbers – Edexcel Further Pure Mathematics 2 (FP2)

What students need to learn:
Euler’s relation $e^{i \theta} = \cos \theta + i \sin \theta$.
Students should be familiar with
$\cos \theta = \dfrac{1}{2}(e^{i \theta} + e^{-i \theta})$ and
$\sin \theta = \dfrac{1}{2i}(e^{i \theta} – e^{-i \theta})$.

De Moivre’s theorem and its application to trigonometric identities and to roots of a complex number.
To include finding $\cos n\theta$ and $\sin m\theta$ in terms of powers of $\sin \theta$ and $\cos \theta$ and also powers of $\sin \theta$ and $\cos \theta$ in terms of multiple angles. Students should be able to prove De Moivre’s theorem for any integer n.

Loci and regions in the Argand diagram.
Loci such as $\lvert z – a \rvert = b$ ,
$\lvert z – a \rvert = k \lvert z – b \rvert$ ,
$\arg (z – a) = \beta$, $\arg \dfrac{z – a}{z – b} = \beta$ and regions such as
$\lvert z – a \rvert \leq \lvert z – b \rvert$, $\lvert z – a \rvert \leq b$.

Elementary transformations from the z-plane to the w-plane.
Transformations such as $w = z^2$ and $w = \dfrac{az+b}{cz+d}$, where $a,b,c,d \in \mathbb{C}$, may be set.

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