#### 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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