# 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.