2015 CJC P2 Q3
A calculator is not to be used in answering this question.
The complex numbers $a$ and $b$ are given by $\frac{1+\mathbf{i}}{1-\mathbf{i}}$ and $\frac{\surd 2}{1-\mathbf{i}}$ respectively.
Find the moduli and arguments of $a$ and $b$.
[3]
In an Argand diagram, the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $a+b$ respectively. The origin is denoted by $O$. By considering the quadrilateral $OACB$ and the argument of $a+b$, show that $\tan \left( \frac{3\text{ }\!\!\pi\!\!\text{ }}{8} \right)=1+\surd 2$.
[3]
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