A calculator is not to be used in answering this question. The complex numbers $a$ and $b$ are g

Question

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 $\tan \left( \frac{3\pi}{8} \right)=1+\surd 2$.[3]

Accepted Answer

$|a|=1$, $\arg(a)=\frac\pi2$, $|b|=1$, $\arg(b)=\frac\pi4$, and $\tan\left(\frac{3\pi}{8}\right)=1+\sqrt2$

2015 CJC P2 Q3

2015 CJC P2 Q3

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