2016 JJC P2 Q4
The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1}{2}\left( 1-\mathbf{i} \right)$.
(i)
Without using a calculator, find the value of ${{a}^{2}}b$ in the form $x+\mathbf{i}y$.
[2]
(ii)
By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$.
[3]
(iii)
Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$.
[2]
(iv)
The diagram below shows an isosceles right triangle $ABC$, where the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $c$ respectively. Find the exact value of $c$.
[2]
Suggested Video and Handwritten Solutions
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