2022 YIJC P1 Q6
(a)
Show that $1+{{\text{e}}^{-\text{i}\alpha }}=2\cos \frac{\alpha }{2}{{\text{e}}^{-\text{i}\frac{\alpha }{2}}}$, where $-\pi <\alpha \le \pi $.
[2]
(b)
Hence or otherwise, show that ${{\left( 1+{{\text{e}}^{-\text{i}\alpha }} \right)}^{3}}-{{\left( 1+{{\text{e}}^{\text{i}\alpha }} \right)}^{3}}=-16\text{i}{{\cos }^{3}}\left( \frac{\alpha }{2} \right)\sin \left( \frac{3\alpha }{2} \right)$.
[3]
(c)
Given further that $0<\alpha <\frac{2}{3}\pi $ and $z={{\left( 1+{{\text{e}}^{-\text{i}\alpha }} \right)}^{3}}-{{\left( 1+{{\text{e}}^{\text{i}\alpha }} \right)}^{3}}$, deduce the modulus and argument of $z$. Express your answers in terms of $\alpha $ whenever applicable.
[2]
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