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202210201553 Solution to 1977-CE-AMATH-I-XX

In the figure below, the complex numbers z_0, z_1, z_2, z_3, and z_4 are represented in the Argand diagram by the vertices of a regular pentagon with centre at the origin O. If z_0=2, write z_2 in polar form and calculate the value of (z_2)^5.

Recall.

A complex number z in Cartesian form z=a+\mathrm{i}b can also be expressed in the polar form

z=re^{\mathrm{i}\varphi}=r(\cos\varphi +\mathrm{i}\sin\varphi )

where r=\sqrt{a^2+b^2} is called the modulus |z|, and \varphi the argument \textrm{arg}(z), of z.

Roughwork.

Observe that

|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=2.

and

\displaystyle{\textrm{arg}(z_2)=2\pi\cdot\frac{2}{5}=\frac{4\pi}{5}}.

Hence

z_2=2e^{\mathrm{i}\frac{4\pi}{5}}.

The remaining are left the reader as an exercise.