Write down the modulus and argument of $z_1$ and $z_2$, where $z_1=1+i\sqrt{3}$ and \(z_2

Question

Write down the modulus and argument of $z_1$ and $z_2$, where $z_1=1+i\sqrt{3}$ and $z_2=\sqrt{3}-i$. In an Argand diagram with origin $O$, the points $P$, $Q$ and $R$ represent the complex numbers $z_1$, $z_2$ and $z_1+z_2$ respectively. Prove that $OPRQ$ is a square. Show geometrically that the angle between $OR$ and the real axis is $15{}^\circ$. By considering the argument of $z_1+z_2$, show that $\tan 15{}^\circ=2-\sqrt{3}$.

Accepted Answer

(a) $|z_1|=2,\arg z_1=\frac\pi3;\ |z_2|=2,\arg z_2=-\frac\pi6$. (b)(i) $OPRQ$ is a square. (ii) $15^\circ$. (c) $\tan15^\circ=2-\sqrt3$.

Geometry and arithmetic of complex numbers

Geometry and arithmetic of complex numbers

Solution:

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