It is given that $$u=\sqrt3+i,\qquad v=-1+\sqrt3\,i.$$ The points $P$, $Q$ and $R$ represent \

Question

It is given that $$u=\sqrt3+i,\qquad v=-1+\sqrt3\,i.$$ The points $P$, $Q$ and $R$ represent $u$, $v$ and $u+v$, respectively. Give all arguments as principal arguments in radians. Sketch $P$, $Q$ and $R$ on an Argand diagram with origin $O$.[3] Show that $OPRQ$ is a square.[2] Hence, by considering the direction of the diagonal $OR$, show that $$\tan\frac{5\pi}{12}=2+\sqrt3.$$[3]

Accepted Answer

(a) $P(\sqrt3,1),\ Q(-1,\sqrt3),\ R(\sqrt3-1,1+\sqrt3)$ (b) Square: equal perpendicular adjacent sides in a parallelogram. (c) $\tan\frac{5\pi}{12}=2+\sqrt3$

A square and the direction of its diagonal

A square and the direction of its diagonal

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