Arguments determine a triangle · source reference

Question

$\arg z=-\pi/3$, $|w|=2$, $\arg w=\pi/4$ and $\arg(w-z)=\pi/3$. Let W and Z represent $w,z$ respectively.

(i) Use an Argand diagram with reference to origin O to show $\angle OWZ=\pi/12$ radians. [2]

(ii) Hence or otherwise find $|z|$. [2]

[2020/DHS/Prelim/I/3 modified]

Source working

Printed working

(i) Let W and Z represent w and z. $\overrightarrow{ZW}$ represents $w-z$.

$$\angle OWZ=\pi/3-\pi/4=\pi/12\quad\text{(shown)}.$$

(ii) The source states $\angle OZW=\pi/3$ “(angle of equilateral $\triangle$)”.

$$\frac2{\sin\angle OZW}=\frac{OZ}{\sin\angle OWZ},\qquad\frac2{\sin(\pi/3)}=\frac{OZ}{\sin(\pi/12)}.$$ $$OZ=0.59772,\qquad |z|=0.598\quad(3\text{ s.f.}).$$

Qualification

OWZ is not equilateral: its angle at W is $\pi/12$. The angle at Z is nevertheless $\pi/3$, because ZO has argument $2\pi/3$ and ZW has argument $\pi/3$. The sine-rule answer remains correct.

Diagram labels

O, W, Z, x, y, length OW $2$, origin angle $\pi/4$, clockwise OZ angle $\pi/3$, direction ZW $\pi/3$ at the axis and at Z, with a horizontal auxiliary ray through Z.

Printed answer key

(ii) $|z|=0.598$ (3 s.f.).

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 4; solution pages 19, 20. These are original document page numbers, retained as text metadata.