A complex square gives an exact tangent

z=3+iz=\sqrt3+i, w=−1+i3w=-1+i\sqrt3. Sketch an Argand diagram with origin O showing Z, W and P representing z,w,z+wz,w,z+w respectively. Describe the geometrical shape OWPZ. marks

By considering the argument of z+wz+w, deduce tan⁡(5π/12)=2+3\tan(5\pi/12)=2+\sqrt3. marks

[2010/MJC/Prelim/I/11]

Total marks: 6

Solution

The shape OWPZ

Sketch and describe

z=3+iz=\sqrt3+i, w=−1+i3w=-1+i\sqrt3. Show Z,W,PZ,W,P representing z,w,z+wz,w,z+w on an Argand diagram and describe the shape OWPZOWPZ.

Explore this idea

Hint

  1. Compare ww with iziz.

  2. Use the parallelogram rule for z+wz+w.

The complex-addition square OWPZ
The complex-addition square OWPZScroll across to see the full diagram.

Answer · The shape OWPZ

O=(0,0),W=(−1,3),P=(3−1,1+3),Z=(3,1)O=(0,0),W=(-1,\sqrt3),P=(\sqrt3-1,1+\sqrt3),Z=(\sqrt3,1); OWPZOWPZ is a square.

Explore the idea

From a rhombus to a square

Turn WW away from ZZ. For any turn, OZPWOZPW is a rhombus; at exactly 90∘90^\circ it becomes a square.

Angle between OZ and OW90°
Try an example
arg(z + w)
75°
tan of that angle
3.732
P
0.732 + 2.732i
|OP|
2.828
  • Moves as the slider changes; a faint path shows where it goes
Key idea

A quarter-turn gives equal perpendicular sides, and with P=z+wP=z+w that is a square.

Ideas to try
  1. At 90∘90^\circ the sides are perpendicular and ∣OP∣=22|OP|=2\sqrt2.
  2. The diagonal always bisects the angle between OZOZ and OWOW, so arg⁡P=30∘+a2\arg P=30^\circ+\frac a2.
tan(5π/12)

Exact trigonometric value from an argument

By considering the argument of z+wz+w, deduce tan⁡5π12=2+3\tan\frac{5\pi}{12}=2+\sqrt3.

Hint

  1. Find arg⁡(z+w)\arg(z+w) geometrically.

  2. Find it again from the coordinates of PP.

Answer · tan(5π/12)

tan⁡(5π/12)=(1+3)/(3−1)=2+3\tan(5\pi/12)=(1+\sqrt3)/(\sqrt3-1)=2+\sqrt3.

Takeaway

Geometry gives the angle, algebra the ratio

The same argument computed two ways, once by geometry and once from coordinates, forces an exact trigonometric identity.