Complex division yields an exact sine

A graphic calculator is not to be used in answering this question. p=1−ip=1-i, q=−1+3iq=-1+\sqrt3i, z=p/qz=p/q.

(i) Express zz as x+yix+yi, where x,yx,y are exact real values to determine. marks

(ii) It is known that ∣z1/z2∣=∣z1∣/∣z2∣|z_1/z_2|=|z_1|/|z_2| and arg⁡(z1/z2)=arg⁡z1−arg⁡z2\arg(z_1/z_2)=\arg z_1-\arg z_2. By considering the moduli and arguments of p,qp,q, find exact ∣z∣,arg⁡z|z|,\arg z with −π<arg⁡z≤π-\pi<\arg z\leq\pi. marks

(iii) Hence show sin⁡(11π/12)=(6−2)/4\sin(11\pi/12)=(\sqrt6-\sqrt2)/4. marks

[modified 2013/TJC/Prelim/I/5]

Total marks: 9

Solution

(i)

Divide using the conjugate

p=1−ip=1-i, q=−1+3iq=-1+\sqrt3i, z=pqz=\frac pq. Express zz as x+yix+yi with exact x,yx,y.

Hint

  1. Multiply top and bottom by the conjugate of qq.

Answer · (i)

z=−(1+3)/4+i(1−3)/4z=-(1+\sqrt3)/4+i(1-\sqrt3)/4.

(ii)

Quotient of moduli and arguments

Using ∣z1z2∣=∣z1∣∣z2∣\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|} and arg⁡z1z2=arg⁡z1−arg⁡z2\arg\frac{z_1}{z_2}=\arg z_1-\arg z_2, find ∣z∣|z| and arg⁡z\arg z with −π<arg⁡z≤π-\pi<\arg z\le\pi.

Explore this idea

Hint

  1. Find the modulus and argument of each of pp and qq.

  2. Divide the moduli and subtract the arguments.

Answer · (ii)

∣z∣=2/2|z|=\sqrt2/2, arg⁡z=−11π/12\arg z=-11\pi/12.

Explore the idea

Dividing shrinks and turns

Slide from pp to z=p/qz=p/q. The point shrinks by a factor 22 and turns clockwise through arg⁡q=120∘\arg q=120^\circ on the way.

Fraction of the division applied1
Try an example
z
-0.683 − 0.183i
|z|
0.707
arg z
-165°
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Division divides lengths and subtracts angles: from pp at −45∘-45^\circ, subtract 120∘120^\circ to land at −165∘-165^\circ.

Ideas to try
  1. At the end, ∣z∣=0.707|z|=0.707 and arg⁡z=−165∘=−11π12\arg z=-165^\circ=-\frac{11\pi}{12}.
  2. The path curves inward: the modulus is divided gradually while the angle is subtracted gradually.
(iii)

Link the two forms of z

Hence show that sin⁡11π12=6−24\sin\frac{11\pi}{12}=\frac{\sqrt6-\sqrt2}4.

Explore this idea

Hint

  1. Equate the imaginary part found in (i) with ∣z∣sin⁡(arg⁡z)|z|\sin(\arg z) from (ii).

Answer · (iii)

sin⁡(11π/12)=(6−2)/4\sin(11\pi/12)=(\sqrt6-\sqrt2)/4.

Explore the idea

The sine is the height on the unit circle

Move the angle θ\theta. The height of the point above the real axis is sin⁡θ\sin\theta.

Angle θ165°
Try an example
sin θ
0.2588
(√6 − √2)/4
0.2588
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Comparing the same complex number in Cartesian and polar form turns a division into an exact sine value.

Ideas to try
  1. At 165∘165^\circ the height is 0.25880.2588, matching 6−24\frac{\sqrt6-\sqrt2}4.
  2. At 15∘15^\circ the height is the same: sin⁡11π12=sin⁡π12\sin\frac{11\pi}{12}=\sin\frac\pi{12}.
Takeaway

Two forms, one number

Cartesian and polar forms of the same number must agree. Matching parts gives exact trigonometric values that a calculator would only approximate.