Complex Numbers – Part 4

Complex Numbers – Part 4

Polar (Trigonometric) Form

🌟 Two Representations of a Complex Number

Until now we have used the Cartesian form: \(z = a + bi\)

Now we learn another form – the polar (or trigonometric) form.

💡 The idea:

Instead of describing a point by (x, y), we describe it by:

  • Distance from the origin (r)
  • Angle from the positive real axis (θ)

📐 Polar Form – Graphically

Re Im z = a + bi a b r θ O
\(r\) Modulus (absolute value) – distance from the origin: \(r = |z| = \sqrt{a^2 + b^2}\)
\(\theta\) Argument – angle from the positive real axis (counter-clockwise)

🔄 Converting Between Forms

Polar to Cartesian:

\(a = r\cos\theta\)

\(b = r\sin\theta\)

Cartesian to Polar:

\(r = \sqrt{a^2 + b^2}\)

\(\tan\theta = \frac{b}{a}\)

⚠️ Care with θ!

When computing \(\theta = \arctan\frac{b}{a}\), you must check which quadrant the point is in!

⭐ The Trigonometric Form

\(z = r(\cos\theta + i\sin\theta)\)

💡 Explanation:

\(z = a + bi = r\cos\theta + i \cdot r\sin\theta = r(\cos\theta + i\sin\theta)\)

Shorthand notation:

\(z = r \cdot \text{cis}\,\theta\)

(cis = cos + i·sin)

🧭 Angle by Quadrant

Quadrant I a > 0, b > 0 θ = arctan(b/a) Quadrant II a < 0, b > 0 θ = π + arctan(b/a) Quadrant III a < 0, b < 0 θ = π + arctan(b/a) Quadrant IV a > 0, b < 0 θ = arctan(b/a) (or 2π + arctan) Re Im

💡 Rule of thumb:

  • Quadrant I: \(\theta\) between 0° and 90°
  • Quadrant II: \(\theta\) between 90° and 180°
  • Quadrant III: \(\theta\) between 180° and 270° (or −180° to −90°)
  • Quadrant IV: \(\theta\) between 270° and 360° (or −90° to 0°)

✏️ Example 1: Cartesian to Polar

Convert to polar form: \(z = 1 + i\)

Solution:

Step 1: Compute r

\(r = \sqrt{1^2 + 1^2} = \sqrt{2}\)

Step 2: Compute θ

\(\tan\theta = \frac{1}{1} = 1\)

Point is in Quadrant I (both components positive)

\(\theta = \frac{\pi}{4}\) (or 45°)

Answer: \(z = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)\)

✏️ Example 2: Point in Quadrant II

Convert to polar form: \(z = -1 + \sqrt{3}i\)

Solution:

Step 1: Compute r

\(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2\)

Step 2: Compute θ

\(\tan\theta = \frac{\sqrt{3}}{-1} = -\sqrt{3}\)

Point is in Quadrant II (a negative, b positive)

Reference angle: \(\arctan(\sqrt{3}) = \frac{\pi}{3}\)

In Quadrant II: \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\) (or 120°)

Answer: \(z = 2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\)

✏️ Example 3: Polar to Cartesian

Convert to Cartesian form: \(z = 4\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\)

Solution:

\(r = 4, \quad \theta = \frac{\pi}{6} = 30°\)

\(a = r\cos\theta = 4 \cdot \cos\frac{\pi}{6} = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\)

\(b = r\sin\theta = 4 \cdot \sin\frac{\pi}{6} = 4 \cdot \frac{1}{2} = 2\)

Answer: \(z = 2\sqrt{3} + 2i\)

📊 Special Angles Table

θ Degrees cos θ sin θ z = cos θ + i sin θ
0 0° 1 0 1
\(\frac{\pi}{6}\) 30° \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2} + \frac{1}{2}i\)
\(\frac{\pi}{4}\) 45° \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\)
\(\frac{\pi}{3}\) 60° \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2} + \frac{\sqrt{3}}{2}i\)
\(\frac{\pi}{2}\) 90° 0 1 i
\(\pi\) 180° -1 0 -1
\(\frac{3\pi}{2}\) 270° 0 -1 -i

⭐ Special Numbers in Polar Form

\(1\)

\(r=1, \theta=0\)

\(-1\)

\(r=1, \theta=\pi\)

\(i\)

\(r=1, \theta=\frac{\pi}{2}\)

\(-i\)

\(r=1, \theta=\frac{3\pi}{2}\)

📋 Summary Table – Part 4

Topic Formula
Polar form \(z = r(\cos\theta + i\sin\theta)\)
Modulus \(r = |z| = \sqrt{a^2 + b^2}\)
Argument \(\tan\theta = \frac{b}{a}\) (check quadrant!)
Polar → Cartesian \(a = r\cos\theta, \, b = r\sin\theta\)

💡 Tips for the Exam

1️⃣ Check the quadrant!

Before computing θ, identify which quadrant the point is in

2️⃣ Special angles

Memorise the table: 30°, 45°, 60°

3️⃣ Check

Convert back to Cartesian and verify you get the same z

4️⃣ r is always positive!

The modulus r ≥ 0 always

📝 Summary – Part 4

\(z = r(\cos\theta + i\sin\theta)\)

\(r = |z|, \quad \theta = \arg(z)\)

In the next part: De Moivre's formula – multiplication, division, powers and roots!