1st Year Math Chapter 1 Exercise 1.5 Notes: Polar Form of a Complex Number (Punjab Board 2026-27)

Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.

Exercise 1.5 closes the Complex Numbers chapter by introducing the polar (trigonometric) form of a complex number, an alternative way to write a + bi using distance and angle instead of just real and imaginary parts.

What Does This Exercise Cover?

This exercise teaches how to convert a complex number between its rectangular (a + bi) form and its polar form, using the modulus and argument, which becomes especially useful for multiplying, dividing, and finding powers of complex numbers.

Key Concepts

1. The Polar Form

Any complex number z = a + bi can be written in polar form as z = r(cos θ + i sin θ), where r is the modulus of z and θ is the argument of z — the angle z makes with the positive real axis.

2. Finding r and θ

The modulus is found the same way as before: r = √(a² + b²). The argument θ is found using tan θ = b/a, adjusted for the correct quadrant based on the signs of a and b.

3. Quadrant Rules for the Argument

Since tan θ = b/a alone cannot distinguish between angles that differ by 180°, the signs of a and b are used to place θ in the correct quadrant before finalizing its value.

4. Converting Polar Form Back to Rectangular Form

Given z = r(cos θ + i sin θ), the rectangular form is recovered using a = r cos θ and b = r sin θ.

Quadrant Reference for the Argument θ

Solved Examples

Example 1: Express z = 1 + i in polar form.

Modulus: r = √(1² + 1²) = √2.

Since a = 1 > 0 and b = 1 > 0, z lies in Quadrant I.

θ = arctan(1/1) = 45°.

Answer: z = √2(cos 45° + i sin 45°)

Example 2: Express z = −1 + i√3 in polar form.

Modulus: r = √((−1)² + (√3)²) = √(1 + 3) = 2.

Reference angle: arctan(√3/1) = 60°.

Since a < 0 and b > 0, z lies in Quadrant II, so θ = 180° − 60° = 120°.

Answer: z = 2(cos 120° + i sin 120°)

Example 3: Convert z = 2(cos 60° + i sin 60°) to rectangular form.

a = r cos θ = 2 × cos 60° = 2 × 0.5 = 1.

b = r sin θ = 2 × sin 60° = 2 × (√3/2) = √3.

Answer: z = 1 + √3 i

Sample MCQs

1. In the polar form z = r(cos θ + i sin θ), r represents:

a) The argument   b) The modulus   c) The real part   d) The imaginary part

Answer: b) The modulus

2. The modulus of z = a + bi is calculated as:

a) a + b   b) a − b   c) √(a² + b²)   d) a² + b²

Answer: c) √(a² + b²)

3. The polar form of z = 1 + i is:

a) √2(cos 45° + i sin 45°)   b) 2(cos 45° + i sin 45°)   c) √2(cos 90° + i sin 90°)   d) 1(cos 45° + i sin 45°)

Answer: a) √2(cos 45° + i sin 45°)

4. The argument θ of a complex number represents:

a) Its distance from the origin   b) The angle it makes with the positive real axis   c) Its real part   d) Its imaginary part

Answer: b) The angle it makes with the positive real axis

5. For z = −1 + i√3, the modulus r equals:

a) 1   b) 2   c) √3   d) 4

Answer: b) 2

Important Short Questions

• Define the polar form of a complex number.
• What do r and θ represent in the polar form of a complex number?
• How is the modulus of a + bi calculated?
• Explain why the correct quadrant matters when finding θ.
• Convert z = 3i into polar form.

Important Long Questions

• Express z = −1 + i√3 in polar form, showing all steps for finding r and θ.
• Convert the polar form z = 2(cos 120° + i sin 120°) into rectangular (a + bi) form.
• Explain the relationship between the rectangular and polar forms of a complex number, including how the correct quadrant is determined.
• Express z = −√3 − i in polar form, showing all working.

How to Approach This Exercise Effectively

1. Always find r first — it never depends on the quadrant, only on a and b.
2. Find the reference angle using positive values of a and b, then adjust for the quadrant using the sign rules.
3. Draw a quick sketch of the point (a, b) whenever you’re unsure which quadrant it falls in.
4. Practice converting in both directions — rectangular to polar and polar to rectangular — since exams test both.
5. Keep standard angle values (30°, 45°, 60°, 90°) and their sine/cosine values memorized, since most exercise problems use these.

FAQs

Q: What is the difference between rectangular and polar form?

A: Rectangular form (a + bi) locates a complex number using horizontal and vertical distances, while polar form (r(cos θ + i sin θ)) locates it using a distance from the origin and an angle.

Q: Why does the argument depend on the quadrant?

A: Because tan θ = b/a gives the same ratio for angles 180° apart, so the actual signs of a and b are needed to pick the correct one of the two possible angles.

Q: Is there only one possible value of θ for a complex number?

A: The principal argument is usually chosen within a set range (such as 0° to 360°, or −180° to 180°), but technically θ has infinitely many values differing by full rotations of 360°.

Q: Why is polar form useful?

A: It makes multiplying, dividing, and finding powers or roots of complex numbers much simpler, since these operations combine moduli and angles directly rather than requiring full algebraic expansion.

Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 1: Complex Numbers, Exercise 1.5.