Polar Form of Complex Numbers
The polar form of a complex number provides a powerful way to represent complex numbers using their magnitude and angle. This representation simplifies many mathematical operations and has applications in fields like electrical engineering and signal processing. In this guide, we’ll explore the definition, derivation, and applications of polar form, along with solved examples and challenges.
Definition of Polar Form
If 
is a complex number. Then z = r ( cosθ + isinθ ) is called polar form or trigonometric form of a complex number.
Derivation of Polar Form Formulas
By comparing real and imaginary parts of a complex number, we get:
x = r cosθ (1)
y = r sinθ (2)
By squaring above equations and adding we get:
Example: Converting a Complex Number to Polar Form
Express in polar form.
Solution:
- Identify x and y: In the complex number -√3 – i, x = -√3 and y = -1.
- Calculate the modulus (r): r = |z| = √(x² + y²) = √((-√3)² + (-1)²) = √(3 + 1) = √4 = 2.
- Find the argument (θ): tan(θ) = y/x = (-1)/(-√3) = 1/√3.
- Determine the quadrant: Since x and y are both negative, the complex number lies in the third quadrant.
- Adjust the argument: The reference angle is π/6, but in the third quadrant, θ = π + π/6 = 7π/6.
- Write the polar form: z = r(cos(θ) + i sin(θ)) = 2(cos(7π/6) + i sin(7π/6)).
Solved Questions of Polar Form
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Challenge Questions
Write the given complex number in polar form first using an argument θ ≠ Arg(z):
- 2
- -10
- -3i
- 6i
- 1 + i
- 5 – 5i
- click here for solution
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