Problem 100 - Algebra and Trigonometry

Question

Explain how to find the power of a complex number in polar form.

Step-by-Step Solution

Answer

To find the power 'n' of a complex number in polar form \(z = r (\cos θ + i \sin θ)\), use the formula \(z^n = r^n (\cos(nθ) + i \sin(nθ))\), where \(r^n\) is the magnitude and nθ is the angular argument of the resultant complex number.

Step 1: Identify the values of r, θ and n

Given a complex number in polar form, \(z = r (\cos θ + i \sin θ)\), and that it needs to be raised to the power of 'n', identify the values of r (magnitude), θ (angle in radians) and 'n' (power). Remember r > 0, 0 ≤ θ < 2π and n is an integer.

Step 2: Use the power of complex number formula

Having identified the values of r, θ, and n, use the formula to find the power of the complex number in polar form, which is \(z^n = r^n (\cos(nθ) + i \sin(nθ))\). Here 'r' raised to the power 'n' represents the magnitude of the resultant complex number, and arguments of the resultant complex numbers are given by nθ.

Step 3: Simplify to find z^n

Simplify the right side of the equation to find the expression for z^n, the power of the complex number in polar form.