Problem 95 - Algebra and Trigonometry

Question

What is the polar form of a complex number?

Step-by-Step Solution

Answer

The polar form of a complex number \(a + bi\) is calculated by finding the magnitude \(r = \sqrt{a^2 + b^2}\) and the angle \(\theta = tan^{-1}\left(\frac{b}{a}\right)\), resulting in the form \(r(cos \(\theta\) + isin\(\theta\)) or r \(\text{cis}\) \(\theta\).

Step 1: Calculate the magnitude of the complex number

To calculate the magnitude of any complex number, \(a + bi\), use Pythagoras' theorem. The magnitude \(r\) is obtained by \(r = \sqrt{a^2 + b^2}\). This represents the distance of the point \((a, b)\) from the origin in a 2D space.

Step 2: Calculate the angle

The angle \(\theta\) can be calculated using \(\theta = tan^{-1}\left(\frac{b}{a}\right)\), as \(b/a\) represents the tangent of the angle formed with the real axis. Use the appropriate quadrant depending on the signs of \(a\) and \(b\).

Step 3: Writing in the polar form

After determining \(r\) and \(\theta\), substitute these values into the polar form \(r(cos \(\theta\) + isin\(\theta\)) or r \(\text{cis}\) \(\theta\). This converts the complex number from rectangular form to polar form.