91Ó°ÊÓ

Complex numbers are numbers that have both a real and an imaginary part. These are usually represented as: \[ z = x + yi \]where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
Another way to represent complex numbers is by using their polar form, which is often more convenient for multiplication and division. The polar form of a complex number \( z \) is given by:\[ z = r (\text{cos} \theta + i \text{sin} \theta) \]Here, \( r \) is the magnitude (or modulus) of the complex number, calculated as:\[ r = \sqrt{x^2 + y^2} \]And \( \theta \) is the argument (or angle) of the complex number, found using:\[ \theta = \tan^{-1} (y/x) \]This form provides an elegant way to express complex numbers, simplifying various operations and helping visualize them geometrically.

The reciprocal of a complex number \( z \), denoted as \( \frac{1}{z} \), is essentially the multiplicative inverse. To find the reciprocal when \( z \) is in its polar form, we start from the given:\[ z = r (\text{cos} \theta + i \text{sin} \theta) \]The reciprocal can thus be expressed as:\[ \frac{1}{z} = \frac{1}{r(\text{cos} \theta + i \text{sin} \theta)} \]To simplify this, we need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. For the complex number in polar form, the conjugate is:\[ \text{cos} \theta - i \text{sin} \theta \]

Rationalizing the denominator involves making the denominator a real number by eliminating any imaginary part. Given the reciprocal form:\[ \frac{1}{z} = \frac{1}{r(\text{cos} \theta + i \text{sin} \theta)} \]we multiply both the numerator and denominator by the conjugate of the denominator:\[ \frac{1}{r(\text{cos} \theta + i \text{sin} \theta)} \times \frac{\text{cos} \theta - i \text{sin} \theta}{\text{cos} \theta - i \text{sin} \theta} \]This yields:\[ \frac{\text{cos} \theta - i \text{sin} \theta}{r((\text{cos} \theta)^2 + (\text{sin} \theta)^2)} \]Since \((\text{cos} \theta)^2 + (\text{sin} \theta)^2 = 1\), the expression simplifies further to:\[ \frac{\text{cos} \theta - i \text{sin} \theta}{r} \]Therefore, the reciprocal of \( z \) in polar form is:\[ \frac{1}{z} = \frac{1}{r} (\text{cos} \theta - i \text{sin} \theta) \]