Problem 87 - Precalculus

Question

Consider the complex number in polar form \(z=r \cos \theta+i \sin \theta .\) What is the polar form of \(-z ?\)

Step-by-Step Solution

Answer

The polar form of \(-z\) is \( r (\cos(\theta + \pi) + i \sin(\theta + \pi)) \).

Step 1: Understanding the Given Complex Number

The complex number is given in polar form as \( z = r \cos \theta + i \sin \theta \), where \( r \) is the magnitude of the complex number and \( \theta \) is the argument or angle.

Step 2: Formulating the Negative of the Complex Number

The negative of the complex number \( z \) is \( -z = - (r \cos \theta + i \sin \theta) = -r \cos \theta - i \sin \theta \).

Step 3: Exploring the Effect of Negation on Polar Coordinates

To express \( -z \) in polar form, explore how negation affects \( r \cos \theta \) and \( i \sin \theta \). The effect of negating a complex number can be thought of as adding \( \pi \) (180 degrees) to the argument \( \theta \).

Step 4: Determine New Polar Form

The polar form of \( -z \) can be rewritten in terms of its magnitude and a new angle. The magnitude remains the same as \( r \), but the angle becomes \( \theta + \pi \). So, the polar form of \( -z \) is \( r (\cos(\theta + \pi) + i \sin(\theta + \pi)) \).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in two main forms: rectangular (or Cartesian) and polar form. In rectangular form, a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. Polar form, however, represents a complex number as a product of its magnitude and a directional component in terms of an angle. This is especially handy when dealing with multiplications or powers of complex numbers.

In polar form, a complex number is typically written as \( z = r(\cos \theta + i\sin \theta) \), where:

• \( r \) is the magnitude or modulus of the complex number, calculated as \( \sqrt{a^2 + b^2} \).
• \( \theta \) is the argument or phase angle, the angle formed with the positive x-axis, usually measured in radians.

This representation leverages Euler's formula, \( e^{i\theta} = \cos \theta + i\sin \theta \), allowing complex numbers to be rewritten as \( z = re^{i\theta} \). The polar form simplifies operations like multiplication and division of complex numbers because the magnitudes are multiplied and angles are added or subtracted.

Negation of Complex Numbers

The negation of a complex number involves multiplying the original number by -1. Understanding this in the context of polar form is crucial. When you negate a complex number \( z = r(\cos \theta + i\sin \theta) \), it becomes \( -z = -r(\cos \theta + i\sin \theta) \).

To comprehend this visually, you can think of negation as reflecting the vector representing the complex number across the origin. This operation does not change the magnitude \( r \), but it affects the direction, specifically the angle \( \theta \). To reflect the vector, you add \( \pi \) (or 180 degrees) to \( \theta \). Hence, the new complex number has the same magnitude but an angle updated to \( \theta + \pi \) in polar coordinates:

\[ -z = r(\cos(\theta + \pi) + i\sin(\theta + \pi)) \]This approach is incredibly useful in complex arithmetic when needing to handle directional changes without a change in magnitude.

Angle Addition in Polar Coordinates

Angle addition in polar coordinates is a key concept when working with complex numbers. It describes how to adjust the angle of a vector (or complex number) in polar form by adding or subtracting another angle.

When you have a complex number in polar form, expressed as \( r(\cos \theta + i\sin \theta) \), and you want to reflect or rotate it, you apply angle addition. Negation of a complex number is a specific case of angle addition where \( \pi \) is added to the angle \( \theta \). This operation alters the direction of the vector without changing its length or magnitude.

Why does adding \( \pi \) work? Adding 180 degrees effectively points the vector in the opposite direction. In terms of the unit circle, if you imagine the angle increasing by \( \pi \), the point (or vector head) shifts to the exact opposite side of the circle. The concept of angle addition is thus valuable in accurately determining the new position of a complex number post-transformation, greatly simplifying complex arithmetic calculations in physics and engineering.