Problem 110 - Precalculus with Limits

Question

Show that \(\overline{z} = r[\cos(-\theta) + i\ \sin(-\theta)]\) is the complex conjugate of \(z = r(\cos\ \theta + i\ \sin\ \theta)\).

Step-by-Step Solution

Answer

The complex conjugate \( \overline{z} \) of \( z = r(\cos\ \theta + i\ \sin\ \theta) \) has been shown to be \( \overline{z} = r[\cos(-\theta) + i\ \sin(-\theta)] \) by using standard trigonometric properties, thus, exhibiting that the initial expression is indeed the complex conjugate.

Step 1: Represent the complex numbers in terms of their components

The given complex number \(z\) can be represented in terms of its components as \(z = r(\cos\ \theta + i\ \sin\ \theta)\) where \(r\) is the modulus of the complex number, and \(\theta\) is the argument of the complex number. The complex conjugate is usually obtained by changing the sign of the imaginary part.

Step 2: Express the conjugate using the components of complex number

The conjugate \(\overline{z}\) can be obtained by changing the sign of the imaginary part. Hence, the conjugate can be expressed initialy as \(\overline{z} = r(\cos\ \theta - i\ \sin\ \theta)\)

Step 3: Use trigonometric property to further simplify expression

However, using trigonometric property sin(-theta)=-sin(theta) and cos(-theta)=cos(theta), the conjugate can be expressed in the form \(\overline{z} = r[\cos(-\theta) + i\ \sin(-\theta)]\)

Key Concepts

Complex Number

A complex number is a mathematical entity consisting of two parts: a real part and an imaginary part. It is typically written in the form of \(a + bi\) , where \(a\) is the real component, \(b\) is the imaginary component, and \(i\) is the imaginary unit, defined by the property that \(i^2 = -1\) .

Complex numbers are essential in various fields of science and engineering because they simplify computations and describe phenomena in two dimensions. They can be added, subtracted, multiplied, and divided, following specific algebraic rules. The concept of complex conjugation is particularly useful, which involves flipping the sign of the imaginary part, transforming \(a + bi\) into \(a - bi\) .

Modulus of a Complex Number

The modulus of a complex number, often represented by \(r\) , is a measure of its size or magnitude, similar to the length of a vector. Mathematically, the modulus of a complex number \(z = a + bi\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\) .

The modulus is always a non-negative real number and provides important information about the complex number, such as its absolute value or distance from the origin in the complex plane. When discussing the geometry of complex numbers, the modulus signifies the radius of a circle centered at the origin, with the complex number lying on the circumference.

Trigonometric Representation of Complex Numbers

Complex numbers can be represented in trigonometric form (also known as polar form) where a complex number \(z\) is described by its modulus \(r\) and angle \(\theta\) , which is known as the argument. This representation is written as \(z = r(\cos \theta + i\sin \theta)\) .

In this form, \(\cos \theta\) and \(\sin \theta\) correspond to the directional components of the complex number on the real and imaginary axes, respectively. The angle \(\theta\) is usually measured from the positive real axis, and trigonometric functions help in describing the rotational aspect of complex numbers in the two-dimensional plane.

Trigonometric Properties

Trigonometric properties play a vital role in understanding complex numbers, especially in the context of calculating their conjugates. Two key properties used in this context are \(\sin(-\theta) = -\sin(\theta)\) and \(\cos(-\theta) = \cos(\theta)\) . These properties reflect the even and odd characteristics of the cosine and sine functions, respectively.

When dealing with complex numbers, these trigonometric properties allow us to convert between the standard form and the conjugate without altering the modulus. They also facilitate transformations in the complex plane, such as reflections across the real axis, leading to deeper insights into the symmetry and behavior of complex functions.