Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

What is the trigonometric form of the complex number \(z=a+b i ?\)

Step-by-Step Solution

Answer

Trigonometric form of the complex number \(z\) is given by, \(r(\cos \theta + i \sin \theta)\).

Step 1: Calculate the magnitude

The magnitude (or modulus) of a complex number in the form \(a+bi\) can be found using Pythagoras’ theorem, \(\sqrt{a^2+b^2}\). Thus, for a complex number \(z\) it is given as \(r = |z| = \sqrt{a^2+b^2}\).

Step 2: Calculate the argument

The argument (or angle) of a complex number is usually denoted by \(\theta\) (theta). It can be calculated using the formula \(\theta = \arctan(\frac{b}{a})\).

Step 3: Write in trigonometric form

After you've calculated both the magnitude \(r\) and the argument \(\theta\), you can write the complex number \(z\) in its trigonometric form, \(z = r(\cos \theta + i \sin \theta)\).

Key Concepts

Complex Number Magnitude

Understanding the magnitude of a complex number is essential when working with these mathematical expressions. The magnitude, also known as the modulus, represents the distance of the complex number from the origin in a two-dimensional plane.

For a complex number written in the form \(z=a+bi\), the magnitude can be calculated by applying the Pythagoras’ theorem to the 'a' and 'b' components, considered as the adjacent and opposite sides of a right triangle respectively. The formula for the magnitude \(r\) is \[r = |z| = \sqrt{a^2+b^2}\]. By using this formula, students can easily determine the 'size' of their complex number, providing a clearer geometric interpretation of complex numbers.

Argument of a Complex Number

The argument of a complex number is quite literally the angle that the line constructed from the origin to the number makes with the positive direction of the x-axis. It's a measure of direction rather than size and is normally denoted by the Greek letter \(\theta\) (theta).

Calculating the argument can be done through the arctangent function, using the formula \(\theta = \arctan(\frac{b}{a})\), where 'a' is the real part and 'b' is the imaginary part of the complex number. This step is crucial for converting the complex number into its trigonometric form as it provides the exact direction of the complex number in the complex plane.

Pythagoras’ Theorem

Pythagoras’ theorem is a fundamental concept in mathematics, particularly when dealing with right-angled triangles. According to this theorem, for a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem is instrumental in calculating the magnitude of complex numbers as it relates the real part \(a\) and the imaginary part \(b\) to the magnitude \(r\), whereby \(r\) acts as the hypotenuse. In essence, \(r = \sqrt{a^2+b^2}\) is derived from this cornerstone of geometry. By visualizing complex numbers as points in the complex plane, students can better understand their positions and values in relation to the coordinate axes.

Trigonometric Representation

Once the magnitude and the argument of a complex number are known, the complex number can be written in its trigonometric form. This form is highly useful for tasks such as multiplication, division, and finding powers and roots of complex numbers.

The trigonometric representation expresses the complex number \(z\) as \[z = r(\cos \theta + i \sin \theta)\], where \(r\) is the magnitude and \(\theta\) is the argument. It's a more geometric way of way of looking at complex numbers and allows a more visual approach to multiplication and division, as it links the algebraic operations with rotations and rescaling in the complex plane. This representation showcases the beautiful interplay between algebra and geometry inherent in complex numbers.