Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 4 i $$

Step-by-Step Solution

Answer

The trigonometric form is \(4(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\).

Step 1: Identify the Complex Number

The given complex number is purely imaginary: \(4i\). In a complex number form, this is expressed as \(0 + 4i\), where the real part \(a = 0\) and the imaginary part \(b = 4\).

Step 2: Calculate Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). For \(0 + 4i\), the magnitude is \(r = \sqrt{0^2 + 4^2} = \sqrt{16} = 4\).

Step 3: Determine Argument

The argument \(\theta\) of the complex number in the form \(a + bi\) is given by \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Since \(a = 0\), we can determine \(\theta\) by analyzing the location on the complex plane. The number \(4i\) is on the positive imaginary axis, giving \(\theta = \frac{\pi}{2}\).

Step 4: Write in Trigonometric Form

The trigonometric form of a complex number is expressed as \(r(\cos \theta + i\sin \theta)\). Substituting \(r = 4\) and \(\theta = \frac{\pi}{2}\) gives: \(4(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\).

Key Concepts

Understanding Complex Numbers

Complex numbers are a fascinating and essential part of mathematics. They are numbers that have both a real part and an imaginary part. The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For example, in the complex number \(4i\), the real part is \(0\) and the imaginary part is \(4\). This number lies on the imaginary axis of the complex plane, which is a two-dimensional plane used to graphically represent complex numbers.
Each complex number can also be visualized as a point or a vector extending from the origin of this plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. Understanding the position of a complex number on this plane is crucial in finding its magnitude and argument.

Calculating the Magnitude of Complex Numbers

The magnitude of a complex number is the distance from the point representing the complex number to the origin on the complex plane. It gives us a measure of the "size" of the complex number. For a complex number \(a + bi\), the magnitude (or modulus) is calculated using the Pythagorean theorem as \(r = \sqrt{a^2 + b^2}\).
In the exercise, for the number \(4i\), with \(a = 0\) and \(b = 4\), the magnitude is calculated by evaluating \(\sqrt{0^2 + 4^2} = \sqrt{16} = 4\).
This process is akin to finding the hypotenuse of a right triangle where one leg is the real part and the other is the imaginary part. Magnitude is always a non-negative quantity and provides a scalar value for comprehensive analysis or further calculations in the trigonometric form.

Exploring the Argument of Complex Numbers

The argument of a complex number, often denoted \(\theta\), is the angle that the line representing the complex number makes with the positive real axis. This angle helps to understand the orientation or direction of the complex number in the complex plane.
To find the argument, we use the inverse tangent function: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). In cases where \(a = 0\), the complex number lies directly on the imaginary axis, which simplifies finding the angle. For \(4i\), since it is on the positive imaginary axis, the argument is \(\theta = \frac{\pi}{2}\).
It's important to note the range of \(\theta\), which is typically from \(0\) to \(2\pi\). Understanding the argument helps to convert complex numbers into their trigonometric, or polar, form using the equation \(r(\cos \theta + i \sin \theta)\). This form is particularly useful in polar coordinates and complex number operations like multiplication and division.