Problem 17 - Precalculus with Limits

Question

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(1 - \sqrt{3}i\)

Step-by-Step Solution

Answer

The trigonometric form of the given complex number \(1 - \sqrt{3}i\) is \(2[\cos(\frac{-\pi}{3})+ i \sin(\frac{-\pi}{3})]\).

Step 1: Identify the real and imaginary parts of the complex number

Here, the given complex number is \(1 - \sqrt{3}i\). So the real part (a) is 1 and the imaginary part (b) is \(-\sqrt{3}\). These can also be seen as the coordinates (a,b) or (1,'-\sqrt{3}') on a two-dimensional plane.

Step 2: Graphical representation of the complex number

Plot a point on a 2-dimensional plane corresponding to the real and imaginary parts of the complex number. That is, draw a dot at x = 1 and y = \(-\sqrt{3}\).

Step 3: Calculate the magnitude (r) of the complex number

The magnitude or modulus of a complex number in the form \(a + bi\) can be calculated as \(r = \sqrt{a^2 + b^2}\). Substituting 1 for 'a' and \(-\sqrt{3}\) for 'b' we get, \(r = \sqrt{1^2 + (-\sqrt{3})^2} = 2\).

Step 4: Calculate the argument (\(\theta\)) of the complex number

The argument or angle made with the positive x-axis can be calculated as \(\theta = arctan(\frac{b}{a})\). Here we use 'arctan' function to find the angle that has tangent equals to \(-\sqrt{3}\). So we have \(\theta = arctan(\frac{-\sqrt{3}}{1}) = \frac{-\pi}{3}\). Take note of the negative sign indicating the angle is measured clockwise from the positive real axis.

Step 5: Write the trigonometric form of the complex number

The trigonometric form of a complex number is \(z = r(\cos(\theta)+ i \sin(\theta))\). Substituting \(r = 2\) and \(\theta = \frac{-\pi}{3}\) we get \(z = 2[\cos(\frac{-\pi}{3})+ i \sin(\frac{-\pi}{3})]\).

Key Concepts

Graphical Representation of Complex Numbers

Understanding the graphical representation of complex numbers is vital to decipher complex analysis intuitively. Imagine a two-dimensional plane, often referred to as the complex plane or Argand diagram, where the horizontal axis represents real numbers, and the perpendicular, vertical axis signifies imaginary numbers.

To represent the complex number \(1 - \sqrt{3}i\), we pinpoint the real part, 1, on the horizontal axis and the imaginary part, \(-\sqrt{3}\), on the vertical axis. The resulting coordinate, \((1, -\sqrt{3})\), is akin to a physical location on a map, marking the exact position of our complex number in the plane.

This graphical approach not only helps in locating complex numbers but also provides a visual tool to comprehend operations like addition, subtraction, and multiplication of complex numbers. Additionally, it lays the groundwork for understanding the next key topics—magnitude and arguments of complex numbers.

Magnitude of Complex Numbers

If you think of a complex number as a point on a plane, the magnitude, also known as modulus, is how far this point is from the origin. In technical terms, for a complex number \(a + bi\), the magnitude is the square root of the sum of squares of its real part, \(a\), and imaginary part, \(b\):

\[ r = \sqrt{a^2 + b^2} \]
Consider our example, \(1 - \sqrt{3}i\). Its magnitude is:
\[ r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2 \]
The magnitude expresses the 'size' of the complex number and is always a non-negative value. In geometrical terms, it represents the length of the line segment (or vector) connecting the origin with the point depicting our complex number. Understanding magnitude is crucial as it facilitates comparisons of sizes between different complex numbers and plays a fundamental role in complex number operations.

Argument of Complex Numbers

The argument of a complex number is the measure of the angle it makes with the positive real axis, akin to a compass showing direction. For a complex number equivalent to \((a, b)\), the argument is calculated using the arctangent function (\(\arctan\)):

\[ \theta = \arctan\left(\frac{b}{a}\right) \]
In our specific case, \(1 - \sqrt{3}i\), the argument is:
\[ \theta = \arctan\left(\frac{-\sqrt{3}}{1}\right) = \frac{-\pi}{3} \]
It is crucial to consider that the sign of the argument indicates the direction of rotation from the positive real axis—with positive values indicating a counter-clockwise rotation, and negative values a clockwise one. The argument plays a central role in understanding the rotational aspect of complex numbers and is essential in the formulation of the trigonometric form, which combines both magnitude and argument to present a complete picture of a complex number's properties.