Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The trigonometric form is \(4\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})\).

Step 1: Identify the real and imaginary parts

The given complex number is \(-4 - 4i\). Here, the real part is \(a = -4\),and the imaginary part is\(b = -4\).

Step 2: Calculate the magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by the formula:\[r = \sqrt{a^2 + b^2}\]Substitute \(a = -4\) and \(b = -4\) into the formula:\[r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]

Step 3: Calculate the argument \(\theta\)

The argument \(\theta\) of a complex number \(a + bi\) is calculated using:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]So we have:\[\theta = \tan^{-1}\left(\frac{-4}{-4}\right) = \tan^{-1}(1)\]This angle \(\theta\) is typically \(\frac{\pi}{4}\), but since the complex number is in the third quadrant (both real and imaginary parts are negative), the actual angle is:\[\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\]

Step 4: Express in trigonometric form

Now, express the complex number in trigonometric form, which is:\[r(\cos \theta + i \sin \theta)\]Substitute \(r = 4\sqrt{2}\) and \(\theta = \frac{5\pi}{4}\):\[-4-4i = 4\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})\]

Key Concepts

Trigonometric Form

To express a complex number in trigonometric form, also known as polar form, we represent it as \(r(\cos \theta + i \sin \theta)\). Here, \(r\) is the magnitude, and \(\theta\) is the argument of the complex number.
The trigonometric form gives a powerful way to visualize and perform operations on complex numbers, particularly multiplication and division, as it translates complex multiplication into simple operations on magnitudes and angles.
The trigonometric form is also extremely useful when plotting complex numbers on the polar coordinate system, which displays the number's distance from the origin (magnitude) and its angle from the positive real axis (argument).
For instance, if you have a complex number \(-4 - 4i\), its trigonometric form can be computed and will help in understanding its placement within the complex plane.

Magnitude of a Complex Number

The magnitude of a complex number, often denoted as \(r\), is the distance of the complex number from the origin in the complex plane. It can be thought of as the "length" of the vector representing the complex number.
To calculate the magnitude, we use the formula \(r = \sqrt{a^2 + b^2}\) where \(a\) is the real part and \(b\) is the imaginary part. The magnitude is always non-negative.
In the example \(-4 - 4i\), this calculation becomes \(r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}\).
A larger magnitude indicates a complex number that is further from the origin, while a smaller magnitude indicates one that is closer.

Argument of a Complex Number

The argument of a complex number defines the angle measured from the positive real axis to the vector representing the complex number in the complex plane.
It is typically denoted by \(\theta\) and can be found using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\), where \(a\) is the real part and \(b\) is the imaginary part. Note that you must consider which quadrant the angle lies in, to adjust \(\theta\) appropriately.
For \(-4-4i\), calculating the argument gives \(\theta = \tan^{-1}\left(\frac{-4}{-4}\right) = \tan^{-1}(1)\). This is initially \(\frac{\pi}{4}\), but since \(-4 - 4i\) is in the third quadrant, the angle is adjusted to \(\frac{5\pi}{4}\).
The correct identification of the argument helps in accurately placing the complex number in the polar coordinate system.

Polar Coordinates

Polar coordinates express a point in terms of its radial distance from the origin and an angle from a reference direction, like a location on a polar grid instead of a more traditional Cartesian grid.
This coordinate system is highly beneficial for handling complex numbers, as it naturally pairs with the trigonometric form \(r(\cos \theta + i \sin \theta)\).
Using polar coordinates, each complex number corresponds to a unique pair \((r, \theta)\), which simplifies the process of multiplication and division of complex numbers. For example, multiplying corresponds to multiplying magnitudes and adding angles.
For the complex number \(-4-4i\), expressed in polar coordinates, you would say it has a magnitude \(4\sqrt{2}\) and an argument \(\frac{5\pi}{4}\). This clear representation aids in performing more complex operations and visualizing transformations in the complex plane.