Problem 18 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-5-2 i $$

Step-by-Step Solution

Answer

\( z = \sqrt{29} ( \cos(\pi + \tan^{-1}(\frac{2}{5})) + i \sin(\pi + \tan^{-1}(\frac{2}{5})) ) \); \( \operatorname{Re}(z) = -5 \), \( \operatorname{Im}(z) = -2 \), \(|z| = \sqrt{29} \), \( \arg(z) = \pi + \tan^{-1}(\frac{2}{5}) \).

Step 1: Identify Real and Imaginary Parts

The complex number given is \( z = -5 - 2i \). We identify the real part as \( \operatorname{Re}(z) = -5 \) and the imaginary part as \( \operatorname{Im}(z) = -2 \).

Step 2: Calculate the Magnitude

The magnitude, or modulus, of a complex number \( z = a + bi \) is given as \( |z| = \sqrt{a^2 + b^2} \). Substitute \( a = -5 \) and \( b = -2 \). We have: \[ |z| = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \]

Step 3: Calculate the Argument

The argument \( \arg(z) \) of a complex number \( a + bi \) is calculated using \( \tan^{-1}\left(\frac{b}{a}\right) \). For \( z = -5 - 2i \), we find: \[ \arg(z) = \tan^{-1}\left(\frac{-2}{-5}\right) = \tan^{-1}\left(\frac{2}{5}\right) \] Since \( z \) is in the third quadrant, add \( \pi \) to the principal value: \[ \arg(z) = \pi + \tan^{-1}\left(\frac{2}{5}\right) \]

Step 4: Identify the Principal Argument

The principal argument \( \operatorname{Arg}(z) \) is the value of the argument that lies in the interval \(( -\pi, \pi] \). In this case, the same as the calculated \( \arg(z) \), which is \( \pi + \tan^{-1}\left(\frac{2}{5}\right) \).

Step 5: Write the Polar Representation

The polar form of a complex number is \( z = r (\cos \theta + i \sin \theta) \), where \( r = |z| \) and \( \theta = \arg(z) \). Substitute the values: \[ z = \sqrt{29} \left( \cos\left( \pi + \tan^{-1}\left(\frac{2}{5}\right) \right) + i \sin\left( \pi + \tan^{-1}\left(\frac{2}{5}\right) \right) \right) \]

Key Concepts

Polar Representation

Complex numbers can be expressed in a form called polar representation, which is an alternative to the standard form \(a + bi\). In polar representation, a complex number \(z = a + bi\) is written in terms of its magnitude \(|z|\) and its argument \(\theta\), as:

• \( z = |z| (\cos \theta + i \sin \theta) \)

This form can also be expressed using Euler's formula as:

• \( z = |z| e^{i\theta} \)

The benefits of polar form include easier multiplication and division of complex numbers. Instead of dealing with real and imaginary parts separately, you simply multiply or divide their magnitudes and add or subtract their arguments. For example, the polar form of \(z = -5 - 2i\) is given by calculating its magnitude \(\sqrt{29}\) and argument \(\pi + \tan^{-1}(\frac{2}{5})\). Therefore, the polar representation is:

• \( z = \sqrt{29} (\cos(\pi + \tan^{-1}(\frac{2}{5})) + i \sin(\pi + \tan^{-1}(\frac{2}{5})) \)

Real and Imaginary Parts

Every complex number consists of two key components: a real part and an imaginary part. These are essential for understanding and manipulating complex numbers. A complex number is commonly expressed as \(z = a + bi\), where:

• \(a\) is the real part, denoted as \(\operatorname{Re}(z)\)
• \(b\) is the imaginary part, denoted as \(\operatorname{Im}(z)\)

For the given exercise with \(z = -5 - 2i\):

• The real part \(\operatorname{Re}(z) = -5\)
• The imaginary part \(\operatorname{Im}(z) = -2\)

These components are fundamental because they allow us to plot complex numbers on the complex plane, where the x-axis represents the real part, and the y-axis represents the imaginary part. Understanding the separation between real and imaginary parts is critical as they contribute differently to operations like addition, subtraction, and conversion to polar form.

Magnitude of Complex Numbers

The magnitude or modulus of a complex number measures the distance from the origin of the complex plane to the point represented by the complex number. It's like finding the length of the hypotenuse in a right triangle formed by the axes.The magnitude is denoted by \(|z|\), and for a complex number \(z = a + bi\), it is calculated as:

• \(|z| = \sqrt{a^2 + b^2}\)

Immediately, you can see from our example where \(a = -5\) and \(b = -2\) that:

• \(|z| = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}\)

The magnitude is essential when transforming complex numbers into their polar form. It represents how "large" or "far away" from the origin the complex number is, which is intrinsically tied to its real and imaginary parts.

Argument of Complex Numbers

The argument of a complex number is a measure of the angle (in radians) formed between the positive real-axis and the line connecting the origin of the complex plane to the point represented by the complex number. It gives directionality to the complex number.For a complex number \(z = a + bi\), the argument \(\arg(z)\) is determined using:

• \( \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \)

However, you must consider the quadrant where \(z\) lies to get the accurate argument. If the complex number is in:

• First quadrant: \(\arg(z)\) remains as calculated.
• Second or third quadrant: add \(\pi\).
• Fourth quadrant: typically subtract \(2\pi\) if needed to stay within \((-\pi, \pi]\).

In our exercise, with \(z = -5 - 2i\) located in the third quadrant, the argument is:

• \( \arg(z) = \pi + \tan^{-1}\left(\frac{2}{5}\right) \)

Understanding \(\arg(z)\) is crucial as it not only tells you the direction of the complex number but is also pivotal for converting to polar coordinates.