Problem 11 - 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 \(\operatorname{Arg}(z)\). $$ z=6 $$

Step-by-Step Solution

Answer

The polar representation of \(z = 6\) is \(6(\cos(0) + i\sin(0))\). \(\operatorname{Re}(z) = 6\), \(\operatorname{Im}(z) = 0\), \(|z| = 6\), \(\arg(z) = 0\), \(\operatorname{Arg}(z) = 0\).

Step 1: Identify the Real and Imaginary Parts of z

Since \(z = 6\), this implies that \(z = 6 + 0i\). Thus, the real part is \(\operatorname{Re}(z) = 6\) and the imaginary part \(\operatorname{Im}(z) = 0\).

Step 2: Calculate the Magnitude of z

The magnitude (or modulus) of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). For \(z = 6 + 0i\), \(|z| = \sqrt{6^2 + 0^2} = \sqrt{36} = 6\).

Step 3: Determine the Argument of z

The argument of a complex number \(z = a + bi\) is calculated using \(\arg(z) = \tan^{-1}(\frac{b}{a})\). For \(z = 6 + 0i\), we have \(\arg(z) = \tan^{-1}(\frac{0}{6}) = \tan^{-1}(0) = 0\). This is due to the fact that the number lies on the positive real axis.

Step 4: Write the Polar Representation of z

The polar form of a complex number is given by \(z = |z| (\cos(\theta) + i\sin(\theta))\) or \(z = |z| e^{i\theta}\). For \(z = 6\), where \(|z| = 6\) and \(\theta = 0\), the polar representation is \(6(\cos(0) + i\sin(0))\) or \(6e^{i \cdot 0} = 6\).

Step 5: Identify Principal Argument of z

The principal argument, \(\operatorname{Arg}(z)\), of a complex number is the value of \(\arg(z)\) that lies in the interval \((-\pi, \pi]\). For \(z = 6\), \(\operatorname{Arg}(z) = 0\), which is within the specified interval.

Key Concepts

Understanding Complex Numbers

Complex numbers are an extension of the real numbers that include a component called the imaginary part. They are expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The unit \(i\) is defined as the square root of \(-1\).

Let's break down the complex number given in our exercise. We have \(z = 6\). This can be rewritten as \(6 + 0i\), which indicates that the real part \(\operatorname{Re}(z) = 6\), and the imaginary part \(\operatorname{Im}(z) = 0\).

Complex numbers are not just for mathematics; they have applications in engineering, physics, and many other fields, particularly where two-dimensional space is involved.

Grasping Polar Coordinates

Polar coordinates offer a different way of representing complex numbers, relying on their distance from the origin (radius) and the angle from the positive real axis (angle). This can be especially helpful when dealing with rotations and oscillations.

A complex number \(z\) can be converted into polar coordinates based on its magnitude \(|z|\) and argument \(\arg(z)\). Polar representation is expressed as:

• \(z = |z| (\cos(\theta) + i\sin(\theta))\)
• Or alternatively, \(z = |z| e^{i\theta}\)

In our case, \(z = 6 + 0i\) is simply \(6\), and since it lies entirely on the real axis, its polar form remains \(6(\cos(0) + i\sin(0))\) or equivalently \(6e^{i \cdot 0} = 6\). This indicates that the angle \(\theta = 0\), confirming it sits on the positive real axis. This form reveals the inherent beauty and symmetry in complex numbers.

Magnitude of a Complex Number

The magnitude, also known as the modulus, of a complex number is a measure of its distance from the origin in the complex plane. This is computed using the formula \(|z| = \sqrt{a^2 + b^2}\), where \(z = a + bi\).

If we consider our example \(z = 6 + 0i\), the magnitude calculation becomes straightforward:

\[|z| = \sqrt{6^2 + 0^2} = \sqrt{36} = 6\]
The magnitude tells us how far away the point representing \(z\) is from the origin. The higher the magnitude, the further the point is, which is a helpful concept when visualizing complex numbers in the plane.

The Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis in the complex plane. It is an essential component of the polar representation. The basic formula for calculating the argument is \(\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)\), valid for \(a eq 0\).

For our specific example, \(z = 6 + 0i\), we calculate:

\[\arg(z) = \tan^{-1}\left(\frac{0}{6}\right) = \tan^{-1}(0) = 0\]
This zero angle reveals that \(z\) lies on the positive real axis. The principal argument, denoted \(\operatorname{Arg}(z)\), is the unique value of the argument that falls within the range \((-\pi, \pi]\).
The argument is crucial because it helps distinguish the direction of the complex number from the origin, making it a fundamental aspect of its polar representation.