Problem 82 - Precalculus

Question

Use a calculator to express the complex number in polar form. $$-3 a-4 a i, \text { where } a>0$$

Step-by-Step Solution

Answer

The polar form is \(5a\left(\cos\left(\pi + \tan^{-1}\left(\frac{4}{3}\right)\right) + i\sin\left(\pi + \tan^{-1}\left(\frac{4}{3}\right)\right)\right)\).

Step 1: Identify the complex number components

The complex number given is \(-3a - 4ai\). Here, the real component is \(-3a\), and the imaginary component is \(-4a\).

Step 2: Calculate the magnitude of the complex number

The magnitude, or modulus, of a complex number \(a + bi\) is calculated using the formula: \(|z| = \sqrt{a^2 + b^2}\). For our complex number \(-3a - 4ai\), substitute \(a\) with \(-3a\) and \(b\) with \(-4a\): \[ |z| = \sqrt{(-3a)^2 + (-4a)^2} = \sqrt{9a^2 + 16a^2} = \sqrt{25a^2} = 5a \]

Step 3: Calculate the argument (angle) of the complex number

The argument (angle) \(\theta\) of a complex number \(a + bi\) is given by \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). For our complex number \(-3a - 4ai\):\[ \theta = \tan^{-1}\left(\frac{-4a}{-3a}\right) = \tan^{-1}\left(\frac{4}{3}\right) \]Since the complex number is in the third quadrant (both \(-3a\) and \(-4a\) are negative), we add \(\pi\) to the angle: \[ \theta = \pi + \tan^{-1}\left(\frac{4}{3}\right) \]

Step 4: Express the complex number in polar form

The polar form of a complex number is given by \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the argument. Substitute the values we have found:\[ z = 5a \left(\cos\left(\pi + \tan^{-1}\left(\frac{4}{3}\right)\right) + i \sin\left(\pi + \tan^{-1}\left(\frac{4}{3}\right)\right)\right) \]

Key Concepts

Polar Form

To express a complex number in polar form, we need to rewrite it using the magnitude and the angle (argument). Complex numbers in polar form are represented as:

• \( r(\cos \theta + i \sin \theta) \)
  
• Here, \( r \) is the magnitude of the complex number, while \( \theta \) is the argument or angle measured from the positive x-axis.

This form is very useful in simplifying complex number arithmetic, especially multiplication and division. It's like converting a number into a geometry-friendly representation that is easier to understand and compute, using radius (magnitude) and angle.

Complex Number Components

A complex number usually consists of a real part and an imaginary part.
For the expression \(-3a - 4ai\), you can identify:

• The real component is \(-3a\).
• The imaginary component is \(-4a\), which is tied to the imaginary unit \(i\).

These components help us visualize the complex number on a two-dimensional plane, called the complex plane. The real component defines the horizontal position, while the imaginary component defines the vertical. This way, complex numbers are easily represented as points or vectors.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number is like its distance from the origin in the complex plane.
It is calculated using the formula:

• \( |z| = \sqrt{a^2 + b^2} \)
  

For our complex number \(-3a - 4ai\), we end up with:

• \( |z| = \sqrt{(-3a)^2 + (-4a)^2} = \sqrt{9a^2 + 16a^2} = \sqrt{25a^2} = 5a \)
  

Understanding magnitude is essential for determining the length or size of a complex number, which helps in quantifying its amplitude if the number represents a signal.

Argument of Complex Numbers

The argument or angle of a complex number provides the direction of the vector on the complex plane.
It is derived using the formula:

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

For \(-3a - 4ai\), both real and imaginary components are negative. This places the complex number in the third quadrant.Since angles are measured against the positive x-axis, you'd adjust by adding \(\pi\) when in the third quadrant:

• \( \theta = \pi + \tan^{-1}\left(\frac{4}{3}\right) \)
  

The angle provides the orientation of the complex number. It’s like the needle on a compass pointing out the direction from the origin to the complex number's location on the plane.