Problem 29 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$-5+12 i$$

Step-by-Step Solution

Answer

The complex number in polar form is \(13(\cos 113.1^\circ + i \sin 113.1^\circ)\).

Step 1: Understand the Components of a Complex Number

A complex number consists of a real part and an imaginary part. For the number \(-5 + 12i\), the real part is \(-5\) and the imaginary part is \(12\).

Step 2: Calculate the Magnitude

The magnitude, or modulus, of a complex number \(a + bi\) is calculated as \(r = \sqrt{a^2 + b^2}\). Here, we have \(a = -5\) and \(b = 12\). Substitute these values into the formula: \[r = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\]So, the magnitude \(r\) is \(13\).

Step 3: Calculate the Argument

The argument \(\theta\) of a complex number is the angle it makes with the positive real axis, calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Here, \(a = -5\) and \(b = 12\). \[\theta = \tan^{-1}\left(\frac{12}{-5}\right)\]This calculation gives an angle in the fourth quadrant, so we add \(180^\circ\) to adjust for being in the second quadrant where \(a < 0\) and \(b > 0\):\[\theta = \tan^{-1}\left(-2.4\right) + 180^\circ \approx 336.8^\circ (20.9^{\circ} + 180^\circ)\]The correct positive angle is \(113.1^\circ\).

Step 4: Write in Polar Form

Now that we have the magnitude \(r = 13\) and angle \(\theta = 113.1^\circ\), we can express the complex number in polar form:\[-5 + 12i = 13\left(\cos 113.1^\circ + i \sin 113.1^\circ\right)\]

Key Concepts

Magnitude of Complex Number

The magnitude of a complex number, also known as its modulus, is a measure of its distance from the origin on a complex plane. This concept can be thought of similarly to the length of the hypotenuse in a right triangle. For a complex number expressed as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component, the magnitude is calculated using the formula

• \( r = \sqrt{a^2 + b^2} \)

This formula results from applying the Pythagorean theorem.
Let's take the example \( -5 + 12i \):

• Real part \( a = -5 \)
• Imaginary part \( b = 12 \)

Substituting these values:\[ r = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]This means the complex number is 13 units from the origin.

Argument of Complex Number

The argument of a complex number is the angle formed with the positive x-axis in a complex plane. It is typically measured in degrees or radians and can direct us toward the number's position around the origin.
When you have a complex number in the form of \( a + bi \), the argument \( \theta \) is determined by the formula:

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

In certain cases, adjusting the angle is necessary to express it as a positive angle or within the standard range:

• For \( -5 + 12i \), \( a = -5 \) and \( b = 12 \)

Substitute the values:\[ \theta = \tan^{-1}\left(\frac{12}{-5}\right)\]This results in an angle in the fourth quadrant. To convert to the second quadrant, where the number truly lies based on its components:\[\theta = 113.1^{\circ} = 20.9^{\circ} + 180^{\circ}\]Thus, the angle or argument is \( \theta = 113.1^{\circ} \). This tells us the direction of the complex number from the origin.

Real and Imaginary Components

Every complex number comprises two important parts: the real component and the imaginary component.
The complex number \(-5 + 12i\) clearly demonstrates this with its distinct sections:

• The real part is \( -5 \), and it dictates the horizontal position on the complex plane.
• The imaginary part is \( 12 \), which determines the vertical position and is represented by the imaginary unit \( i \), indicating a 90-degree rotation.

Consider these components as forming a 2D vector, where:

• The x-coordinate is given by the real part \(-5\).
• The y-coordinate is provided by the imaginary part \(12\).

Thus, a complex number elegantly combines both real and imaginary components, functioning within a realm that extends beyond our typical understanding of numbers. Appreciate how these components interact to place the number in a nearly three-dimensional space, influencing both the magnitude and direction.