Problem 27 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$-6+5 i$$

Step-by-Step Solution

Answer

The polar form is \(\sqrt{61} \text{cis} 140.19^\circ\).

Step 1: Identify the Real and Imaginary Parts

For the complex number \(-6+5i\), the real part is \(-6\) and the imaginary part is \(5\). We will use these values to convert the complex number into polar form.

Step 2: Calculate the Magnitude

The magnitude of a complex number \(a + bi\) is determined by the formula: \(|z| = \sqrt{a^2 + b^2}\). Here, \(a = -6\) and \(b = 5\). Compute the magnitude: \(|-6+5i| = \sqrt{(-6)^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61}\).

Step 3: Calculate the Argument

The argument \(\theta\) is found using the arctangent function: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\), taking into account the sign of the real part. We use \(b = 5\) and \(a = -6\): \(\theta = \tan^{-1}\left(\frac{5}{-6}\right)\). Using a calculator, find the principal value: \(\theta \approx 140.19^\circ\) or \(\theta \approx 2.44\text{ radians}\) because the complex number is in the second quadrant.

Step 4: Formulate in Polar Form

The polar form of a complex number is expressed as: \(z = r\text{cis}\,\theta\), where \(r\) is the magnitude calculated in step 2, and \(\theta\) is the argument from step 3. Thus, the polar form of \(-6+5i\) is: \[\sqrt{61} \text{cis} 140.19^\circ\] or \[\sqrt{61} \text{cis} 2.44\]\text{ radians}.

Key Concepts

Complex Numbers

Complex numbers are fascinating because they extend our familiar number system to include numbers that can express both real and imaginary parts. A standard complex number is written in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part multiplied by the imaginary unit \(i\), defined as \(i = \sqrt{-1}\).
For example, the complex number \(-6 + 5i\) has a real part \(-6\) and an imaginary part \(5\). This type of number helps solve equations that do not have solutions in the real number set alone.
One key feature of complex numbers is their ability to be expressed in different forms, such as rectangular form \(a + bi\) and polar form \(r\text{cis}\,\theta\), offering versatile ways to represent and manipulate these numbers.

Magnitude

Magnitude in complex numbers refers to the length of the vector representing the complex number in the complex plane. This can be thought of as the distance from the origin \((0, 0)\) to the point \((a, b)\) represented by the complex number \(a + bi\).
The magnitude is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\).
For the complex number \(-6 + 5i\), its magnitude is:

• Calculate \((-6)^2 = 36\)
• Calculate \(5^2 = 25\)
• Sum the squares \(36 + 25 = 61\)
• Take the square root: \(|z| = \sqrt{61}\)

This value \(|z| = \sqrt{61}\) represents how far the number \(-6 + 5i\) is from the origin on the complex plane.

Argument

The argument of a complex number is the angle created with the positive x-axis of the complex plane. It's essentially how much you need to "rotate" to get from the positive x-axis to the direction of the vector representing the complex number.
For a complex number \(a + bi\), the argument \(\theta\) is obtained from the arctangent function: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).
In the case of \(-6 + 5i\):

• Use the values \(a = -6\) and \(b = 5\)
• Compute \(\theta = \tan^{-1}\left(\frac{5}{-6}\right)\)
• Considering the complex number is in the second quadrant, the calculated argument is approximately \(140.19^\circ\) or \(2.44\) radians

Knowing the quadrant helps define the correct direction of the angle, which is crucial in placing the complex number accurately.

Conversion to Polar Form

Converting a complex number to polar form involves expressing it as \(z = r\text{cis}\,\theta\), where \(r\) is the magnitude, and \(\theta\) is the argument.
This form highlights both the distance from the origin \(r\) and the angle of rotation \(\theta\). This method is particularly useful in multiplication and division of complex numbers.
For \(-6 + 5i\), follow these steps:

• Find the magnitude: \(r = \sqrt{61}\)
• Find the argument: \(\theta \approx 140.19^\circ\) or \(2.44\) radians
• Express in polar form: \(z = \sqrt{61} \text{cis} \, 140.19^\circ\) or \(z = \sqrt{61} \text{cis} \, 2.44\) radians

This polar representation offers a clear way to visualize and calculate manipulations involving the complex number.