Problem 93 - Precalculus

Question

Another way of using a graphing calculator to represent complex numbers in rectangular form is to enter the real and imaginary parts as a list of two numbers and use the \([\text { sUM }]\) command to find the modulus. Write \(28-21 i\) in polar form using the \([\mathrm{SUM}]\) command to find its modulus, and round the angle to the nearest degree.

Step-by-Step Solution

Answer

The polar form of \(28 - 21i\) is \(35 \text{cis}(-37^\circ)\).

Step 1: Identify Real and Imaginary Parts

The given complex number is \(28 - 21i\). Here, 28 is the real part and -21 is the imaginary part.

Step 2: Create a List

Input the real and imaginary parts as a list: \([28, -21]\). This list represents the components of the complex number.

Step 3: Calculate the Modulus Using SUM Command

Use the SUM command to calculate the modulus (magnitude) of the complex number. The modulus \([\text{SUM}]\) is calculated as \(\sqrt{28^2 + (-21)^2}\).

Step 4: Compute the Modulus

Calculate \(\sqrt{28^2 + (-21)^2}\) which simplifies to \(\sqrt{784 + 441} = \sqrt{1225} = 35\). So, the modulus is 35.

Step 5: Calculate the Argument

Find the argument (angle) \(\theta\) using \(\tan^{-1}\left(\frac{-21}{28}\right)\). First compute \( \frac{-21}{28} = -0.75 \).

Step 6: Compute and Round the Angle

Compute \( \theta = \tan^{-1}(-0.75) \), which approximately equals -36.87 degrees. Round to the nearest degree: -37 degrees.

Step 7: Express in Polar Form

The polar form of the complex number is \(35(\cos(-37^\circ) + i\sin(-37^\circ))\) or \(35 \text{cis}(-37^\circ)\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. The general form of a complex number is given by:\[a + bi\]where \(a\) is the real part, and \(b\) is the imaginary part. Imaginary numbers have the unit \(i\), which is defined by the property \(i^2 = -1\). This allows us to deal with the square roots of negative numbers. For example, \(28 - 21i\) is a complex number where 28 is the real part and \(-21i\) is the imaginary part.
Complex numbers are a natural extension of the real numbers and are essential in various fields of science and engineering. They are used for solutions to polynomial equations, impedance in electrical engineering, and in many areas of applied mathematics.

• Real Part: The non-imaginary component of a complex number.
• Imaginary Part: The coefficient of \(i\) in a complex number.

Understanding complex numbers sets the stage for diving deeper into more advanced topics like the polar form.

Rectangular Form

Rectangular form is simply another way to present complex numbers, utilizing the real and imaginary parts directly. You can visualize this on a complex plane, similar to a Cartesian coordinate system, where the x-axis represents the real part and the y-axis represents the imaginary part.
The rectangular form expresses complex numbers as:\[a + bi\]
In this form, each complex number is represented as a point or a vector from the origin to the point \((a, b)\) on the complex plane. For instance, the complex number \(28 - 21i\) corresponds to the point \((28, -21)\).
This form is useful for addition and subtraction of complex numbers, as it directly relates to operations on vectors:

• Addition: \((a + c) + (b + d)i\)
• Subtraction: \((a - c) + (b - d)i\)

By understanding the rectangular form, you can seamlessly transition to the polar form of complex numbers.

Modulus

The modulus of a complex number is a measure of its "length" or "magnitude" on the complex plane. Given a complex number \(a + bi\), the modulus \(|z|\) is found using the formula:
\[|z| = \sqrt{a^2 + b^2}\]In our example \(28 - 21i\), calculating the modulus involves finding:\[\sqrt{28^2 + (-21)^2} = \sqrt{784 + 441} = \sqrt{1225} = 35\]
The modulus signifies how far the point \((a, b)\) is from the origin. In physical terms, it can be likened to finding the length of the vector in the coordinate plane. Measuring the modulus is an important step when converting a complex number into its polar form, as it represents the radial distance in the polar coordinates.

Argument

The argument of a complex number is the angle it makes with the positive real axis on the complex plane. This angle is crucial in representing complex numbers in polar form, which provides an alternative way to express them.
The argument \(\theta\) is calculated as:

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

For \(28 - 21i\), we find:
\[\theta = \tan^{-1}\left(\frac{-21}{28}\right) = \tan^{-1}(-0.75) \approx -36.87^\circ\]
When rounding this angle, we get \(-37^\circ\). The argument helps locate the direction of the complex number vector from the positive x-axis moving counterclockwise. Like the modulus, it plays a vital role in the polar representation,

• Polar Form: \(|z|(\cos\theta + i\sin\theta)\)

Understanding the argument is key to mastering complex number operations, especially in polar form.