Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is \( \sqrt{2} + \sqrt{2}i \).

Step 1: Identify the Magnitude and Angle

The given complex number is in polar form: \(2(\cos 45^{\circ} + i \sin 45^{\circ})\). Here, 2 is the magnitude (or modulus), and \(45^{\circ}\) is the angle (or argument).

Step 2: Convert the Angle to Radians

Since many problems use radians, it's helpful to convert the angle. The angle \(45^{\circ}\) in radians is \(\frac{\pi}{4}\).

Step 3: Compute the Cosine and Sine Values

For \(\theta = 45^{\circ} = \frac{\pi}{4}\), we have \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 4: Write the Complex Number in Rectangular Form

The rectangular form of a complex number is \(a + bi\). Here, \(a = 2 \times \cos 45^{\circ}\) and \(b = 2 \times \sin 45^{\circ}\). Substitute the values: \(a = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\) and \(b = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\).

Step 5: Combine Real and Imaginary Parts

Now, the complex number in rectangular form is \(\sqrt{2} + \sqrt{2}i\). Both the real and imaginary parts are well-defined.

Key Concepts

Rectangular Form

The rectangular form of a complex number is the most common way to express complex numbers by using real and imaginary components. This form simplifies understanding and calculations, especially in adding and subtracting complex numbers. Complex numbers in rectangular form are represented as \(a + bi\), where:

• \(a\) represents the real part of the complex number.
• \(b\) is the coefficient of \(i\), which is the imaginary part.
• \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

For example, the complex number \(\sqrt{2} + \sqrt{2}i\) is written in rectangular form where both the real part \(a\) and the imaginary part \(b\) are equal to \(\sqrt{2}\). It perfectly combines the two dimensions of complex numbers: the real axis and the imaginary axis. Rectangular form is particularly useful when performing operations on complex numbers like addition and subtraction because you can simply add or subtract the corresponding real and imaginary parts.

Polar Form

Polar form provides an alternative way to express complex numbers using the magnitude and angle from the positive x-axis in the complex plane. It's especially useful for multiplication and division tasks. A complex number in polar form is denoted as \(r(\cos \theta + i \sin \theta)\), where:

• \(r\) is the magnitude or modulus of the complex number. It's equivalent to the distance from the origin in a polar coordinate system.
• \(\theta\) is the argument or angle with the positive x-axis. It can be measured in degrees or radians.

Converting from polar to rectangular form involves using trigonometric functions: find \(r \cos \theta\) to determine the real part, and \(r \sin \theta\) for the imaginary part. For instance, with the angle \(45^{\circ}\) (or \(\frac{\pi}{4}\) radians), both cosine and sine equal \(\frac{\sqrt{2}}{2}\). Thus, a complex number in polar form such as \(2(\cos 45^{\circ} + i \sin 45^{\circ})\) transforms into \(\sqrt{2} + \sqrt{2}i\) when converted to rectangular form.

Trigonometric Form

The trigonometric form of a complex number closely resembles the polar form and is another way to express complex numbers using trigonometric terms. It is expressed as \(r(\cos \theta + i \sin \theta)\), which is sometimes referred to as "CIS" form, where:

• \(r\) represents the modulus, capturing the distance from the origin to the point in the complex plane.
• \(\theta\) is the argument, defining the direction of the line formed with the x-axis by the complex number in radians or degrees.

The trigonometric form of a complex number focuses on its geometric representation and is practical for multiplying or dividing complex numbers because multiplication involves multiplying the magnitudes and adding the angles. For the given example, \(2(\cos 45^{\circ} + i \sin 45^{\circ})\) is its trigonometric form, highlighting not only the magnitude of 2 but also the direction at an angle of \(45^{\circ}\). Recognizing the trigonometric form allows students to transition easily between different forms, showcasing the number's position in the complex plane.