Problem 42 - 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. $$6 \text { cis } \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The rectangular form is \(-3\sqrt{2} + 3\sqrt{2}i\).

Step 1: Understanding the problem

We need to express the complex number given in polar form, \( 6 \text{ cis } \frac{3\pi}{4} \), into its rectangular form \((a + bi)\). This requires identifying the values of \(a\) and \(b\).

Step 2: Using the formula for rectangular form

The formula for converting a complex number from polar to rectangular form is \( z = r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument. Here, \( r = 6 \) and \( \theta = \frac{3\pi}{4} \).

Step 3: Calculate the real part

Substitute \( \theta = \frac{3\pi}{4} \) into \( \cos \theta \). Knowing \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), the real part becomes \( 6 \times \left(-\frac{\sqrt{2}}{2}\right) = -3\sqrt{2} \).

Step 4: Calculate the imaginary part

Substitute \( \theta = \frac{3\pi}{4} \) into \( \sin \theta \). Knowing \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), the imaginary part is \( 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} \).

Step 5: Express the complex number in rectangular form

Combine the real and imaginary parts to express the complex number: \(-3\sqrt{2} + 3\sqrt{2}i\). Thus, the complex number in rectangular form is \(-3\sqrt{2} + 3\sqrt{2}i\).

Key Concepts

Polar to Rectangular Conversion

The process of converting a complex number from polar to rectangular form involves translating the magnitude and direction of the number into a standard format with real and imaginary components. This conversion allows us to express the complex number in the more commonly used rectangular form, often denoted as \( a + bi \).
Polar form expresses complex numbers as \( r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus representing the length of the vector from the origin, and \( \theta \) is the argument describing the angle with respect to the positive real axis. To convert, we use the formulas:

• Real part: \( a = r \cos \theta \)
• Imaginary part: \( b = r \sin \theta \)

In our example, \( r = 6 \) and \( \theta = \frac{3\pi}{4} \). By computing these expressions, we map the abstract information from the polar form to a tangible position on the complex plane in rectangular form.

Real and Imaginary Parts

The identification of real and imaginary parts is crucial in the conversion process and in understanding the geometry of complex numbers on the complex plane. When a complex number is expressed in rectangular form as \( a + bi \), \( a \) and \( b \) are the real and imaginary parts, respectively.
Finding these parts involves simple trigonometric evaluations:

• For the real part, we substitute the argument \( \theta \) into \( \cos \theta \), then multiply by the modulus. So, with \( \theta = \frac{3\pi}{4} \), we find \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), making the real part \( -3\sqrt{2} \).
• For the imaginary part, we use \( \sin \theta \). At \( \theta = \frac{3\pi}{4} \), \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), thus the imaginary part is \( 3\sqrt{2} \).

These calculated values combine to give us the rectangular form of the complex number: \( -3\sqrt{2} + 3\sqrt{2}i \). Understanding these components helps in visualizing where the number lies on the plane.

Trigonometric Functions

A pivotal aspect of converting complex numbers is the role trigonometric functions play in finding exact values of the real and imaginary components. These functions bridge the gap between the geometric representation of a vector in the complex plane and the algebraic form of complex numbers.
In our scenario, we need exact values of trigonometric functions for specific angles. The angle \( \theta = \frac{3\pi}{4} \) is a common angle where:

• \( \cos \theta = -\frac{\sqrt{2}}{2} \)
• \( \sin \theta = \frac{\sqrt{2}}{2} \)

Calculating these involves understanding unit circle principles, where the cosine of an angle corresponds to the x-coordinate and sine corresponds to the y-coordinate of a point on the unit circle. These coordinates help translate the angle and radius from polar form into precise rectangular coordinates. For students, mastering these trigonometric values is fundamental in ensuring smooth and exact conversions across different mathematical problems involving complex numbers.