Problem 28 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=3 \operatorname{cis}\left(\frac{4 \pi}{3}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \(-\frac{3}{2} - \frac{3\sqrt{3}}{2}i\).

Step 1: Understand the given complex number in polar form

The given complex number is in polar form expressed as \( z = r \text{cis}(\theta) \) where \( r = 3 \) and \( \theta = \frac{4 \pi}{3} \). "cis" stands for "cosine plus i sine", meaning \( \text{cis}(\theta) = \cos(\theta) + i \sin(\theta) \).

Step 2: Convert the complex number to rectangular form

To convert the number from polar to rectangular form, use the identities \( \cos(\theta) = \cos\left(\frac{4 \pi}{3}\right) \) and \( \sin(\theta) = \sin\left(\frac{4 \pi}{3}\right) \). Recall the known values: \( \cos\left(\frac{4 \pi}{3}\right) = -\frac{1}{2} \) and \( \sin\left(\frac{4 \pi}{3}\right) = -\frac{\sqrt{3}}{2} \).

Step 3: Calculate each component of the rectangular form

Using the values from Step 2, calculate the real part: \( x = 3 \times \cos\left(\frac{4 \pi}{3}\right) = 3 \times -\frac{1}{2} = -\frac{3}{2} \). Calculate the imaginary part: \( y = 3 \times \sin\left(\frac{4 \pi}{3}\right) = 3 \times -\frac{\sqrt{3}}{2} = -\frac{3\sqrt{3}}{2} \).

Step 4: Write the final rectangular form of the complex number

Combine the real and imaginary components to express the complex number in rectangular form: \( z = -\frac{3}{2} - \frac{3\sqrt{3}}{2} i \).

Key Concepts

Rectangular Form

Complex numbers can be expressed in different forms, and the rectangular form is one of the most straightforward. It represents a complex number in terms of its real and imaginary components. A complex number in rectangular form is written as:

• \( z = x + yi \)

where \( x \) is the real part and \( y \) is the imaginary part. The letter \( i \) represents the imaginary unit, which satisfies \( i^2 = -1 \). The rectangular form is particularly useful for performing arithmetic operations like addition and subtraction with complex numbers. To find the rectangular form from another format, such as polar, it involves calculating both parts separately using trigonometric functions.

Polar Form

The polar form of a complex number provides an alternative to the rectangular form. It focuses more on the magnitude and angle of the complex number relative to the origin in the complex plane. A complex number in polar form is expressed as:

• \( z = r \text{cis}(\theta) \)

Here, \( r \) is the magnitude (or modulus) of the complex number, calculated as \( r = \sqrt{x^2 + y^2} \), and \( \theta \) is the angle (or argument) measured in radians from the positive x-axis to the line connecting the origin to the point representing the complex number. The term "cis" is a shorthand for "cosine plus i sine"; hence,

• \( \text{cis}(\theta) = \cos(\theta) + i \sin(\theta) \)

The polar form is particularly helpful when multiplying or dividing complex numbers, as it simplifies the operations by handling magnitudes and angles separately.

Trigonometric Identities

Trigonometric identities play a vital role when converting complex numbers from polar to rectangular form, or vice versa. These identities provide exact values for sine and cosine functions, which are often critical in accurately finding components of complex numbers. Two significant identities frequently used include:

• \( \cos(\theta) \)
• \( \sin(\theta) \)

For example, the identities \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \) and \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \) are utilized to compute the components of complex numbers when converting from the polar form \( z = r \text{cis}(\theta) \). By substituting these exact trigonometric values, we can effectively find the real and imaginary parts of the complex number as in the original solution.