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Chapter 6: Problem 24

Write each complex number in trigonometric form, using degree measure for the argument. \(\frac{\sqrt{3}}{6}+\frac{i}{6}\)

Short Answer

Expert verified
\[z = \frac{1}{3} (\text{cos} 30^\text{o} + i \text{sin} 30^\text{o} )\]

Step by step solution

01

- Identify Real and Imaginary Parts

Given the complex number \(\frac{\frac{\text{鈭殅 3}{6}+\frac{i}{6}}\), identify the real part (x) and the imaginary part (y). Here, \(x = \frac{\text{鈭殅 3}{6}\) and \(y = \frac{1}{6}\).
02

- Calculate Magnitude

To find the magnitude \(r\), use the formula \(r = \text{鈭殅(x^2 + y^2)\). Substitute the values of \(x\) and \(y\): \[ r = \text{鈭殅 \bigg( \bigg( \frac{\text{鈭殅 3}{6} \bigg)^2 + \bigg( \frac{1}{6} \bigg)^2 \bigg) = \text{鈭殅 \bigg( \frac{3}{36} + \frac{1}{36} \bigg) = \text{鈭殅 \bigg( \frac{4}{36} \bigg) = \text{鈭殅 ( \frac{1}{9} ) = \frac{1}{3} \]
03

- Determine Argument (Angle)

To find the argument \(胃\) in degrees, use the tangent function: \[ \theta = \text{tan}^{-1} \bigg( \frac{y}{x} \bigg) \]Substitute the values of \(x\) and \(y\): \[ \theta = \text{tan}^{-1} \bigg( \frac{\frac{1}{6}}{\frac{\text{鈭殅 3}{6}} \bigg) = \text{tan}^{-1} \bigg( \frac{1}{\text{鈭殅 3} \bigg) = \text{tan}^{-1} (\frac{1}{\text{鈭殅 3}) = 30^\text{o} \]
04

- Write in Trigonometric Form

Combine the magnitude \(r\) and the argument \(胃\) to express the complex number in trigonometric form: \[ z = r (\text{cos} 胃 + i \text{sin} 胃) \] Substituting \(r = \frac{1}{3} \) and \(胃 = 30^\text{o}\): \[ z = \frac{1}{3} (\text{cos} 30^\text{o} + i \text{sin} 30^\text{o}) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. An imaginary number is a multiple of \( i \), where \( i \) is defined as \( \sqrt{-1} \).

For example, in the exercise, the complex number \( \frac{\sqrt{3}}{6} + \frac{i}{6} \) has a real part \( \frac{\sqrt{3}}{6} \) and an imaginary part \( \frac{1}{6} \).

Understanding the composition of a complex number is the first step towards working with them effectively.
Calculating the Magnitude of Complex Numbers
The magnitude, or modulus, of a complex number is a measure of its size or length. It is the distance from the origin to the point representing the complex number in the complex plane.

To find the magnitude \( r \) of the complex number \( x + yi \), we use the formula:
\[ r = \sqrt{x^2 + y^2} \] In our exercise, \( x = \frac{\sqrt{3}}{6} \) and \( y = \frac{1}{6} \). So the magnitude would be:
\[ r = \sqrt{ \left( \frac{\sqrt{3}}{6} \right)^2 + \left( \frac{1}{6} \right)^2 } = \sqrt{ \frac{3}{36} + \frac{1}{36} } = \sqrt{ \frac{4}{36} } = \frac{1}{3} \]
Calculating the magnitude is an essential step for converting a complex number to its trigonometric form.
Finding the Argument of Complex Numbers
The argument of a complex number is the angle it forms with the positive direction of the x-axis in the complex plane. This angle is often denoted by \( \theta \).

To find the argument, we use the tangent function:
\[ \theta = \tan^{-1} \bigg( \frac{y}{x} \bigg) \] For our exercise, \( x = \frac{\sqrt{3}}{6} \) and \( y = \frac{1}{6} \):
\[ \theta = \tan^{-1} \bigg( \frac{\frac{1}{6}}{\frac{\sqrt{3}}{6}} \bigg) = \tan^{-1} \bigg( \frac{1}{\sqrt{3}} \bigg) = \tan^{-1} \bigg( \frac{1}{\sqrt{3}} \bigg) = 30^{\circ} \].
The argument tells us the direction of the complex number in the complex plane.
Converting to Trigonometric Form
Writing a complex number in trigonometric form makes it easier to perform certain operations, such as multiplication and division.

The trigonometric form of a complex number is given by:
\[ z = r ( \cos \theta + i \sin \theta ) \]
Here, \( r \) is the magnitude and \( \theta \) is the argument.
In our exercise with \( r = \frac{1}{3} \) and \( \theta = 30^{\circ} \), the trigonometric form is:
\[ z = \frac{1}{3} ( \cos 30^{\circ} + i \sin 30^{\circ} ) \].
This conversion facilitates a better understanding and handling of complex numbers in various mathematical contexts.

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