Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![On this page, we will explore how to represent a complex number in trigonometric form.
The image features two Cartesian coordinate systems (graphs), each containing a complex number. Each graph has a horizontal Real axis and a vertical Imaginary axis, both ranging from -10 to 10.
In the first graph:
- The complex number \(2 - 2\sqrt{3}i\) is plotted.
- The representation of this number is marked with a black dot.
- The coordinates of this point are at (2, -2√3) on the graph, indicating it's 2 units to the right of the origin and approximately -3.46 units down along the imaginary axis.
In the second graph:
- The same complex number \(2 - 2\sqrt{3}i\) is displayed (identical coordinates as the first).
- Similarly, the point is marked along the position (2, -2√3) with a dotted line indicating the direction from the origin to the point.
Next, we are asked to write the trigonometric form of the complex number. The general trigonometric form of a complex number is given by:
\[ z = r (\cos \theta + i \sin \theta) \]
where:
- \( r \) is the modulus of the complex number, calculated as \( r = \sqrt{x^2 + y^2} \)
- \( \theta \) is the argument of the complex number, calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)
We are given:
\[ z = 2 - 2 \sqrt{3} i \]
Here:
- \( x = 2 \)
- \( y = -2\sqrt{3} \)
To find the modulus \( r \):
\[ r = \sqrt{2^2 + (-2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \]
To find the argument \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{-2\sqrt{3}}{2}\right) = \tan^{-1}(-\sqrt{3}) \]
The value of \( \tan^{-1}(-\sqrt{3}) \) is \(\frac{5\pi}{3}\), as we're considering the principal value between \(0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fab864d-7419-4b8c-bc50-25971d273c3c%2Fc3250f90-571c-4713-8250-0ea2539bbfe8%2Fg6nv8y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this page, we will explore how to represent a complex number in trigonometric form.
The image features two Cartesian coordinate systems (graphs), each containing a complex number. Each graph has a horizontal Real axis and a vertical Imaginary axis, both ranging from -10 to 10.
In the first graph:
- The complex number \(2 - 2\sqrt{3}i\) is plotted.
- The representation of this number is marked with a black dot.
- The coordinates of this point are at (2, -2√3) on the graph, indicating it's 2 units to the right of the origin and approximately -3.46 units down along the imaginary axis.
In the second graph:
- The same complex number \(2 - 2\sqrt{3}i\) is displayed (identical coordinates as the first).
- Similarly, the point is marked along the position (2, -2√3) with a dotted line indicating the direction from the origin to the point.
Next, we are asked to write the trigonometric form of the complex number. The general trigonometric form of a complex number is given by:
\[ z = r (\cos \theta + i \sin \theta) \]
where:
- \( r \) is the modulus of the complex number, calculated as \( r = \sqrt{x^2 + y^2} \)
- \( \theta \) is the argument of the complex number, calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)
We are given:
\[ z = 2 - 2 \sqrt{3} i \]
Here:
- \( x = 2 \)
- \( y = -2\sqrt{3} \)
To find the modulus \( r \):
\[ r = \sqrt{2^2 + (-2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \]
To find the argument \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{-2\sqrt{3}}{2}\right) = \tan^{-1}(-\sqrt{3}) \]
The value of \( \tan^{-1}(-\sqrt{3}) \) is \(\frac{5\pi}{3}\), as we're considering the principal value between \(0

Transcribed Image Text:### Plotting Complex Numbers
#### Example: Plot the Complex Number \( 2 - 2\sqrt{3}i \)
In this example, we are tasked with plotting the complex number \( 2 - 2\sqrt{3}i \) on the complex plane.
A complex number can be expressed in the form \( a + bi \), where:
- \( a \) is the real part
- \( bi \) is the imaginary part
For the given number \( 2 - 2\sqrt{3}i \):
- The real part \( a \) is \( 2 \)
- The imaginary part \( bi \) is \( -2\sqrt{3}i \)
To plot this on a complex plane, follow these steps:
1. **Identify Real and Imaginary Components**:
- Real part \( a = 2 \)
- Imaginary part \( b = -2\sqrt{3} \)
2. **Mark the Real Component**:
- Along the Real axis, move 2 units to the right from the origin (since \( a \) is positive).
3. **Mark the Imaginary Component**:
- Along the Imaginary axis, move \( 2\sqrt{3} \) units downward (since \( b \) is negative).
The complex plane plots in the images show two equivalent points for \( 2 - 2\sqrt{3}i \) in different quadrants due to the sign changes:
- **Left Graph Explanation**:
- The point \( 2 - 2\sqrt{3}i \) is plotted by moving 2 units to the right on the Real axis and then moving \( 2\sqrt{3} \approx 3.46 \) units down on the Imaginary axis.
- The dotted line traces the position from the origin \((0, 0)\) to the point \((2, -3.46)\)
- **Right Graph Explanation**:
- The point \( -2 + 2\sqrt{3}i \) is wrongly plotted and discussed in this context; it distinctly visualizes a common error.
- Here, the point is mistakenly plotted erroneously by reflecting the coordinates due to a common sign mistake.
Each plot correctly positions the complex number \( 2 - 2\sqrt{3}i \) to demonstrate how it appears on a
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