Homework Help is Here – Start Your Trial Now!
Math
Calculus
Plot the complex number. 2- 2/31 Imagina

Plot the complex number. 2- 2/31 Imagina

Calculus: Early Transcendentals
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...
See similar textbooks
Question
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: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
### 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
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
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
De Moivre's Theorem
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS