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 partFor 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

Answered: Plot the complex number. 2- 2/31…

Math

Calculus

Plot the complex number. 2- 2/31 Imagina

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

Related questions

Q: Find the sum. Graph both complex numbers and their sum: 2+3i,-4-i

A: Given: (2+3i), (-4-i)

Q: Evaluate the complex number 36, showing all of your work. Evaluate the complex number 8-2, showing…

A: using Euler's formula for complex exponentiation eix=cos(x)+isin(x)a)First, express 36i in terms of…

Q: Find the difference of the complex numbers in the complex plane. (-8+) - (8 + 1) Need Help? Read It…

A: Given that we need to find the difference of complex numbers.

Q: Write the complex number in trigonome begn by sketching the argument the graph to help find 8.…

A: Given complex number is z=-5-53i The general form of complex number is z=x+iy where x is real part…

Q: Write the Complex num ber 603 in the Simplest form

Q: Match each complex number to its conjugate. Write the letters of the conjugate pairs in the given…

A: The solution are next step

Q: Consider the complex number 3+ 2i. What is the meaning of i in the number?

A: To find the correct option

Q: Plot the complex number -4i. Then write the complex number in polar form. You may express the…

Q: 2. Perform the following operations: a. (-1+ i)8 1+i b. 3-i

Q: Write the standard form of the complex number. (Round numerical values to four decimal places.)…

Q: Plot and label each complex number in the complex plane. 1) z = -5i 2) z = -2+i 3) z = 3- i

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

### 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

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

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.

Similar questions

Graph the complex number on the complex plane. -5 -4 -3 -2 5 2- -1 0 -1 -2 M -3- 1 2 لها 3 4 -2+3i 5 Submit Assignment Quit & Save
Write the complex number in trigonometric form. (Let 0 < 0 < 2r.) z = Imaginary axis 10 5 - Real axis -10 -5 10 -5 -10
Write the complex number in standard form. (Simplify your answer completely.) 3 cis 210°
Give an example of a complex number where z7 = 1 but z ≠ 1.
(show all the steps) 1. Write the complex number 2+ iv3 in the form of rele
Plot the complex number on the complex plane and write it in polar form and in exponential form. AImaginary axis 12- - 8 +9 i 10– Plot the complex number on the complex plane. Real axis Write the complex number -8 +9 i in polar form. Select the correct choice below 40 12 and fill in the answer box(es) within your choice. (Simplify your answer. Type an exact answer for r, using radicals as needed. Type any angle measures in radians, rounding to three decimal places as needed. Use angle measures greater than or equal to 0 and less than 2n.) -121-10 -8 -4 to O A. z= V145 (cos 2.297 + i sin 2.297 40 (sin + i sin) 42 B. Z= C. Z= cos + i cos O D. z= (sin + i cos) Write the complex number - 8+ 9 i in exponential form. i 2.297 z= /145 e (Simplify your answer. Type an exact answer for r, using radicals as needed. Type any angle measures in radians, rounding to three decimal places as needed. Use angle measures greater than or equal to 0 and less than 2n.)
Write the complex numbers in binomial form
Which option
Write the complex number in polar form with argument 0 between 0 and 2n. -1 + i Submit Answer
Use a calculator to help write the complex number in standard form. Round the numbers in your answer to the nearest hundredth. 10 cis 5.8 8.86- 4.65i 8.86-4.69i 8.90- 4.69i 8.94- 4.57i 8.90- 4.65i