Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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
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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
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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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