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Chapter 8: Problem 29

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$2 \sqrt{3}-2 i$$

Short Answer

Expert verified
The polar form is \(4\left(\cos\frac{11\pi}{6} + i\sin\frac{11\pi}{6}\right)\).

Step by step solution

01

Identify the Rectangular Form

The given complex number is \(2 \sqrt{3} - 2i\). In rectangular form, this is expressed as \(a + bi\), where \(a = 2 \sqrt{3}\) and \(b = -2\).
02

Determine the Modulus

The modulus \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Plugging in the values, we get \[r = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4.\]
03

Calculate the Argument

The argument \(\theta\) of a complex number is found using \(\tan \theta = \frac{b}{a}\). Substitute \(a = 2\sqrt{3}\) and \(b = -2\) to get \[\tan \theta = \frac{-2}{2\sqrt{3}} = -\frac{1}{\sqrt{3}}.\] This implies that \(\theta = -\frac{\pi}{6}\). Since arguments are typically between 0 and \(2\pi\), we add \(2\pi\) to the negative angle: \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).
04

Write in Polar Form

The polar form of a complex number is given by \(r(\cos \theta + i\sin \theta)\). Using \(r = 4\) and \(\theta = \frac{11\pi}{6}\), the polar form is \[4\left(\cos\frac{11\pi}{6} + i\sin\frac{11\pi}{6}\right).\]

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Key Concepts

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

complex numbers
A complex number is a number that includes both a real and an imaginary part. It's expressed in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, while \(i\) is the imaginary unit equivalent to \(\sqrt{-1}\).
  • Real numbers are simply numbers without an imaginary part, such as 3 or -7.
  • Imaginary numbers are multiples of \(i\), like \(5i\) or \(-2i\).
Complex numbers are vital in various fields of science and engineering because they allow for the expression and manipulation of quantities that are otherwise not possible to express using just real numbers.
rectangular form
The rectangular form of a complex number is the most straightforward representation, written as \(a + bi\). This form highlights both the real and imaginary components, making it easier to perform basic arithmetic operations like addition and subtraction.
To transition from numbers to complex arithmetic, express a complex number in rectangular form:
  • Identify the real component \(a\) (e.g., \(2\sqrt{3}\)).
  • Note the imaginary component \(bi\) (e.g., \(-2i\)).
The rectangular form provides a simple, intuitive framework for thinking about complex numbers and their interactions.
modulus and argument
The modulus and argument of a complex number transform it from its rectangular form to the polar form.
  • The modulus \(r\) is the distance from the origin to the point \((a, b)\) in the complex plane and is calculated through \(r = \sqrt{a^2 + b^2}\).
  • The argument \(\theta\) is the angle that the line from the origin to the point makes with the positive x-axis, calculated using \(\tan \theta = \frac{b}{a}\).
Understanding the modulus and argument provides insight into the complex number's magnitude and directional orientation in the complex plane. These concepts are pivotal in converting between rectangular and polar forms.
trigonometric representation
The trigonometric representation, or polar form, presents a complex number as \(r(\cos \theta + i\sin \theta)\). This form emphasizes the use of trigonometry to express complex numbers, providing an alternative to the rectangular form that often simplifies multiplication and division processes.
  • In polar form, \(r\) is the modulus determining the length from the origin, indicating the number's size.
  • The angle \(\theta\), the argument, illustrates the direction of this modulus in the complex plane.
Expressing complex numbers in polar form is especially advantageous in calculations involving powers and roots, due to its simplification of the multiplicative properties.

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Most popular questions from this chapter

Sketch the graph of the polar equation. $$r=-2 \cos \theta$$

Sketch the graph of the polar equation. $$r=1+\sin \theta$$

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(i\)

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and \(1 / z_{1}\). $$z_{1}=\sqrt{2}-\sqrt{2} i, \quad z_{2}=1-i$$

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2} .\) Express your answer in polar form. $$z_{1}=3\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right), \quad z_{2}=5\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)$$
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