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Chapter 7: Problem 27

Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\) $$-4-4 i$$

Short Answer

Expert verified
The trigonometric form of \(-4 - 4i\) is \[4\sqrt{2}(\cos \frac{5\pi}{4} + i\sin \frac{5\pi}{4})\].

Step by step solution

01

Identify Real and Imaginary Parts

The complex number given is \(-4 - 4i\). This can be broken down into its real part, \(-4\), and its imaginary part, \(-4\).
02

Calculate the Magnitude

The magnitude (modulus) of the complex number is given by the formula:\[ r = \sqrt{a^2 + b^2} \]Substituting the values \(a = -4\) and \(b = -4\):\[ r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]
03

Find the Argument

The argument \(\theta\) of a complex number is found using the tangent function: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]Plug in the values: \[ \theta = \tan^{-1}\left(\frac{-4}{-4}\right) = \tan^{-1}(1)\]For the fourth quadrant, the angle \(\theta = \frac{5\pi}{4}\) satisfies the condition \(0 \leq \theta < 2\pi\) because the complex number is in the second quadrant due to the negative real and imaginary components. Thus,\(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\).
04

Express in Trigonometric Form

Express the complex number \(-4 - 4i\) in trigonometric form:\[-4 - 4i = r(\cos \theta + i\sin \theta)\]Substitute \(r = 4\sqrt{2}\) and \(\theta = \frac{5\pi}{4}\):\[-4 - 4i = 4\sqrt{2}(\cos \frac{5\pi}{4} + i\sin \frac{5\pi}{4})\]

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Key Concepts

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

Trigonometric Form
The trigonometric form, also known as the polar form, of a complex number provides a unique way to express complex numbers using both distance and angle. This is useful because it translates a complex number into a circle format within the complex plane. The general form is:
\[ z = r(\cos \theta + i\sin \theta) \]where:
  • \( z \) is the complex number.
  • \( r \) is the magnitude of the complex number.
  • \( \theta \) is the argument, representing the angle.
For instance, a complex number such as \(-4 - 4i\) can be expressed in trigonometric form by determining both its magnitude and argument. This expression, known as the trigonometric form, allows for straightforward multiplication, division, and exponentiation of complex numbers.
Magnitude
The magnitude of a complex number, often referred to as the modulus, measures its distance from the origin of the complex plane. You calculate the magnitude using the formula:
\[ r = \sqrt{a^2 + b^2} \]where \( a \) and \( b \) are the real and imaginary components, respectively. So, for \(-4 - 4i\):
  • Substitute \(a = -4\) and \(b = -4\) into the magnitude formula.
  • The value is \( r = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \).
Understanding the magnitude of complex numbers gives insights into their position and influence on the complex plane.
Argument
The argument is the angle a complex number makes with the positive real axis within the complex plane. This angle provides information about the direction of the number. To find it, use:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]When calculating the argument for \(-4 - 4i\):
  • Since both real and imaginary parts are negative, the number lies in the third quadrant.
  • The angle is \( \tan^{-1}(1)\), which is traditionally \( \frac{\pi}{4}\).
  • In the correct quadrant, adjust it to \( \frac{5\pi}{4} \).
Understanding the argument helps clarify the orientation of the complex number in the plane and aids in conversions between forms.
Complex Plane
The complex plane is an essential concept when working with complex numbers. It allows a visual representation where each complex number corresponds to a unique point. The horizontal axis (real axis) represents the real part of the complex number, and the vertical axis (imaginary axis) represents its imaginary part. In this plane:
  • A point like \(-4 - 4i\) is represented by moving 4 units left on the real axis and 4 units down on the imaginary axis.
  • The position is seen as a vector from the origin \((0,0)\) to the point \((-4, -4)\).
  • The trigonometric form helps depict this vector as having both direction (argument) and length (magnitude).
Utilizing the complex plane facilitates operations on complex numbers using geometrical interpretations and contributes to solving problems in physics and engineering.

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