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Chapter 7: Problem 44

Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\) $$-2-7 i$$

Short Answer

Expert verified
The trigonometric form is \(\sqrt{53}(\cos(\pi + \arctan(\frac{7}{2})) + i\sin(\pi + \arctan(\frac{7}{2})))\).

Step by step solution

01

Identify the complex number components

The given complex number is \[-2 - 7i\] where the real part \(a = -2\) and the imaginary part \(b = -7\).
02

Calculate the modulus

The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). Substitute the values: \(|z| = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}\).
03

Determine the argument

The argument (\(\theta\)) for a complex number in the form \(a + bi\) is calculated using \(\tan(\theta) = \frac{b}{a}\). Here, \(\tan(\theta) = \frac{-7}{-2} = \frac{7}{2}\).Since the complex number lies in the third quadrant,\(\theta = \pi + \arctan\left(\frac{7}{2}\right)\).
04

Express in trigonometric form

The trigonometric form of a complex number is \(z = |z|(\cos\theta + i\sin\theta)\).Substitute the values: \(-2 - 7i = \sqrt{53}(\cos(\pi + \arctan(\frac{7}{2})) + i\sin(\pi + \arctan(\frac{7}{2})))\).

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Key Concepts

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

Trigonometric Form
Complex numbers can be represented in various forms. One of the most intuitive and useful ways is the trigonometric form. This approach expresses a complex number using its modulus (size) and argument (angle). Here's how it works:
  • The trigonometric form of a complex number is expressed as \( z = |z| (\cos \theta + i \sin \theta) \).
  • \(|z|\) is the modulus, which tells you how far the number is from the origin on the complex plane.
  • The \(\theta\) (theta) is the argument, showing the direction of the complex number in relation to the positive real axis.
This form makes operations like multiplication and division easier, as you can simply use trigonometric identities and angle addition or subtraction formulas.
Modulus of a Complex Number
The modulus of a complex number provides a measure of its magnitude on the complex plane. It represents the "length" of the line connecting the number to the origin. It's similar to the absolute value for real numbers.
  • The modulus \(|z|\) of a complex number \(z = a + bi\) is calculated using the formula: \(|z| = \sqrt{a^2 + b^2}\).
  • This formula comes from the Pythagorean theorem, as the real and imaginary parts \(a\) and \(b\) are akin to the lengths of the sides of a right triangle.
  • For the complex number \(-2 - 7i\), substituting \(a = -2\) and \(b = -7\) gives us \(|z| = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}\).
Determining the modulus is essential for transitioning a complex number from its standard form to its trigonometric form.
Argument of a Complex Number
The argument of a complex number reveals its angle in the complex plane, relative to the positive real axis. This angle fully specifies the direction of the complex number.
  • The argument \(\theta\) is found using the formula: \(\tan(\theta) = \frac{b}{a}\), where \(a\) and \(b\) are the real and imaginary components respectively.
  • It's crucial to consider the quadrant in which the complex number lies to determine the correct angle.
  • Given the complex number \(-2 - 7i\), the tangent is calculated as \(\tan(\theta) = \frac{-7}{-2} = \frac{7}{2}\).
  • Since \(-2 - 7i\) falls in the third quadrant, the angle \(\theta\) is \(\theta = \pi + \arctan\left(\frac{7}{2}\right)\).
Knowing the argument completes the information needed to express the complex number in its trigonometric form.
Quadrant Analysis
To properly represent a complex number in trigonometric form, it is essential to understand which quadrant the number is located in on the complex plane. Each quadrant affects how you calculate the argument.
  • The complex plane is divided into four quadrants:
    • First Quadrant: Both real part \(a\) and imaginary part \(b\) are positive.
    • Second Quadrant: \(a\) is negative, \(b\) is positive.
    • Third Quadrant: Both \(a\) and \(b\) are negative, as is the case for \(-2 - 7i\).
    • Fourth Quadrant: \(a\) is positive, \(b\) is negative.
  • Quadrant analysis influences the formula used to find \(\theta\). For the third quadrant: \(\theta = \pi + \arctan\left(\frac{|b|}{|a|}\right)\).
  • This ensures the correct directional angle is taken into account when plotting the complex number.
By understanding the quadrant, you ensure the trigonometric form accurately represents the complex number.

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