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Chapter 8: Problem 350

Find the powers of each complex number in polar form. Find \(z^{4}\) when \(z=\operatorname{cis}\left(\frac{3 \pi}{16}\right).\)

Short Answer

Expert verified
The fourth power is \(z^4 = \operatorname{cis}\left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \).

Step by step solution

01

Understanding the Given Complex Number

The given complex number is in the form \( z = \operatorname{cis}\left( \frac{3 \pi}{16} \right) \). In polar form, this is written as \( z = \cos\left( \frac{3 \pi}{16} \right) + i\sin\left( \frac{3 \pi}{16} \right) \).
02

Applying the Power of a Complex Number Formula

To find the power of a complex number in polar form, we use De Moivre's Theorem: \( (\operatorname{cis}(\theta))^n = \operatorname{cis}(n\theta) \). Here, we need to find \( z^4 = \operatorname{cis}^4\left( \frac{3 \pi}{16} \right) \).
03

Finding the Angle

Apply De Moivre's Theorem: multiply the angle by the power. So, we calculate \( n \times \theta = 4 \times \frac{3 \pi}{16} = \frac{12 \pi}{16} = \frac{3 \pi}{4} \).
04

Writing the Final Expression

Substitute back into the polar format, so \( z^4 = \operatorname{cis}\left( \frac{3 \pi}{4} \right) \). In rectangular form, this is \( z^4 = \cos\left( \frac{3 \pi}{4} \right) + i\sin\left( \frac{3 \pi}{4} \right) = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \).

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Key Concepts

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

Complex Numbers
Complex numbers are a fascinating extension of the real number system. Essentially, they are numbers that have both a real part and an imaginary part. The general form for a complex number is:
  • Where the real part is indicated by "a" and the imaginary part by "b" alongside the imaginary unit "i," which represents \( \sqrt{-1} \).
Complex numbers are typically represented in two main forms:
  • Rectangular form: Expressed as \( a + bi \)
  • Polar form: Expressed using polar coordinates based on a magnitude and an angle, such as \( r \operatorname{cis}(\theta) \)
These diverse representations make complex numbers versatile and invaluable in various fields such as engineering, physics, and computer science.
Polar Form
The polar form of complex numbers offers an alternative way to represent them using magnitudes and angles. Instead of using \( a + bi \), we can use the "magnitude" (distance from the origin) and "angle" (direction from the positive x-axis) to define a complex number. The magnitude is found by \( r = \sqrt{a^2 + b^2} \), and the angle, \( \theta \), is given by \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Thus, in polar form, a complex number is represented as:
  • \( z = r \operatorname{cis}(\theta) \)
The polar representation is particularly useful for multiplying and dividing complex numbers, as it simplifies these operations compared to their rectangular counterparts.
cis Function
The "cis" function is a shorthand notation used within the polar form of complex numbers. In mathematical terms:
  • \( \operatorname{cis}(\theta) = \cos(\theta) + i\sin(\theta) \)
This convenient expression compacts the usual way of representing complex numbers.Using "cis" simplifies expressions, especially when using De Moivre's Theorem for powers and roots of complex numbers. This simplification allows us to focus mainly on the angle, \( \theta \), which can then be multiplied or divided easily as part of transformations like rotation or scaling in the complex plane.
Powers of Complex Numbers
Calculating powers of complex numbers is much more straightforward in polar form thanks to De Moivre's Theorem. The theorem states:
  • For a complex number \( z = r \operatorname{cis}(\theta) \), raising it to a power \( n \) gives \( z^n = r^n \operatorname{cis}(n\theta) \).
This approach dramatically simplifies exponentiation for complex numbers by focusing on the angle, \( \theta \), and scaling the magnitude by \( n \).For instance, raising a complex number to the fourth power, as in the exercise example, involves multiplying the angle by 4. This application of De Moivre's Theorem allows for a clearer and simpler method of determining powers in both mathematical theory and practical calculations.

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Most popular questions from this chapter

For the following exercises, determine whether the two vectors, \(u\) and \(v,\) are equal, where \(u\) has an initial point \(P_{1}\) and a terminal point \(P_{2},\) and \(v\) has an initial point \(P_{3}\) and a teminal point \(P_{4}\) . $$P_{1}=(-1,4), P_{2}=(3,1), P_{3}=(5,5) \text { and } P_{4}=(9,2)$$

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