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Chapter 8: Problem 91

In Exercises 91 and 92, express the complex number in polar form. $$ a-2 a i, \text { where } a>0 $$

Short Answer

Expert verified
The polar form is \( a\sqrt{5}(\cos (\tan^{-1}(-2)) + i \sin (\tan^{-1}(-2))) \).

Step by step solution

01

Identify the Real and Imaginary Parts

The given complex number is \( a - 2ai \). Here, \( a \) is the real part and \( -2a \) is the imaginary part.
02

Determine the Magnitude

The magnitude \( r \) of the complex number is given by the formula \( r = \sqrt{a^2 + (-2a)^2} \). Simplifying, \( r = \sqrt{a^2 + 4a^2} = \sqrt{5a^2} = a\sqrt{5} \).
03

Find the Argument

The argument \( \theta \) is found using the formula \( \theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right) \). Thus, \( \theta = \tan^{-1}\left(\frac{-2a}{a}\right) = \tan^{-1}(-2) \).
04

Express in Polar Form

The polar form of a complex number is \( r(\cos \theta + i \sin \theta) \). Substituting the values gives: \( a\sqrt{5}(\cos (\tan^{-1}(-2)) + i \sin (\tan^{-1}(-2))) \).

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Key Concepts

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

Real Part and Imaginary Part
To express a complex number in polar form, we must first understand its structure, which consists of a real part and an imaginary part. A complex number is typically written as \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part.
In the provided exercise, the complex number is \( a - 2ai \). Here, the real part is \( a \) and the imaginary part is \(-2a\).
Identifying these components is crucial when converting to polar form as it helps in determining other necessary properties like magnitude and argument.
Magnitude of a Complex Number
The magnitude (or modulus) of a complex number, represented as \( r \), is a measure of its size, regardless of its direction on the complex plane. It's calculated using the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts respectively.
For the complex number in the exercise, \( r = \sqrt{a^2 + (-2a)^2} \).
This simplifies to \( r = \sqrt{5a^2} \), which further reduces to \( a\sqrt{5} \).
Thus, the magnitude tells us how far away the number is from the origin in the complex plane.
Argument of a Complex Number
The argument of a complex number is the angle formed with the positive direction of the real axis. It helps in understanding the direction of the complex number in the complex plane.
The argument \( \theta \) can be found using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \), where \( b \) is the imaginary part and \( a \) is the real part.
For the exercise's complex number, the argument is calculated as \( \theta = \tan^{-1}\left(\frac{-2a}{a}\right) = \tan^{-1}(-2) \).
This value, \( -2 \), shows that the angle is not only about direction but also involves rotation about the origin, indicating whether the imaginary part is positive or negative.

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Most popular questions from this chapter

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=\sqrt{t}, y=t+2 $$

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=2 \sin ^{2} t, y=2 \cos ^{2} t, t \text { in }[0,2 \pi] $$

For Exercises 77 and 78, refer to the following: The sword artistry of the samurai is legendary in Japanese folklore and myth. The elegance with which a samurai could wield a sword rivals the grace exhibited by modern figure skaters. In more modern times, such legends have been rendered digitally in many different video games (e.g., Ominusha). In order to make the characters realistically move across the screen-and in particular, wield various sword motions true to the legends-trigonometric functions are extensively used in constructing the underlying graphics module. One famous movement is a figure eight, swept out with two hands on the sword. The actual path of the tip of the blade as the movement progresses in this figure-eight motion depends essentially on the length \(L\) of the sword and the speed with which the figure is swept out. Such a path is modeled using a polar equation of the form \(r^{2}(\theta)=L \cos (A \theta)\) or \(r^{2}(\theta)=L \sin (A \theta), \theta_{1} \leq \theta \leq \theta_{2}\). Video Games. Write a polar equation that would describe the motion of a sword 12 units long that makes 8 complete motions in \([0,2 \pi]\).

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