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Chapter 8: Problem 24

In Exercises 21-40, find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$ z_{1}=4\left(\cos 280^{\circ}+i \sin 280^{\circ}\right) \text { and } z_{2}=4\left(\cos 55^{\circ}+i \sin 55^{\circ}\right) $$

Short Answer

Expert verified
The quotient \(\frac{z_1}{z_2}\) in rectangular form is approximately \(-0.707 - 0.707i\).

Step by step solution

01

Understand the Problem

We are given two complex numbers in polar form: \(z_1 = 4(\cos 280^{\circ} + i \sin 280^{\circ})\) and \(z_2 = 4(\cos 55^{\circ} + i \sin 55^{\circ})\). We need to find \(\frac{z_1}{z_2}\) and express the result in rectangular form \(a + bi\).
02

Simplify the Expression Using Polar Form

The division of two complex numbers \(\frac{z_1}{z_2}\) in polar form is given by the formula: \[\frac{z_1}{z_2} = \frac{r_1}{r_2}\left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)\]where \(r_1 = 4\), \(r_2 = 4\), \(\theta_1 = 280^{\circ}\), and \(\theta_2 = 55^{\circ}\).
03

Calculate the Magnitude of the Quotient

The magnitudes are \(r_1 = 4\) and \(r_2 = 4\). Therefore, the magnitude of \(\frac{z_1}{z_2}\) is:\[\frac{r_1}{r_2} = \frac{4}{4} = 1\]
04

Calculate the Angle of the Quotient

Calculate the difference in angles:\[\theta_1 - \theta_2 = 280^{\circ} - 55^{\circ} = 225^{\circ}\]
05

Convert to Rectangular Form

The rectangular form is obtained using the cosine and sine of the angle:\[\frac{z_1}{z_2} = 1(\cos 225^{\circ} + i \sin 225^{\circ})\]Calculate these values:\(\cos 225^{\circ} = -\frac{1}{\sqrt{2}}\) and \(\sin 225^{\circ} = -\frac{1}{\sqrt{2}}\).Now,\[\frac{z_1}{z_2} = -\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}\].
06

Simplify the Rectangular Form

Given that \(\sqrt{2}\approx1.414\), we can convert \(-\frac{1}{\sqrt{2}}\) to a simpler form:\(-\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \approx -0.707\).Thus, the rectangular form of the quotient is:\[\frac{z_1}{z_2} = -0.707 - 0.707i\].

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Key Concepts

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

Polar to Rectangular Conversion
Complex numbers can be represented in different forms. The most intuitive is the rectangular form, but sometimes it's easier to work with them in polar form. The polar form represents complex numbers as a combination of a magnitude (or distance from the origin) and an angle from the positive x-axis (often in degrees or radians).

Converting from polar to rectangular form involves transforming this combination into a standard complex number format, expressed as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. This is achieved using the trigonometric identities:
  • \(a = r \cos \theta\)
  • \(b = r \sin \theta\)
Thus, for a polar complex number \(z = r(\cos \theta + i \sin \theta)\), its rectangular form can be calculated as follows:
  • Find \(a\) using \(a = r \cos \theta\).
  • Find \(b\) using \(b = r \sin \theta\).
  • Combine the parts to get \(z = a + bi\).
In the given exercise, by following these steps after computing the necessary trigonometric values for the angle 225掳, we derived the rectangular form, eventually simplifying it to approximately \(-0.707 - 0.707i\).
Complex Number Division
Dividing complex numbers isn't as straightforward as division with real numbers. However, using polar form makes this process much simpler. When given two complex numbers \(z_1\) and \(z_2\) in polar form, their division \(\frac{z_1}{z_2}\) involves:
  • Dividing the magnitudes: \(\frac{r_1}{r_2}\).
  • Subtracting the angles: \(\theta_1 - \theta_2\).
This results in a new complex number still in polar form. The formula is:\[\frac{z_1}{z_2} = \frac{r_1}{r_2}\left(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right)\]

In our case, the magnitudes of both \(z_1\) and \(z_2\) were equal, simplifying the division of magnitudes to 1. The angle subtraction was straightforward: \(280掳 - 55掳 = 225掳\), giving us the polar representation \(1(\cos 225掳 + i \sin 225掳)\). The result becomes clearer when further converted into rectangular form.
Trigonometric Form of Complex Numbers
A powerful aspect of complex numbers is their trigonometric form, also known as polar form. In this form, each complex number is described by a magnitude \(r\) and an angle \(\theta\) using trigonometric functions of cosine and sine.
  • Magnitude \(r\) represents the distance from the origin to the point in the complex plane.
  • Angle \(\theta\) indicates the direction of the line connecting this point to the origin, originating from the positive x-axis.
  • The expression becomes \(z = r(\cos \theta + i \sin \theta)\).
This form makes multiplication and division of complex numbers more intuitive. By handling the magnitudes and angles separately, computations become streamlined.

For the given exercise, it helped us effortlessly manage the operation \(\frac{z_1}{z_2}\) by allowing us to divide by simply using these characteristics and make appropriate trigonometric calculations. This emphasizes why polar coordinates are so useful in complex number arithmetic.

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Most popular questions from this chapter

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. Suppose the ride conductor was rather sinister and speeded up the ride to twice the speed. How would you modify the parametric equations to model such a change? Now vary the values of \(A\) and \(B\). What do you conjecture these parameters are modeling in this problem?

Determine an algebraic method for testing a polar equation for symmetry to the \(x\)-axis, the \(y\)-axis, and the origin. Apply the test to determine what symmetry the graph with equation \(r=\sin (3 \theta)\) has.

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r=-2 \cos \theta $$

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Baseball. A baseball is hit at an initial speed of \(105 \mathrm{mph}\) and an angle of \(20^{\circ}\) at a height of 3 feet above the ground. If home plate is 420 feet from the back fence, which is 15 feet tall, will the baseball clear the back fence for a home run?

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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