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Chapter 8: Problem 23

In Exercises 21-40, find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$ z_{1}=10\left(\cos 200^{\circ}+i \sin 200^{\circ}\right) \text { and } z_{2}=5\left(\cos 65^{\circ}+i \sin 65^{\circ}\right) $$

Short Answer

Expert verified
Divide magnitudes as \( \frac{10}{5} = 2 \) and subtract angles: \( 200^\circ - 65^\circ = 135^\circ \). Convert to rectangular form: result is approximately \(-1.414 + 1.414i\).

Step by step solution

01

Review Formula for Dividing Complex Numbers

For two complex numbers given in polar form, \( z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \), the quotient \( \frac{z_1}{z_2} \) can be found by dividing their magnitudes and subtracting their arguments: \( \frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \).

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Key Concepts

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

Polar Form
Complex numbers can be represented in various ways. One of the most intuitive is the polar form. In the polar representation, a complex number is expressed using a magnitude and an angle. This form looks like:
  • A magnitude, often labeled as \( r \), shows how far the number is from the origin on a complex plane.
  • An angle, \( \theta \), specifies the direction in which the number is located.
This can be written as \( z = r(\cos \theta + i\sin \theta) \).

Polar form is particularly useful in multiplication and division of complex numbers. By handling complex numbers in this way, it becomes straightforward to multiply or divide them by simply manipulating their magnitudes and angles, simplifying complex computations.

In our exercise, we start with complex numbers expressed in polar form for both \( z_1 \) and \( z_2 \). This makes it easy to find their quotient.
Quotient of Complex Numbers
Finding the quotient of two complex numbers is markedly simplified when using polar form. To divide two complex numbers, \( \frac{z_1}{z_2} \), in polar form:
  • First, divide their magnitudes: \( \frac{r_1}{r_2} \).
  • Then, subtract their angles: \( \theta_1 - \theta_2 \).
After computing these, the resultant complex number is also neatly expressed in polar form. In our given case:
  • The magnitudes are \( r_1 = 10 \) and \( r_2 = 5 \), so their quotient is \( \frac{10}{5} = 2 \).
  • The angles are \( \theta_1 = 200^{\circ} \) and \( \theta_2 = 65^{\circ} \), leading to the angle \( 200^{\circ} - 65^{\circ} = 135^{\circ} \).
So, the quotient in polar form becomes \( 2(\cos 135^{\circ} + i\sin 135^{\circ}) \).

It's important to understand this method as it leverages the symmetry of trigonometric functions, making it very powerful and elegant for these kinds of calculations.
Rectangular Form
Once we have the quotient of two complex numbers in polar form, converting it to rectangular form is the next step. Rectangular form represents a complex number as \( a + bi \), where \( a \) is the real part, and \( b \) is the imaginary part of the complex number.

To convert from polar to rectangular form, use:
  • \( a = r \cos \theta \)
  • \( b = r \sin \theta \)
For the quotient we found, \( 2(\cos 135^{\circ} + i\sin 135^{\circ}) \):
  • Calculate \( a = 2 \cos 135^{\circ} = 2(-\frac{\sqrt{2}}{2}) = -\sqrt{2} \).
  • Similarly, \( b = 2 \sin 135^{\circ} = 2(\frac{\sqrt{2}}{2}) = \sqrt{2} \).
Thus, the rectangular form of the quotient is \( -\sqrt{2} + \sqrt{2}i \).

Understanding how to switch between these forms is crucial because both provide different insights into the nature and behavior of complex numbers.

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Most popular questions from this chapter

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. $$ r^{2} \sin ^{2} \theta+2 r \cos \theta=3 $$

Bicycle Racing. A boy on a bicycle racing around an oval track has a position given by the equations \(x=-100 \sin \left(\frac{t}{4}\right)\) and \(y=75 \cos \left(\frac{t}{4}\right)\), where \(x\) and \(y\) are the horizontal and vertical positions in feet relative to the center of the track \(t\) seconds after the start of the race. Find the boy's position at \(t=10,20\), and 30 .

For Exercises 75 and 76, refer to the following: Many microphone manufacturers advertise their exceptional pickup capabilities that isolate the sound source and minimize background noise. The name of these microphones comes from the pattern formed by the range of the pickup. Cardioid Pickup Pattern. Graph the cardioid curve to see what the range of a microphone might look like: \(r=-4-4 \sin \theta\).

For Exercises 73 and 74, refer to the following: The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with different wing profiles. Flapping Wings of Birds. Compare the two following possible lemniscate patterns by graphing them on the same polar graph: \(r^{2}=4 \cos (2 \theta)\) and \(r^{2}=\frac{1}{4} \cos (2 \theta)\).

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. $$ x=\sec ^{2} t, y=\tan ^{2} t $$
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