/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Find the product \(z_{1} z_{2}\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 57

Find the product \(z_{1} z_{2}\) and the quotient \(z_{1} / z_{2}\) . Express your answer in polar form. $$ \begin{array}{l}{z_{1}=4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)} \\\ {z_{2}=2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)}\end{array} $$

Short Answer

Expert verified
Product: \(8(\cos 150^\circ + i \sin 150^\circ)\); Quotient: \(2(\cos 90^\circ + i \sin 90^\circ)\).

Step by step solution

01

Understanding the Problem

We are given two complex numbers in polar form: \( z_1 = 4(\cos 120^\circ + i \sin 120^\circ) \) and \( z_2 = 2(\cos 30^\circ + i \sin 30^\circ) \). We need to find the product \( z_1 z_2 \) and the quotient \( \frac{z_1}{z_2} \), both in polar form.
02

Find the Product

To find \( z_1 z_2 \), multiply the magnitudes and add the angles: \( |z_1| \cdot |z_2| = 4 \cdot 2 = 8 \). The angle is \( 120^\circ + 30^\circ = 150^\circ \). Thus, the product is \( 8(\cos 150^\circ + i \sin 150^\circ) \).
03

Find the Quotient

To find \( \frac{z_1}{z_2} \), divide the magnitudes and subtract the angles: \( \frac{|z_1|}{|z_2|} = \frac{4}{2} = 2 \). The angle is \( 120^\circ - 30^\circ = 90^\circ \). Thus, the quotient is \( 2(\cos 90^\circ + i \sin 90^\circ) \).
04

Solution in Polar Form

The product \( z_1 z_2 \) in polar form is \( 8(\cos 150^\circ + i \sin 150^\circ) \) and the quotient \( \frac{z_1}{z_2} \) in polar form is \( 2(\cos 90^\circ + i \sin 90^\circ) \).

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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 numbers that have both a real part and an imaginary part. They are usually expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \).

A complex number can also be expressed in polar form, which is especially useful for multiplication and division. The polar form is \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle). Writing a complex number in this form makes it easy to interpret geometrically and to perform operations on complex numbers.

Polar coordinates are an incredibly helpful way of working with complex numbers, especially when considering their multiplication and division, because it simplifies the operations into simpler arithmetic on the magnitudes and angles.
Product and Quotient of Complex Numbers
The product and quotient of complex numbers become much simpler to handle when these numbers are in polar form.

  • **Product**: To find the product of two complex numbers \( z_1 \) and \( z_2 \) given in polar form as \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \), you multiply their magnitudes and add their angles:\[ |z_1 z_2| = r_1r_2 \]\[ \text{Angle of } z_1 z_2 = \theta_1 + \theta_2 \]This results in the product \( r_1 r_2(\cos(\theta_1+\theta_2) + i \sin(\theta_1+\theta_2)) \).
  • **Quotient**:To find the quotient \( \frac{z_1}{z_2} \) of two complex numbers, you divide their magnitudes and subtract their angles:\[ |\frac{z_1}{z_2}| = \frac{r_1}{r_2} \]\[ \text{Angle of } \frac{z_1}{z_2} = \theta_1 - \theta_2 \]This results in the quotient \( \frac{r_1}{r_2}(\cos(\theta_1-\theta_2) + i \sin(\theta_1-\theta_2)) \).

Simplifying these operations becomes very handy when dealing with complex numbers in practical applications.
de Moivre's Theorem
De Moivre's Theorem is a powerful tool in complex number analysis that links complex numbers with trigonometry in a strategic manner. This theorem states:If \( z = r(\cos \theta + i \sin \theta) \) is a complex number in polar form and \( n \) is any integer, then:\[(z)^n = r^n(\cos(n\theta) + i \sin(n\theta))\]This formula is exceptionally useful for computing powers and roots of complex numbers without reverting to the standard rectangular form. Applying de Moivre's Theorem can simplify calculations significantly, making it a favorite for dealing with complex exponential growth and solving power equations. It highlights the beauty of polar coordinates by demonstrating how elegantly and efficiently they allow for operations on complex numbers, especially in exponential contexts.

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Most popular questions from this chapter

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(1+i\)

Spiral Path of a Dog \(A\) dog is tied to a circular tree trunk of radius 1 ft by a long leash. He has managed to wrap the entire leash around the tree while playing in the yard, and he finds himself at the point \((1,0)\) in the figure. Seeing a squirrel, he runs around the tree counterclockwise, keeping the leash taut while chasing the intruder. (a) Show that parametric equations for the dog's path (called an involute of a circle) are $$ x=\cos \theta+\theta \sin \theta \quad y=\sin \theta-\theta \cos \theta $$ [Hint: Note that the leash is always tangent to the tree, so OT is perpendicular to \(T D .\) I (b) Graph the path of the dog for \(0 \leq \theta \leq 4 \pi\)

Complex Coefficients and the Quadratic Formula The quadratic formula works whether the coefficients of the equation are real or complex. Solve these equations using the quadratic formula and, if necessary, De Moivre's Theorem. $$ \begin{array}{l}{\text { (a) } z^{2}+(1+i) z+i=0} \\ {\text { (b) } z^{2}-i z+1=0} \\ {\text { (c) } z^{2}-(2-i) z-\frac{1}{4} i=0}\end{array} $$

Orbit of a Satellite Scientists and engineers often use polar equations to model the motion of satellites in earth orbit. Let's consider a satellite whose orbit is modeled by the equation \(r=22500 /(4-\cos \theta),\) where \(r\) is the distance in miles between the satellite and the center of the earth and \(\theta\) is the angle shown in the following figure. (a) On the same viewing screen, graph the circle \(r=3960\) (to represent the earth, which we will assume to be a sphere of radius 3960 \(\mathrm{mi}\) and the polar equation of the satellite's orbit. Describe the motion of the satellite as \(\theta\) increases from 0 to \(2 \pi .\) (b) For what angle \(\theta\) is the satellite closest to the earth? Find the height of the satellite above the earth's surface for this value of \(\theta .\)

Solve the equation. $$ z^{8}-i=0 $$
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