Explanation Given information: z 1 = 1 + i 3 , z 2 = 2 i Concept Involved: If z 1 = r 1 ( cos θ 1 + i sin θ 1 ) = r 1 c i s θ 1 and z 2 = r 2 ( cos θ 2 + i sin θ 2 ) = r 2 c i s θ 2 are two complex numbers in trigonometric form, then their product, z 1 / z 2 , is z 1 z 2 = r 1 ( cos θ 1 + i sin θ 1 ) r 2 ( cos θ 2 + i sin θ 2 ) = r 1 r 2 [ ( cos ( θ 1 − θ 2 ) + i sin ( θ 1 − θ 2 ) ) ] = r 1 r 2 c i s ( θ 1 − θ 2 ) In words: To divide two complex numbers in trigonometric form, dividing the moduli and subtracting the arguments. In other words, to divide two complex numbers in trigonometric form, we divide absolute values, r 1 r 2 and subtract angles, θ 1 − θ 2 . Modulus: The absolute value or modulus of the complex number z = x + y i is the distance from the origin to the point ( x , y ) in the complex plane. If this distance is denoted by r , then r = | z | = | x + y i | = x 2 + y 2 Argument: The argument of the complex number z = x + y i , denoted a r g ( z ) , is the smallest positive angle θ from the positive real axis to the graph of z. Trigonometric Form: If z = x + y i is a complex number in standard form, then the trigonometric form of z is given by: z = r ( cos θ + i sin θ ) = r c i s θ Where r is the modulus of z and θ is the argument of z. We can convert back and forth between standard form and trigonometric form using the relationships that follow: For z = x + y i = r ( cos θ + i sin θ ) = r c i s θ r = x 2 + y 2 and θ is such that cos θ = x r , sin θ = y r , a n d tan θ = y x The sign of trigonometric ratio in different quadrants Formula Used: ( a + b ) ( a + b ) = a 2 + b 2 + 2 a b Calculation: TABLE 1 Steps Description z 1 / z 2 = ( 1 + i 3 ) / ( 2 i ) Substituting z 1 = 3 + i , z 2 = 2 i = ( 1 + i 3 ) ( − 2 i ) ( 2 i ) ( − 2 i ) = − 2 i − 2 3 i 2 − 4 i 2 = − 2 i + 2 3 4 = 3 2 − 1 2 i z 1 / z 2 = 3 2 − 1 2 i Dividing common factor Calculation: TABLE 2 Steps

8th Edition

ISBN: 9781337320733

Author: Mckeague

Publisher: Cengage Learning

See similar textbooks

Question

Chapter 8.3, Problem 52PS

To determine

To find:

The quotient z1/z2 in standard form, then write z1 and z2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 51PS

Chapter 8.3, Problem 53PS

Students have asked these similar questions

Solve the oblique triangle with sides, a - 13.2 m, b = 37.3 m, c = Assume that, a is the side opposite to ZA, b is the side opposite to B, c is the side opposite to C. Enter answers accurate to . If there is no solution, enter DNE. ZA = 38.6 m. LB = ZC = O O

Please solve with steps

Find all solutions of the equation answering as an expression, using n as the variable, in the form a+bn

Chapter 8 Solutions

ACP TRIG. CUESTA COLL.>CUSTOM< - TEXT

Ch. 8.1 - For Questions 1 through 10, fill in the blank with...Ch. 8.1 - Prob. 2PS Ch. 8.1 - Prob. 3PS Ch. 8.1 - Prob. 4PS Ch. 8.1 - Prob. 5PS Ch. 8.1 - Prob. 6PS Ch. 8.1 - Prob. 7PS Ch. 8.1 - Prob. 8PS Ch. 8.1 - Prob. 9PS Ch. 8.1 - Prob. 10PS

Knowledge Booster

The quotient z 1 / z 2 in standard form, then write z 1 and z 2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.

Solution Summary: The author explains how to find the quotient in trigonometric form, then convert it to standard form to show that they are equal.

To find:

The quotient z1/z2 in standard form, then write z1 and z2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.

Want to see the full answer?

Chapter 8 Solutions

The quotient z 1 / z 2 in standard form, then write z 1 and z 2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.

Solution Summary: The author explains how to find the quotient in trigonometric form, then convert it to standard form to show that they are equal.

To find: The quotient z1/z2 in standard form, then write z1 and z2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.

Want to see the full answer?

Chapter 8 Solutions

To find:

The quotient z1/z2 in standard form, then write z1 and z2 in trigonometric form, find their quotient again and finally convert that in trigonometric form to standard form to show that the quotients are equal.