Explanation Given information: z 1 = 1 + i , z 2 = 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 multiply two complex numbers in trigonometric form, multiply the moduli and add the arguments. As we can see, to multiply two complex numbers in trigonometric form, we multiply absolute values, r 1 r 2 , and add 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 ) ( 2 + 2 i ) Substituting z 1 = 1 + i , z 2 = 2 + 2 i ( 1 + i ) ( 2 + 2 i ) = ( 1 + i ) ⋅ 2 ( 1 + i ) = 2 ( i + 1 ) ( i + 1 ) = 2 ( i 2 + 2 i + 1 ) = 2 ( − 1 + 2 i + 1 ) = 4 i z 1 z 2 = 4 i Applying algebraic identity ( a + b ) ( a + b ) = a

8th Edition

ISBN: 9781337320733

Author: Mckeague

Publisher: Cengage Learning

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Chapter 8.3, Problem 12PS

To determine

To find:

The product z1z2 in standard form, then write z1 and z2 in trigonometric form, find their product again and finally convert that in trigonometric form to standard form to show that the products are equal.

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Chapter 8.3, Problem 11PS

Chapter 8.3, Problem 13PS

Chapter 8 Solutions

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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

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

Solution Summary: The author explains that if z_12 are two complex numbers in trigonometric form, then their product is

To find:

The product z1z2 in standard form, then write z1 and z2 in trigonometric form, find their product again and finally convert that in trigonometric form to standard form to show that the products are equal.

Want to see the full answer?

Chapter 8 Solutions

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

Solution Summary: The author explains that if z_12 are two complex numbers in trigonometric form, then their product is

To find: The product z1z2 in standard form, then write z1 and z2 in trigonometric form, find their product again and finally convert that in trigonometric form to standard form to show that the products are equal.

Want to see the full answer?

Chapter 8 Solutions

To find:

The product z1z2 in standard form, then write z1 and z2 in trigonometric form, find their product again and finally convert that in trigonometric form to standard form to show that the products are equal.