PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 6.6, Problem 28E
Solution

a)

To find: The product z1z2 and quotient z1/z2 using the polar form for z1 and z2 .

The product is 2.730.73i and quotient is 0.180.68i .

Given information:

It is given that z1=1i and z2=3+i .

Formula used:

The product of two complex numbers is given by,

  z1z2=r1r2cosθ1+θ2+isinθ1+θ2

The quotient of two complex numbers is given by,

  z1z2=r1r2cosθ1θ2+isinθ1θ2

Calculation:

Consider the first complex number.

  z1=1i

From the given complex number,

  a=1b=1

Substitute 1 for a and 1 for b in equation r=a2+b2 to find modulus r .

  r=12+12=1+1=2

Substitute 1 for a and 1 for b in equation tanθ=ba to find argument.

  tanθ=11θ=tan11=2πtan11=7π4

Substitute 2 for r and 7π4 for θ in equation z=rcosθ+isinθ .

  z1=2cos7π4+isin7π4

Consider the second complex number.

  z2=3+i

From the given complex number,

  a=3b=1

Substitute 3 for a and 1 for b in equation r=a2+b2 to find modulus r .

  r=32+12=4=2

Substitute 3 for a and 1 for b in equation tanθ=ba to find argument.

  tanθ=13θ=tan113=π6

Substitute 2 for r and π6 for θ in equation z=rcosθ+isinθ .

  z2=2cosπ6+isinπ6

Find the product of two complex numbers.

  z1z2=22cos7π4+π6+isin7π4+π6=22cos23π12+isin23π12=2.730.73i

Find the quotient of two complex numbers.

  z1z2=22cos7π4+π6+isin7π4+π6=22cos19π12+isin19π12=0.180.68i

Therefore, the product is 2.730.73i and quotient is 0.180.68i .

b)

To find: The product z1z2 and quotient z1/z2 using the standard form for z1 and z2 .

The product is 2.730.73i and quotient is

  0.180.68i .

Given information:

It is given that z1=32i and z2=1+i .

Calculation:

Find the product of two complex numbers.

  z1z2=1i3+i=13+1i+i3+ii=3+ii3i2=3+ii31

Simplify further.

  z1z2=1+3+13i=2.730.73i

Find the quotient of two complex numbers.

  z1z2=1i3+i3i3i=13+1i+i3+ii32i2=3ii3+i23i2=3ii3+12312

Simplify further.

  z1z2=311+3i4=3141+34i=0.180.68i

Therefore, the product is 2.730.73i and quotient is 0.180.68i .

Chapter 6 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

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