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Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter I, Problem 32E
To determine

To find: The polar form of zw,zwand1z by putting z and w in polar form.

Expert Solution

Answer to Problem 32E

The polar form of the complex number zw is 242(cos17π12+isin17π12).

The polar form of the complex number zw is 423(cos(13π12)+isin(13π12)).

The polar form of the complex number 1z is 1z=18[cosπ6isinπ6].

Explanation of Solution

Formula used:

Let z1=r1(cosθ1+isinθ1) and z2=r2(cosθ2+isinθ2) polar form of the two complex number then,

z1z2=r1r2[cos(θ1+θ2)+isin(θ1+θ2)], z1z2=r1r2[cos(θ1θ2)+isin(θ1θ2)]wherez20 and 1z1=1r1[cosθ1isinθ1]

Calculation:

It is given that z=4(3+i)andw=33i.

Rewrite the complex number z=4(3+i)andw=33i in polar form.

The polar form of the complex number z=a+bi is z=r(cosθ+isinθ) where r=|z|=a2+b2  and tanθ=ba.

Consider the complex number z=4(3+i).

Obtain the argument of the complex number z=4(3+i).

tanθ=443=13

Thus, the argument of argument of the complex number z=4(3+i) is θ=tan1(13)=π6

Obtain the modulus of the complex number z=4(3+i).

r=|4(3+i)|=|43+4|=(43)2+(4)2=64=8

Thus, the value of r=8.

Thus, the polar form of the complex number z=4(3+i) is 8(cosπ6+isinπ6).

Similarly obtain the polar form of the complex number w=33i.

The argument of argument of the complex number w=33i is,

θ=tan1(33)=tan1(1)=5π4

The modulus of the complex number w=33i is,

r=|33i|=(3)2+(3)2=18=32

Thus, the polar form of the complex number w=33i is 32(cos5π4+isin5π4).

Use the above formula to obtain the polar of the complex number zw,zwand1z respectively.

Here, z=8(cosπ6+isinπ6) and w=32(cos5π4+isin5π4).

zw=8(32)[cos(π6+5π4)+isin(π6+5π4)]=242(cos17π12+isin17π12)

Thus, the polar form of the complex number zw is 242(cos17π12+isin17π12).

Compute the polar form of the complex number zw.

zw=832[cos(π65π4)+isin(π65π4)]=423(cos(13π12)+isin(13π12))

Thus, the polar form of the complex number zw is 423(cos(13π12)+isin(13π12)).

Obtain the polar form of the complex number 1z.

1z=18[cosπ6isinπ6]

Thus, the polar form of the complex number 1z is 1z=18[cosπ6isinπ6].

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