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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 29E
To determine

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

Expert Solution

Answer to Problem 29E

The polar form of the complex number zw is zw=4(cosπ2+isinπ2).

The polar form of the complex number zw is zw=cos(π6)+isin(π6).

The polar form of the complex number 1z is 1z=12[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=3+iandw=1+3i.

Rewrite the complex number z=3+iandw=1+3i 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=3+i.

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

tanθ=13

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

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

r=|3+i|=32+(1)2=4=2

Thus, the value of r=2.

Thus, the polar form of the complex number z=3+i is 2(cosπ6+isinπ6).

Similarly, obtain the polar form of the complex number w=1+3i.

The argument of argument of the complex number w=1+3i is θ=tan1(3)=π3

The modulus of the complex number w=1+3i is,

r=|1+3i|=32+(1)2=4=2

Thus, the polar form of the complex number w=1+3i is 2(cosπ3+isinπ3).

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

Here, z=2(cosπ6+isinπ6) and w=2(cosπ3+isinπ3).

zw=2(2)[cos(π6+π3)+isin(π6+π3)]=4(cosπ2+isinπ2)

Thus, the polar form of the complex number zw is zw=4(cosπ2+isinπ2).

Compute the polar form of the complex number zw.

zw=22[cos(π6π3)+isin(π6π3)]=cos(π6)+isin(π6)

Thus, the polar form of the complex number zw is zw=cos(π6)+isin(π6).

Obtain the polar form of the complex number 1z.

1z=12[cosπ6isinπ6]

Thus, the polar form of the complex number 1z is 1z=12[cosπ6isinπ6].

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