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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 30E
To determine

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

Expert Solution

Answer to Problem 30E

The polar form of the complex number zw is zw=64(cos7π3+isin7π2).

The polar form of the complex number zw is zw=cos(4π3)+isin(4π3).

The polar form of the complex number 1z is 1z=18[cos11π6isin11π6].

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=434iandw=8i.

Rewrite the complex number z=434iandw=8i 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=434i.

Obtain the argument of the complex number z=434i.

tanθ=443=13

Thus, the argument of argument of the complex number z=434i is θ=tan1(13)=11π6

Obtain the modulus of the complex number z=434i.

r=|434i|=(43)2+(4)2=48+16=8

Thus, the value of r=8.

Thus, the polar form of the complex number z=434i is 8(cos11π6+isin11π6).

Similarly, obtain the polar form of the complex number w=8i.

The argument of argument of the complex number w=8i is,

θ=tan1(80)=tan1()=π2

Thus, the argument of argument of the complex number w=8i is π2.

The modulus of the complex number w=8i is,

r=|8i|=02+82=64=8

Thus, the polar form of the complex number w=8i is 8(cosπ2+isinπ2).

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

Here, z=8(cos11π6+isin11π6) and w=8(cosπ2+isinπ2).

zw=8(8)[cos(11π6+π2)+isin(11π6+π2)]=64(cos7π3+isin7π2)

Thus, the polar form of the complex number zw is zw=64(cos7π3+isin7π2).

Compute the polar form of the complex number zw.

zw=88[cos(11π6π2)+isin(11π6π2)]=cos(4π3)+isin(4π3)

Thus, the polar form of the complex number zw is zw=cos(4π3)+isin(4π3).

Obtain the polar form of the complex number 1z.

1z=18[cos11π6isin11π6]

Thus, the polar form of the complex number 1z is 1z=18[cos11π6isin11π6].

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