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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 31E
To determine

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

Expert Solution

Answer to Problem 31E

The polar form of the complex number zw is zw=42(cos7π12+isin7π12).

The polar form of the complex number zw is zw=22(cos(11π12)+isin(11π12)).

The polar form of the complex number 1z is 1z=14[cos(π6)isin(π6)].

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=232iandw=1+i.

Rewrite the complex number z=232iandw=1+i 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=232i.

Obtain the argument of the complex number z=232i.

tanθ=223=13

Thus, the argument of argument of the complex number z=232i is θ=tan1(13)=π6

Obtain the modulus of the complex number z=232i.

r=|232i|=(23)2+(2)2=16=4

Thus, the value of r=4.

Thus, the polar form of the complex number z=232i is 4(cos(π6)+isin(π6)).

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

The argument of argument of the complex number w=1+i is θ=tan1(1)=3π4.

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

r=|1+i|=(1)2+(1)2=2

Thus, the polar form of the complex number w=1+i is 2(cos3π4+isin3π4).

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

Here, z=4(cos(π6)+isin(π6)) and w=2(cos3π4+isin3π4).

zw=4(2)[cos(π6+3π4)+isin(π6+3π4)]=42(cos7π12+isin7π12)

Thus, the polar form of the complex number zw is zw=42(cos7π12+isin7π12).

Compute the polar form of the complex number zw.

zw=42[cos(π63π4)+isin(π63π4)]=22(cos(11π12)+isin(11π12))

Thus, the polar form of the complex number zw is zw=22(cos(11π12)+isin(11π12)).

Obtain the polar form of the complex number 1z.

1z=14[cos(π6)isin(π6)]

Thus, the polar form of the complex number 1z is 1z=14[cos(π6)isin(π6)].

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