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High School Math Calculus EBK PRECALCULUS W/LIMITS

The trigonometric forms of thecomplex numbers.

The trigonometric forms of thecomplex numbers.

EBK PRECALCULUS W/LIMITS

EBK PRECALCULUS W/LIMITS

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 6.6, Problem 46E

(a)

To determine

The trigonometric forms of thecomplex numbers.

(a)

Expert Solution

Answer to Problem 46E

The trigonometric formis $(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]$

Explanation of Solution

Given info:

$(3+i)(1+i)$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

$z=r(cosθ+isinθ)$

Where $r=a2+b2$ and $θ=tan−1(ba)$

Calculation:

We have

$z=(3+i)(1+i)3+i_a=3,b=1r=a2+b2=(3)2+12=2θ=tan−1(13)=π6z=r(cosθ+isinθ)3+i=2(cosπ6+isinπ6)1+i_a=1,b=1r=a2+b2=12+12=2θ=tan−1(11)=π4z=r(cosθ+isinθ)1+i=2(cosπ4+isinπ4)z=(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]$

The trigonometric form is $(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]$

Conclusion:

Thus,the trigonometric form is $(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]$

(b)

To determine

The product using the trigonometric forms.

(b)

Expert Solution

Answer to Problem 46E

The product using the trigonometric form is $z=0.732+2.73i$

Explanation of Solution

Given info:

$(3+i)(1+i)$

Formula used:

Let $z1=r1(cosθ1+isinθ1) and z2=r2(cosθ2+isinθ2)$ we have

$z1z2=r1r2(cos(θ1+θ2)+isin(θ1+θ2))z1z2=r1r2(cos(θ1−θ2)+isin(θ1−θ2))$

Calculation:

From part a we have $(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]$

We have

$z=(3+i)(1+i)=[2(cosπ6+isinπ6)][2(cosπ4+isinπ4)]z1=r1(cosθ1+isinθ1)=2(cosπ6+isinπ6)z2=r2(cosθ2+isinθ2)=2(cosπ4+isinπ4)z=z1z2=r1r2(cos(θ1+θ2)+isin(θ1+θ2))z=22(cos(π6+π4)+isin(π6+π4))z=22(cos5π12+isin5π12)z=0.732+2.73i$

The product using the trigonometric formis $z=0.732+2.73i$

Conclusion:

Thus,the product using the trigonometric form is $z=0.732+2.73i$

(c)

To determine

The product using the standard forms, and checkyour result with that of part (b).

(c)

Expert Solution

Answer to Problem 46E

The product using standard form and trigonometric form are same.

Explanation of Solution

Given info:

$(3+i)(1+i)$

Formula used:

We have the results

$i2=−1$

Calculation:

Product from part b is $z=0.732+2.73i$

The product using the standard forms is given by

$z=(3+i)(1+i)z=3+3i+i−i2=0.732+2.73i$

Product using standard form and trigonometric form are same.

Conclusion:

Thus,the product using standard form and trigonometric form are same.

Chapter 6.6, Problem 45E

Chapter 6.6, Problem 47E

Chapter 6 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - Prob. 55E Ch. 6.1 - Prob. 56E Ch. 6.1 - Prob. 57E Ch. 6.1 - Prob. 58E Ch. 6.1 - Prob. 59E Ch. 6.1 - Prob. 60E Ch. 6.1 - 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Prob. 28E Ch. 6.6 - Prob. 29E Ch. 6.6 - Prob. 30E Ch. 6.6 - Prob. 31E Ch. 6.6 - Prob. 32E Ch. 6.6 - Prob. 33E Ch. 6.6 - Prob. 34E Ch. 6.6 - Prob. 35E Ch. 6.6 - Prob. 36E Ch. 6.6 - Prob. 37E Ch. 6.6 - Prob. 38E Ch. 6.6 - Prob. 39E Ch. 6.6 - Prob. 40E Ch. 6.6 - Prob. 41E Ch. 6.6 - Prob. 42E Ch. 6.6 - Prob. 43E Ch. 6.6 - Prob. 44E Ch. 6.6 - Prob. 45E Ch. 6.6 - Prob. 46E Ch. 6.6 - Prob. 47E Ch. 6.6 - Prob. 48E Ch. 6.6 - Prob. 49E Ch. 6.6 - Prob. 50E Ch. 6.6 - Prob. 51E Ch. 6.6 - Prob. 52E Ch. 6.6 - Prob. 53E Ch. 6.6 - Prob. 54E Ch. 6.6 - Prob. 55E Ch. 6.6 - Prob. 56E Ch. 6.6 - Prob. 57E Ch. 6.6 - Prob. 58E Ch. 6.6 - Prob. 59E Ch. 6.6 - Prob. 60E Ch. 6.6 - Prob. 61E Ch. 6.6 - Prob. 62E Ch. 6.6 - Prob. 63E Ch. 6.6 - Prob. 64E Ch. 6.6 - Prob. 65E Ch. 6.6 - Prob. 66E Ch. 6.6 - Prob. 67E Ch. 6.6 - Prob. 68E Ch. 6.6 - Prob. 69E Ch. 6.6 - Prob. 70E Ch. 6.6 - Prob. 71E Ch. 6.6 - Prob. 72E Ch. 6.6 - Prob. 73E Ch. 6.6 - Prob. 74E Ch. 6.6 - Prob. 75E Ch. 6.6 - Prob. 76E Ch. 6.6 - Prob. 77E Ch. 6.6 - 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