9-5 Complex Numbers and De Moivre's Theorem · Moiver ¶s Theorem to find the third power. 62/87,21...

Graph each number in the complex plane and find its absolute value.

1.z = 4 + 4i

SOLUTION:For z = 4 + 4i, (a, b) = (4, 4). Graph the point (4, 4)in the complex plane.

Use the absolute value of a complex number formula.

2.z = 3 + i

SOLUTION:For z = 3 + i, (a, b) = (3, 1). Graph the point (3,1) in the complex plane.


3.z = 4 6i

SOLUTION:For z = 4 6i, (a, b) = (4, 6). Graph the point (4, 6) in the complex plane.


4.z = 2 5i



5.z = 3 + 4i



6.z = 7 + 5i

SOLUTION:For z = 7 + 5i, (a, b) = (7, 5). Graph the point (7, 5) in the complex plane.


7.z = 3 7i



8.z = 8 2i



9.VECTORS The force on an object is given by z = 10 + 15i, where the components are measured in newtons (N). a. Represent z as a vector in the complex plane. b. Find the magnitude and direction angle of the vector.

SOLUTION:a. For z = 10 + 15i, (a, b) = (10, 15). Graph the point (10, 15) in the complex plane. Then draw a vector with an initial point at the origin and a terminal point at (10, 15).

b. Use the absolute value of a complex number formula.

Find the measure of the angle thatthevectormakes with the positive real axis.

The magnitude of the force is about 18.03 newtons atanangleofabout56.31.

Express each complex number in polar form.10.4 + 4i

SOLUTION:4 + 4i

Find the modulus r and argument .

The polar form of 4+ 4i is .

11.2 + i

SOLUTION:2 + i Find the modulus r and argument .

The polar form of 2 + i is (cos 2.68 + i sin 2.68).

12.4 i

SOLUTION:

4 i


The polar form of 4 i is

3 (cos 0.34 + i sin 0.34).

13.2 2i

SOLUTION:2 2i Find the modulus r and argument .

The polar form of 2 2i is

.

14.4 + 5i

SOLUTION:4 + 5i


The polar form of 4+ 5i is (cos0.90+i sin 0.90).

15.2 + 4i

SOLUTION:2 + 4i Find the modulus r and argument .

The polar form of 2 + 4i is 2 (cos 2.03 + i sin 2.03).

16.1 i

SOLUTION:

1 i



.

17.3 + 3i

SOLUTION:3 + 3i



Graph each complex number on a polar grid. Then express it in rectangular form.

18.10(cos 6 + i sin 6)

SOLUTION:

The value of r is 10, and the value of is6.Plotthe polar coordinates (10, 6).

To express the number in rectangular form, evaluate the trigonometric values and simplify.

The rectangular form of is

.

19.2(cos 3 + i sin 3)

SOLUTION:

The value of r is 2, and the value of is3.Plotthepolar coordinates (2, 3). Notice that 3 radians is

slightly greater than butlessthan.


The rectangular form of 2(cos 3 + i sin 3) is 1.98 + 0.28i.

20.

SOLUTION:

The value of r is 4, and the value of is . Plot

the polar coordinates .



.

21.

SOLUTION:





i.

22.

SOLUTION:

The value of r is 1, and the value of is .

Plot the polar coordinates .



.

23.

SOLUTION:




The rectangular form of is1

i.

24.3(cos 180 + i sin 180)

SOLUTION:

The value of r is 3, and the value of is180.Plot the polar coordinates (3,180).


The rectangular form of 3(cos 180 + i sin 180) is3.

25. (cos 360 + i sin 360)

SOLUTION:

The value of r is , and the value of is360.

Plot the polar coordinates


The rectangular form of (cos 360 + i sin 360) is

.

Find each product or quotient and express it in rectangular form.

26.

SOLUTION:Use the Product Formula to find the product in polar form.

Now find the rectangular form of the product.

The polar form is . The

rectangular form is .

27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)



The polar form is 10(cos 180 + i sin 180). The rectangular form is 10.

28.

SOLUTION:Use the Quotient Formula to find the quotient in polar form.

Now find the rectangular form.



29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)




rectangular form is 4.

30.



The polar form of the quotient is

. The rectangular form

of the quotient is .

31.



The polar form of the quotient

is . The rectangular form of the

quotient is .

32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)





33.



The polar form of the quotient is .

The rectangular form of the quotient is .

34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)





35.




. The rectangular form of the

quotient is .

Find each power and express it in rectangular form.

36.(2 + 2 i)6

SOLUTION:

First, write 2 + 2 i in polar form.

The polar form of 2 + 2 i is .

Now use De Moivres Theorem to find the sixth power.

Therefore, .

37.(12i 5)3

SOLUTION:First, write 12i 5 in polar form.

The polar form of 12i 5 is . Now use De Moivres Theorem to find the third power.

Therefore, .

38.

SOLUTION:

is already in polar form. Use De

Moivres Theorem to find the fourth power.

39.( i)3

SOLUTION:

First, write i in polar form.

The polar form of i is . Now use De Moivres Theorem to find the third power.

Therefore, .

40.(3 5i)4

SOLUTION:First, write 3 5i in polar form.


. Now use De


Therefore, .

41.(2 + 4i)4

SOLUTION:First, write 2 + 4i in polar form.

The polar form of 2 + 4iis

. Now use De


Therefore, .

42.(3 6i)4


The polar form of 3 6i

is . Now use De


Therefore, .

43.(2 + 3i)2


The polar form of 2 + 3i

is . Now use De

Moivres Theorem to find the second power.

Therefore, .

44.

SOLUTION:


Moivers Theorem to find the third power.

45.

SOLUTION:


Moivers Theorem to find the fourth power.

46.DESIGN Stella works for an advertising agency. She wants to incorporate a design comprised of regular hexagons as the artwork for one of her proposals. Stella can locate the vertices of one of the central regular hexagons by graphing the

solutions to x6 1 = 0 in the complex plane. Find

the vertices of this hexagon.

SOLUTION:

The equation x6 1 = 0 can be written as x

6 = 1.

To find the location of the six vertices for one of the hexagons, find the sixth roots of 1. First, write 1 in polar form.

Thepolarformof1is1(cos0+i sin 0). Now write an expression for the sixth roots.

Let n = 0 to find the first root of 1.

Notice that the modulus of each complex number is

1. The arguments are found by , resulting in

increasingby foreachsuccessiveroot.

Therefore, we can calculate the remaining roots by

adding toeachprevious .

n = 1

n = 2

n = 3

n = 4

n = 5

The vertices are located at

.

Find all of the distinct p th roots of the complex number.

47.sixth roots of i

SOLUTION:First, write i in polar form.

The polar form of i is 1(cos + i sin ). Now

write an expression for the sixth roots.

Let n = 0, 1, 2, 3, 4 and 5 successively to find the sixth roots. Let n = 0.

Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

Let n = 4

The sixth roots of i are approximately 0.97 + 0.26i, 0.26 + 0.97i, 0.71 + 0.71i, 0.97 0.26 i, 0.26 0.97i, 0.71 0.71i.

48.fifth roots of i

SOLUTION:


The polar form of i is .

Now write an expression for the fifth roots.

Let n = 0, 1, 2, 3 and 4 successively to find the fifthroots. Let n = 0.

Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

The sixth roots of i are approximately 0.59 + 0.81i, 0.59 + 0.81i, 0.95 0.31i, i, 0.95 0.31i.

49.fourth roots of 4 4i

SOLUTION:

First, write 4 4i in polar form.


is . Now write an

expression for the fourth roots.

Let n = 0, 1, 2 and 3 successively to find the fourth roots. Let n = 0.

Let n = 1.

Let n = 2.

Let n = 3.

The fourth roots of 4 4i are approximately 0.22 + 1.67i, 1.67 + 0.22i, 0.22 1.67i, 1.67 0.22i.

50.cube roots of 117 + 44i


The polar form of 117 + 44i is 125(cos 2.78 + i sin2.78). Now write an expression for the cube roots.

Let n = 0, 1 and 2 successively to find the cube roots. Let n = 0.

Let n = 1.

Let n = 2.

The cube roots of 117 + 44i are approximately 3 +4i, 4.96 + 0.60i, 1.96 4.60i.

51.fifth roots of 1 + 11 i

SOLUTION:


The polar form of 1 + 11 i is (cos 1.63 + i sin 1.63). Now write an expression for the fifth roots.

Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots. Let n = 0.

Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

The fifth roots of 1 + 11 i are approximately 1.64 + 0.55i, 0.02 + 1.73i, 1.65 + 0.52i, 1.00 1.41i, and 1.03 1.39i.

52.square root of 3 4i


The polar form of 3 4i is 5(cos 4.07 + i sin 4.07). Now write an expression for the square roots.

Let n = 0 and 1 successively to find the square roots. Let n = 0.

Let n = 1.

The square roots of 3 4i are approximately 1 +2i and 1 2i.

53.find the square roots of unity

SOLUTION:First, write 1 in polar form.

Thepolarformof1is1(cos0+i sin 0). Now write an expression for the square roots.


Notice that the modulus of each complex number is1. The arguments are found by n, resulting in increasingbynfor each successive root.

Therefore, we can calculate the remaining root by

adding nto the previous . n = 1

The square roots of 1 are 1.

54.find the fourth roots of unity


Thepolarformof1is1(cos0+i sin 0). Now write an expression for the fourth roots.







n = 1

n = 2

n = 3

The fourth roots of 1 are 1 and i.

55.ELECTRICITY The impedance in one part of a series circuit is 5(cos 0.9 + j sin 0.9) ohms. In the second part of the circuit, it is 8(cos 0.4 + j sin 0.4) ohms. a. Convert each expression to rectangular form. b. Add your answers from part a to find the total impedance in the circuit. c. Convert the total impedance back to polar form.

SOLUTION:a. Evaluate the trigonometric values and simplify.

b. Find the sum. (3.11 + 3.92j ) + (7.37 + 3.12j ) = 10.48 + 7.04j ohms c. Find the modulus r and argument .

The polar form of 10.48 + 7.04j is 12.63(cos 0.59 +j sin 0.59) ohms.

Find each product. Then repeat the process by multiplying the polar forms of each pair of complex numbers using the Product Formula.

56.(1 i)(4 + 4i)

SOLUTION:First, find each product.

Express each complex number in polar form.

For 1 i, find the modulus r and argument .

The polar form of 1 i is .

For 4 + 4i, find the modulus r and argument .

The polar form of 4 + 4i is .

Use the Product Formula to find the product in polar form.


57.(3 + i)(3 i)


Express each complex number in polar form. For 3 + i, find the modulus r and argument .

The polar form of 3 + i is

.





58.(4 + i)(3 i)


Express each complex number in polar form. For 3 i, find the modulus r and argument .


For 4 + i, find the modulus r and argument .


.



59.(6 + 5i)(2 3i)


Express each complex number in polar form. For 6 + 5i, find the modulus r and argument .

The polar form of 6 + 5i is

.

For 2 3i, find the modulus r and argument .

The polar form of 2 3i is .



60.( + 2i)(1 + i)



For +2i, find the modulus r and argument .

The polar form of +2i is .



.



61.(3 2i)(1 + i)





.



.



62.FRACTALS A fractal is a geometric figure that is made up of a pattern that is repeated indefinitely on successively smaller scales, as shown below. Refer to the image on Page 578. In this problem, you will generate a fractal through

iterations of f (z) = z2. Consider z0 = 0.8 + 0.5i.

a. Calculate z1, z2, z3, z4, z5, z6, and z7 where z1 = f

(z0), z2 = f (z1), and so on.

b. Graph each of the numbers on the complex plane. c. Predict the location of z100. Explain.

SOLUTION:a. Calculate z1.

Use the expression that you found for z1 to find z2.





1.z = 4 + 4i



2.z = 3 + i



3.z = 4 6i



4.z = 2 5i



5.z = 3 + 4i



6.z = 7 + 5i



7.z = 3 7i



8.z = 8 2i









SOLUTION:4 + 4i



11.2 + i



12.4 i

SOLUTION:

4 i



3 (cos 0.34 + i sin 0.34).

13.2 2i



.

14.4 + 5i

SOLUTION:4 + 5i



15.2 + 4i



16.1 i

SOLUTION:

1 i



.

17.3 + 3i

SOLUTION:3 + 3i




18.10(cos 6 + i sin 6)

SOLUTION:




.

19.2(cos 3 + i sin 3)

SOLUTION:





20.

SOLUTION:





.

21.

SOLUTION:





i.

22.

SOLUTION:





.

23.

SOLUTION:





i.

24.3(cos 180 + i sin 180)

SOLUTION:




25. (cos 360 + i sin 360)

SOLUTION:





.


26.





27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)




28.





29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)





30.






31.





quotient is .

32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)





33.





34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)





35.





quotient is .


36.(2 + 2 i)6

SOLUTION:




Therefore, .

37.(12i 5)3



Therefore, .

38.

SOLUTION:



39.( i)3

SOLUTION:



Therefore, .

40.(3 5i)4



. Now use De


Therefore, .

41.(2 + 4i)4



. Now use De


Therefore, .

42.(3 6i)4



is . Now use De


Therefore, .

43.(2 + 3i)2



is . Now use De


Therefore, .

44.

SOLUTION:



45.

SOLUTION:






SOLUTION:


6 = 1.









n = 1

n = 2

n = 3

n = 4

n = 5


.


47.sixth roots of i





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

Let n = 4


48.fifth roots of i

SOLUTION:





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.



SOLUTION:



is . Now write an



Let n = 1.

Let n = 2.

Let n = 3.






Let n = 1.

Let n = 2.



SOLUTION:




Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.






Let n = 1.



















n = 1

n = 2

n = 3







56.(1 i)(4 + 4i)









57.(3 + i)(3 i)




.





58.(4 + i)(3 i)






.



59.(6 + 5i)(2 3i)




.





60.( + 2i)(1 + i)







.



61.(3 2i)(1 + i)





.



.






(z0), z2 = f (z1), and so on.







eSolutions Manual - Powered by Cognero Page 1

9-5 Complex Numbers and De Moivre's Theorem


1.z = 4 + 4i



2.z = 3 + i



3.z = 4 6i



4.z = 2 5i



5.z = 3 + 4i



6.z = 7 + 5i



7.z = 3 7i



8.z = 8 2i









SOLUTION:4 + 4i



11.2 + i



12.4 i

SOLUTION:

4 i



3 (cos 0.34 + i sin 0.34).

13.2 2i



.

14.4 + 5i

SOLUTION:4 + 5i



15.2 + 4i



16.1 i

SOLUTION:

1 i



.

17.3 + 3i

SOLUTION:3 + 3i




18.10(cos 6 + i sin 6)

SOLUTION:




.

19.2(cos 3 + i sin 3)

SOLUTION:





20.

SOLUTION:





.

21.

SOLUTION:





i.

22.

SOLUTION:





.

23.

SOLUTION:





i.

24.3(cos 180 + i sin 180)

SOLUTION:




25. (cos 360 + i sin 360)

SOLUTION:





.


26.





27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)




28.





29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)





30.






31.





quotient is .

32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)





33.





34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)





35.





quotient is .


36.(2 + 2 i)6

SOLUTION:




Therefore, .

37.(12i 5)3



Therefore, .

38.

SOLUTION:



39.( i)3

SOLUTION:



Therefore, .

40.(3 5i)4



. Now use De


Therefore, .

41.(2 + 4i)4



. Now use De


Therefore, .

42.(3 6i)4



is . Now use De


Therefore, .

43.(2 + 3i)2



is . Now use De


Therefore, .

44.

SOLUTION:



45.

SOLUTION:






SOLUTION:


6 = 1.









n = 1

n = 2

n = 3

n = 4

n = 5


.


47.sixth roots of i





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

Let n = 4


48.fifth roots of i

SOLUTION:





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.



SOLUTION:



is . Now write an



Let n = 1.

Let n = 2.

Let n = 3.






Let n = 1.

Let n = 2.



SOLUTION:




Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.






Let n = 1.



















n = 1

n = 2

n = 3







56.(1 i)(4 + 4i)









57.(3 + i)(3 i)




.





58.(4 + i)(3 i)






.



59.(6 + 5i)(2 3i)




.





60.( + 2i)(1 + i)







.



61.(3 2i)(1 + i)





.



.






(z0), z2 = f (z1), and so on.








1.z = 4 + 4i



2.z = 3 + i



3.z = 4 6i



4.z = 2 5i



5.z = 3 + 4i



6.z = 7 + 5i



7.z = 3 7i



8.z = 8 2i









SOLUTION:4 + 4i



11.2 + i



12.4 i

SOLUTION:

4 i



3 (cos 0.34 + i sin 0.34).

13.2 2i



.

14.4 + 5i

SOLUTION:4 + 5i



15.2 + 4i



16.1 i

SOLUTION:

1 i



.

17.3 + 3i

SOLUTION:3 + 3i




18.10(cos 6 + i sin 6)

SOLUTION:




.

19.2(cos 3 + i sin 3)

SOLUTION:





20.

SOLUTION:





.

21.

SOLUTION:





i.

22.

SOLUTION:





.

23.

SOLUTION:





i.

24.3(cos 180 + i sin 180)

SOLUTION:




25. (cos 360 + i sin 360)

SOLUTION:





.


26.





27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)




28.





29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)





30.






31.





quotient is .

32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)





33.





34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)





35.





quotient is .


36.(2 + 2 i)6

SOLUTION:




Therefore, .

37.(12i 5)3



Therefore, .

38.

SOLUTION:



39.( i)3

SOLUTION:



Therefore, .

40.(3 5i)4



. Now use De


Therefore, .

41.(2 + 4i)4



. Now use De


Therefore, .

42.(3 6i)4



is . Now use De


Therefore, .

43.(2 + 3i)2



is . Now use De


Therefore, .

44.

SOLUTION:



45.

SOLUTION:






SOLUTION:


6 = 1.









n = 1

n = 2

n = 3

n = 4

n = 5


.


47.sixth roots of i





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

Let n = 4


48.fifth roots of i

SOLUTION:





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.



SOLUTION:



is . Now write an



Let n = 1.

Let n = 2.

Let n = 3.






Let n = 1.

Let n = 2.



SOLUTION:




Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.






Let n = 1.



















n = 1

n = 2

n = 3







56.(1 i)(4 + 4i)









57.(3 + i)(3 i)




.





58.(4 + i)(3 i)






.



59.(6 + 5i)(2 3i)




.





60.( + 2i)(1 + i)







.



61.(3 2i)(1 + i)





.



.






(z0), z2 = f (z1), and so on.







eSolutions Manual - Powered by Cognero Page 2

9-5 Complex Numbers and De Moivre's Theorem


1.z = 4 + 4i



2.z = 3 + i



3.z = 4 6i



4.z = 2 5i



5.z = 3 + 4i



6.z = 7 + 5i



7.z = 3 7i



8.z = 8 2i









SOLUTION:4 + 4i



11.2 + i



12.4 i

SOLUTION:

4 i



3 (cos 0.34 + i sin 0.34).

13.2 2i



.

14.4 + 5i

SOLUTION:4 + 5i



15.2 + 4i



16.1 i

SOLUTION:

1 i



.

17.3 + 3i

SOLUTION:3 + 3i




18.10(cos 6 + i sin 6)

SOLUTION:




.

19.2(cos 3 + i sin 3)

SOLUTION:





20.

SOLUTION:





.

21.

SOLUTION:





i.

22.

SOLUTION:





.

23.

SOLUTION:





i.

24.3(cos 180 + i sin 180)

SOLUTION:




25. (cos 360 + i sin 360)

SOLUTION:





.


26.





27.5(cos 135 + i sin 135) 2 (cos 45 + i sin 45)




28.





29.2(cos 90 + i sin 90) 2(cos 270 + i sin 270)





30.






31.





quotient is .

32. (cos 60 + i sin 60) 6(cos 150 + i sin 150)





33.





34.5(cos 180 + i sin 180) 2(cos 135 + i sin 135)





35.





quotient is .


36.(2 + 2 i)6

SOLUTION:




Therefore, .

37.(12i 5)3



Therefore, .

38.

SOLUTION:



39.( i)3

SOLUTION:



Therefore, .

40.(3 5i)4



. Now use De


Therefore, .

41.(2 + 4i)4



. Now use De


Therefore, .

42.(3 6i)4



is . Now use De


Therefore, .

43.(2 + 3i)2



is . Now use De


Therefore, .

44.

SOLUTION:



45.

SOLUTION:






SOLUTION:


6 = 1.









n = 1

n = 2

n = 3

n = 4

n = 5


.


47.sixth roots of i





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.

Let n = 4


48.fifth roots of i

SOLUTION:





Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.



SOLUTION:



is . Now write an



Let n = 1.

Let n = 2.

Let n = 3.






Let n = 1.

Let n = 2.



SOLUTION:




Let n = 1.

Let n = 2.

Let n = 3.

Let n = 4.






Let n = 1.



















n = 1

n = 2

n = 3







56.(1 i)(4 + 4i)









57.(3 + i)(3 i)




.





58.(4 + i)(3 i)


Express each complex number in polar form. For 3 i, find the modulus r and argu

9-5 Complex Numbers and De Moivre's Theorem · Moiver ¶s Theorem to find the third power. 62/87,21...

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Transcript of 9-5 Complex Numbers and De Moivre's Theorem · Moiver ¶s Theorem to find the third power. 62/87,21...