Complexos I

Question 1

i^n = i^k

Onde k é o ____ da ____ de n por ____.

Answer 1

Onde k é o resto da divisão de n por 4.

Question 2

(1 + i) ^ 2 =

Answer 2

2i

Question 3

(1 - i) ^ 2 =

Answer 3

-2i

Question 4

(1 + i) / (1 - i) =

Answer 4

i

Question 5

1 / i =

Answer 5

-i

Question 6

__

z . z =

Answer 6

z | ^ 2

Question 7

______

(z + w) =

Answer 7

_ _

z + w

Question 8

_____

(z . w) =

Answer 8

_ _

z . w

Question 9

_____

(z / w) =

Answer 9

_ _

z / w

Question 10

Expresse de quatro outras formas o número i.

Answer 10

[(1 + i) ^ 2] / 2

[(1 - i) ^ 2] / (-2)

(1 + i) / (1 - i)

-1 / i

Question 11

z | ^ 2 =

Answer 11

__

z . z

Question 12

_ _

z + w =

Answer 12

______

z + w

Question 13

_ _

z . w =

Answer 13

_____

z . w

Question 14

_ _

z / w

Answer 14

_____

z / w

Question 15

Cis(x) =

Answer 15

cos(x) + i.sen(x)

Question 16

1 + cis(x) =

Answer 16

2 . cos(x / 2) . cis(x / 2)

Question 17

1 - cis(x) =

Answer 17

-2 . i . sen(x / 2) . cis(x / 2)

Question 18

Cos(x) + i.sen(x) =

Answer 18

cis(x)

Question 19

2 . cos(x / 2) . cis (x / 2) =

Answer 19

1 + cis(x)

Question 20

-2 . i . sen(x / 2) . cis (x / 2)

Answer 20

1 - cis(x)

Question 21

O complexo z = 1 + cis(x) possui que módulo e que argumento?

Answer 21

1 + cis(x) = 2 . cos(x / 2) . cis (x / 2)

Logo o módulo é 2 . cos(x / 2)
e o argumento é x / 2.

Question 22

Cis(x) + cis(-x)

Answer 22

2.cos(x)

Question 23

Cis(x) - cis(-x)

Answer 23

2.i.sen(x)

Question 24

[ e^(i.x) + e^(-i.x) ] / 2

Answer 24

Cos(x)

Question 25

[ e^(i.x) - e^(-i.x) ] / ( 2.i )

Answer 25

Sen(x)

Question 26

2.cos(x)

Answer 26

Cis(x) + cis(-x)

Question 27

2.i.sen(x)

Answer 27

Cis(x) - cis(-x)

Question 28

z = r.cis(x)
w = s.cis(y)

Que conclusões tiramos se z = w?

Answer 28

r = s.
x = y + 2.k.pi.
Onde k é um inteiro.

Question 29

Se z^n = r.cis(x),

qual é o valor de z?

Answer 29

z = sqrt(r).cis( (x + 2.k.pi) / n ).

Onde k é um inteiro e 0 < n.