Question

How would you go about doing this question without a calculator. I know I have to know my unit circle for this. but yea how would one go about solving this problem without a calculator?

Answer 1

So for the first question I got into polar form using oilers formula
\[e ^{\frac{ 4\pi i }{ 3 }}\] Its kind of hard to see that but its says e^(4pi(i)/3)

Answer 2

\[(e ^{\frac{ 4\pi \iota }{ 3}})^{111}\]

Answer 3

how would I get the exact value for this in the form a + bi

Answer 4

@freckles

Answer 5

put it into trig form

Answer 6

at least thats my first thought

Answer 7

bah, im not real sure what the question is asking for i spose ...

Answer 8

its asking to raise it to the power of 111 and simply if it to put into a + bi form

Answer 9

(r,t)^n = (r^n, nt) right?

Answer 10

r,t being the ordered pair for

(r,t) -> r(cos(t) + i sin(t))

Answer 11

yes

Answer 12

it appears r = 1, since its the unit circle, what is t?

Answer 13

4pi/3 I believe

Answer 14

something about 60 yes, and the other is something about 45

Answer 15

111 (4pi/3) ... reduces to what in periodic terms?

Answer 16

148pi?

Answer 17

can we simplify that to between 0 and 2pi?

Answer 18

thats where im getting stuck, how would you do that?

Answer 19

2k = 148, when k=74

so 74 revolutions gets us to 148 right?

Answer 20

yes

Answer 21

then 148 pi is just a large version of 2pi

Answer 22

which is just a slightly larger version of 0

Answer 23

cos(0) + i sin(0)

Answer 24

1+0i

Answer 25

444pi/3 = 148pi = k*2pi

Answer 26

yes

Answer 27

what I dont understand is that how did you know 148pi is equivalent to 2pi?

Answer 28

becuase a circle repeats itself every 2pi ...

so 2pi = 4pi = 6pi = ...

how many times does 2pi go into 148pi? how many times does the circle turn around on itself?

Answer 29

74 times

Answer 30

then 74 revolutions just gets us back to the start of the circle

Answer 31

ok so say the angle was 195pi how would you get the equivalent angle between 0 and 2pi?

Answer 32

divide it by 2pi, whats the remainder?

Answer 33

1/2

Answer 34

then 1/2 of 2pi is pi

Answer 35

if its not a full rotation of 2pi, then its a partial rotation ...

Answer 36

after 97 rotation, we have 1/2 a rotation

Answer 37

so an angle of 873 pi would also be equivalent to pi?

Answer 38

cuz 873/2= 436.5. and remainder is 0.5

Answer 39

spose we have 23.3247 rotations of 2pi ...

then we have 23 + 0.3247 rotations ...