Question

D

Answer 1

so you would start by using de moivres to re-write the problem r^n (cos(n*theta) + isin(n*theta)) then use the polar to rectangular conversion

Answer 2

yeah I got something other than D

Answer 3

still there?

when you re-write the formula with de moivres you get

[sqrt(2)] ^ 6 * cos(6*20) for the x-coordinate, simply evaluate that to see what it would be in rectangular form

Answer 4

-4

Answer 5

B

Answer 6

good and there's only one choice where the x-coordinate is -4

Answer 7

good, B

Answer 8

D

Answer 9

need to re-teach myself this ;_; might take a bit

Answer 10

@Shadow have you learned this yet? (nth roots)

Answer 11

ah no. But it's summer and I've already turned off my brain so who knows.

Answer 12

what I have so far: you can find the angle of the original expression by taking arctan(y/x) as usual, that gives arctan(-sqrt(3)) = -60 degrees, converting this to a positive angle gives 300 since we must add 360 degrees

from there you're supposed to divide by the root # to get 75 degrees but that's not one of the choices :S

I guess I can try plugging them into a calculator to see which one works

Answer 13

C mahybe

Answer 14

had to use a calculator for this tbh but I end up with theta = 30?

Answer 15

SO D

Answer 16

Ugh thank you , you really didnt have to do that

Answer 17

B

Answer 18

good attempt but not quite

first you'd have to convert to polar coordinates

r = sqrt(x^2 + y^2) = ?

Answer 19

remember that x is the first term sqrt(3) and y is the the coefficient of i (1)

Answer 20

2

Answer 21

good, and if we look at our de moivre formula we can see that r^n is the coefficient where n is the exponent, so the coefficient in front must be 2^3 or 8, not 2

Answer 22

C

Answer 23

then you need to find arctan(y/x) to find the angle then multiply by 3 to find the angle

arctan(1/sqrt(3)) gives theta = pi/6, multiply that by 3 to get pi/2, so yes C

Answer 24

B and C

Answer 25

C yes, still double checking B

Answer 26

hm, not quite

for the angle you can find possible angles by taking

(theta + 2*pi*k)/n

where theta is the original angle (pi/3), n is the root number (5), and k is a constant integer from 0,1,2, etc.

what happens if you plug in k = 0?

Answer 27

0

Answer 28

(pi/3 + 0*pi)/5 = ?

Answer 29

pi/15 so A

Answer 30

good, A + C only

Answer 31

C

Answer 32

hm. not quite. "sixth root of unity" means that number can be raised to the power of 6 and will result in 1 (not -1)

so which answer choice, when raised to the sixth power, gives 1 as the result?

Answer 33

A

Answer 34

check your calculations again

be very careful with negative signs and parentheses

Answer 35

-1

Answer 36

good so B

Answer 37

that's the same question as before

Answer 38

five

Answer 39

T

Answer 40

oh whoops got distracted by something, yes I believe those are both correct

Answer 41

F

Answer 42

this one is actually true, looking at the formula gives us

r^n and 2theta as the things to plug in, giving only one possible solution

Answer 43

technically you could do the angle + 2pi thing but for all intents and purposes they're equivalent angles

Answer 44

try plugging in a few n values for 1^(1/n), you'll see that you will get the same result each time

Answer 45

1

Answer 46

good, 1 is your solution

Answer 47

C

Answer 48

excellent