Question

De Moivres theorem
(5-4i)^10
no need to solve this
the question is if i'm expressing this as r^10(cos(10a)+isin(10a))

how do i find a?
is it just tan-1(-4/5)? or do i have to think about the quadrants?

Answer 1

you have to think about quadrants

Answer 2

range of arctan is from -pi/2 to pi/2 so arctan will not give the right argument if you are in quadrant 2 or 3

Answer 3

actually in this case you are ok because you are in quadrant 4 so you can just use arctan(-4/5)

Answer 4

but 4th quadrant arguments ar expressed as - tan(|y/x|)^-1

Answer 5

what if it was (-5-4i) and i was in quadrant 3?

Answer 6

not sure what you mean. arctan(-4/5) is the number (angle) between -pi/2 and p/2 whose tangent if -4/5. this

Answer 7

you are in quadrant three right? so if you take arctan(b/a) you get something in quadrant 1, directly across the circle.
add pi to your answer in that case

Answer 8

in third quadrant general argument representation is tan(|y/x)^-1 -pi

Answer 9

i am user vicky007 is correct as well. there is no unique answer.

Answer 10

@vicky your argument will be negative right?

Answer 11

yes

Answer 12

if it give the same result, should i just do tan-1(y/x) in each case and not bother with quadrants?

Answer 13

@tsolrm the long and the short of it is since the range of arctan is restricted to -pi/2 to pi/2 you do have to check the quadrant.

Answer 14

oh no not the same result!

Answer 15

easy example take
\[-1+i\]

Answer 16

you are in quadrant 2. but if just use
\[\tan^{-1}(\frac{b}{a})\] you get
\[tan^{-1}(-1)=-\frac{\pi}{4}\]

Answer 17

and that is definitely wrong.

Answer 18

especially since
\[\sqrt{2}(\cos(-\frac{\pi}{4})+i\sin(-\frac{\pi}{4}))=1-i\] not \[-1+i\]

Answer 19

i've looked at past papers and they've never sticked a minus sign in the demoivres question it's always positive and therefore it's in quadrant 1. so i got used to doing tan-1(y/x) and that's it. so now i'm terrified that there will be a minus sign tomorrow

Answer 20

but what you've said really cleared this up. thank you so much

Answer 21

i am not sure what you mean by "it is alway positive" , but good luck, and yes make sure to know what quadrant you are in!

Answer 22

i mean that they've always presented it as (something + i(something)) ^ something
so the x and y have always been positive :)

Answer 23

ooh well hope they do tomorrow too, but it is not such a big deal so long as you remember to check

Answer 24

i'll need a miracle to pass this exam haha
thanks again!