Question

If \[z, \omega \ne 0.\] & \[|z|= |\omega|.\]
and \[\arg.(z)+\arg.(\omega)=\pi \space \space then \space \space z=...?\]

Answer 1

Answer is \[- \bar \omega\]

Answer 2

seems that they are conjugate of each other. what is 'w' BTW??

Answer 3

omega

Answer 4

i know ... i mean what is it's value?? like is it root of unity or something??

Answer 5

or does it have some specific value??

Answer 6

cube root of unity

Answer 7

if not specified we take cube root of unity.

Answer 8

\[ \omega^3 = 1 = e^{2 \pi} \implies \omega = \cos \left ( 2 \pi \over 3 \right ) - \sin \left ( 2 \pi \over 3 \right )\]

Answer 9

z is it's conjugate.

Answer 10

how we come to know that "z is it's conjugate." ?

Answer 11

i think there should be \[ \omega = \cos \left ( 2 \pi \over 3 \right ) - i \sin \left ( 2 \pi \over 3 \right )\]

Answer 12

i think a typing mistake:)

Answer 13

Nop ... +
sorry i did not pay attention.

Answer 14

but np:)

Answer 15

the argument of w is 2pi/3
the sum of argument is pi ... so find the argument of z

Answer 16

pi/3

Answer 17

so z = cos pi/3 + i sin pi/3

Answer 18

ya i think too:)

Answer 19

the modulus of z should be 1 .. so only value you can have is the above value.

Answer 20

Isn't there any logical & stepwise method?????

Answer 21

i don't know.

Answer 22

np:)
but thanx a lot for help:)