Hello.
when you see the\theta which is unknown
i need to solve the following by Moivres theorem
r^3(cos 3\theta + i sin 3\theta )=1..how can i find the \theta ?
i thought that the Real part equals 1 as you can see...and the imaginary part is 0 since is not there...so
r^3 cos3\theta =1
r^3 sin3\theta =0
so to find the angle \theta i tried r^3 sin3\theta devided by r^3 cos3\theta equals 0 devided by 1...then the next step would be sin3\theta devided cos3\theta equals 0...therefore tan3\theta =0...from now on i am stuck...but i am not sure it was the right step to follow...any advices how to get ahead?
Since r^3 cos(3theta)= 1, r cannot be 0 and so from r^3 sin(theta)= 0, you have sin(theta)= 0. Either theta= 0 or theta= pi. If theta pi, then r^3 cos(theta)= -r^3= 1 which is impossible because r is never negative: theta= 0 so that r^3 cos(theta)= r^3= 1 and r= 1. That, is one solution to z^3= 1 is z= 1!!! If you found that difficult then finding the other two solutions is going to be really difficult.