Ok so the problem is as follows:
Find all real number solutions y of:
e^(iy) + e^(2iy) + e^(3iy)=0
Firstly I tried letting e^(iy) equal some variable X. Then:
X + X^2 + X^3=0 ==> X(1 + X + X^2)=0.
I used the quadratic formula on (1 + X + X^2) to get X=-1/2 ± (sqrt(3)/2)*i
So e^(iy)=-1/2 ± (sqrt(3)/2)*i
Where do I go from here? Am I even on the right track? Any help much appreciated!
Idea and Plato have used different methods but what you have done is perfectly valid.
Of course, , in the first case, and . To find that logarithm, write the numbers in "polar" or exponential form: and so that we have so that and then .
The logarithm, like most functions extended to the complex numbers, is not "single valued". , for any integer k, so that . that is, we can add any multiple of to the solution above.
When I plug in pi/3 into the polar formula (cosy+cos2y+cos3y+i(siny+sin2y+sin3y)=0), the result is not zero. What am I missing? I've tried working with the cosines and sines separately, making them equal to zero, but I'm not getting anywhere with that either.