Given the rotation matrix (cos x -sin x)

sinx cosx

(supposed to by a 2x2 matrix)

I have to find the complex eigenvalues and eigenvectors when sin x not=0

Dont know if i'm going the right way but I have p(L) = det(cosx-L -sin x)

sin x cosx-L

= (cosx-L)^2 - (-sinx)^2 = (cosx)^2-2Lcosx+L^2+(sinx)^2

=L^2-2Lcosx+1

I dont know where to go from here, thanks