Matrix Spectra with Troublesome Trigonometric Term

Hello!

I’m trying to find the spectra (eigenvalues) of matrix

I start by generating its characteristic polynomial which yields:

Applying the Euler Formula:

At this point, I want to extract roots, the solutions for eigenvalues . One, from the LH term, is clearly . But that in the middle is stopping me from factorizing it via any way I can work out. I expect, since A was orthogonal, that the other two roots will be a complex conjugate pair but am not 100% on that – in any case, I can’t see how I can legally break that trig term into a complex one.

Any ideas?

Cheers!

Alexandicity