Symmetric part of an orthogonal matrix ($\lambda (\mathrm{sym}A)=\cos \theta \text{when}\lambda (A)=e^{i\theta }$)

For a real orthogonal matrix $A$, $A^{\mathsf{T}}=A^{-1}$. If $Av=\lambda v$ for a nonzero complex eigenvector, then $A^{\mathsf{T}}v={\lambda }^{-1}v$. Therefore the same eigenvector has eigenvalue $(\lambda +{\lambda }^{-1})/2$ for the symmetric part of a matrix $(A+A^{\mathsf{T}})/2$. Orthogonality implies $|\lambda |=1$, so writing $\lambda =e^{i\theta }$ gives $\cos \theta$. The complete complex orthonormal eigenbasis of an orthogonal matrix shows that all the eigenvalues of its symmetric part lie in $[-1,1]$.