Eigenvalues of a rotation

Characteristic polynomial.
$$\det(R_\theta-\lambda I)=\lambda^2-2\cos\theta\cdot\lambda+1.$$
Discriminant: $\Delta=4\cos^2\theta-4=-4\sin^2\theta$.

Case $\sin\theta=0$ (θ=0 or π).
- $\theta=0$: $R=I$, $\lambda=1$ (double).
- $\theta=\pi$: $R=-I$, $\lambda=-1$ (double).

Case $\sin\theta\neq0$. Complex eigenvalues:
$$\lambda_{\pm}=e^{\pm i\theta}=\cos\theta\pm i\sin\theta.$$

Conclusion. A rotation of angle $\theta\notin\{0,\pi\}$ has no real eigenvalues but complex eigenvalues $e^{\pm i\theta}$. This reflects the fact that a non-trivial rotation has no invariant line.