Chebyshev Root

The roots of the Chebyshev polynomial $T_N(x)$ are $$\xi_k = \cos\!\left(\frac{(2k+1)\pi}{2N}\right), \quad k = 0, 1, \ldots, N-1.$$ These are $N$ distinct real numbers in the interval $(-1, 1)$. They satisfy $T_N(\xi_k) = 0$ by the defining relation $T_N(\cos\theta) = \cos(N\theta)$.