  For the class of sine polynomials $b_1\sin t+b_2\sin2t+...+b_N\sin Nt,\;
(b_N\not= 0),$ which are nonnegative on $(0,\pi)$, W. Rogosinski and G. Szeg\"o
derived, among other things, exact bounds for $|b_2|$ via the Luk\'acs
presentation of nonnegative algebraic polynomials and a variational type
argument for exact bounds, but they did not find the extremizers. Within this
algebraic framework, we construct explicit polynomials which attain these
bounds and prove their uniqueness.
  The proof uses the Fej\'er - Riesz representation of nonnegative
trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite
dimension, and Chebyshev polynomials of the second kind and their derivatives.
