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In this paper, we study the theory of orthogonal trigonometric polynomials(OTP). We obtain asymptotics of OTP with positive and analytic weight functionsby Riemann-Hilbert approach and find they have relations with orthogonalpolynomials on the unit circle (OPUC). By the relations and the theory of OPUC,we also get four-terms recurrent formulae, Christoffel-Darboux formula and someproperties of zeros for orthogonal trigonometric polynomials.

Orthogonal Trigonometric Polynomials: Riemann-Hilbert Analysis and Relations with OPUC