We prove a Polya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" I subset of F-p, provided (p(1/2) log p)/|I| = o(1). Applications include density results for irreducible trinomials in F-p[x], i.e. the number of irreducible polynomials in the set {f(x) = x(d) + a(1)x + a(0) is an element of F-p[x]}a(0) is an element of I-0,I- a(1) is an element of I-1 is similar to |I-0|.|I-1|/d provided |I-0| > p(1/2+is an element of), |I-1| > p(is an element of), or |I-1| > p(1/2+is an element of), |I-0| > p