We investigate the set of all distribution functions of two spe- cial sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterize the set of all distribution functions G(x n) for (x n)n≥1 = ({cos(αn)n})n≥1 and arbitrary α, where {x} denotes the fractional part of x. Furthermore we give a sufficient number-theoretic condition on α for which (x n)n≥1 = ({log(n) cos(αn)})n≥1 is uniformly distributed. Finally we cal- culate G(x n) in the case when 2 α π ∈ Q.