There exists a huge variety of algorithms for approximating the roots of a complex polynomial \(p(z)\). But in several applications one is interested in the location of the roots with respect to a given set. For example, for the analysis of discrete linear time invariant systems one is interested in the location of the roots with respect to the unit circle. In an earlier paper, F. Locher [Numer. Math. 66, No. 1, 33-40 (1993; Zbl 0797.65041)] studied real polynomials using Sturm sequences and Chebyshev polynomials. In the present paper the authors are dealing with complex polynomials. The starting point is to look at the curve \(p (\exp (it))\), where \(t \in [0,2 \pi]\). First, they establish a representation of the real resp. imaginary part of \(p(\exp (it))\) with the help of Chebyshev polynomials. This leads to an Euclidean algorithm determining either the unimodal roots of \(p\) or by Sturm sequence arguments the number of roots of \(p\) in the unit disc.