Summary: Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max \(k\)-Cut, Max-Sat, Max-Dicut, Max-bisection, \(k\)-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this paper, we give techniques to derandomize the above class of randomized algorithms, thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems. At the heart of our technique is the use of spherical symmetry to convert a nested sequence of \(n\) integrations, which cannot be approximated sufficiently well in polynomial time, to a nested sequence of just a constant number of integrations, which can be approximated sufficiently well in polynomial time.