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MAX SAT (the maximum satisfiability problem) is stated as follows: given a set of clauses with weights, find a truth assignment that maximizes the sum of the weights of the satisfied clauses. In this paper, we consider approximation algorithms for MAX SAT proposed by Goemans and Williamson and present a sharpened analysis of their performance guarantees. We also show that these algorithms, combined with recent approximation algorithms for MAX 2SAT, MAX 3SAT, and MAX SAT due to Feige and Goemans, Karloff and Zwick, and Zwick, respectively, lead to an improved approximation algorithm for MAX SAT. By using the MAX 2SAT and 3SAT algorithms, we obtain a performance guarantee of 0.7846, and by using Zwick's algorithm, we obtain a performance guarantee of 0.8331, which improves upon the performance guarantee of 0.7977 based on Zwick's conjecture. The best previous result for MAX SAT without assuming Zwick's conjecture is a 0.770-approximation algorithm of Asano. Our best algorithm requires a new family of 3/4-approximation algorithms that generalize a previous algorithm of Goemans and Williamson.