Taliaferro was my teacher in a freshman seminar in number theory and set theory, so I have some partiality for this translation, but it is widely acknowledged that the following translation by Toomer is superior.
Translated from the Greek, annotated, with a preface, introduction and appendices by G. J. Toomer. Revised reprint of the 1984 translation. With a foreword by Owen Gingerich. Princeton University Press, Princeton, NJ, 1998. xviii + 693 pp. ISBN: 0-691-00260-6. MR1656361 (99g:01004).
Van Brummelen, Glen, The mathematics of the heavens and the earth: the early history of trigonometry, Princeton: Princeton University Press, c2009. [Have book]
Ptolemy's work is discussed on pp. 68-93, first the creation of trigonometric tables, then the application of trigonometry to the motion of the planets. Not surprisingly, some earlier work on trigonometry was used by Ptolemy. An excellent introduction.
Jones, Alexander, "Ptolemy's mathematical models and their meaning," Mathematics and the historian's craft, 23--42, CMS Books Math./Ouvrages Math. SMC, 21, Springer, New York, 2005. MR2149142 (2006d:01005) has no information. [I have this volume]