Trinitatis III is the third Sunday after Trinitatis. Bach wrote two cantatas for this Sunday.
The first one, Ich hatte viel Bekümmernis, BWV 21, is one of Bach''s most performed cantatas, and he also considered it a fine work since he performed it several times. Created originally in Weimar in 1714, it is partially based on a funeral cantata he wrote in 1713. It was performed not only on Trinitatis III, as Bach designated it as a cantata ""per ogni tempore"" - for various occasions.
When Bach went to Hamburg in 1720 to apply for the position of organist in the Jacobikirche he also performed this cantata. Little did he know that the title (""I had a lot of sorrow"") would prove visionary... Upon his return in Köthen he is told that his wife Maria Barbara passed away during his absence. Hat tip to the outstanding cantata website by Eduard Van Hengel - article in Dutch - https://www.eduardvanhengel.nl/werken/BWV_21 .
Ach Herr, mich armen Sünder, BWV 135, is the last of the four cantatas with which Bach opened his second Leipzig cantata cycle, the chorale cantatas. While the first liturgical cycle in Leipzig contains cantatas Bach already composed in Weimar, this second cycle was even more ambitious: a new cantata every Sunday (and other liturgical holidays), based on a protestant hymn, here a hymn by Cyriakus Schneegaß (1597), inspired by the parable of the shepherd and the lost sheep.