A study of 1095 objects located towards the Sagittarius dwarf spheroidal (Sgr dSph)has been undertaken. 19 of these stars were classified as C-type stars, 341 were classifiedas M-type stars and 733 were classified as K-type stars (one of the objects wasfound to be the galaxy 2MASX J18521545-2948214). These stars belong to two radialvelocity populations; the Galactic stars with an average radial velocity peak of -13 kms-1 and a dispersion of 53 km s-1, and the Sgr dSph stars with an average radial velocitypeak of 141 km s^-1 and a dispersion of 15 km s^-1. The determined radial velocitiesare accurate to ± 10 km s^-1. Stars with (J - K) values placing them on the observedgiant branch are likely to belong to the Sgr dSph.Five observed stars are suggested to belong to the metal-poor population of the SgrdSph, and from these observations it is determined that the metal-poor population ofthe Sgr dSph is 1/11 the size of the bulk population.The onset of thermal pulses is found to coincide with the luminosity of the RGBtip. Four carbon stars are located at this luminosity and it is suggested that, rather thanstars in the Sgr dSph becoming carbon-rich very early on the TP-AGB, these stars infact are in the helium burning phase of the thermal pulse cycle.An average carbon-rich lifetime of between 1.07 and 2.26 × 10^5 years is determinedand it is calculated that carbon stars in the Sgr dSph may only undergo one or twothermal pulses before high mass-loss rates terminate their AGB lifetimes. Total mass lossrates of 10^-5 M⊙ yr^-1 are found for the most reddened carbon stars.An approximate C/M ratio for the Sgr dSph is determined to be 0.54, indicatingthat carbon stars are produced by the bulk population.