A detailed spectroscopic study of the ISO-SWS data of the red giant Alpha Tau is presented, which enables not only the accurate determination of the stellar parameters of Alpha Tau, but also serves as a critical review of the ISO-SWS calibration. This study is situated in a broader context of an iterative process in which both accurate observations of stellar templates and cool star atmosphere models are involved to improve the ISO-SWS calibration process as well as the theoretical modelling of stellar atmospheres. Therefore a sample of cool stars, covering the whole A0 -- M8 spectral classification, has been observed in order to disentangle calibration problems and problems in generating the theoretical models and corresponding synthetic spectrum. By using stellar parameters found in the literature large discrepancies were seen between the ISO-SWS data and the generated synthetic spectrum of Alpha Tau. A study of the influence of various stellar parameters on the theoretical models and synthetic spectra, in conjunction with the Kolmogorov-Smirnov test to evaluate objectively the goodness-of-fit, enables us to pin down the stellar parameters with a high accuracy: Teff = 3850 +/- 70 K, log g = 1.50 +/- 0.15, M = 2.3 +/- 0.8 Msun, z = -0.15 +/- 0.20 dex, microturbulence = 1.7 +/- 0.3 km/s, 12C/13C= 10 +/- 1, abundance of C = 8.35 +/- 0.20 dex, abundance of N= 8.35 +/- 0.25 dex, abundance of O = 8.83 +/- 0.15 dex and the angular diameter is 20.77 +/- 0.83 mas. These atmospheric parameters were then compared with the results provided by other authors using other methods and/or spectra.
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