The giant planet orbiting tau Bootis was among the first extrasolar planets to be discovered through the reflex motion of its host star. It is one of the brightest known and most nearby planets with an orbital period of just a few days. Over the course of more than a decade, measurements of its orbital inclination have been announced and refuted, and have subsequently remained elusive until now. Here we report on the detection of carbon monoxide absorption in the thermal day-side spectrum of tau Bootis b. At a spectral resolution of R~100,000, we trace the change in the radial velocity of the planet over a large range in phase, determining an orbital inclination of i=44.5+-1.5 degrees and a true planet mass of 5.95+-0.28 MJup. This result extends atmospheric characterisation to non-transiting planets. The strong absorption signal points to an atmosphere with a temperature that is decreasing towards higher altitudes. This is a stark contrast to the temperature inversion invoked for other highly irradiated planets, and supports models in which the absorbing compounds believed to cause such atmospheric
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