He also made important contributions to pure mathematics; he was the first to replace the use of Greek chords by singes, with a clear understanding of their superiority in mathematical calculations, computed a table of cotangents and formulated certain propositions in spherical trigonometry. Battānī produced a number of trigonometrical relationships:
He also used al-Marwazi's idea of tangents ("shadows") to develop equations for calculating tangents and cotangents, compiling tables of them. He also discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants, which he referred to as a "table of shadows" (in reference to the shadow of a gnomon), for each degree from 1° to 90°.
One of his best-known achievements in astronomy was the determination of the solar year as being 365 days, 5 hours, 46 minutes and 24 seconds.
During his observations for his improved tables of the Sun and the Moon, he discovered that the direction of the Sun's eccentric was changing, which in modern astronomy equivalent to the Earth is moving in an elliptical orbit around the Sun. His times for the new moon, lengths for the solar year and sidereal year, prediction of eclipses, and work on the phenomenon of parallax, carried astronomers "to the verge of relativity and the space age.