Nasir al-Din al-Tusi was a Persian scholar who invented Tusi couples and worked with trigonometry outside of just the astronomical context. Tusi couples explain how two circular motions can create a linear motion. When a circle “rolls around” on the inside of a circle twice as big, each point on that circle would travel in a straight line. This notion later helped Copernicus in creating the heliocentric model because it demonstrates heliocentric models’ strength in explaining planetary motion. Unlike the sun, the moon or the stars, planets seem to wander around and sometimes even move backwards. Ptolemy tried to explain this phenomenon by stacking circles on top of circles, but that made the model much more complicated. Although al-Tusi only attempted to simplify the model to only having two circles, his work is even more effective for a heliocentric model. A heliocentric model naturally puts two orbits in opposite directions, the orbit of the Earth and the orbit of another planet. Using Tusi couples, the heliocentric model can then explain why planets move in the peculiar patterns we observe.
Al-Tusi also established trigonometry "as a mathematical discipline on its own right" (O'Connor). In his book "Treatise on the quadrilateral", Al-Tusi explored plane and spherical trigonometry, featuring theorems such as the sine formula for plane triangles. Previous to the book, trigonometry was only used in Astronomy to compute the positions of stars and planets on the celestial sphere. Al-Tusi's work established that trigonometry has uses outside of Astronomy and propelled the field forward.
Al-Tusi’s discovery was aided by his decision to join the Mongol troops as their scientific adviser. Al-Tusi lived in the time of Mongol conquest, when Genghis-Khan and his descendants conquered the Islamic empire. He was forced to move between different strongholds to do continue his work, but when the Mongols finally conquered the stronghold he is in, “he seemed enthusiastic in joining the victorious Mongols who appointed him as their scientific adviser” (O’Connor). Because of his revered scientific reputation, Hulegu, Genghis-Khan’s grandson and leader of the Mongol army, treated him well and offered enormous resources for his research. Hulegu gladly agreed when al-Tusi asked Hulegu to build an observatory after the army conquered Baghdad. Technicians from Islamic world and China came together to construct the observatory, and al-Tusi further furnished the observatory with books on science and astronomy. However, the support of the Mongols was not the only contribution to his work.
Al-Tusi's family background helped him study the basics of science and do scientific research that established him as an outstanding scholar. Al-Tusi's father was a jurist in the Twelfth Iman School. Although the school taught mainly religion, he also studied logic and physics with his uncle and mathematics with other teachers. Al-Tusi was then able to move from his hometown Tus to Nishapur which was an important center of learning. Al-Tusi had opportunities to learn about science and mathematics from experts of his time that helped him in his pursuit of Astronomy.
Reference: O'Connor, J.J. and Robertson, E.F. Al-Tusi_Nasir biography. University of St. Andrews, Scotland. 1999.