Al-Battani

Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī () (Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was a Syrian Arab astronomer, and mathematician. He introduced a number of trigonometric relations, and his "Kitāb az-Zīj" was frequently quoted by many medieval astronomers, including Copernicus.
Often called the "Ptolemy of the Arabs", al-Battani is perhaps the greatest and best known astronomer of the medieval Islamic world.
Life.
Little of al-Battānī's life is known other than his birthplace in Harran near Urfa, in Upper Mesopotamia, (today in Turkey) and his father's fame as a maker of scientific instruments. Jabir ibn Sinan al-Harrani was likely this famous instrument maker, although this is not something that has been proven. Ibn Khallikaniaani expresses ignorance on the question of his Muslim faith, and points out that his epithet "aṣ-Ṣabi’" suggests possible Sabian-sect ancestry. Although his ancestors were likely of the Sabian-sect, he was a Muslim, as his name was Muhammad, with his kunya being Abu Abd Allah. The "Encyclopædia Britannica Eleventh Edition" stated that he had noble origins as an Arab prince, but traditional Arabic biographers make no mention of this. Between 1877 and 1918/19, over a forty-year period, he lived in the ancient city of Raqqa, in north central Syria, recording his astronomical observations. He died at Qasr al-Jiss, which is located near Samarra, Iraq. He was returning from Baghdad to al-Raqqa after resolving a dispute on behalf of the people of al-Raqqa.
Astronomy.
One of al-Battānī's best-known achievements in astronomy was the determination of the solar year as being 365 days, 5 hours, 46 minutes and 24 seconds, which is only 2 minutes and 22 seconds off.
Another of al-Battānī's accomplishments is that he concluded how an annular solar eclipse occurs. He did this by observing that the radius between the Earth and the Sun changes throughout the year. This led him to arrive to the conclusion that when the Sun is farthest from the Earth, an annular solar eclipse occurs. He was the first to make this observation and inference.
The twelfth-century Egyptian encyclopedist al-Qifṭī, in his biographical history "Ta’rīkh al-Ḥukamā’", mentions al-Battānī’s contribution to advances in astronomical observation and calculations based on Ptolemy’s "Almagest".
Al-Battānī amended some of Ptolemy's results and compiled new tables of the Sun and Moon, long accepted as authoritative. Some of his measurements were more accurate than ones taken by Copernicus many centuries later and some ascribe this phenomenon to al-Battānī's location lying closer to the equator such that the ecliptic and the Sun, being higher in the sky, are less susceptible to atmospheric refraction. Al-Battānī observed that the direction of the Sun's apogee, as recorded by Ptolemy, was changing. 
Al-Battānī's work was instrumental in the development of science and astronomy. Copernicus, in his book that initiated the Copernican Revolution, the "De Revolutionibus Orbium Coelestium", quotes his name no fewer than 23 times, and also mentions him in the "Commentariolus". Tycho Brahe, Riccioli, Kepler, Galileo and others frequently cited him or his observations. He also influenced Jewish rabbis and philosophers such as Abraham ibn Ezra, Abraham bar Hiyya, and Maimonides. His data is still used in geophysics.
The major lunar crater Albategnius is named in his honor.
Mathematics.
In mathematics, al-Battānī produced a number of trigonometrical relationships:
He also solved the equation sin "x" = "a" cos "x" discovering the formula:
He gives other trigonometric formulae for right-angled triangles such as:
Al-Battānī 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°.
Using these trigonometrical relationships, Al-Battānī created an equation for finding the qibla, which Muslims must face in each of the five prayers they practice everyday. The equation he created did not give accurate directions, as it did not take into account the fact that Earth is a sphere. The relationship Al-Battānī used was fairly precise when a person is in Mecca, or close to Mecca, but resulted in more and more inaccurate results as one gets more distant from Mecca. However, it was still a widely used method at the time. The equation is as follows:
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In popular culture.
A ship in "" is named after Al-Battānī, known as the USS "Al-Batani", on which Kathryn Janeway originally served.