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The birth of trigonometry lies in astronomy, one of the sciences studied most vigorously by the Muslims, particularly due to its relevance in determining the exact times of the ritual prayer. But even before the Muslims, Greek astronomers were calculating the sides and angles of certain triangles, given the value of the remaining sides or angles, in order to understand the motions of the sun, the moon, and the then known five planets.
Motivated by questions such as the position of the sun, moon, and planets, the Greek composed tables and rules that enabled geometric problems to be tackled. The most thorough treatment of the subject is found in the work Almagest by Ptolemy, who was an astronomer working in Alexandria in the early part of the second century C.E. Ptolemy’s treatise reached European scholars via Muslim hands, who translated the original Greek title, which meant The Great Arrangement, into more succinct terms to product Al-Majisti, simply meaning The Greatest.
Astronomers from late antiquity would draw principally upon a table found in Book I of Almagest, which was called A Table of Chords in a Circle, to solve all their plane trigonometric problems. For arcs at angles in increments of half a degree up to 180 degrees, the table gives the lengths of the chords subtending the angles in a circle of radius 60 units.
In his work The Transversal Figure, 13th century Muslim astronomer Al-Tusi explains how this table of chord lengths was employed to solve problems relating to right angled triangles. Al-Tusi made the crucial observation that established the blink between triangles and arcs of circles: Any triangle may be inscribed in a circle; therefore, its sides maybe viewed as the chords subtending the arcs opposite the angles of the triangle.
A chain of Muslim scholars had already laid the foundations of trigonometry before the 10th century, paving the way for Al-Tusi to collect, organise, and elaborate on their contributions. It was Al-Battani, born in Harran, Turkey, who was one of the most influential figures in trigonometry. He is considered to be one of the greatest Muslim astronomers and mathematicians, eventually dying in Samarra, now in Iraq, in 929. Al-Battani was the first to use the expressions sine and cosine, defining them as lengths, rather than the ratios we know them as today. The tangent was referred to by Al-Battani as the “extended shadow,” the shadow of a notional horizontal rod mounted on a wall. In the 11th century, Al-Biruni defined the trigonometric functions of tangent and cotangent, which were inherited in a tentative form from the Indians.
Al-Biruni, born in 973, was among those who laid the foundation for modern trigonometry; Al-Khwarizmi, born in 780, developed the sine, cosine, and trigonometric tables, which were later translated to the West.