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Published 2010
This is the third of three articles on the History of Trigonometry.
Part 2 (Sections 5  7) can be found here. Part 1 (Sections 1  4) can be found here.
In the diagram above, the blue triangle with sides $a$, $b$, and $c$ represents the distances between stars on the inside of a sphere. The apex where the three angles $\alpha$, $\beta$, and $\gamma$ are marked, is the position of the observer. The blue curves are Great Circles on the sphere, and by measuring the angles, finding more accurate values for their functions, and assuming a value for $R$ the radius of the sphere, it became possible to find the greatcircle distances between the stars.
Abu alRayhan Muhammad ibn Ahmad AlBiruni (9731050) was an outstanding scholar reputed to have written over 100 treatises on astronomy, science, mathematics, geography, history, geodesy and philosophy. Only about twenty of these works now survive, and only about a dozen of these have been published.
AlBiruni's treatise entitled Maqalid 'ilm alhay'a (Keys to the Science of Astronomy) ran to over one thousand pages and contained extensive developments in on trigonometry. Among many theorems, he produced a demonstration of the tangent formula, shown below.

AlTusi invented a new geometrical technique now called the 'AlTusi couple' that generated linear motion from the sum of two circular motions. He used this technique to replace the equant used by Ptolemy, and this device was later used by Copernicus in his heliocentric model of the universe. 
AlTusi was one of the greatest scientists of Mediaeval Islam and responsible for some 150 works ranging from astronomy, mathematics and science to philosophy and poetry. 
Richard's early work was a series of instructions (canons) for the use of astronomical tables that had been drawn up by John Maudith, the Merton College Astronomer. Later he wrote an important work, the Quadripartitum, on the fundamentals of trigonometry needed for the solution of problems of spherical astronomy. The first part of this work is a theory of trigonometrical identities, and was regarded as a basis for the calculation of sines, cosines, chords and versed sines. The next two parts of the Quadripartitum dealt with a systematic and rigorous exposition of Menelaos' theorem. The work ends with an application of these principles to astronomy. The main sources of the work appear to be Ptolemy's Almagest, and Thabit ibn Qurra (826901 CE). 
His early work, Tabulae Eclipsium circulated in manuscript was not published until 1514, contained tables of his eclipse calculations that were based on the Alfonsine Tables. He calculated sines for every minute of arc for a radius of 600,000 units and he introduced the HinduArabic numerals in his tables. [See Note 9 below] 
Regiomontanus had become a pupil of Peuerbach at the University of Vienna in 1450. Later, he undertook with Peuerbach to correct the errors found in the Alfonsine Tables. He had a printing press where he produced tables of sines and tangents and continued Puerbach's innovation of using HinduArabic numerals.  
As promised, he finished Peuerbach's Epitome of the Almagest, which he completed in 1462 and was printed in Venice. The Epitome was not just a translation, it added new observations, revised calculations and made critical comments about Ptolemy's work. 
"For no one can bypass the science of triangles and reach a satisfying knowledge of the stars .... You, who wish to study great and wonderful things, who wonder about the movement of the stars, must read these theorems about triangles. Knowing these ideas will open the door to all of astronomy and to certain geometric problems. For although certain figures must be transformed into triangles to be solved, the remaining questions of astronomy require these books." [See Note 10 below]
The first book gives the basic definitions of quantity, ratio, equality, circles, arcs, chords and the sine function. Next come a list of axioms he will assume, and then $33$ theorems for right, isosceles and scalene triangles. The formula for the area of a triangle is given followed by the sine rule giving examples of its application. Books III to V cover the allimportant theory of spherical trigonometry. The whole book is organised in the style of Euclid with propositions and theorems set out in a logical hierarchical manner. This work, published in 1533 was of great value to Copernicus.
Regiomontanus also built the first astronomical observatory in Germany at Nuremburg with a workshop where he built astronomical instruments. He also took observations on a comet in 1472 that were accurate enough to allow it to be identified as Halley's Comet that reappeared 210 years later.
Regiomontanus died during an outbreak of plague in Rome in 1476.
Copernicus wrote a brief outline of his proposed system called the Commentariolus that he circulated to friends somewhere between 1510 and 1514. By this time he had used observations of the planet Mercury and the Alfonsine Tables to convince himself that he could explain the motion of the Earth as one of the planets. The manuscript of Copernicus' work has survived and it is thought that by the 1530s most of his work had been completed, but he delayed publishing the book.
His student, Rheticus read the manuscript and made a summary of Copernicus' theory and published it as the Narratio Prima (the First Account) in 1540. Since it seemed that the Narratio had been well accepted by colleagues, Copernicus was persuaded to publish more of his main work, and in 1542 he published a section on his spherical trigonometry as De lateribus et angulis traingulorum (On the sides and angles of triangles). Further persuaded by Rheticus and others, he finally agreed to publish the whole work, De Revolutionibus Orbium Coelestium (The Revolutions of the Heavenly Spheres) and dedicated it to Pope Paul III. It appeared just before Copernicus' death in 1543. [See Note 11 below]
Rheticus had facilitated the publication of Copernicus' work, and had clearly understood the basic principles of the new planetary theory.
In 1551, with the help of six assistants, Rheticus recalculated and produced the Opus Palatinum de Triangulis (Canon of the Science of Triangles) which became the first publication of tables of all six trigonometric functions. This was intended to be an introduction to his greatest work, The Science of Triangles.
When he died his work was still unfinished, but like Copernicus, Rheticus acquired a student, Valentinus Otho who supervised the calculation (by hand) of some one hundred thousand ratios to at least ten decimal places filling some 1,500 pages. This was finally completed in 1596. These tables were accurate enough to be used as the basis for astronomical calculations up to the early 20th
century.

The term trigonometry is due to Pitiscus and as first appeared in his Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, published in 1595. A revised version in1600 was the Canon triangularum sive tabulae sinuum, tangentium et secantium ad partes radii 100000 (A Canon of triangles, or tables of sines tangents and secants with a radius of 100,000 parts.) The book shows how to construct sine and other tables, and presents a number of theorems on plane and spherical trigonometry with their proofs. [See Note 11 below] 
However, soon after Rheticus' Opus Palatinum was published, serious inaccuracies were found in the tangent and secant tables at the ends near $1^\circ$ and $90^\circ$. Pitiscus was commissioned to correct these errors and obtained a manuscript copy of Rheticus' work. Many of the results were recalculated and new pages were printed incorporating the corrections. Eventually, Pitiscus published a new work in 1613 incorporating that of Rheticus with a table of sines calculated to fifteen decimal places entitled the Thesaurus Mathematicus.
By the beginning of the seventeenth century, the science of trigonometry had become a sophisticated technique used in calculating more and more accurate tables for use in astronomy and navigation, and had been instrumental in fundamentally changing man's concept of his world.
1. See Part 1 section 3 on the Sulbasutras.
2. See Note 4 in Part1.
The use of the capital S in Sine is to show that the radius of the circle used is not unity, or the same as $\sin\theta$ in our system, but could be an arbitrary length R. This means that Sin$\theta$ is equal to R sin$\theta$ . In the Indian texts, different astronomers took different values for R, and in most cases the value had to be deduced from the context.
3. The advantage of the 'versine' (or reversed sine) is that it's value is always positive and so its logarithm is defined everywhere (except at $0^\circ$ and $180^\circ$). A positive logarithm was necessary when calculations had to be done using tables. The most important use was in navigation, for calculating the distance between two points on a sphere. The perpendicular distance from the mid point of a chord to a curve is still used as a measure of 'deviation from straightness', for example, by railway engineers. It is used also in optics for measuring the curvature of lenses and mirrors, where he versine is sometimes called the sagitta from the Latin for arrow.
4. Compare the sine curve from $0^\circ$ to $180^\circ$ with $y = a(x \pi/2)^2 + c$. By adjusting the values of $a$ and $c$, it is possible to produce a curve of 'best fit' inside the sine curve. You can obtain a remarkably good fit for $0 < x < \pi$.
5. The Hindu word jiya for the sine was adopted by the Arabs who called the sine jiba. Eventually jiba became jaib and this word actually meant a 'fold'. When Europeans translated the Arabic works into Latin they translated jaib into the word sinus meaning a fold in Latin. In his Practica Geometriae (1220) Fibonacci uses the term sinus rectus arcus which soon encouraged the universal use of the word sine.
When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.