TITLE: When the trig functions moved from the right triangle to the unit circle?
QUESTION [26 upvotes]: I have to write a paper about the unit circle and I'm trying to uncover some of its origins. 
Also, when the trig functions were expanded to angles greater than $90^{\circ}$ and what was the rationale behind it? Also, why mirror the right triangle along the axis instead of just moving the base of the right triangle to the next axis $(+x\to+y\to-x\to-y\to+x)$?
I had no luck trying to find that online, and all the books that I looked into would jump right into the practical part of it.

REPLY [9 votes]: @Luiz:   Early trigonometry was based on a large circle (to reduce fractions). Sines and cosines were line segments in that circle. Tables listed the lengths of those lines for angles between 0 and 90 degrees. For a detailed history of trigonometry up to 1550, see The Mathematics of the Heavens and the Earth: The Early History of Trigonometry by Glen van Brummelen, Princeton University Press, 2009. 
The year 1748 was a watershed year for the history of trigonometry, for in that year Euler published his Introductio in analysin infinitorum (English translation by John Blanton, 
Introduction to Analysis of the Infinite, Springer 1988). Euler changed the objects studied in calculus from curves to functions. In Chapter VIII, he introduces trigonometric FUNCTIONS on the unit circle. If you read this chapter, it all seems ho-hum; that is because we have adopted Euler's approach. At the time it was revolutionary. 
This change allowed Euler to differentiate the trigonometric functions. See his Institutiones calculi differentialis, 1755, section 201. (English translation by Blanton, Foundations of Differential Calculus Springer 2000). Amazingly, although both Newton and Leibniz had power series for sine and cosine, they did not differentiate; there was no curve, so no tangent line to find. 
Euler's approach was quickly adopted by the research mathematicians and led to new fields of mathematics such as Fourier series and Cantor's work on set theory.