# When did trigonometry start using negative numbers?

I'm asking this question looking at the unit circle, and thinking that greek mathematicians didn't use negative numbers. Maybe that can give enough insight into what I'm asking?

Seventeenth century mathematicians were still hesitatnt to use negative numbers even when it came to Cartesian coordinates. According to Tannery:

"One frequently attributes wrongly to Descartes the introduction of the convention of reckoning coordinates positively and negatively, in the sense in which we start them from the origin. The truth is that in this respect the Geometrie of 1637 contains only certain remarks touching the interpretation of real or false (positive or negative) roots of equations.

"...If then we examine with care the rules given by Descartes in his Geometrie as well as his application of them, we notice that he adopts as a principle that an equation of a geometric locus is not valid except for the angle of the coordinates (quadrant) in which it was established, and all his contemporaries do likewise. The extension of an equation to other angles (quadrants) was freely made in particular cases for the interpretation of the negative roots of equations; but while it served particular conventions (for example for reckoning distances as positive and negative), it was in reality quite long in completely establishing itself, and one cannot attribute the honor for it to any particular geometer." [quoted from Cajori, History of Mathematics, p.175-6]

Even Newton and Leibniz, although they knew the power series for sine and cosine, did not treat them as functions of a real variable. The view of them as functions (in the analytic sense of 18th century) derives from Euler's Introductio in Analysin Infinitorum (1748), chapter VIII, where he uses not only negative but even complex values as arguments, adopts the $$\pi$$ symbol, and introduces the unit circle conventions. There is an English translation by Blanton. At the beginning of the chapter Euler writes:

"After having considered logarithms and exponentials, we must now turn to circular arcs with their sines and cosines. This is not only because these are further genera of transcendental quantities, but also since they arise from logarithms and exponentials when complex values are used."

Later he notes explicitly that formulas like $$\sin\left(\frac{4n+1}2\pi+z\right)=+\cos z$$ "hold whether $$n$$ is a positive or a negative integer".

• Thanks for this. Were mathematicians before Euler able observe the repetitive [ trying to avoid using cyclical! ] nature of sine & cosine while disregarding negative numbers? – A Citizen of The World Jan 27 '20 at 22:55
• @ACitizenofTheWorld It was not really something to observe, it is a convention, and an artificial one in either geometric or computational contexts where they were used. Periodicity only makes sense when one treats them as functions on the real line, or its positive part at least, which was done by Euler. – Conifold Jan 28 '20 at 0:18