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In the work of early physicists like Newton, everything is explained in terms of cumbersome (in today's standards) geometry. They don't talk about "cosines" of certain angle, but about proportions between the sides of triangles. But aren't sines and cosines known since antiquity? So why Newton, Copernicus, etc didn't make use of them.

When and how did trigonometric functions become mainstream?

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Trigonometric functions became "mainstream" since the publication by Ptolemy (II AD) of trigonometric tables. To be sure he did not use our modern sine and cosine, but a single trigonometric function, the chord ($=2\sin(t/2)$). Modern definitions of sine and cosine were introduced by Indian mathematicians (Surya Siddhanta (V century AD), and reached Europe through Middle Eastern mathematicians. After the fall of Constantinople, this knowledge quickly spread in Europe. The first treatises on trigonometry in Europe were written by Peuerbach and Johann Muller (Regiomontanus) in 1464. Copernicus of course used trigonometry extensively, as well as Kepler.

Theory of proportions has nothing to do with this, by the way.

You may look to Wikipedia: https://en.wikipedia.org/wiki/Trigonometry#History to begin with.

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    $\begingroup$ It is debatable whether the spread of trigonometry is due to the fall of Constantinople or to the invention of the printing press. Europeans did have trigonometry before 1453, eg the Alphonsine tables in 1250. $\endgroup$
    – Chrystomath
    Oct 24, 2019 at 16:55
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    $\begingroup$ I did not claim "it was due to the fall of Constantinople". But it roughly coincided in time. Besides this, there is some indirect evidence: Peuerbach was a student of Bessarion who emigrated from Constantinople. $\endgroup$
    – Alexandre Eremenko
    Oct 24, 2019 at 19:01

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