“How were sine, cosine, and tangent derived?” Discovery, without using calculator

The ancient Babylonians and Egyptians knew some about trigonometry. As did the ancient Greeks.

The Greeks, Euclid and Archimedes, had the law of cosines and law of sines in their work.

Often thought of, it seems, in terms of chords of triangles.

Here’s an example,

chords trigonometry

I saw that Hipparchus of Nicaea did quite a bit with early trigonometry.

You can measure lengths of triangles within a circle using an instrument to measure distance. They could make tables of values.

Later on, with angles, you can still calculate sine, cosine, and tangent of angles using the ratios of triangles.

For a right triangle,

sinθ = O/H

cosθ = A/H

tanθ = O/A

opposite, adjacent, hypotenuse