The main part of trigonometry is the study of triangles on spheres. Also, another part of trigonometry is the constant positive curvature.
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Not specifically trigonometry, but functions in general. As a general rule, functions must be evaluated before using the results in other parts of the calcuation.
plane trigonometry spherical trigonometry
The main kinds are plane trigonometry and solid trigonometry. The latter will include trigonometry in hyper-spaces.
Hipparchus is the father of trigonometry.
Trigonometry based on a unit circle and radians and trigonometry based on a right triangle.