There are more than two but the two most common ones are plane trig and spherical trig.
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The main kinds are plane trigonometry and solid trigonometry. The latter will include trigonometry in hyper-spaces.
adjacent opposite hypotenuse
Right, acute, and obtuse...
The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.
Trigonometry based on a unit circle and radians and trigonometry based on a right triangle.