to give the basics for astronomy, distance calculation etc
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Start by teaching the basic properties of different triangles because trigonometry is the science of triangles.
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.