Authors: Steven Kenneth Kauffmann
Trigonometry studies the properties of the cosine and sine functions, which relate a contiguous arc of the unit-radius circle centered on the origin of coordinates to the rectangular Cartesian coordinates of the arc's endpoints. Since the Pythagorean theorem underlies the concept of Cartesian coordinates, this tutorial commences with a plane-geometry recapitulation of that theorem. In the non-calculus treatment of the cosine and sine, their demonstrable properties are encompassed by the unit length of unit-radius circle vectors and the "angle-addition formula" which relates the rectangular coordinates of the endpoints of two immediately successive arcs of the unit-radius circle to the rectangular coordinates of the endpoints of the combined contiguous arc. Those properties are insensitive, however, to simultaneous single-parameter rescaling of all of the arc lengths involved, and so don't unambiguously characterize the cosine and sine functions of directed arc length. Unambiguous determination of the cosine and sine hinges on whether their derivatives with respect to directed arc length are well-defined, which presents no issues for arcs of the unit-radius circle. In fact the cosine and sine functions fascinatingly are the real and imaginary parts of the hyper-well-behaved exponential function of imaginary argument.
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