Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García's user avatar
Emmanuel José García
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If the half-angle formulas aren't fundamental, then how did Viète manage to derive his formula for $\pi$ using them? And why is it that the Weierstrass substitution, hailed as the most ingenious substitution in the world by Spivak, is actually based on those very formulas? If these half-angle formulas lack significance, then how on earth have I successfully derived the formulas of Heron, Brahmagupta, and Bretschneider, as well as the laws of cosines, sines, and tangents, the Mollweide's formula (or perhaps we should credit Newton), the angle bisector formula, the inradius formula for mixtillinear incircle, and the sum and difference of angle identities, not to mention Euler's remarkable triangular inequality, all stemming from these supposedly non-fundamental formulas? And if that's not enough, how have I managed to extend the Pythagorean trigonometric identity (with over 2000 years of history), the Mollweide formula (or Newton's, if you prefer) (with over 300 years of existence), and even the half-angle formulas themselves (likely over 2000 years old) into broader generalizations? Still skeptical? Well, I invite you to click on the link below and see for yourself.

Link: The theoretical importance of half-angle formulas

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Jul 26, 2022