Let me tell you about an exciting discovery I made while exploring the theoretical importance of half-angle formulas in trigonometry. These formulas, often overlooked and underappreciated in comparison to the more well-known laws of sines, cosines, and tangents, have a wealth of untapped potential waiting to be discovered.
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| The standard half-angle formulas |
I'm not talking about the standard version of the half-angle formulas that appear in most introductory trigonometry textbooks. No, I'm referring to a version that relates the sides of a triangle, the perimeter, and its angles. These formulas have been largely ignored by the mathematical community, with even the almighty Wikipedia lacking an article on them.
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| The half-angle formulas for a triangle |
My journey began when I stumbled upon (1) while attempting to prove the law of cosines by contradiction. I later discovered that this formula was already known, but that didn't stop me from exploring its potential. I went on to generalize the formula and discovered (2), which applies to cyclic quadrilaterals. When I found out that (2) was also already known, I refused to give up and continued to push the limits of these formulas.
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| The half-angle formulas for a cyclic quadrilateral |
What I discovered next was truly mind-blowing. I managed to generalize (2) even further to (3), which applies to general quadrilaterals. Surprisingly, (3) also generalizes the Pythagorean trigonometric identity! These formulas not only provide a new framework for the Heron-Brahmagupta-Bretschneider development (see Two Identities and their Consequences, pp. 5), but they also have the potential to derive a plethora of other formulas and theorems, including the law of cosines, the law of sines, the law of tangents, and Stewart's theorem.
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| The half-angle formulas for a general quadrilateral |
And it doesn't stop there. Using these half-angle formulas, I also managed to derive a generalization of Mollweide's formula and a generalization of the law of tangents. The possibilities are endless, and I'm only scratching the surface of what these formulas can do.
But what excites me the most about these formulas is their potential to revolutionize the way we teach trigonometry to high school students. By introducing the half-angle formulas, we can help students better understand the development of the formulas that derive from them. This is where the half-angle formulas truly shine and can make a significant impact on math education.
In conclusion, the theoretical importance of the half-angle formulas cannot be overstated. They have the potential to unlock a whole new world of mathematical discoveries and have already proven to be a powerful tool for deriving a wide range of formulas and theorems. And with the right approach, they can also be a game-changer for teaching trigonometry to the next generation of students. The possibilities are truly endless, and I can't wait to see what else these formulas have in store for us.
See also The theoretical importance of the half-angle formulas.