And she says "the way of mathematics is to MAKE STUFF UP until it makes sense"?!
I'd rather say it's using a common terminology as well as axioms, measurements, observations, and Cartesian common sense to understand and define problems and ideas which are already present.
LOL... Nowhere in the video does she try to hit home the point that π is the relation of diameter (not radius) and circumference. And she doesn't mention Archimedes and his method of deriving π - she's wrong when she says the radius and not the diameter is the fundamental idea you use in determining the circumference.
Well, technically she's not wrong, but she's making a observation which is stating the same thing as what Archimedes said, just making it less simple. She's just using a simple conversion to justify her method. In fact, in the beginning she rants about the simple "conversion" (which technically isn't a conversion but the way it's been understood, derived, and done for thousands of years) - but then she goes on to use tau as HER conversion.
AND like you said (essentially), if you're dealing with a plane centered at zero, with radius 1 (from -1 to 1), your value of the circumference would be 2π.
And when calculating sinθ, cosθ, and tanθ on a unit circle... You want the radius to be 1 and the circumference to be 2π for simplicity because sinθ = y/r, cosθ = x/r, and tanθ = y/x.
When a circle's radius is 1, its circumference is 2π. When a circle's diameter is 1, its circumference is π.
Which is also why it's common practice to define 2π as the circumference in radians - it's completely commonsense and more mathematically simple than her method.
I'm sorry... As a dual math and physics major and someone who has always got a great deal of sense out of mathematics... This video and her methodology - which was ultimately correct - was just a conversion which tends to complicate things (at least from my viewpoint) when dealing with calculations.
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