Let's talk Right Triangles:
contain one angle of 90 degrees.
triangles can only contain one. why?
Pythagorean Theorem
one of the most "used" items.
let's prove it. once.
want more? here: http://www.cut-the-knot.org/pythagoras/
useful ... Pythagorean Triples
3, 4, 5 is the most common one ... but there are an infinite amount of them
5, 12, 13
here is a link to them: https://www.wikiwand.com/en/Pythagorean_triple (easiest reading)
Primitive means they can't all be divided by n.
we can make as many as we want ... check out this graphic:
we notice a pattern between the primitives and the non-primitives
multiplied by value k ... but the angles shouldn't change at all.
the ratio remains the same, which means our movement remains the same.
this is true of ALL right triangles, whether or not they are triples.
Which leads us to irrational numbers, and the fact they exist
Did not jibe with what Pythagoras knew, but that's okay.
Hipparsus (maybe) was the one who took this to the logical conclusion and proved irrational numbers
Here's a super deep and super fun paper outline: http://math.uga.edu/~pete/4400irrationals.pdf
A quote: "This is also one of the most historically important theorems in mathematics. History tells us that the result was discovered by Pythagoras, or at least someone in his school, and it was quite a shocking development (some sources say that the unnamed discoverer was feted, others that he was cast into the sea). It caused Greek mathematicians to believe that geometric reasoning was more reliable than numerical, or quantitative reasoning, so that geometry became extremely well-developed in Greek mathematics at the expense of algebra"
Two "Brilliant" links (they call themselves that):
Switch place and time. We are now geocentric.
this creates an interesting idea--the shell, with stars as the end of the universe
1 Celestial Unit is key here
eventually, the sine is the height of the star, and the cosine is how far away the star is on the horizontal plane.
we've created the unit circle. Which I've adapted here: https://www.desmos.com/calculator/fyzainx7b5
Some of these Triangles are "easy" to find some values for:
45 - 45 - 90
30 - 60 - 90
But all triangles have these relationships between size and angles ... and if we can find the ratio, we can find the other sides.
THIS ALLOWS US TO SOLVE THE TRIANGLE.
we have tables with these solved out ratios for us (let us NASA because we can):
as the ratio values are consistent, so too is our ability to find the values, the heights ... or the angle.
This is what our calculators do, and it's awesome.