I've been looking for a simple way to solve for square roots for a while now, and also nth roots. But I was curious how we know the value for the square root of 2. I got messing around (I like figuring things out on my own) and I arrived at this:

Since the cos(x) = a/h
Then cos(x)h = a

So the adjacent side of a right angle triangle is just cos(x)h

In the right triangle that the Greeks used with side lengths 1, the hypotenuse is square root 2. Thus cos(x)?2 = 1
Basic Algebra will get us to ?2 = 1/cos(45)

Note: I'm using degrees here instead of radians.

I tried applying this to ?3 but since all the side lengths are unknown it is extremely difficult. Does anyone have any idea how to solve this using just trigonometry?