So, I was working with a theorem that I had been working on about more properties of a circle, more invisible properties. Here is my work for the theorem. It isn't much, but it is something: I noticed that when I did sin of 45, I got the value 0.707106, which I knew was close to π / 4, so I continued to increase the value. When I got to sin of 51, I got 0.777, which was really close, so I tried 52. Unfortunately, it would have been 3.15, which I knew wasn't π, but I knew something. I tried what(if I remember who it was) Archemidies, who specified a values that was higher and lower than pi. So, I did sin(51) < π < sin(52). I then began adding decimals after 51. I got to sin(51.7575) * 4, which was equal to 3.14159. I then noticed that ϕ * 32 was similar to this value. So, I replaced the value with ϕ * 32, which is approximately 51.777, and I got the value 3.142438 approximately. From here, I noticed that maybe this formula could have calculate pi faster than any other method before, because when I tried using the Fibonacci numbers to calculate phi for this formula, one time the value was close to 3.14159265, ect, but it kept going, then when it reached 3.1424 it stopped and went from there. Does this mean anything?
You can stop right here, as this is manifestly not true. For any real number x, -1 <= sin(x) <= 1. What you seem to be doing is attempting to solve the equation 4 sin(x) = [itex]\pi/4[/itex]. A solution to this equation is x = sin^{-1}([itex]\pi/4[/itex]) ≈ .903339 radians ≈ 51.75°.
It is true that, for x measured in radians, [itex]\lim_{x\to 0}sin(x)/x= 1[/itex]. That is, for x very close to 0, sin(x) will be very close to x. Once you get up to things like 45 degrees, which is equivalent to [itex]\pi/4[/itex] radians, they are still "close" but not "very close". The farther from 0 you get, the farther apart those get.