I want to know why the equation gives an approximate square.
I was just playing around with functions and I wanted to see if (radians) would give a semicircle for the interval [0,2] as the distance of (1,0) is the same from (0,0), (2,0) and (1,1), all of which will lie on the curve. The equation of a unit semicircle with its centre at (1,0) is
I know that the curves of both the equations don't resemble each other much but I still thought of approximating the sine function using this because I thought that it could still be combined with another approximation to make a better approximation. Anyway, I did it and for , the value of sinϕ can to be approximately . It looked like a semi-ellipse and so I verified it to find that it was a semi-ellipse. I thought of using this to derive the equation for an ellipse with it's centre at the origin and the value of and b being and 1 respectively.
The equation came out to be :
Finally, I thought of playing with this equation and changed the exponent of x. I observed that as I increased the power, keeping it even, the figure got closer and closer to a square.
gave a good approximation of a square. For the exponent of x being some power of 10 greater than 1012, a part of the curve began to disappear.