Taylor series are great approximations of complicated functions using polynomials. This is done by replacing the actual function with polynomials that have the same derivatives as the original function. As the number of derivatives that a polynomial has in common with a specific function increases, so does the accuracy of the representation. Here I look at a very popular use of a Taylor series: the approximation of sine or sinus. All of the regular calculus functions can be approximated this way around the point x=0. For sine, we can get a fairly accurate representation of the actual function by using a polynomial at the 7th power of x for the range between -π/2 to π/2. The picture of the function covers the entire picture of sine, so by moving and/or mirroring the values by multiples of π, we can calculate sine for any value.
The Taylor series for sine looks like this: Y = X - X^3/ 3! + X^5/ 5! - ... + (-1)^(n+1) * X^(2*n-1)/ (2n-1)!