Taylor's theorem uses function notation, so we describe functions as f(X) = X^ 2 - 3X + 2 instead of Y = X^ 2 - 3X + 2. This notation makes it easier to describe multiple derivatives. Factorials are denoted with the exclamation symbol. An example is 5! = 5 x 4 x 3 x 2 x 1 = 120. Another example is 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040.
The derivative of a function is another function that describes how the function changes. The value of a derivative at a point gives the slope of the line that is tangent to the curve at that point. The derivative of a polynomial is found by deleting the constant term and doing this transformation to each remaining term, e.g., aX^n goes to anX^(n - 1). For example, the derivative of X^2 - 3X + 2 is 2X - 3. The derivatives of polynomials always decrease the degree by one so after a few derivatives it becomes zero. Other functions can have an infinite number of derivatives. The first derivative of f(x) is denoted f ' (x) and the derivative of this is denoted f ''(x).
Taylor's theorem for the polynomial approximation of a function is f(X) = f(X0) + (f ' (X0)/1!)(X - X0) + (f ''' (X0)/2!)(X - X0)^2 + (f ''' (X0)/3!)(X - X0)^3 + and so on. You can continue the series for as many terms as you need to get a good approximation. If you approximate a polynomial, it will only continue for a few terms before going to zero. With other functions it can theoretically go on forever. Usually only a few terms are needed to get a very good approximation.
For the polynomial approximation of Y = Sin X, note that f(X) = Sin X. f ' (X) = Cos X, f '' (X) = -Sin X, and so on. Sin (X) = f(X) = f(X0) + (f ' (X0)/1!)(X - X0) + (f '' (X0)/2!)(X - X0)^2 + (f ''' (X0)/3!)(X - X0)^3 = Sin (0) + (Cos(0)/1!)(X - 0) + (-Sin(0)/2!)(X - 0)^2 + (-Cos(0)/3!)(X - 0) + and so on = 0 + (1/1)X + 0 - X^3/3! + 0 + X^5/5! + 0 and so on = X - X^3/3! + X^5/5! - X^7/7! + and so on. This means that Sin X = X - X^3/3! + X^5/5! - X^7/7! + and so on.