Derive Sine Function (Taylor Series)

This example derives the power series expansion of the sine function using a Taylor Series Expansion.

1. Start with the general form of the taylor series where . Note, the variable has been replaced with the variable , because sine takes in an angle as input.

   For this expansion, we are going to use to circle definition of sine for the function.

   This definition corresponds to the geometry of a point on the circle shown below.

   

   We are also going to use the derivatives of sine and cosine^[1].

2. Next, we need to choose a point to expand around. Observe that for the input angle of radians the sine function returns making it a likely point to expand around.

   Substitute into the formula.

   Simplify.

3. Take the first, second, third and etc. derivatives of the function.

4. Evaluate the functions at the point .

5. Substitute the values into the expression.

6. Remove the expressions that go to zero.

7. Simplify the numerator and denominator.

   Substituting the name of the function back into the formula .

   The taylor series expansion of the sine function is given by the series above.

1. Prove the Derivative of Sine Example
   https://wumbo.net/examples/prove-derivative-of-sine/