The issue is that the Maclauren Series uses the derivatives of a function to develop the series:

For the case of f(x) = sin(x) this yields:

which becomes the series you are familiar with:

This formula works because if x is in radians then the derivative of sin(x) is cos(x), the second derivative of sin(x) is -sin(x), etc. If x is in units of degrees then the derivative would be . The second derivative is , etc.

The infinite series for x in degrees then becomes: