Taking Trigonometry Series-ly

Part 1: Numerical

A mathematician friend mentioned to me that for small values of $x$ we might closely approximate $\sin(x)$ and $\cos(x)$ by cubic polynomials. Was she correct? To answer this question, use your calculator in degree mode to try to find the coefficients $a, \dots, h$ in the following suggested polynomial approximations:

$$\sin(x) = a +bx+cx^2+dx^3\quad\quad \cos(x) = e+fx+gx^2+hx^3\quad\quad -0.1\leq x\leq 0.1$$ Do you find that there is a best answer to finding these coefficients? How do the coefficients change when you try out the problem in radian mode on the calculator? Can you find a good choice of the coefficients of a possible fourth order term? Don't forget that you already possibly know certain values of the trig functions which might give you a good starting point for a search for these functions.

Part 2: Using calculus

My mathematician friend now tells me that she thinks of trigonometrical functions in terms of solutions to differential equations, and not in terms of triangles. She says that $\sin(x)$ and $\cos(x)$ are both solutions to the second order differential equation:

$$\frac{d^2 f}{dx^2}+f =0\;.$$ Using this approach, what values for the coefficients $a, \dots, g$ emerge for the polynomial approximations from the first part? Does this correspond to your degree or radians expression? Or something different?

If the polynomial approximation continued to an arbitrarily high order, what would the coefficients be?

If you find the approximation to the sixth power of $x$ you can now estimate trigonometrical values without using the $\sin$ or $\cos$ button on your calculator. Test the accuracy of your series for various values of $x$ between $0$ and $\pi$/2.

Discussion points: Do you think that your calculator stores values of sin and cos, or works them out on demand? Would your series provide an efficient way of evaluating the numerical values of sin(x) and cos(x)?