Question

If we approximate the function y=sin(x) with a0+a1 x a2 x^2 +a3 x^3, what is a0,a2,a2,a3?

Answer 1

The coefficients $a_0,a_1,a_2,a_3$ could be chosen to be the coefficients in the Maclaurin series of $\sin(x)$.

We have

$y = \sin(x) \approx a_0 + a_1 x + a_2 x^2 + a_3 x^3 \\\\ \implies y(0) = 0 = a_0$

$y' = \cos(x) \approx a_1 + 2a_2 x + 3a_3 x^2 \\\\ \implies y'(0) = 1 = a_1$

$y'' = -\sin(x) \approx 2a_2 + 6a_3 x \\\\ \implies y''(0) = 0 = 2a_2$

$y''' = -\cos(x) \approx 6a_3 \\\\ \implies y'''(0) = -1 = 6a_3$

It follows that $a_0=0$, $a_1=1$, $a_2=0$, and $a_3 = -\frac16$.