Newton's Divided Difference Formula without given data point

Question

This is the original question:

Let f(x)= sin((pi*x)/6) and P(x) a quadratic polynomial such that f(x) = P(x) at x=0,1, and 2. Find P(x) using Newton's Divided Difference Formula

Since Newton's Divided Difference Formula required the data of x0, x1, x2, but it is missing in the question. Should I first convert the trigonometry question (sin((pi*x)/6)) into a quadratic polynomial, then find the data point of x0, x1,x2 by the quadratic equation?

But I am confusing of how to convert it into quadratic equation. I only can solve sin((pi*x)/6)=0, x= 12n or 6+12n . And I am confusing of the next step.

Thank you so much

Answer 1

You have the data... $$ y_0=f(x_0)= \sin 0 = 0, \quad, y_1 = f(x_1) = \sin \frac{\pi}{6} = \frac 12,\quad y_2=f(x_2)=\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}. $$

With this information, you can compute $f[x_0], f[x_0,x_1]$ and $f[x_0,x_1,x_2]$ and assemble the interpolating polynomial.