construct the interpolation polynomial for the function y=sin ^x,choosing the nodes :0,1/6,1/2?

Answer 1

I'm going to guess that you mean sin(pi*x), since then the function values can easily be computed at the given nodes, without using a calculator.

You can use divided differences, but for a low degree such as this, it's not hard to use Lagrange interpolation either. The polynomial then has the form

P(x) = sin(0) (x-1/6)(x-1/2)/[(0-1/6)(0-1/2)]
+ sin(pi/6) (x-0)(x-1/2)/[(1/6-0)(1/6-1/2)]
+ sin(pi/2) (x-0)(x-1/6)/[(1/2-0)(1/2-1/6)]
= (1/2) x(x-1/2)/(-1/18) + x(x-1/6)/1/6
= -9x(x-1/2) + x(6x-1)
= -3x^2 + 3.5x.

Answer 2

What is the meaning of "^"? Do you simply mean sin(x)?

See the example in Wikipedia for constructing the interpolating polynomial to tan(x) using divided differences, at 5 points.