Let $\alpha \in \mathbb{R}$. If $\alpha x$ is the polynomial which interpolates the function $f(x) = \sin \pi x$ on $[-1, 1]$ at all the zeroes of the polynomial $4x^3-3x$, then $\alpha$ is ?
As- $0.47$
My Approach:
Roots of $4x^3-3x= 0$ are $x = 0,\pm \frac{\sqrt{3}}{2}$. Now at $x =0$, $\sin \pi x = 0$. Also at $x = \frac{\sqrt{3}}{2}$, we have $\sin \pi \frac{\sqrt{3}}{2} = 0.0474$ (in radians). Where I am wrong?
Your are wrong on $2$ counts: first $\sin\left(\pi\frac{\sqrt3}2\text{ degrees}\right)=0.0474673131$; you want $\sin\left(\pi\frac{\sqrt3}2\text{ radians}\right)=0.4085762330$.
Next what you want is $\sin\pi x=\alpha x$ for $x\in\{-\frac{\sqrt3}2,0,\frac{\sqrt3}2\}$. Since they are both odd functions, all you need to make this happen is $$\sin\left(\pi\frac{\sqrt3}2\right)=0.4085762330=\alpha\frac{\sqrt3}2=0.8660254038\,\alpha$$ So $\alpha=0.4717831963$.