Is the vector space of R vector system sinx, sin(x-1), cos(x+2) linearly independent? Is the vector space of $R^R$, vector system $\sin{x}$, $\sin(x-1)$, $\cos(x+2)$ linearly independent?

You can use the Wronskian to verify that the functions are linearly dependent.

Alternatively, using trigonometric formulas, note that:
$\quad\quad \sin(x-1) = \cos(1) \sin(x) - \sin(1) \cos(x)$
$\quad\quad \cos(x+2) = \cos(2) \cos(x) - \sin(2) \sin(x)$
So (multiply by $\cos(2)$ and $\sin(1)$ respectively):
$\quad\quad \cos(2)\sin(x-1) = \cos(1)\cos(2) \sin(x) \color{red}{- \sin(1)\cos(2) \cos(x)}$
$\quad\quad \sin(1)\cos(x+2) = \color{red}{\sin(1)\cos(2) \cos(x)} - \sin(1)\sin(2) \sin(x)$
Add both equations:
$\quad\quad \cos(2)\sin(x-1)+\sin(1)\cos(x+2) = \left(\cos(1)\cos(2) - \sin(1)\sin(2)\right)\sin(x)$.
So it is possible to write $\sin x$ as a linear combination of $\sin(x-1)$ and $\cos(x+2)$.
