Help me understand how to solve an inequality using Lagrange theorem

[xll11]

I'm asked to prove the following inequality: (pi/6) + 1/(5*sqrt(3)) < arcsin(0.6) < (pi/6) + (1/8) and I'm instructed to use Lagrange theorem on a similar function, now I know what Lagrange theorem (mean value theorem) means, but I fail to understand how to relate it to this inequality