$4\mathrm{.}19$ To calculate a planet's space coordinates, we have to solve the function

$f(x)=x-1-0\mathrm{.}5sinx$

Let the base point be $a=x_i=\frac{}{2}$ on the interval $0,.$ Determine the highest$-$order Taylor series expansion resulting in a maximum error of $0\mathrm{.}015$ on the specified interval. The error is equal to the absolute value of the difference between the given function and the specific Taylor series expansion. $($Hint: Solve graphically.$)$