\geq
\overline{T}_{n+4}(t)\geq \sin t,$$ $$\label{Ineq_11}
\Big{(}\forall t \in \big{[}\mbox{\small $-\sqrt{(n+3)(n+4)}$},0\big{]}\Big{)}\,\underline{T}_n(t)
\leq
\underline{T}_{n+4}(t)\leq \sin t.$$

For the value $t=0$ the inequalities in [$($\[Ineq\_10\]$)$]{} and [$($\[Ineq\_11\]$)$]{} turn into equalities. For the values $t\!=\!\mbox{\small $\pm\sqrt{(n+3)(n+4)}$}$ the equalities $\overline{T}_n(t)\!=\!\overline{T}_{n+4}(t)$ and $\underline{T}_n(t)\!=\!\underline{T}_{n+4}(t)$ are true, respectively.

[**(ii)**]{} For the polynomial $T_n(t)=\!\!\!\displaystyle\sum\limits_{i=0}^{(n-1)/2} \!\!\dfrac{(-1)^it^{2i+1}}{(2i+1)!}$, where $n=4k+3$, $k\in \mathbb{N}_{0}$, it is\

valid:

$$\label{Ineq_12}
\Big{(}\forall t \in \big{[}0,\mbox{\small $\sqrt{(n+3)(n+4)}$}\,\big{]}\Big{)}\,\underline{T}_n(t)
\leq
\underline{T}_{n+4}(t)\leq \sin t,$$ $$\label{Ineq_13}
\Big{(}\forall t \in \big{[}\mbox{\small $-\sqrt{(n+3)(n+4)}$},0\big{]}\Big{)}\,\overline{T}_n(t)
\geq
\overline{T}_{n+4}(t)\geq \sin t.$$

For the value $t=0$ the inequalities in [$($\[Ineq\_12\]$)$]{} and [$($\[Ineq\_13\]$)$]{} turn into equalities. For the values $t=\mbox{\small $\pm\sqrt{(n+3)(n+4)}$}$ the equalities $\underline{T}_n(t)=\underline{T}_{n+4}(t)$ and $\overline{T}_n(t)=\overline{T}_{n+4}(t)$ are true, respectively.

Let us notice that for the function $\sin x$ we have following order: $$\label{Ineq_14}
\begin{array}{ll}
\underline{T}_{3}^{\sin ,0}(x) \leq \underline{T}_{7}^{\sin ,0}(x) \leq \underline{T}_{11}^{\sin ,0}(x) \leq \underline{T}_{15}^{\sin ,0}(x)
\,\leq\,\ldots\,\leq\,\sin x\,\leq
\ldots &                                                                                                 \\[1.5 ex]
\leq
\overline{T}_{13}^{\sin ,0}(x) \leq \overline{T}_{9}^{\sin ,0}(x) \leq \overline{T}_{5}^{\sin ,0}(x) \leq \overline{T}_{1}^{\sin ,0}(x)
\;\; \mbox{for} \;\; x\!\in\!\left[\,0,\sqrt{20}\,\right].
\end{array}$$

[**(i)**]{} For the polynomial $T_n(t)=\displaystyle\sum\limits_{i=0}^{n/2}\dfrac{(-1)^it^{2i}}{(2i)!}$, where $n\!=\!4k$, $k\!\in\!\mathbb{N}_{0}$,\

it is valid: $$\label{Ineq_15}
\Big{(}\forall t \in \big{[}\mbox{\small $-\sqrt{(n+3)(n+4)},\sqrt{(n+3)(n+4)}$}\,\big{]}\Big{)} \,\,\,\overline{T}_n(t)\geq \overline{T}_{n+4}(t)\geq \cos t.$$

For the value $t=0$ the inequality in [$($\[Ineq\_15\]$)$]{} turns into equality. For the values $t=\mbox{\small $\pm\sqrt{(n+3)(n+4)}$}$ the equality $\overline{T}_n(t)=\overline{T}_{n+4}(t)$ is true.

[**(ii)**]{} For the polynomial $T_n(t)=\displaystyle\sum\limits_{i=0}^{n/2}\dfrac{(-1)^it^{2i}}{(2i)!}$, where $n\!=\!4k\!+\!2$, $k\!\in\!\mathbb{N}_{0}$, it is 