In 1974, Askey and Steinig proved that for all
n
≥
0
n\geq 0
and
x
∈
(
0
,
2
π
)
x\in (0,2\pi )
the trigonometric sums
sin
(
x
/
4
)
1
+
sin
(
5
x
/
4
)
2
+
⋯
+
sin
(
(
4
n
+
1
)
x
/
4
)
n
+
1
\begin{equation*} \frac {\sin (x/4)}{1}+\frac {\sin (5x/4)}{2}+\cdots + \frac {\sin ((4n+1)x/4)}{n+1} \end{equation*}
and
cos
(
x
/
4
)
1
+
cos
(
5
x
/
4
)
2
+
⋯
+
cos
(
(
4
n
+
1
)
x
/
4
)
n
+
1
\begin{equation*} \frac {\cos (x/4)}{1}+\frac {\cos (5x/4)}{2}+\cdots + \frac {\cos ((4n+1)x/4)}{n+1} \end{equation*}
are positive. Recently, the Askey-Steinig inequalities were improved by the present authors. In this paper, we further improve these inequalities and provide new sharp upper and lower bounds for the two sums given above.