Bounds on $\sum\limits_{k=1}^n \frac{\sin(k)}{k}$ $\sum\limits_{k=1}^n \frac{\sin(k)}{k}$ converges as $n$ increases, to a limit of $\frac12(\pi-1) \approx 1.0708$
Empirically, it seems to be bounded by about $\frac12(\pi-1)  \pm \frac{1.043}{n}$, as shown in the chart below.
What is the precise value of this $1.043$ term?
Added: I think the three answers from Oliver Diaz, Gary, and Gabriel Romon, plus the comments, have demonstrated that the bounds are $$\frac12(\pi-1)  \pm \frac1{2\sin(1/2)}\frac{1}{n}$$ not only asymptotically but also as tight actual bounds; $\frac1{2\sin(1/2)}\approx 1.0429148214667441$

created in R with
n <- 1:11000
plot(n, cumsum(sin(n)/n), ylim=c(1.07,1.0715), xlim=c(1,10000), pch=46)
abline(h=(pi-1)/2, col="red")
curve((pi-1)/2 + 1.043/x, from=1, to=max(n), add=TRUE, col="red")
curve((pi-1)/2 - 1.043/x, from=1, to=max(n), add=TRUE, col="red")

 A: This is a twice the bound of the OP:
Suppose $c_n\searrow0$.
Fix $M\in\mathbb{N}$ and for $n\geq N$ let
$$P_n(x)=\sum^n_{k=M}e^{ikx}=\frac{e^{iMx}-e^{i(n+1)}}{1-e^{ix}}=e^{iMx}\frac{1-e^{(n-M+1)ix}}{1-e^{ix}}=\frac{\sin\big((n-M+1)x/2\big)}{\sin(x/2)}e^{i(n-M)x/2}$$
From this, we get that
$$ S_n(x)=\sum^n_{k=M}\sin kx =\frac{\sin\big((n-M+1)x/2\big)}{\sin(x/2)}\sin\big((n-M)x/2\big)
$$
Then by summation by parts
$$\begin{align}
\sum^N_{n=M} c_n e^{inx} &=c_NP_N(x) +\sum^{N-1}_{n=M}(c_n-c_{n+1})P_n(x)\\
\sum^N_{n=M} c_n \sin nx & = c_NS_N(x) +\sum^{N-1}_{n=M}(c_n-c_{n+1})S_n(x)
\end{align}$$
Then
$$
\begin{align}
\Big|\sum^N_{n=M} c_n e^{inx}\Big| &\leq c_N|P_N(x)| +\sum^{N-1}_{n=M}(c_n-c_{n+1})|P_n(x)|\\
\Big|\sum^N_{n=M} c_n \sin nx\Big| &\leq c_N|S_N(x)| +\sum^{N-1}_{n=M}(c_n-c_{n+1})|S_n(x)|
\end{align}
$$
For $P_n$ and $S_n$ we have the estimates
$$\begin{align}
|P_n(x)|&= \big|\sin\big((n-M+1)x/2\big)\big|\csc(x/2)|\leq  |\csc(x/2)| \\
|S_n(x)|&=\Big|\sin\big((n-M+1)x/2\big)\sin\big((n-M)x/2\big)\Big||\csc(x/2)|\leq|\csc(x/2)|
\end{align}$$
for all $x\neq0$ and $n$,
\begin{align}
\Big|\sum^N_{n=M} c_n e^{inx}\Big|&\leq \Big(c_N +\sum^{N-1}_{n=M}(c_n-c_{n+1})\Big)|\csc(x/2)|\leq c_M |\csc(x/2)|\tag{1}\label{one}\\
\Big|\sum^N_{n=M} c_n \sin nx\Big|&\leq \Big(c_N +\sum^{N-1}_{n=M}(c_n-c_{n+1})\Big)|\csc(x/2)|\leq  c_M|\csc(x/2)|\tag{2}\label{two}
\end{align}
For the OP's series
$$
\sum^\infty_{n=1}\frac{\sin nx}{n}=\frac{1}{2i}\sum_{|n|\geq1}\operatorname{sign}(n)\frac{e^{inx}}{n}=\frac12(\pi-x),\qquad 0<x<2\pi
$$
From  \eqref{one} or \eqref{two} we obtain bound
$$\Big|\sum^{N-1}_{n=1}\frac{\sin nx}{n}-\frac{\pi-x}{2}\Big|=\Big|\sum^\infty_{n=N}\frac{\sin nx}{n}\Big|\leq \frac{1}{N}|\csc(x/2)|
$$
For $x=1$, we get twice the OP's bound. I ignore at the moment how to improve the bound in \eqref{one} by a factor of $1/2$ which is what the OP is observing.
A: Note that $\displaystyle\left|\sum_{k=1}^n \frac{\sin k}k - \frac12(\pi-1)\right| = \left|\sum_{k=n+1}^\infty \frac{\sin k}k \right|$.
We prove the following inequality:
$$
\left|\sum_{k=n+1}^\infty \frac{\sin k}k \right|
\leq 
\frac{1}{2 \sin(\frac 12)} \frac 1{n+1}
+
\frac{3}{4\sin^2(\frac 12)} \frac 1{(n+1)(n+2)}
$$

Let $\displaystyle S_n = \sum_{k=1}^n \sin k = \frac{\cos(1/2)-\cos(n+\frac 12)}{2\sin(\frac 12)}$ and
$\displaystyle T_n 
= \sum_{k=1}^n S_k 
= \frac{n}{2\tan(\frac 12)} - \frac{\sin(n+1)-\sin (1)}{4\sin^2(\frac 12)}$.
Performing summation by parts twice yields
$$\sum_{k=n+1}^\infty \frac{\sin k}k 
= \sum_{k=n+1}^\infty \Big(\frac{S_k}{k(k+1)}\Big) - \frac{S_n}{n+1}
=\sum_{k=n+1}^\infty \Big(\frac{2T_k}{k(k+1)(k+2)}\Big) - \frac{T_n}{(n+1)(n+2)} - \frac{S_n}{n+1}.
$$
Note that
$$\sum_{k=n+1}^\infty \frac{2T_k}{k(k+1)(k+2)} = \frac{1}{\tan(\frac 12)} \frac{1}{n+2} - \frac{1}{2 \sin^2(\frac 12)} \sum_{k=n+1}^\infty \frac{\sin(k+1)-\sin(1)}{k(k+1)(k+2)} $$
and
$$\begin{align}
\frac{T_n}{(n+1)(n+2)} + \frac{S_n}{n+1}
= &\phantom{-}\Big(\frac{1}{\tan(\frac 12)}-\frac{\cos(n+\frac 12)}{2 \sin(\frac 12)} \Big)
\frac 1{n+1} \\ &- 
\Big(\frac{1}{\tan(\frac 12)} +  \frac{\sin(n+1)-\sin (1)}{4\sin^2(\frac 12)} \Big) \frac 1{(n+1)(n+2)}.
\end{align}
$$
Therefore
$$\begin{aligned}
\sum_{k=n+1}^\infty \frac{\sin k}k = 
&\phantom{+}\frac{\cos(n+\frac 12)}{2 \sin(\frac 12)} \frac 1{n+1}
+
\frac{\sin(n+1)-\sin (1)}{4\sin^2(\frac 12)} \frac 1{(n+1)(n+2)}
\\
&-\frac{1}{2 \sin^2(\frac 12)} \sum_{k=n+1}^\infty \frac{\sin(k+1)-\sin(1)}{k(k+1)(k+2)}
\end{aligned}
$$
and
$$\begin{aligned}
\left|\sum_{k=n+1}^\infty \frac{\sin k}k \right|
&\leq \frac{|\cos(n+\frac 12)|}{2 \sin(\frac 12)} \frac 1{n+1}
+
\frac{|\sin(n+1)-\sin (1)|+1}{4\sin^2(\frac 12)} \frac 1{(n+1)(n+2)} \\
&\leq 
\frac{1}{2 \sin(\frac 12)} \frac 1{n+1}
+
\frac{3}{4\sin^2(\frac 12)} \frac 1{(n+1)(n+2)}
\end{aligned}
$$
A: A slightly worse estimate may be obtained as follows. If $|\arg z|<\pi$ then
$$
\log (1 + z) = \sum\limits_{k = 1}^N {( - 1)^{k + 1} \frac{{z^k }}{k}}  + ( - 1)^N z^{N + 1} \int_0^1 {\frac{{t^N }}{{1 + zt}}dt} 
$$
for any $N\geq 0$. Substituting $z =  - e^i$, taking the imaginary part of each side and rearranging, we find
$$
\sum\limits_{k = 1}^N {\frac{{\sin (k)}}{k}}  - \frac{{\pi  - 1}}{2} = \Im \left[ { - e^{i(N + 1)} \int_0^1 {\frac{{t^N }}{{1 - e^i t}}dt} } \right].
$$
Here
\begin{align*}
\left| {\Im \left[ { - e^{i(N + 1)} \int_0^1 {\frac{{t^N }}{{1 - e^i t}}dt} } \right]} \right| & \le \int_0^1 {\frac{{t^N }}{{\left| {1 - e^i t} \right|}}dt}  \le \frac{1}{{\sin 1}}\int_0^1 {t^N dt} \\ & = \frac{1}{{(N + 1)\sin 1}} < \frac{{1.1884}}{{N + 1}}.
\end{align*}
This is larger than the expected result by a factor of $\frac{1}{{\cos (1/2)}}=1.13949\ldots$. An exact formula for the remainder can be obtained by computing the imaginary part explicitly, but it results in a complicated formula which does not seem to be useful for better estimates.
Addendum. From the link you posted, the remainder term is
$$
{\int_0^1 {\left( {\frac{1}{{2\sin \frac{t}{2}}} - \frac{1}{t}} \right)\sin \left( {\left( {n + \tfrac{1}{2}} \right)t} \right)dt}  - \int_{n + 1/2}^\infty  {\frac{{\sin t}}{t}dt} }.
$$
Integration by parts gives
$$
\int_0^1 {\left( {\frac{1}{{2\sin \frac{t}{2}}} - \frac{1}{t}} \right)e^{i\left( {n + \frac{1}{2}} \right)t} dt}  = ie^{i\left( {n + \frac{1}{2}} \right)} \left( {1 - \frac{1}{{2\sin \left( {\frac{1}{2}} \right)}}} \right)\frac{1}{n} + \mathcal{O}\!\left( {\frac{1}{{n^2 }}} \right).
$$
Taking imaginary parts
$$
\int_0^1 {\left( {\frac{1}{{2\sin \frac{t}{2}}} - \frac{1}{t}} \right)\sin \left( {\left( {n + \tfrac{1}{2}} \right)t} \right)dt}  = \left( {1 - \frac{1}{{2\sin \left( {\frac{1}{2}} \right)}}} \right)\frac{{\cos \left( {n + \frac{1}{2}} \right)}}{n} + \mathcal{O}\!\left( {\frac{1}{{n^2 }}} \right).
$$
By another integration by parts
$$
\int_{n + 1/2}^\infty  {\frac{{\sin t}}{t}dt}  =  \frac{1}{n}\cos \left( {n + \tfrac{1}{2}} \right) + \mathcal{O}\!\left( {\frac{1}{{n^2 }}} \right).
$$
Thus, your remainder term is
$$
-\frac{1}{{2\sin \left( {\frac{1}{2}} \right)}}\frac{1}{n}\cos \left( {n + \tfrac{1}{2}} \right) +  \mathcal{O}\!\left( {\frac{1}{{n^2 }}} \right),
$$
which for large $n$ is at most
$$
 \approx \frac{1}{{2\sin \left( {\frac{1}{2}} \right)}}\frac{1}{n}
$$
in absolute value. Because of the big-$\mathcal{O}$, in principle, it could be slightly larger, so this is not a complete proof.
A: Comment: The value of this sum is indeed (pi-1)/2. This is confirmed by noting that the function f(x) = pi-x on (0, pi), extended oddly to (-pi, 0), and then extended to all real numbers periodically and with the usual convention, has Fourier series = sum(sin(nx)/n; n=1,2,...). The extended function is continuous except at integer multiples of pi. Hence, we have equality at x=1, so the sum in question = f(1) = (pi-1)/2.
For more details, see:
Michael W. Ecker, Combining Mathematics and Technology to Sum Certain Series, Mathematics and Computer Education, Winter 2013, Vol. 47, No. 1, p. 19-23.

