The \systeme
command cannot cope with this kind of object: it wants coefficients to be in front of the variable, as far as I know.
If you accept this, you can do
documentclass{article}
\usepackage{amsmath}
\usepackage{systeme}
\usepackage{siunitx}
\begin{document}
\[
\protected\def\c{\cos60^\circ}
\protected\def\s{\sin60^\circ}
\sisetup{parse-numbers=false}
\syssubstitute{{x_1}{F_1}{x_2}{F_2}{x_3}{F_3}{x_4}{F_4}{x_5}{F_5}}
\systeme{
\c x_1+x_3=\qty{2\cos60^\circ}{kN},
\s x_2=\qty{2}{kN},
-\c x_1+\c x_2+x_4=0,
-\s x_1-\s x_2=0,
-x_4-\c x_5=\qty{-\cos60^\circ}{kN}
}
\]
\end{document}

Alternatively, do it the hard way.
\documentclass{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{siunitx}
\begin{document}
\[
\sisetup{parse-numbers=false}
\renewcommand{\arraystretch}{1.5}
\setlength{\arraycolsep}{0pt}
\protected\def\c{\cos60^\circ}
\protected\def\s{\sin60^\circ}
\left\lbrace
\begin{array}{
*{5}{
r % unknown
>{{}}l % coefficient
>{{}}c<{{}} % operation or relation
}
l % constant term
}
% F_1 & c & + & F_2 & c & + & F_3 & c & + & F_4 & c & + & F_5 & c & = & c
F_1 & \c & & & & + & F_3 & & & & & & & & = & \qty{2\cos60^\circ}{kN} \\
& & & F_2 & \s & & & & & & & & & & = & \qty{2}{kN} \\
-F_1 & \c & + & F_2 & \c & & & & + & F_4 & & & & & = & 0 \\
-F_1 & \s & - & F_2 & \s & & & & & & & & & & = & 0 \\
& & & & & & & & - & F_4 & & - & F_5 & \c & = & \qty{-\cos60^\circ}{kN}
\end{array}
\right.
\]
\end{document}

systeme
is designed to cope with such cases.F
byf
works. Side note:\rm{...}
isn't doing what you probably think.