<s>[INST] if $\sin{(aw)}+\sin{(bw)}+\sin{(cw)}=3$ find $w$ range let $w$ is postive integer,if there exist $a,b,c(\pi\le a<b<c\le 2\pi)$such
$$\sin{(aw)}+\sin{(bw)}+\sin{(cw)}=3$$
Find the $w$ range.
My attempt：
since
$$\sin{(wa)}\le 1,\sin{(wb)}\le 1,\sin{(wc)}\le 1$$
then $$\sin{(wa)}=\sin{(wb)}=\sin{(wc)}=1$$
$$aw=\dfrac{\pi}{2}+2k_{1}\pi,k_{1}\in Z$$
$$bw=\dfrac{\pi}{2}+2k_{2}\pi,k_{2}\in Z$$
$$cw=\dfrac{\pi}{2}+2k_{3}\pi,k_{3}\in Z$$
what approaches do you think, I could take to solving the next step?
 [/INST] we also solve if $w$ be postive real numbers,I have done This result is $$w\in [\dfrac{17}{4},\dfrac{9}{2}]\bigcup [\dfrac{21}{4},+\infty) $$
This problem is equivl this:there $p,q,r\in N^{+}$,such
$$w\pi\le 2p\pi+\dfrac{\pi}{2}<2q\pi+\dfrac{\pi}{2}<2r\pi+\dfrac{\pi}{2}\le 2w\pi$$
case1 if $w\ge 6$ it is clear,because $(w\pi,2w\pi)\subset (6\pi,12\pi)$,and $\dfrac{12\pi-6\pi}{3}=2\pi$,
case 2: if $0<w<6$,then $(w\pi,2w\pi)\subset (0,12\pi)$,
then
2.1 if 
$$w\pi\le\dfrac{\pi}{2}<\dfrac{5\pi}{2}<\dfrac{9\pi}{2}\le 2w\pi$$
then $$\Longrightarrow  w\le\dfrac{1}{2},\rm{and} w\ge\dfrac{9}{4}$$
that's impossible.
2.2 if $$w\pi\le\dfrac{5\pi}{2}<\dfrac{9\pi}{2}\le\dfrac{13\pi}{2}\le 2w\pi$$
then 
$$\Longrightarrow w\ge \dfrac{13}{4},\rm{and} w\le \dfrac{5}{2}$$
also impossible
$2.3$  if
$$w\pi\le\dfrac{9\pi}{2}<\dfrac{13\pi}{2}<\dfrac{17}{2}\pi\le 2w\pi$$
then
$$\Longrightarrow \dfrac{17}{4}\le w\le\dfrac{9}{2}$$
$2.4$ if 
$$w\pi\le\dfrac{13\pi}{2}<\dfrac{17\pi}{2}<\dfrac{21\pi}{2}\le 2w\pi$$
then
$$\dfrac{21}{4}\le w\le\dfrac{13}{2}$$
so
$$w\in [\dfrac{17}{4},\dfrac{9}{2}]\bigcup [\dfrac{21}{4},+\infty) $$
Note: if $w\in Z^{+}$,then $w\ge 6,w\in Z^{+}$$
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