Find the $\frac mn$ if $T=\sin 5°+\sin10°+\sin 15°+\cdots+\sin175°=\tan \frac mn$ It's really embarrassing to be able to doesn't solve this simple-looking trigonometry question.

$$T=\sin(5^\circ) +\sin(10^\circ) + \sin(15^\circ)  + \cdots +\sin(175^\circ) =\tan \frac mn$$
Find the $\frac mn=?$, where $m$ and $n$ are positive integer numbers.

Attepmts:
$$T=2\Big(\sin(5°)+\sin(10°) + \cdots + \sin(85°)\Big) + 1 = 2\Big((\sin(5°) + \cos(5°))+(\sin(10°)+ \cos(10°))+\cdots + (\sin(40°)+\cos(40°))\Big)+1 = 2\Big(\sqrt 2((\sin50°+\sin55°)+\cdots+\sin(80°))\Big)+1.$$
and then I can not see an any way...

By geometric series,
$$T=\sin 5°+\sin10°+\sin 15°+...+\sin175° = \\
\operatorname{Im} (\sum_{n=1}^{35}\exp(i n 5 \pi/180)) =\\
\operatorname{Im} \exp(i 5 \pi/180) \frac{\exp(i 35 \cdot 5 \pi/180)-1}{\exp(i 5 \pi/180)-1} =\\
\operatorname{Im} \exp(i 5 \pi/180) \exp(i 34 \cdot 5 \pi/360) \frac{\sin( 35 \cdot 5 \pi/360)}{\sin( 5 \pi/360)} =
\\
\operatorname{Im}  \exp(i 36 \cdot 5 \pi/360) \frac{\sin( 35 \cdot 5 \pi/360)}{\sin( 5 \pi/360)}  =
\\ \frac{\sin( 35 \cdot 5 \pi/360)}{\sin( 5 \pi/360)} \\
 $$
and
$$
\arctan(\frac{\sin( 35 \cdot 5 \pi/360)}{\sin( 5 \pi/360)} ) = \\
\arctan(\frac{\sin( 35 \cdot 5 \pi/360)}{\cos(  \frac{\pi}{2}  - 5 \pi/360)} ) = \\
\arctan(\frac{\sin( 35 \cdot 5  \pi/360)}{\cos( 35 \cdot 5 \pi/360)} ) = 
\frac{35 \pi}{72}
$$
or, in degrees, $\frac{35 \cdot 180}{72} =\frac{175}{2} = 87.5 $
