Evaluating $sin 10^circ + sin 20^circ + sin 30^circ + cdots +sin 90^circ$

Best it is to follow symbolic derivation of sums of sine of $n$ angles in A.P, of common difference $beta= 10^{circ}.$ And then apply it.

$$ S = sin ( average; angle)cdot dfrac{sin n beta/2}{sin beta/2}$$

$$={ sin 50^{circ}}dfrac{ sin 9times 5^{circ}}{sin 5^{circ}}.$$