Explanation

(a)
Let $S = \sin{10{^\circ} + \sin{20{^\circ} + \sin{30{^\circ} + ... + \sin{360{^\circ}}}}}$
Here, the angles $10{^\circ},\ 20{^\circ},\ 30{^\circ},\ldots,360{^\circ}$ are in AP.
Where first term $(\alpha) = 10{^\circ}$, Common difference $(\beta) = 10{^\circ}$
Let number of terms be $n$
$\therefore\ \ 360{^\circ} = 10{^\circ} + (n - 1)10{^\circ}$
$\Rightarrow \ \ n - 1 = 35\ \ \ \Rightarrow \ \ \ n = 36$
$\therefore\ \ S = \frac{\sin\frac{360{^\circ}}{2}}{\sin{5{^\circ}}}\sin{\lbrack 10{^\circ} + (36 - 1)5{^\circ}\rbrack}$
$= \frac{\sin{180{^\circ}}}{\sin{5{^\circ}}} \times \sin{(180{^\circ} + 5{^\circ})} = - \sin{180{^\circ} = 0}$