Trigonometry

$\begin{aligned} S & = \sin^2 3^\circ + \sin^2 9^\circ + \sin^2 15^\circ + \cdots + \sin^2 165^\circ + \sin^2 171^\circ + \sin^2 177^\circ & \small \color{#3D99F6} \text{Note: } \sin (180^\circ - \theta) = \sin \theta \\ & = \sin^2 3^\circ + \sin^2 9^\circ + \sin^2 15^\circ + \cdots + \sin^2 15^\circ + \sin^2 9^\circ + \sin^2 3^\circ \\ & = 2 \underbrace{\left( \sin^2 3^\circ + \sin^2 9^\circ + \sin^2 15^\circ + \cdots + \sin^2 87^\circ\right)}_{\text{15 terms}} & \small \color{#3D99F6} \text{Note: } \sin^2 \theta = \frac 12 (1-\cos \theta) \\ & = 15 - \left( \cos 6^\circ + \cos 18^\circ + \cos 30^\circ + \cdots + \cos 174^\circ\right) & \small \color{#3D99F6} \text{Note: } \cos (180^\circ - \theta) = - \cos \theta \end{aligned}$

$\begin{aligned} \ \ \ & = 15 - \left( \cos 6^\circ + \cos 18^\circ + \cos 30^\circ + \cdots + \cos 90^\circ + \cdots - \cos 30^\circ - \cos 18^\circ - \cos 6^\circ\right) \\ & = \boxed{15} \end{aligned}$

In general, we have $\displaystyle \sin^2 \left(x^{\circ} + 90^{\circ} \right) = \cos^2 x^{\circ}$ . Well, our sum can be written as follows:

$\displaystyle \sin^2 \left(3^{\circ} \right) + \sin^2 \left(93^{\circ} \right)$

$\displaystyle + \sin^2 \left(9^{\circ} \right) + \sin^2 \left(99^{\circ} \right)$

$\displaystyle + \dots \$

$\displaystyle + \sin^2 \left(87^{\circ} \right) + \sin^2 \left(177^{\circ} \right)$

$\displaystyle = \sin^2 \left(3^{\circ} \right) + \cos^2 \left(3^{\circ} \right)$

$\displaystyle + \sin^2 \left(9^{\circ} \right) + \cos^2 \left(9^{\circ} \right)$

$\displaystyle + \dots \$

$\displaystyle + \sin^2 \left(87^{\circ} \right) + \cos^2 \left(87^{\circ} \right)$

$\displaystyle = 15$ ,

counting the terms and using the classic formula $\displaystyle \cos^2 x + \sin^2 x = 1$ .

e^(i x)=cosx + i sinx e^(-i x)=cosx-i sinx sinx=(e^(i x)-e^(-i x))/(2*i)

Killing a fly with a canon is at the same time unnecessary and effective. See that it forms a geometric progression and done. P.S:I know it isn’t the most eloquent of solutions

The angles follows the patterns : 6x -3 So the sum can be written as 2(sin^2 (3°) + sin^2 (9°) + . ...+ sin^2 (87°)) = 2(7) + 2sin^2 45°= 15