240 Followers Problem - 4

$The ~ sum ~ given~ = - 1\frac 1 2,~~ |Cos\theta|~is~ never~ greater~ than~ 1,~~ the~ sum~~ is~~ negative,\\ we~also~ know~~ |Cos(60, ~120~240,OR~~300)|=\dfrac 1 2.\\ \therefore~ - 1\frac 1 2=- \frac 1 2 - \frac 1 2 - \frac 1 2, ~~~OR~~~ 0, - \frac 1 2, - 1 ~~~OR~~~-1,~-1, +\frac 1 2.\\ Let~us~try~the~1^{st}.\text{ Hence differences must be -120 and -240.}\\ \text{Let us take one angle as 60. The other two must be } 60\pm 120,~~~ or~~~60\pm 240. \\ \therefore~\theta_1=60^o, ~~\theta_2=180^o, ~~\theta_3=300^o,\\ !!!! ~It~works!\\ Sin^2 60+Sin^2 180+Sin^2 300=\Large ~~~~\color{#D61F06}{ 1\frac 1 2}$

I am coming up with more explanation shortly.