Question

sqrt sin(30)+sin(45)+sin(60)+sin(75).....infnity

Answer 1

since sin(180+x) = -sin(x) this series cannot be evaluated. Or, it can be evaluated in a variety of different ways, giving different answers.

The ratio test fails.

Answer 2

first assumption: the square root is over the whole thing

2nd:
it would be nice if the series has included the terms
sin 0 and sin 15
well, let's put them in

so we have
√ ( -sin 0 - sin 15 + (sin 0 + sin15 + sin30 + ... sin 345) + (sin 360 + sin 375 + .... sin 705) + (......) )

as Steve noted , we have opposite terms in each of the set of brackets
e.g.
sin 0 + sin 15 + .... + sin 345) = 0
so is each of the rest of the brackets
since the series goes to infinity,
I see the series as
√( 0 - sin 15 + 0+0+0+....)
which of course is not a real number, since -sin15° is negative,
and as Steve said , it cannot be evaluated.