Question

find the set of values of theta lying in the interval -1/2 pi < theta < 1/2pi such that the sum to infinity of the geometric series 1+sintheta+sin^2theta +... is greater than 2

Answer 1

since you have a GP with
a = 1
r = sinθ
we want
1/(1-sinθ) > 2
sinθ > 1/2
so π/6 < θ < π/2

Answer 2

sum(all terms) = a/(1-r)
1/(1 - sinθ) > 2
1 > 2 - 2sinθ
2sinθ > 1
sinθ > 1/2

Now look at the graph of y = sinx
and knowing that sin π/6 = 1/2

www.wolframalpha.com/input/?i=graph+y+%3D+sinx+from+-%CF%80%2F2+to+%CF%80%2F2

π/6 < θ < π/2

Answer 3

thank you so much!!