Answer:
see the attached for a graph
the range is (-∞, 1] ∪ [1, ∞)
Explanation:
You know the range of the sine function is [-1, 1], including zero. So, the minimum absolute value of 1/sin(x) will be 1/1 = 1, and it will go up from there (or down from -1) toward infinity. The vertical asymptotes will be where sin(x) = 0, at multiples of π.
To graph it by hand, you can pick a few points in any quadrant of the sine function and find their inverses. Plot those. You can take advantage of the symmetry about π/2 and about the origin. You can also take advantage of the periodicity.
A graph is attached.