Find the range of [1/sin{x}]

[.] denotes GIF, {.} denotes fractional part.

Hi Speed Racer,

Clearly 0≤{x}<1.

So 0≤sin{x}<sin1

Now 1<pie/2, but 1>pie/6.

So sin1<1. but sin1>1/2

So sin{x} can take all values from (0,1/2+delta]...... (where delta is >0)

And hence the range of this function would be all positive integers.

As sin{x} can be = 1/n where n=2,3,4,...... (also sin{x} = 1/2+delta, would give 1/sin{x} lies between 1 to 2)

So [1/sin{x}] = 1.

And hence range is all positive integers.

Hope that helps.

All the best.

Regards,

Ashwin (IIT Madras).