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Find the range of [1/sin{x}] [.] denotes GIF, {.} denotes fractional part.
Find the range of [1/sin{x}]
[.] denotes GIF, {.} denotes fractional part.
Dear Speed,
f(x)=[1/sin{x}],0≤{x}<1,
0≤sin{x}<sin(1 radian)
1/(sin(1 radian))<1/sin{x}<∞
f(x) is any integer greater than [1/sin(1 radian)]
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CHETAN MANDAYAM NAYAKAR
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).
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.
Ashwin (IIT Madras).
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