# where f(x)= [1/ sin{x} ]   and  [.]=greatest integer and {}=fractional part function? explain and solve.

290 Points
12 years ago

Hi Shelley,

We know 0≤{x}<1

But as {x} is in the Denominator in this problem, we have 0<{x}<1

So sin0<sin{x}<sin1

or 0<sin{x}<sin1.

Also sin1>sin(pi/6)

or sin1>0.5.

Hence 0<sin{x}< (a number that is more than 0.5)

Hence 1/sin{x} can take all values more than 1/sin1

1/sin1 = 1.xyz....

Hence [1/sin{x}] can take all positive integral values.

And Hence the Range of this function is the set of all Natural Numbers.

Regards,

Mir Noaman Ali IIT-Roorkee
57 Points
12 years ago

fractional part of x means x is always between 0 and 1 radian meaning always between 0 and 57.2 degrees so as x increases from 0 to 57.2 sin x increases and 1/ sin x decreases so the integral part of it also decreases ...so its a decreasing function,as the final function is in integral part , the range will be only integers , now when x = 0 f (x) becomes infinite and when x keeps  thaon increasin from 0 to 57.2 degrees the integral values keep decreasin form infinite to ... [1/sin(57.2) degrees] thats integral part of 1.19 = 1 , so the range is from infinite to 1 all the integers between them,meaning the range is natural number..... APPROVE IF YOU HAVE UNDERSTOOD THE SOLUTION

Regards

Noaman Ali

IIT-Roorkee