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Mathematics Question on Trigonometric Functions

The range of the function f(x)=sin⁡[x],−π4<x<π4f(x)= \sin [x],-\frac {\pi}{4} < x < \frac {\pi}{4} where [x][x] denotes the greatest integer ≤x\leq x, is ________

A

0\\{0\\}

B

0,−1\\{0,-1\\}

C

0,±sin 1\\{0, \pm sin\, 1\\}

D

0,−sin1\\{0, - sin 1\\}

Answer

0,−sin1\\{0, - sin 1\\}

Explanation

Solution

Given, f(x)=sin⁡[x],−π4<x<π4f(x)=\sin [x],-\frac{\pi}{4}< x< \frac{\pi}{4}
Clearly, sin⁡0=0\sin 0=0
and [π4]=[3.144]=0\left[\frac{\pi}{4}\right]=\left[\frac{3.14}{4}\right]=0
∴∀x∈[0,π4],sin⁡[x]=0\therefore \forall x \in\left[0, \frac{\pi}{4}\right], \sin [x]=0
∀x∈[−π4,0),[x]=−1\forall x \in\left[-\frac{\pi}{4}, 0\right),[x]=-1
∴sin⁡[x]=sin⁡(−1)=−sin⁡1\therefore \sin [x]=\sin (-1)=-\sin 1
So, the range of f(x)f(x) is 0,−sin⁡1\\{0,-\sin 1\\}.