Question

If \( y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) \), where \(\operatorname{sgn}(p)\) denotes the signum function of \(p\), then the sum of elements in the range of \(y\) is:

• A. \(4\)
• B. \(-2\)
• C. \(0\)
• D. \(2\)

Hint:

When dealing with signum of trigonometric functions, always analyze the problem {quadrant-wise} for one complete cycle.

Answer 1

The Correct Option is D

Concept:
The signum function is defined as: \[ \operatorname{sgn}(p) = \begin{cases} +1, & p>0\\ 0, & p=0\\ -1, & p<0 \end{cases} \] For this problem, we only consider intervals where the trigonometric functions are defined and non-zero.
Step 1: Analyze by Quadrants
Consider one full cycle \(0<x<2\pi\). First Quadrant \((0, \frac{\pi}{2})\):
\[ \sin x>0,\ \cos x>0,\ \tan x>0,\ \cot x>0 \] \[ y = 1+1+1+1 = 4 \] Second Quadrant \((\frac{\pi}{2}, \pi)\):
\[ \sin x>0,\ \cos x<0,\ \tan x<0,\ \cot x<0 \] \[ y = 1-1-1-1 = -2 \] Third Quadrant \((\pi, \frac{3\pi}{2})\):
\[ \sin x<0,\ \cos x<0,\ \tan x>0,\ \cot x>0 \] \[ y = -1-1+1+1 = 0 \] Fourth Quadrant \((\frac{3\pi}{2}, 2\pi)\):
\[ \sin x<0,\ \cos x>0,\ \tan x<0,\ \cot x<0 \] \[ y = -1+1-1-1 = -2 \] 
Step 2: Determine the Range of \(y\)
Possible values of \(y\): \[ \{4,\ -2,\ 0\} \] 
Step 3: Sum of Elements in the Range
\[ 4 + (-2) + 0 = 2 \] \[ \boxed{2} \]