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

To solve the given problem, we need to understand the role of the signum function, denoted as \operatorname{sgn}(p).

The signum function indicates:

• \operatorname{sgn}(p) = 1 if p \gt 0
• \operatorname{sgn}(p) = 0 if p = 0
• \operatorname{sgn}(p) = -1 if p \lt 0

Given:

y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x)

Let's evaluate the sign for each trigonometric function individually based on values of x:

• \sin x and \cos x range between -1 and 1. Their sign alternates based on the quadrant of x.
• \tan x = \frac{\sin x}{\cos x}. Hence, its sign depends on the signs of both \sin x and \cos x.
• \cot x = \frac{\cos x}{\sin x}. Similarly, its sign also depends on the signs of \cos x and \sin x.

Key observations for each quadrant:

1. In the first quadrant: \sin x \gt 0, \cos x \gt 0 \Rightarrow \tan x \gt 0, \cot x \gt 0
2. In the second quadrant: \sin x \gt 0, \cos x \lt 0 \Rightarrow \tan x \lt 0, \cot x \lt 0
3. In the third quadrant: \sin x \lt 0, \cos x \lt 0 \Rightarrow \tan x \gt 0, \cot x \gt 0
4. In the fourth quadrant: \sin x \lt 0, \cos x \gt 0 \Rightarrow \tan x \lt 0, \cot x \lt 0

Now, calculating y in each quadrant:

• 1st quadrant: y = 1 + 1 + 1 + 1 = 4
• 2nd quadrant: y = 1 - 1 - 1 - 1 = -2
• 3rd quadrant: y = -1 - 1 + 1 + 1 = 0
• 4th quadrant: y = -1 + 1 - 1 - 1 = -2

Hence, the range of y is: \{4, -2, 0\}

The sum of elements in this range is:

4 + (-2) + 0 = 2

Thus, the correct answer is 2.