The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$

where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]

Question

The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$

where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]

Smart Achievers · Accepted Answer

(3)

$x \in (0, \frac{π}{2}) \Rightarrow y = 1 + 1 + 1 + 1 = 4$

$x \in (\frac{π}{2}, π) \Rightarrow y = 1 - 1 - 1 - 1 = - 2$

$x \in (π, \frac{3 π}{2}) \Rightarrow y = - 1 - 1 + 1 + 1 = 0$

$x \in (\frac{3 π}{2}, 2 π) \Rightarrow y = - 1 + 1 - 1 - 1 = - 2$

$∴ Range of y is {- 2, 0, 4}$

$Required sum = - 2 + 0 + 4 = 2$

Solution

Q.
The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$

where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]

1 4

2 -2

3 2

4 0

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Solution

Q. The sum of all the elements in the range of f(x)=Sgn(sinx)+Sgn(cosx)+Sgn(tanx)+Sgn(cotx), x≠nπ2, n∈ℤ where sgn(t)={1if-1ift>0t<0 is [2026]

1 4

2 -2

3 2

4 0

Ans. (3) x∈(0,π2)⇒y=1+1+1+1=4 x∈(π2,π)⇒y=1-1-1-1=-2 x∈(π,3π2)⇒y=-1-1+1+1=0 x∈(3π2,2π)⇒y=-1+1-1-1=-2 ∴ Range of y is {-2,0,4} Required sum=-2+0+4=2

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Q.
The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$

where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]