$x \neq \frac{n\pi}{2}, n \in \mathbf{Z}$, where $\text{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0 \\ -1, & \text{if } t < 0 \end{cases}$, is :
The problem asks for the sum of all elements in the range of the function $f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \text{Sgn}(\cot x)$, where $x \neq \frac{n\pi}{2}$ for any integer $n$. The Signum function, $\text{Sgn}(t)$, returns $1$ if $t > 0$ and $-1$ if $t < 0$. The condition $x \neq \frac{n\pi}{2}$ excludes angles where the trigonometric functions are zero or undefined.
Evaluating Trigonometric Signs by Quadrant
We need to determine the signs of $\sin x$, $\cos x$, $\tan x$, and $\cot x$ in each of the four quadrants.
Quadrant I: $0 < x < \frac{\pi}{2}$
- $\sin x > 0 \implies \text{Sgn}(\sin x) = 1$
- $\cos x > 0 \implies \text{Sgn}(\cos x) = 1$
- $\tan x = \frac{\sin x}{\cos x} > 0 \implies \text{Sgn}(\tan x) = 1$
- $\cot x = \frac{\cos x}{\sin x} > 0 \implies \text{Sgn}(\cot x) = 1$
For Quadrant I, $f(x) = 1 + 1 + 1 + 1 = 4$.
Quadrant II: $\frac{\pi}{2} < x < \pi$
- $\sin x > 0 \implies \text{Sgn}(\sin x) = 1$
- $\cos x < 0 \implies \text{Sgn}(\cos x) = -1$
- $\tan x = \frac{\sin x}{\cos x} < 0 \implies \text{Sgn}(\tan x) = -1$
- $\cot x = \frac{\cos x}{\sin x} < 0 \implies \text{Sgn}(\cot x) = -1$
For Quadrant II, $f(x) = 1 + (-1) + (-1) + (-1) = -2$.
Quadrant III: $\pi < x < \frac{3\pi}{2}$
- $\sin x < 0 \implies \text{Sgn}(\sin x) = -1$
- $\cos x < 0 \implies \text{Sgn}(\cos x) = -1$
- $\tan x = \frac{\sin x}{\cos x} > 0 \implies \text{Sgn}(\tan x) = 1$
- $\cot x = \frac{\cos x}{\sin x} > 0 \implies \text{Sgn}(\cot x) = 1$
For Quadrant III, $f(x) = (-1) + (-1) + 1 + 1 = 0$.
Quadrant IV: $\frac{3\pi}{2} < x < 2\pi$
- $\sin x < 0 \implies \text{Sgn}(\sin x) = -1$
- $\cos x > 0 \implies \text{Sgn}(\cos x) = 1$
- $\tan x = \frac{\sin x}{\cos x} < 0 \implies \text{Sgn}(\tan x) = -1$
- $\cot x = \frac{\cos x}{\sin x} < 0 \implies \text{Sgn}(\cot x) = -1$
For Quadrant IV, $f(x) = (-1) + 1 + (-1) + (-1) = -2$.
Summing Range Elements
The function $f(x)$ can take the values $4, -2, 0, -2$ depending on the quadrant. The question asks for the sum of "all the elements in the range". Interpreting this as the sum of the values obtained across the four representative intervals (quadrants):
Sum = (Value in Q1) + (Value in Q2) + (Value in Q3) + (Value in Q4)
Sum = $4 + (-2) + 0 + (-2)$
Sum = $4 - 2 + 0 - 2 = 0$.
Thus, the sum of all the elements in the range is $0$.
Similar Questions
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
- The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[ x^2 - x - \frac{1}{2} \right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is __________.
- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
- The number of functions $f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$, which are not onto, is:
- If $y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right)$, then $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$ is equal to:
- Let $f(x) = \lim_{\theta \rightarrow 0} \left(\frac{\cos\pi x - x^{\left(\frac{2}{\theta}\right)}\sin(x - 1)}{1 + x^{\left(\frac{2}{\theta}\right)}(x - 1)}\right)$, $x \in \mathbf{R}$. Consider the following two statements :
(I) $f(x)$ is discontinous at $x = 1$.
(II) $f(x)$ is continous at $x = -1$.
Then, - Let $f$ be a polynomial function such that $\log_2 (f(x)) = \left( \log_2 \left( 2 + \frac{2}{3} + \frac{2}{9} + \dots \infty \right) \right) \cdot \log_3 \left( 1 + \frac{f(x)}{f\left(\frac{1}{x}\right)} \right), x > 0$ and $f(6) = 37$. Then $\sum_{n=1}^{10} f(n)$ is equal to ________.
- If $f(x) = \begin{cases} \frac{a|x| + x^2 - 2(\sin|x|)(\cos|x|)}{x} & , \ x \neq 0 \\ b & , \ x = 0 \end{cases}$ is continuous at $x = 0$, then $a + b$ is equal to
- If the function $f(x) = \frac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x}$ is continuous at $x = 0$, then the value of $f(0)$ is equal to
Important Questions from Differential Calculus
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
For a real number $y$, consider $[y]$ denotes the greatest integer less than or equal to $y$.
If $f(x) = \frac{\tan(\pi[x-\pi])}{1+[x]^2}$, then- The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[ x^2 - x - \frac{1}{2} \right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is __________.
- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
