Inverse Trigonometry Questions - JEE

Range of $f(x)=\cot^{-1}(\log_e(1-x^2))$ is:

Range of $f(x)=\sin^{-1}x+\tan^{-1}x+\sec^{-1}x$ is:

$\cot^{-1}\!\left(\dfrac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right)$, \(\pi/2:

$\sin^{-1}(\sin 5)>x^2-4x$ holds for:

If $\sin(2\cos^{-1}\frac{1}{\sqrt{5}})+\cos(2\tan^{-1}\frac{1}{3})=\frac{p}{q}$ (coprime), units digit of $(p-q)^{2k+1}$, $k\in\mathbb{N}$ can be:

If $\sin^{-1}\!\sqrt{x/2}+\sin^{-1}\!\sqrt{1-x/4}+\tan^{-1}y=\frac{2\pi}{3}$, then which are true?

If $ x = \sin^{-1}(\sin 10) $ and $ y = \cos^{-1}(\cos 10) $, then $ y - x $ is equal to:

For $\alpha,\beta,\gamma\neq0$. If $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals

If we consider only the principal values of the inverse trigonometric functions, then the value of $\tan^{-1}\left(\cos^{-1}\frac{1}{5} - \sin^{-1}\frac{4}{17}\right)$ is

The complete set of values of $x$ satisfying the inequality $\sin^{-1}(\sin 5) > x^2 - 4x$ is $(2 - \sqrt{\lambda - 2\pi}, 2 + \sqrt{\lambda - 2\pi})$, then $\lambda =$

If $\sin^{-1}a + \sin^{-1}b + \sin^{-1}c = \pi$, then the value of $a\sqrt{1-a^2} + b\sqrt{1-b^2} + c\sqrt{1-c^2}$ will be

Ex. 34. Statement I: If α, β are roots of 6x² + 11x + 3 = 0, then cos α exists but not cos⁻¹β (α > 0).Statement II: Domain of cos⁻¹x is [−1, 1].

$\sin^{-1}x > \cos^{-1}x$ holds for

Let $\cos^{-1}(4x^3 - 3x) = a + b\cos^{-1}x$.If $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$, then $\sin^{-1}\left(\sin\frac{a}{b}\right)$ is:

Let $f(x) = \sin^{-1}x - \cos^{-1}x$, then the set of values of $k$ for which $|f(x)| = k$ has exactly two distinct solutions is:

The sides of triangle ABC satisfy the equation $2a^2 + 4b^2 + c^2 - 4ab - 2ac = 0$. Then

All $ x $ satisfying the inequality $ (\cot^{-1}x)^2 - 7(\cot^{-1}x) + 10 > 0 $, lie in the interval:

$\cot^{-1}9+\csc^{-1}\!\dfrac{\sqrt{41}}{4}=$

$\displaystyle\sum_{r=0}^{\infty}\tan^{-1}\!\frac{1}{r^2+3r+3}=$

If $\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi$, then $xy+yz+zx=$

Range of $f(x)=\sin^{-1}x+\cos^{-1}x+\tan^{-1}x$ is:

$\tan\!\left[\frac{\pi}{4}+\frac{1}{2}\cos^{-1}x\right]+\tan\!\left[\frac{\pi}{4}-\frac{1}{2}\cos^{-1}x\right]$ equals:

Roots $r,s,t$ of $x(x-2)(3x-7)=2$ are real and positive. $\tan^{-1}r+\tan^{-1}s+\tan^{-1}t=$

Domain of $f(x)=\log_e(\cos^{-1}\{\sqrt{x}\})$ where $\{\cdot\}$ denotes fractional part:

$\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\tan^{-1}\!\frac{2r+1}{r^4+2r^3+r^2+1}=$

Range of $f(x)=\sin^{-1}(\log_2(-x^2+2x+3))$ is:

$f(x)=\cot^{-1}\!\sqrt{x(x+3)}+\cos^{-1}\!\sqrt{x^2+3x+1}$ is defined on set $S$. $S$ equals:

Total symmetric relations on $A$ ($|A|=n$):

In triangle ABC, if ∠B = sec⁻¹(5/4) + cosec⁻¹(5/3), ∠C = cosec⁻¹(25/7) + cot⁻¹(4/3), and c = 3, then tan A, tan B, tan C are in:

If $\alpha=2\tan^{-1}\!\frac{1+x}{1-x}$ and $\beta=\sin^{-1}\!\frac{1-x^2}{1+x^2}$ for $x>1$, then $\alpha+\beta=$

Find the value of $ \sum_{m=1}^{\infty} \tan^{-1}\!\left(\dfrac{3m^2 - 3m + 1}{m^6 - 3m^5 + 3m^4 - m^3 + 1}\right) $.

Given $\sin^{-1}\left(\dfrac{12}{13}\right) - \sin^{-1}\left(\dfrac{3}{5}\right) = ?$

Solve the equation $\cos^{-1}\sqrt{x^2 - 25} = \cos^{-1}\frac{\sqrt{x^2-25}}{x}$ for $x > 12$

If sides AB, BC and CA of a triangle ABC are represented by x + 2 = 0, 3x + y = 0 and x + 3y + 2 = 0 respectively, then identify the correct statement.(a) \sum \tan A = -\frac{4}{3}(b) \pi - (\tan^{-1}2 + \tan^{-1}3)(c) \frac{\pi}{4}(d) \pi + (\tan^{-1}2 + \tan^{-1}3)

If $\cot^{-1} x + \cot^{-1} y + \cot^{-1} z = \dfrac{\pi}{2}$, then $x + y + z$ equals

In an acute angled triangle ABC, ∠A = 20°, let DEF be the feet of altitudes through A, B, C respectively and H is the orthocentre of △ABC. Find AH/AD + BH/BE + CH/CF.

The number of solutions of $\sin^{-1}\left(\frac{1+x^2}{2x}\right) = \frac{\pi}{2}\sec(x-1)$ is

If $ (\sin^{-1}x)^3 + (\cos^{-1}x)^3 - a\pi^2 = 0 $, then $ a $ belongs to:

The value of $\displaystyle\sum_{m=1}^{\infty}\left(\tan^{-1}\!\left(\dfrac{3m^2 - 3m + 1}{m^6 - 3m^5 + 3m^4 - m^3 + 1}\right)\right)$ equals:

Number of values of $x$ satisfying simultaneously $\sin^{-1} x = 2\tan^{-1} x$ and $\tan^2\sqrt{x(x-1)} + \csc^{-1}\sqrt{1+x-x^2} = \frac{\pi}{2}$

The value of $\cos\{\tan^{-1}(\tan 2)\}$ is

If the equation $5 \arctan(x^2 + x + k) + 3 \text{arccot}(x^2 + x + k) = 2\pi$ has two distinct solutions, then the range of $k$ is

$\cos^{-1}(\cos \theta) = \theta$, for all $\theta$ belonging to

If $\sin^{-1}\left(x - \frac{x^2}{2} + \frac{x^3}{4} - K\right) + \cos^{-1}\left(x^2 - \frac{x^2}{2} + \frac{x^4}{4} - K\right) = \frac{\pi}{2}$ for \(0

A value of $ x $ satisfying the equation $ \sin[\cot^{-1}(1+x)] = \cos[\tan^{-1}x] $ is

Consider the equation $\tan^{-1}x + \cos^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right) = \sin^{-1}\left(\frac{3}{\sqrt{10}}\right)$. Let $\alpha =$ sum of positive integral solutions of $x$ and $\beta =$ sum of positive integral solutions of $y$. Then $\beta - \alpha$ = _______.

If $x$ takes negative permissible value then $\sin^{-1} x$ is

The value of $ \cot\!\left(\displaystyle\sum_{n=1}^{19}\cot^{-1}\!\left(1+\displaystyle\sum_{p=1}^{n}2p\right)\right) $ is:

If $f(x) = \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right) - 2\tan^{-1}x$ and $g(x) = \sin^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right) + 4\tan^{-1}x$, then range of $(f(x) - g(x))$ for $x \in (-\infty, -1]$ is:

The domain of f(x) = sin⁻¹(2x) is

Inverse Trigonometry Questions (1043)