Trigonometry Questions - JEE

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:

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

If a root of the equation $n^2\sin^2 x - 2\sin x - (2n+1) = 0$ lies in $\left[\dfrac{\pi}{2}, \pi\right]$, then the minimum positive integer value of $n$ is

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

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

If $\sum_{n=0}^{\infty} 2\cot^{-1}\left(\frac{n^2 + n + 4}{2}\right) = k\pi$, then find the value of $k$.

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:

Complete solution set of [\cot^{-1}x] + 2[\tan^{-1}x] = 0, where [\cdot] denotes the greatest integer function, is equal to

The value of $ \displaystyle\sum_{\omega=1}^{\infty} \sin^{-1}\left[\dfrac{2\omega+1}{\omega(\omega+1)(\sqrt{\omega^2+2\omega}+\sqrt{\omega^2-1})}\right] $ is equal to:

$2\cos x - 3\sin x = a$ has real solutions for x if:

If $2\tan^{-1}\frac{1}{5} - \sin^{-1}\frac{1}{5} = -\cos^{-1}\frac{63}{\lambda}$, then $\lambda =$

Given $0 \leq x \leq \frac{1}{2}$, then the value of $\sin^{-1}\left(\frac{x + \sqrt{1-x^2}}{2}\right) - \sin^{-1}x$ is

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

Let a_1 = 1, a_2, a_3, a_4, \ldots be consecutive natural numbers. Then \tan^{-1}\left(\frac{1}{1+a_1 a_2}\right) + \tan^{-1}\left(\frac{1}{1+a_2 a_3}\right) + \ldots + \tan^{-1}\left(\frac{1}{1+a_{2021}a_{2022}}\right) \text{ is equal to}

The upper $\left(\dfrac{3}{4}\right)$th portion of a vertical pole subtends an angle $\tan^{-1}\left(\dfrac{3}{5}\right)$ at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is:

If the sum and product of four positive consecutive terms of a G.P. are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

Let (a, b) \subset (0, 2\pi) be the largest interval for which \sin^{-1}(\sin\theta) - \cos^{-1}(\sin\theta) > 0, \theta \in (0, 2\pi) holds. If \alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0 and \alpha - \beta = b - a, then \alpha is equal to:

The sum of the roots of the equation \[\cos^{-1}(\cos x) = [x]\] where $[x]$ denotes the greatest integer function, is

Trigonometry Questions (1127)