Inverse Trigonometry Questions - JEE

Let $\frac{5}{6}\cos^{-1}\sqrt{\dfrac{3}{3+\pi^2}}+\frac{1}{3}\sin^{-1}\dfrac{2\sqrt{3}\pi}{3+\pi^2}+\frac{1}{6}\tan^{-1}\dfrac{\sqrt{3}}{\pi}=a$ and $\cos^{-1}\!\left[\frac{13}{40}\cos\!\left(\cot^{-1}\frac{5}{12}\right)+\frac{13}{32}\sin\!\left(\cos^{-1}\frac{5}{13}\right)\right]=b$. Then $\csc\!\left(\displaystyle\int_b^a\left[\frac{\tan x}{\sqrt{3}}\right]dx\right)$ is ($[\cdot]$ = GIF)

Find the range of f(x) = \sin^{-1} x + \tan^{-1} x + \sec^{-1} x

Find the number of solutions of $\cos x = |1 + \sin x|$, \(0

$\text{cosec}^{-1}(\cos x)$ exists if:

If a = \tan x, then the value of \cot\left(\frac{\pi}{4} - a\right) is

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

In $\triangle ABC$, if incircle touches the sides $AB$, $BC$ and $CA$ at $P$, $Q$ and $R$ respectively and $s - a = 3$, $s - b = 5$ and $s - c = 7$, then area of the quadrilateral $QCRI$ is, where $I$ is incentre of $\triangle ABC$:[Note: Symbols used have usual meaning in $\triangle ABC$.]

If \sin\theta + \sqrt{3}\cos\theta = 6x - x^2 - 11, 0 \leq \theta \leq 4\pi, x \in \mathbb{R}, then:

If A = \cos(\cos x) + \sin(\cos x), then the least and greatest value of A are

Find the value of $\cos 3A + \cos 3B + \cos 3C$ given that $A + B + C = 180°$ (angles of a triangle), and determine under what conditions the expression equals $1 + \cos(3A + 3B)$. Specifically, evaluate: $\cos 3A + \cos 3B = 1 - \cos(3C)$, i.e., $2\cos\dfrac{3}{2}(A+B)\cos\dfrac{3}{2}(A-B) = 2\cos^2\dfrac{3}{2}(A+B)$. If $\cos\dfrac{3}{2}(A+B) = 0$, then $\dfrac{3}{2}(A+B) = 90°$, $A + B = 60°$, so $C = 120°$. What is the answer?

The principal value of \cos^{-1}\left(\cos\left(2\cot^{-1}(\sqrt{2}-1)\right)\right) is equal to

Ex. 35. Statement I: If tan⁻¹x + tan⁻¹y + tan⁻¹z = π/4 and x + y + z = 1, then arithmetic mean of odd powers of x, y, z is equal to 1/3.Statement II: For any x, y, z we have xyz − xy − yz − zx + x + y + z = 1 + (x − 1)(y − 1)(z − 1)

Find the number of solutions of the equation in the interval [0, 2π] where the graph of y = tan x and y = \frac{71}{x} intersect.

The number of integral values of k for which the equation 7\cos x + 5\sin x = 2k + 1 has a solution is

If f(x) = \cos[p^2] x + \cos[-p^2], where [\cdot] = G.I.F., then which statement is true?

If angle \theta\ be divided into two parts such that the tangent of one part is k\ times the tangent of the other and \phi\ is their difference, then \sin\phi\ is equal to

The number of values of $x$, for which $\tan^{-1}\!\left(\dfrac{1}{x}\right) = \pi + \tan^{-1} x$, \(0

In a triangle ABC, if tan B + C - A}{4} tan C + A - B}{4} tan A + B - C}{4} = 1, then find the value of cos A + cos B + cos C.

If $ \alpha = \cos^{-1}\!\left(\dfrac{3}{5}\right) $, $ \beta = \tan^{-1}\!\left(\dfrac{1}{3}\right) $, where \( 0

The value of $ \tan^{-1}\!\left[\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right] $, \( |x|

Sides of a triangle ABC are in AP. If \(a

Ex. 22: Statement I In a triangle ABC, if $aStatement II For triangle ABC, \(r_1r_2 + r_2r_3 + r_3r_1 = r$

If $a = 2$, then obviously $c = a - 1$, and then, from Eq. (i), $(a + 2)^2 = a(2a + 1)$. Find the value of $a$.

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

If $\alpha$ is a root of $5\sin^2 x + 3\sin x \cos x - 3\cos^2 x = 2$ and $\beta$ is a root of $\sin 2x - \cos 2x = 2 - \sin 2x$, then $\tan \alpha + \tan \beta$ can be equal to

Find the number of integral values of x satisfying $x! - (x-1)! > 0$ and \(\left(2^{\tan^{-1}x} - 4\right)(x-4)(x-10)

In a cyclic quadrilateral with one angle being $60°$, find the area given $\cos 60° = \frac{4 + 25 - c^2}{2 × 5}$.

Maximum value of $\cos x (\sin x + \cos x)$ is equal to:

There is a unique angle $\theta$ between $0^\circ$ and $90^\circ$ such that for non-negative integers $n$, the value of $\tan(2^n\theta)$ is positive when $n$ is a multiple of 3, and negative otherwise. The degree measure of $\theta$ is $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p + q$.

If \tan^{-1}\frac{x}{2} , x \in \mathbb{N}, then the maximum value of x is

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

If $\cos^{-1}x-\dfrac{y}{2}=\alpha$, then $4x^2-4xy\cos\alpha+y^2=$

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

If $\sin^{-1}\left(\frac{k}{4}\right) + \cos^{-1}\left(\frac{k}{2}\right) = \frac{\pi}{4}$, then the value of x is

$(G) \sqrt{3} - 1$

782. Find the number of integral values of x satisfying the inequality\[\frac{\left(2^{\tan^{-1}x} - 4\right)(x-4)(x-10)}{x! - (x-1)!}

(D) $10\sqrt{3}(2 + \sqrt{3})$

The number of solutions of the equation $1 + \sin^4 x = \cos^2(3x)$, $x \in \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]$ is

If the equation $\cos 3x + \cos 2x = \sin \frac{x}{2} + \sin \frac{3x}{2}$ is satisfied for $0 \leq x \leq 2\pi$, then the number of values of $x$ is

The most general values of $\theta$ satisfying $\tan\left(\theta + \frac{3\pi}{4}\right) + \tan\theta = 2$ is/are

The lengths of the sides CB and CA of a triangle ABC are given by a and b and the angle C is $\frac{2\pi}{3}$. The line CD bisects the angle C and meets AB at D. Then the length of CD is:

If $\tan x = -\frac{4}{3}$, $\frac{3\pi}{2} , find the value of \(9\sec^2 x - 4\cot x$.

If PQR is a triangle of area Δ with a = 2, b = 7/2, and c = 5/2, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively, then $\frac{2\sin P - \sin 2P}{2\sin P + \sin 2P}$ equals

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

If $\frac{\sin 3A}{\sin A} = k$, show that $\frac{\sin 3A}{\sin A} = \frac{2k}{k-1}$ and k cannot lie between $\frac{1}{3}$ and 3.

Given $1 + \sin^4 x = \cos^2 3x$, find the number of solutions for $x \in \left[-\dfrac{5\pi}{2}, \dfrac{5\pi}{2}\right]$.

If $u_n = \sin(n\theta)\sec^n \theta$, $v_n = \cos(n\theta)\sec^n \theta$, $n \in \mathbb{N}$, $n \neq 1$, then $\frac{v_n - v_{n-1}}{u_{n-1}} + \frac{1}{n}\frac{u_n}{v_n} =$

The maximum value of $\cos a_1 \cos a_2 \cdots \cos a_n$ under the restriction $0 and \(\cot a_1 \cot a_2 \cdots \cot a_n = 1$ is

172. The value of $\cos\!\left[\log_5\!\left(\dfrac{\sin^2 A + \cos^2 A + \tan^2 A - \sec^2 A \cdot \sin^2 A}{(1+\tan^2 A)(1-\sin^2 A)}\right)\right]$ is equal to:

The number of integer x satisfying \sin^{-1}|x-2| + \cos^{-1}(1-|3-x|) = \frac{\pi}{2} is

Inverse Trigonometry Questions (1043)