Inverse Trigonometry Questions - JEE

Let m and n be positive real numbers such that m + n = 3. If $\frac{m}{s} = \sin^2\theta$ and $\frac{n}{t} = \cos^2\theta$, then the minimum value of $s + t$ is:

If $E = (3\sqrt{5} - 4\cos x + \sqrt{13 - 12\sin x})$, find the minimum value of $E^2$.

The number of solutions of the equation $\sqrt{1 + \cos 2x} = \sqrt{2}\sin^{-1}(\sin x)$ for $x \in [-\pi, \pi]$ is:

The value of \[ I = \sum_{r=0}^{10} \frac{1}{4}\left(\cos\frac{3\pi r}{3} + 3\cos\frac{\pi r}{3}\right) \] is equal to ___.

If T(n) = cos²(30° − n°) − cos(30° − n°)cos(30° + n°) + cos²(30° + n°), find the value of $4\sum_{n=1}^{30} nT(n)$.

262. If $\sec^{-1}(x) + \tan^{-1}\sqrt{9y^2 - 1} + \sin^{-1}(x^2 + y^2) = \lambda$ has no solution, then exhaustive set of values of $\lambda$ is equal to:

If $(\sin^{-1}x)^2 + (\sin^{-1}y)^2 + 2\sin^{-1}x\sin^{-1}y = \pi^2$, then $x^2 + y^2$ is equal to:

The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

If sum of all the solutions of the equation $8\cos x\left[\cos\left(\frac{\pi}{6}+x\right)\cdot\cos\left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right]=1$ in $[0,\pi]$ is $k\pi$, then $k$ is equal to

Find the number of solutions of the equations $(\sin x - 1)^3 + (\cos x - 1)^3 + (\sin x)^3 = (2\sin x + \cos x - 2)^3$ in $[0, 2\pi]$.

If $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ in triangle $PQR$ is

In a triangle with sides $a, b, c$ where $s - a + s - b + s - c = 15$ (so $s = 15$) and the incircle touches side $BC$ at $Q$ and side $CA$ at $C'$ with $QC = s - c$. If $s - a = 3,\; s - b = 5,\; s - c = 7$, find the area of quadrilateral $QCRI$ (where $I$ is the incentre and $R$ is the point of tangency on $CA$).

Given the angle of elevation of a cloud from a point P which is 25 m above a lake is $30°$ and the angle of depression of the reflection of the cloud in the lake from P is $60°$. Find the height of the cloud from the surface (in metres).

Statement I: $y = \tan^{-1}(\tan x)$ and $y = \cos^{-1}(\cos x)$ are not the same functionStatement II: The range of $\tan^{-1}(\tan x)$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and the range of $\cos^{-1}(\cos x)$ is $[0, \pi]$

The number of solutions of the pair of equations 2 sin²θ − cos 2θ = 0 and 2 cos²θ − 3sin θ = 0 in the interval [0, 2π] is:

The value of $\sin\left[\tan^{-1}\left(\tan\dfrac{7\pi}{6}\right) + \cos^{-1}\left(\cos\dfrac{7\pi}{3}\right)\right]$ is

The range of values of $k$ for which the equation $2\cos 4x - \sin 4x + k = 0$ has at least one solution is $[l, m]$. Find the value of $9m + l$.

Total number of solutions of $\sin^4 x + \cos^4 x = \sin x \times \cos x$ in $[0, 2\pi]$ is equal to

The general solution of $e^x - 1 = 2(e^{\sin x} + e^{\cos x}) = 2$ is

If $\tan\frac{a}{2}$ and $\tan\frac{b}{2}$ are the roots of the equation $8x^2 - 26x + 15 = 0$, then $\cos(a + b)$ is equal to

The number of solutions of the equation $\sin^{-1}\left(x + \frac{2}{3}\right) + \cos^{-1}\left(x - \frac{2}{3}\right) = x^2$ for x ∈ [−1, 1], where [x] denotes the greatest integer less than or equal to x

$\sin 47° + \sin 61° - \sin 11° - \sin 25°$ is equal to

The value of $\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}$ is

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression $\frac{a}{c}\sin 2C + \frac{c}{a}\sin 2A$ is

If cos(α + β) = 4/5, sin(α - β) = 5/13 and α, β lie between 0 and π/4, then tan 2α is equal to

If (sin A - sin C)/(cos C - cos A) = cot B, then A, B and C are in

The value of $(\cos^4 1° + \cos^4 2° + \cos^4 3° + \ldots + \cos^4 179°) - (\sin^4 1° + \sin^4 2° + \sin^4 3° + \ldots + \sin^4 179°)$ equals

If $\sin(x\cos\theta) = \cos(x\sin\theta)$ then $\sin 2\theta$ is equal to

Let the maximum value of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$ for $x\in\left[-\dfrac{\sqrt{3}}{2},\dfrac{1}{\sqrt{2}}\right]$ be $\dfrac{m}{n}\pi^2$, where $\gcd(m,n)=1$. Then $m+n$ is equal to _____.

If $k=\tan\!\left(\dfrac{\pi}{4}+\dfrac{1}{2}\cos^{-1}\!\dfrac{2}{3}\right)+\tan\!\left(\dfrac{1}{2}\sin^{-1}\!\dfrac{2}{3}\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x$ is _____.

$\tan^{-1}\left(\frac{c_1 x - y}{c_1 y + x}\right) + \tan^{-1}\left(\frac{c_2 - c_1}{1 + c_2 c_1}\right) + \tan^{-1}\left(\frac{c_3 - c_2}{1 + c_3 c_2}\right) + \ldots + \tan^{-1}(1)$ is equal to

Find the value of $\frac{\cos A + \cos B}{\sin A - \sin B} + \frac{\sin A + \sin B}{\cos A - \cos B}$ (where $n$ is even).

The value of $\tan^{-1} \left[ \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right]$ is

Suppose that $a$ is a non-zero real number for which $\sin x + \sin y = a$ and $\cos x + \cos y = 2a$. The value of $\cos(x - y)$ is

$\sin(\alpha - \beta)$ is equal to

In an acute angled triangle $ABC$, $\angle A = 20°$, let $DEF$ be the feet of altitudes through $A, B, C$ respectively and $H$ is the orthocentre of $\triangle ABC$. Find $$\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF}$$

The minimum value of the function $f(x) = (3\sin x - 4\cos x - 10)(3\sin x + 4\cos x - 10)$ is

A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 m and the distance between B and C is 100 m, then the height of the balloon is given by

A variable triangle ABC is circumscribed about a fixed circle of unit radius. Side BC always touches the circle at D and has fixed direction. If B and C vary in such a way that (BD)·(CD) = 2, then the locus of vertex A will be a

Draw the curves p|cos θ| and q|sin θ| and find the number of intersection points.

If $t = x + y + z$, then $\sin x + \sin y + \sin z - \sin t$ equals:

Let $x + \frac{1}{x} = 2, y + \frac{1}{y} = -2$ and $\sin^{-1} \left(\frac{1}{x}\right) + y = mx$, then the value of $m$ is

Let A(7,0), B(4,4) and C(0,0) and triangle DEF is isosceles with DE = DF. Then, the curve on which F may lie

$\sin^{-1}(3x - 4x^3) = \lambda \sin^{-1} x$ then $\lambda = \underline{\quad}$.

If inside triangle ABC, a, b, c and angle A are given and \(c\sin A

The value of $\sqrt{\sin^2\frac{2\pi}{11} - \cos\frac{8\pi}{11}}$ is equal to

If $P = \frac{\tan(3^n + 10) - \tan\theta}{\cos(3^n \theta)}$ and $Q = \text{(expression)}$, then

Given, $a^2 + 2a + \csc^2\frac{x}{2} - (a+x) = 0$, then which of the following holds good?

Let f(x) = sin²³x + cos²²x and g(x) = 1 − $\frac{1}{2}$ tan⁻¹|x|. The number of values of x in interval [−100°, 200°] satisfying the equation f(x) = sgn(g(x)), is 5a. Then, a is equal to ……

The complete set of values of a for which the function f(x) = \tan^{-1}(x^2 - 18x + a) \geq 0, \forall x \in \mathbb{R}, is

Inverse Trigonometry Questions (1043)