Trigonometry & Inverse Trigonometry Questions - JEE

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

Let a and b be the lengths of the legs of a right triangle with the following properties: (a) All 3 sides of the triangle are integers. (b) The perimeter of the triangle is numerically equal to area of the triangle, it is given that a < b. The number of ordered pairs (a, b) will be :

The value of $S = \dfrac{\sin^2\dfrac{2\pi}{7}}{\sin^2\dfrac{\pi}{7}} + \dfrac{\sin^2\dfrac{4\pi}{7}}{\sin^2\dfrac{2\pi}{7}} + \dfrac{\sin^2\dfrac{\pi}{7}}{\sin^2\dfrac{4\pi}{7}}$ is:

If ABCD is a cyclic quadrilateral then $\cos A + \cos B + \cos C + \cos D$ is equal to

A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (= a) subtends an angle of 60° at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30°. The height of the tower is

A $150\left(\sqrt{3}+1\right)$ ft.

Which are correct?(A) $\cot^{-1}x=\tan^{-1}(1/x)\ \forall x\in\mathbb{R}\setminus\{0\}$(B) $f(x)=\text{sgn}(e^x)$ is into(C) $f:\mathbb{R}^+\to\mathbb{R},\,f(x)=\sin x+x$ is odd(D) $f(x)=e^x/e^{[x]}$ is periodic

$\sin[\cot^{-1}\{\tan(\cos^{-1} x)\}]$ is equal to

The number of possible solutions of x such that $\sin^2 x + \cos^2 x = 1$ is:

Let $a \in \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$ such that $\tan^{-1}\!\left(\dfrac{\tan\alpha}{3 + 2\tan^2\alpha}\right) + \tan^{-1}\!\left(\dfrac{2\tan\alpha}{3}\right) = \dfrac{\pi}{12}$, then $\alpha$ equals:

811. In $\triangle ABC$, if $\sin A \sin B \sin C + \cos A \cos B = 1$ then the value of $\cos^2 A + \sin^2 B + 2\sin^2 \dfrac{C}{2}$ is:

If \(2\le a

If \(0 \leq x

The sum of all values of $\theta \in \left(0, \dfrac{\pi}{2}\right)$ satisfying $\sin^2 2\theta + \cos^4 2\theta = \dfrac{3}{4}$ is:

If \(0

Suppose 3\sin^{-1}(\log_2 x) + \cos^{-1}(\log_2 y) = \frac{\pi}{2} and \sin^{-1}(\log_2 x) + 2\cos^{-1}(\log_2 y) = \frac{11\pi}{6} then the value of x^2 + y^2 equals

If $\cos^{-1}(2x^2-1)=2\pi-2\cos^{-1}x$, then:

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

Trigonometry & Inverse Trigonometry Questions (1013)