Trigonometry & Inverse Trigonometry Questions - JEE

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

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

A tower of height $h$ stands at point $O$. Points $A$, $B$ are on the ground such that $OA = OC = h$, angle of elevation from $B$ to top of tower is $30°$, from $A$ is $45°$, and $AB = 54\sqrt{2}$. Find the height $h$ of the tower.

In the following incomplete sentences, fill in the blanks so that the resulting sentences may become true.(d) If \(|\tan x|

$T_1$ is an isosceles triangle with circumcircle K. Let $T_2$ be another isosceles triangle inscribed in K whose base is one of the equal side of $T_1$ and which overlaps the interior of $T_1$. Similarly create isosceles triangles $T_3$ from $T_2$, $T_4$ from $T_3$ and so on to the triangle $T_n$. Then the base angle of the triangle $T_n$ as $n \to \infty$ is

Circum radius of a △ABC is 3 units; let O be the circum centre and H be the orthocentre then the value of $\frac{1}{64}(AH^2 + BC^2)(BH^2 + AC^2)(CH^2 + AB^2)$ equals:

If $2[x + 32] = 3[x - 64]$ and $y = \displaystyle\prod_{j=1}^{9} \sin\left(\dfrac{2j-1}{18}\right)\pi$, then find the value of $\left[\dfrac{1}{x}\right] + \left[\dfrac{1}{16y}\right]$.[Note: Where [k] denotes greatest integer function less than or equal to k.]

If $\tan 2x \cdot \tan x = 1$, then $x$ is

If $\tan\theta = -\dfrac{4}{3}$, then $\sin\theta$ is

Let $p = \sin 1 \sin 3 \sin 5 \cdots \sin 89$. We have \[ p = \sqrt{\sin 1 \sin 3 \sin 5 \cdots \sin 177 \sin 179} \] and after simplification, $\dfrac{1}{p^2} = 2^{89}$. Find $2 + 89$.

The equation $2\cos^2\frac{x}{2} - \sin^2 x = x^2 + x - 2$, where \(x

The value of $4\cos\frac{\pi}{10} - 3\sec\frac{\pi}{10} - \tan\frac{\pi}{10}$ is equal to(a) $1$(b) $\sqrt{5} - 1$(c) $2$(d) $0$

Given, $\cos(a - b) = 1$ and $\cos(a + b) = \frac{1}{e}$. Find the number of ordered pairs $(a, b)$ satisfying the relation $\cos(2a) = \frac{1}{e}$ where \(-\pi

(A) $\sqrt{3}$

98. If in a △ABC, b : c = 2 : 1 and $\sin\left(B - C\right) = \dfrac{1}{2}$ then the △ABC is

Let $y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \cos ec^{-1}(\cos ec 7)$. If $y$ simplifies to $ar + b$ then $(a - b) =$

Let $\sec x + \tan x = \frac{22}{7}$, where \(0

Find the number of solutions to the equation involving inverse trigonometric functions where positive values of x and y satisfy the given constraint.

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60°. If the area of the quadrilateral is $4\sqrt{3}$, find the value of $n$ where $\angle D = 120°$, $AB = 2$, $BC = 5$ and $\angle B = 60°$.

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression of the car to change from 30° to 45°; then after this, the time taken (in min) by the car to reach the foot of the tower, is

The value of the expression $\sin\left(2\tan^{-1}\frac{1}{3}\right) + \cos\left(\tan^{-1}2\sqrt{2}\right)$ is

Considering only the principal values of inverse functions, the set $A = \{x \geq 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$

If $\dfrac{\csc\theta}{1} = \dfrac{p+q}{p-q}$, then $\left|\cot\left(\dfrac{\pi}{4} + \dfrac{\theta}{2}\right)\right|$ equals

The equation $4\cos 2x = 7 - 4x + 4x^2$ has no solution for $\theta$ if $x \in \mathbb{R}$.State whether the statement is true or false.

Number of integral solutions of the equation $\log_{\sin x} \sqrt{\sin^2 x} + \log_{\cos x} \sqrt{\cos^2 x} = 2$, where $x \in [0, 6\pi]$

Let $f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, \ldots$ Then for all $x \in R$, the value of $f_4(x) - f_6(x)$ is equal to __________ (up to four decimal places).

If a\sin^{-1}x - b\cos^{-1}x = c, then a\sin^{-1}x + b\cos^{-1}x is equal to

If $a \tan \alpha + \sqrt{a^2 - 1} \tan \beta + \sqrt{a^2 + 1} \tan \gamma = 2a$, where $a$ is constant and $\alpha, \beta, \gamma$ are variable angles, then find the least value of $2727(\tan^2 \alpha + \tan^2 \beta + \tan^2 \gamma)$.

The sides of a triangle are $\sin a$, $\cos a$, $\sqrt{1 + \sin a \cos a}$ for some $0

(h) $\frac{20}{\sqrt{3}}$ meters

If $a_1, a_2, a_3, \ldots, a_n$ are in AP with common difference $5$ and if $a_i a_j \neq -1$ for $i, j = 1, 2, \ldots, n$, then $\tan^{-1}\left(\frac{5}{1 + a_1 a_2}\right) + \tan^{-1}\left(\frac{5}{1 + a_2 a_3}\right) + \tan^{-1}\left(\frac{5}{1 + a_{n-1}a_n}\right) + \ldots + \tan^{-1}\left(\frac{5}{1 + a_n a_1}\right)$ is equal to

Note that if $\tan\theta$ is positive, then $\theta$ is in the first or third quadrant, so \(0°

In a triangle $ABC$, $\angle AGB = \dfrac{\pi}{2}$, $AD = 4$, $AG = \dfrac{2}{3} \times AD$. If $\angle BAG = \dfrac{\pi}{6}$ and $\angle ABC = \dfrac{\pi}{3}$, then the area of $\triangle ABC$ is:

With the usual notation, in $\triangle ABC$, if $\angle A + \angle B = 120°$, $a = \sqrt{3} + 1$ and $b = \sqrt{3} - 1$, then the ratio $\angle A : \angle B$ is

Ex. 34. Statement I: x=kπ2, k∈I does not represent the general solution of trigonometric equation.Statement II: Both x=nπ, n∈I and x=kπ2, k∈I satisfy the trigonometric equation sin13x−sin13xcos2x=0.

The complete set of values of $x$ satisfying \(\frac{2\sin 6x}{\sin x - 1}

Trigonometry & Inverse Trigonometry Questions (1013)