Trigonometry & Inverse Trigonometry Questions - JEE

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

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

Trigonometry & Inverse Trigonometry Questions (1013)