Trigonometry & Inverse Trigonometry Questions (1013)

The value of (sin 70 ) (cot 10 cot 70 - 1) is ∘ ∘ ∘

If sin x + sin x = 1, x \in (0, 2 \pi ) , then (cos 12 x + tan 12 x) + 3 (cos 10 x + tan 10 x+ 2 cos 8 x + tan 8 x) + (cos 6 x + tan 6 x) is equal to :

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum 2 2 values of 16 ((sec -1 x) + (cosec -1 x) ) is :

cos(sin -1 3 + sin -1 5 + sin -1 33 ) is equal to: 5 13 65

154. Let \(f: R \to \left(0, \dfrac{2\pi}{3}\right]\) defined as \(f(x) = \cot^{-1}(x^2 - 4x + \alpha)\). The smallest integral value of \(\alpha\) such that \(f(x)\) is into function, is equal to:

Let $x=\dfrac{m}{n}$ ($m$, $n$ are co-prime natural numbers) be a solution of the equation $\cos(2\sin^{-1}x)=\dfrac{1}{9}$ and let $\alpha,\beta$ ($\alpha>\beta$) be the roots of the equation $mx^2-nx-m+n=0$. Then the point $(\alpha,\beta)$ lies on the line

If $a=\sin^{-1}(\sin5)$ and $b=\cos^{-1}(\cos5)$, then $a^2+b^2$ is equal to

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:

If $\dfrac{\pi}{2} \leq x \leq \dfrac{3\pi}{4}$, then $\cos^{-1}\!\left(\dfrac{12}{13}\cos x + \dfrac{5}{13}\sin x\right)$ is equal to

If for some $\alpha,\beta$; $\alpha \leq \beta$, $\alpha+\beta = 8$ and $\sec^2(\tan^{-1}\alpha)+\operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2+\beta$ is ________.

If $\alpha>\beta>\gamma>0$, then the expression $\cot^{-1}\!\left\{\beta+\dfrac{1+\beta^2}{\alpha-\beta}\right\}+\cot^{-1}\!\left\{\gamma+\dfrac{1+\gamma^2}{\beta-\gamma}\right\}+\cot^{-1}\!\left\{\alpha+\dfrac{1+\alpha^2}{\gamma-\alpha}\right\}$ is equal to:

Let $S=\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\}$. Then $\displaystyle\sum_{x\in S}(2x-1)^2$ is equal to ________.

The integral $\displaystyle\int_{1/4}^{3/4}\cos\!\left(2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)dx$ is equal to

For $n\in\mathbb{N}$, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\dfrac{\pi}{4}$, then $n$ is equal to

Given a = 6, b = 3 and \(\cos(A - B) = -\frac{1}{4}\), find \(\sin A\).

If \(k_1 = \tan 27\theta - \tan 9\theta + \tan 9\theta - \tan 3\theta + \tan 3\theta - \tan \theta\) and \(k_2 = \frac{\sin 3\theta}{\cos 3\theta} + \frac{\sin 9\theta}{\cos 9\theta} + \frac{\sin 27\theta}{\cos 27\theta}\), then

If the value of $\dfrac{3\cos36^\circ+5\sin18^\circ}{5\cos36^\circ-3\sin18^\circ}$ is $\dfrac{a\sqrt{5}-b}{c}$, where $a,b,c$ are natural numbers and $\gcd(a,c)=1$, then $a+b+c$ is equal to:

If $\sin x=-\dfrac{3}{5}$, where $\pi<x<\dfrac{3\pi}{2}$, then $80\left(\tan^2 x-\cos x\right)$ is equal to

The set of real numbers a such that \(a^2 + 2a\), \(2a + 3\), \(a^2 + 3a + 8\) are the sides of a triangle, is:

The minimum and maximum values of \(a\sin x + b\sqrt{1 - a^2}\cos x + c\) (where \(|a| 0\)) respectively are

If \(\frac{\cos 0 \cos 2\theta}{1 - \sin \theta} + \frac{\sin \theta \sin 2\theta}{1 + \cos \theta} = 1 + \cos \theta\), then number of possible values of \(\theta\) is (where \(\theta \in [0, 2\pi]\))

Let $S = \{x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1)\}$. Then $\sum_{x \in S} (2x - 1)^2$ is equal to ______.

If \sin\theta = 3\sin(\theta + 2\alpha)\, then the value of \tan(\theta + \alpha) + 2\tan\alpha\ is

Let \(f(x) = 1 + 2\sin\left(\frac{\pi x}{e^x+1}\right)\), \(x > 0\), then \(f^{-1}(x)\) is equal to (assuming \(f\) is bijective)

Let \(g: \mathbb{R} \to \left[0, \frac{7\pi}{2}\right)\) is defined by \(g(x) = \cos^{-1}\frac{x}{1+x^2}\). Then the possible values of \(k\) for which \(g\) is a surjective function, is

Find the number of solutions to the equation \(y = |x^2 - 1| = |\tan^{-1}|x||\)

If \(\alpha = \frac{1}{3}\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{1}{3}\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) where \(x \geq \frac{4}{3}\), then the value of \(\dfrac{\cos 2\alpha + \sec\alpha + 3\sqrt{3}}{\sqrt{3}}\) is equal to:

If f(x) = ∑r=1n [tan−1(x+r) − tan−1(x+r−1)], then limx→0 f'(x) is

The value of x satisfying (cot−1x)(tan−1x) + 2(\(\frac{π}{2}\) − cot−1x) − 3tan−1x − 3(\(\frac{π}{2}\)) ≥ 0 is

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in meters) above the line AC is (JEE Main 2020)

If \(\dfrac{\cos x + \cos y + \cos z}{\cos(x+y+z)} = 2\) and \(\dfrac{\sin x + \sin y + \sin z}{\sin(x+y+z)} = 2\), then the value of \(\cos(x+y) + \cos(y+z) + \cos(z+x)\) is equal to: (where \(x, y, z \in R\))

If the solution of the equation $\log_{\cos x}\cot x + 4\log_{\sin x}\tan x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, is $\sin^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha$, $\beta$ are integers, then $\alpha + \beta$ is equal to:

The number of solutions of the equation: $x^2 + (x+1)\sin\frac{\pi x}{6} = \frac{3+x}{2}$; $-2 \leq x \leq 0$

The period of the function $f(x) = e^{\sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right)}$ is:

If $u=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}+\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}$, then the difference between maximum and minimum values of $u^2$ is given by:

A quadrilateral $ABCD$ in which $AB = a$, $BC = b$, $CD = c$ and $DA = d$ is such that one circle can be inscribed in it and another circle can be circumscribed about it. $\cos A =$

The number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\dfrac{\pi}{6}$, where $-\dfrac{1}{2\sqrt{6}}<x<\dfrac{1}{2\sqrt{6}}$, is equal to

If the angles of elevation of the top of a tower from three collinear points $A$, $B$ and $C$ on a line leading to the foot of the tower are $30°$, $45°$ and $60°$ respectively, then the ratio $AB : BC$ is

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

If $[\sin^{-1}\cos x - \sin^{-1}\tan x - 1] = 1$ whose $[.]$ denotes the greatest integer function, then $x$ belongs to:

Roots \(r,s,t\) of \(x(x-2)(3x-7)=2\) are real and positive. \(\tan^{-1}r+\tan^{-1}s+\tan^{-1}t=\)

If $\dfrac{\tan(A-B)}{\tan A}+\dfrac{\sin^2C}{\sin^2A}=1$, $A,B,C\in\left(0,\dfrac{\pi}{2}\right)$, then

Two rays are drawn through a point $A$ at an angle of $30°$. A point $B$ is taken on one of them at a distance $a$ from the point $A$. A perpendicular is drawn from the point $B$ to the other ray and another perpendicular is drawn from its foot to $AB$ to meet $AB$ at another point from where the similar process is repeated indefinitely. The length of the resulting infinite polygon line is:

Let $\dfrac{\pi}{2}<\theta<\pi$ and $\cot\theta=-\dfrac{1}{2\sqrt{2}}$. Then the value of $\sin\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta+\sin8\theta)+\cos\!\left(\dfrac{15\theta}{2}\right)(\cos8\theta-\sin8\theta)$ is equal to

The least value of $\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}$ is: (Where $A, B, C$ are interior angles of a triangle)

If the equation $a_1 + a_2 \cos 2x + a_3 \sin^2 x = 1$ is satisfied by every real value of $x$, then the number of possible values of the triplet $(a_1, a_2, a_3)$ is:

For a triangle ABC, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

If $\pi < \theta < \frac{3\pi}{2}$ and $\cos \theta = -\frac{3}{5}$, then $\tan \left(\frac{\theta}{2}\right)$ is equal to

A tower of height 50 m is located on top of a hill opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_1$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is

Let S = \left\{x \in \mathbb{R} : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}. If n(S) denotes the number of elements in S then: