Trigonometry & Inverse Trigonometry Questions (1013)

169. If \(\sin\alpha + \sin\beta + \sin\gamma = -3\), \(\alpha, \beta, \gamma \in (0, 2\pi)\), then \(\cos 2\alpha + \cos 4\beta + \cos 6\gamma\) is equal to:

703. Let \(f(x) = \cos^{-1}\!\left(\sqrt{\sin^{-1}\!\left(\sec\!\left(\ln\!\left(\dfrac{2x^2+3x-2}{x^2-3x+2}\right)\right)\right)}\right)\). Find the value of \(1 + \left(\displaystyle\sum \alpha_i^2\right)\), where \(\alpha_i\) represents the integers in the range of \(f(x)\). If there are no integers in the range of \(f(x)\), then enter your answer as zero.

175. If \(A\) lies in the fourth quadrant and \(3\tan A + 4 = 0\), then \(5\sin 2A + 2\sin A + 4\cos A\) is equal to:

The angles A, B and C of a triangle ABC are in arithmetic progression. If \(2b^2 = 3c^2\) then the angle A is:

Let $|\cos\theta\cos(60^\circ-\theta)\cos(60^\circ+\theta)|\leq\dfrac{1}{8}$, $\theta\in[0,2\pi]$. Then, the sum of all $\theta\in[0,2\pi]$, where $\cos3\theta$ attains its maximum value, is:

The angle of elevation of the top $P$ of a tower from the feet of one person standing due south of the tower is $45°$ and from the feet of another person standing due west of the tower is $30°$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

If $S=\left\{x\in\mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^2+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^2+1}}\right)=\dfrac{\pi}{4}\right\}$, then $\displaystyle\sum_{x\in S}\left(\sin\frac{(x^2+x+5)\pi}{2}-\cos\frac{(x^2+x+5)\pi}{2}\right)$ is equal to _________.

For $x\in(-1,1]$, the number of solutions of the equation $\sin^{-1}x=2\tan^{-1}x$ is equal to

In $\triangle ABC$, if $\cos A+2\cos B+\cos C=2$ and the sides opposite to $A$ and $C$ are $3$ and $7$ respectively, then $\cos A-\cos C$ is equal to

The value of $\tan 9°-\tan 27°-\tan 63°+\tan 81°$ is _____.

In △ABC, a = 4, b = 12 and B = 60°, then the value of sin A is

The principal value of \(\tan^{-1}(-\sqrt{3})\) is ______.

If \sum , then a + b is equal to : 13 1 2 2 { } = a\sqrt3 + b, a, b \in Z r=1 \pi \pi \pi r\pi sin( +(r-1) ) sin( + ) 4 6 4 6

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