Trigonometry Questions - JEE

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 ________.

If $2(\sin A-\sin^3 A)=\cos B$ and $2(\cos A+\cos^3 A)=\sin B$; $0<A,B<\pi/2$, then $\cos B=\sqrt{\frac{m}{n}}$ where $m,n$ are co-prime and $m+n$ is

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 value of $x \in \left(0,\dfrac{\pi}{2}\right)$ satisfying $\dfrac{\sqrt{5}-1}{\sin x} + \dfrac{\sqrt{10+2\sqrt{5}}}{\cos x} = 8$ is

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 $y = \tan^{-1}\dfrac{4x}{1+5x^2} + \tan^{-1}\dfrac{2+3x}{3-2x}$, find $\dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2}$. Find $\alpha$.

Let $T(\theta) = \cos^2(30°-\theta) - \cos(30°-\theta)\cos(30°+\theta) + \cos^2(30°+\theta)$. Then the value of $4\displaystyle\sum_{\theta=1}^{30} \theta\, T(\theta)$ is

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

Value of $\sin32°\cdot\sin88°\cdot\sin152°$ equals

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

Let set $A$ denote the solutions of $\cos^{-1}(4x^3-3x)=\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$. Then

The upper four-fifth portion of a vertical tower subtends an angle $\tan^{-1}\frac{8}{21}$ at a point $A$ in the horizontal plane through its foot and at a distance 50 m from the foot. If the angle subtended by the lower one fifth of tower at point $A$ is $\beta$, then the height of the tower can be

Let $\frac{5}{6}\cos^{-1}\sqrt{\dfrac{3}{3+\pi^2}}+\frac{1}{3}\sin^{-1}\dfrac{2\sqrt{3}\pi}{3+\pi^2}+\frac{1}{6}\tan^{-1}\dfrac{\sqrt{3}}{\pi}=a$ and $\cos^{-1}\!\left[\frac{13}{40}\cos\!\left(\cot^{-1}\frac{5}{12}\right)+\frac{13}{32}\sin\!\left(\cos^{-1}\frac{5}{13}\right)\right]=b$. Then $\csc\!\left(\displaystyle\int_b^a\left[\frac{\tan x}{\sqrt{3}}\right]dx\right)$ is ($[\cdot]$ = GIF)

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 $\cos^{-1}\!\sqrt{p}+\cos^{-1}\!\sqrt{1-p}+\cos^{-1}\!\sqrt{1-q}=\dfrac{3\pi}{4}$, then $q$ is

If $x=\cos1°\cos2°\cos3°\cdots\cos89°$ and $y=\cos2°\cos6°\cos10°\cdots\cos86°$, then $\dfrac{2}{7}\log_2\!\left(\dfrac{y}{x}\right)$ is equal to

Number of ordered pairs $(x,y)$ satisfying $\dfrac{4^{\sin x}\cdot16^{\sin y}}{(1+16^{\sin x})(1+256^{\sin y})}=\dfrac{1}{4}$ and $3^{1+\sqrt{\cos^2x}}+3^{1+\cos y}=10$; $x,y\in[0,2\pi]$ is

Let $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ be quadratic polynomials with real coefficients. If $\tan\theta_1, \tan\theta_2$ are the roots of $P(x)$ and $\cot\theta_1, \cot\theta_2$ are the roots of $Q(x)$ for some $\theta_1, \theta_2 \in (0, \pi/2)$, such that $\theta_1 + \theta_2 = \pi/4$ and $P(1)\cdot Q(1) = 16$, then the value of $\dfrac{a}{c+d}$ is equal to:

Let $\dfrac{\pi}{2}<x<\pi$ such that $\cot x = -\dfrac{5}{\sqrt{11}}$. Then $\left(\sin\dfrac{11x}{2}\right)(\sin^6 x-\cos^6 x)+\left(\cos\dfrac{11x}{2}\right)(\sin^6 x+\cos^6 x)$ equals:

Suppose $a$ is a real number such that the equation $a(\sin x+\sin 2x)=\sin 3x$ has more than one solution in the interval $(0,\pi)$. The number of integral values of $a$ satisfying the given condition is:

Let $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ be quadratic polynomials with real coefficients. If $\tan\theta_1, \tan\theta_2$ are the roots of $P(x)$ and $\cot\theta_1, \cot\theta_2$ are the roots of $Q(x)$ for some $\theta_1, \theta_2 \in (0, \pi/2)$, such that $\theta_1 + \theta_2 = \pi/4$ and $P(1)\cdot Q(1) = 16$, then the value of $\dfrac{a}{c+d}$ is equal to:

Considering only principal values of inverse trigonometric functions, the value of $\tan\!\left(\sin^{-1}\frac{3}{5}-2\cos^{-1}\frac{2}{5}\right)$ is:

Match List-I (number of solutions in given intervals) with List-II (counts): P) $\cos3x\cos6x=\cos4x\cos7x$, $x\in[0,\pi/2)$ Q) $\sin2x\sin6x=\cos x\cos3x$, $x\in[0,\pi/2]$ R) $\cos3x\cos7x=\cos2x\cos8x$, $x\in[0,\pi/2]$ S) $\sin5x\cos3x=\sin6x\sin2x$, $x\in[0,\pi/2)$ List-II: 1)1, 2)2, 3)3, 4)4, 5)5

$\displaystyle\sum_{n=1}^\infty\cot^{-1}\!\left(\frac{(2n^2+2n+1)(n^2+n+1)}{n^4+2n^3+2n^2+2n+2}\right)=\sec^{-1}\!\left(\frac{5}{\lambda}\right)$. Then $\lambda$ equals:

Considering only principal values of inverse trigonometric functions, the value of $\tan\!\left(\sin^{-1}\frac{3}{5}-2\cos^{-1}\frac{2}{5}\right)$ is:

Trigonometry Questions (1127)