Trigonometry Questions (1127)

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:

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:

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:

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

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

The value of $\displaystyle\prod_{r=1}^{7} \cos\frac{r\pi}{15}$ is

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: