Inverse Trigonometry Questions - JEE

28. If the equation $\sum_{n=0}^{10} \text{arc cot}\left(\frac{1+2^{2n+1}}{2^n}\right) = \text{arc cot}\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. The value of $\log_2\left(\frac{b+a}{a-b}\right)$, is:

10. For a triangle ABC with $\cot A + \cot B + \cot C = \cot \theta$, find $\sin(A - \theta)\sin(B - \theta)\sin(C - \theta)$:

The range of the function f(x) = sec−1(x) + tan−1(x) is

Statement I: If $a, b, c \in \mathbb{R}$ and not all equal, then $\frac{bc + ca + ab}{a^2 + b^2 + c^2} Statement II: \(\sec \theta 1$

Let $\cos(\alpha+\beta)=-\dfrac{1}{10}$ and $\sin(\alpha-\beta)=\dfrac{3}{8}$, where $0<\alpha<\dfrac{\pi}{3}$ and $0<\beta<\dfrac{\pi}{4}$. If $\tan2\alpha=\dfrac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$, $r,s\in\mathbb{N}$, then $r+s$ is equal to _____.

If ∑∞n=0 2 cot-1(n² + n + 4)/2 = kπ, then find the value of k.

In the given figure, if AB = AC, ∠BAD = 30° and AE = AD, then x is equal to

In a triangle ABC, \angle C = \frac{\pi}{4}, a = \sqrt{2} and b = \sqrt{2 + \sqrt{2}}. Find the sum of digits in the measure of angle A (in degrees).

Let $\triangle ABC$ be inscribed in a circle having radius unity. The three internal bisectors of the angles $A, B$ and $C$ are extended to intersect the circumcircle of $\triangle ABC$ at $A_1, B_1$ and $C_1$ respectively. Find $$\frac{AA_1\cos\frac{A}{2} + BB_1\cos\frac{B}{2} + CC_1\cos\frac{C}{2}}{\sin A + \sin B + \sin C}$$

From point $ D $, 40 m away from the base $ A $ of a vertical tower $ BC $ of height $ h $, the angle of elevation of the top $ C $ is $ 30^\circ $. From point $ B $ (at the base of the tower), the angle of elevation of $ C $ is $ 60^\circ $, and $ B $ is at a horizontal distance $ x $ from $ A $. Find $ x $ (in metres).

ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are $\cot^{-1}(3\sqrt{2})$ and $\csc^{-1}(2\sqrt{2})$ respectively, then the height of the tower (in m) is (JEE Main 2019)

If OA = r cot(π/4 - q/2) = 2r cot(q/2), and tan(q/2) = t, find a + b + c where (1+t²)/(1-t) = t and tan(q/2) = (√17-3)/2.

If $\frac{\sin \theta}{a} + \frac{\cos \theta}{b} = 1$, then $\frac{\sin^3 \theta}{a^3} + \frac{\cos^3 \theta}{b^3}$ is

If $x + \sin y = 2014$ and $x + 2014\cos y = 2013, 0 \leq y \leq \frac{\pi}{2}$, then find the value of $[x + y] - 2005$ (where $[.]$ denotes greatest integer function)

The range of value's of $k$ for which the equation $2\cos^4 x - \sin^4 x + k = 0$ has atleast one solution is $[\lambda, \mu]$. Find the value of $(9\mu + \lambda)$

Given $5\cos A + 3 = 0$, the roots of the equation $9x^2 + 27x + 20 = 0$ are:

The value of $\cos\left(\frac{\pi}{14}\right)\cos\left(\frac{3\pi}{14}\right)\cos\left(\frac{5\pi}{14}\right)$ is

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \displaystyle\sum_{k=1}^{n} \sin k$, where $k$ represents radian measure. Find the index of the 100th term for which \(a_n

If $\frac{1}{a+c} + \frac{1}{b+c} = \frac{1}{a+b+c}$, then $\angle C$ is

If \(0 \leq x

If the angles A, B and C of triangle ABC are in arithmetic progression and a, b, c represents length of sides opposite to angles A, B and C respectively, then the value of $\dfrac{a+c}{\sqrt{(a^2 - ac + c^2)}}$ is:

The number of solutions of $\sin x \cdot \tan 4x = \cos x$ in $\left(0, \pi\right)$ is:

If $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi$, then $x^2 + y^2 + z^2 + 2xyz$ equals

In a triangle $PQR$, $\angle R = \dfrac{\pi}{2}$. If $\tan\left(\dfrac{P}{2}\right)$ and $\tan\left(\dfrac{Q}{2}\right)$ are the roots of $ax^2 + bx + c = 0,\ a \neq 0$ then:

If $A + B = \frac{\pi}{3}$, $(\cot A - 1)(\cot B - 1)$ is equal to

The number of solutions of |cos x| > 1 in (0, 2013π) is

If $c^4 - 2(a^2 + b^2)c^2 + a^4 + a^2b^2 + b^4 = 0$, then the angle $C$ is

Assertion (A): The value of tan 3α · cot α cannot lie between 3 and 1/3.Reason (R): In a triangle ABC, the maximum value of sin(A/2) sin(B/2) sin(C/2) is 1/8.

Is \(|\tan x + \cot x|

The possible value(s) of \theta satisfying the equation \sin 2\theta \tan\theta + \cos 2\theta \cot\theta - \sin 2\theta = 1 + \tan\theta + \cot\theta where \theta \in [0, \pi] is/are:

The equation whose roots are $\tan^2\left(\frac{\pi}{7}\right)$, $\tan^2\left(\frac{3\pi}{7}\right)$, $\tan^2\left(\frac{5\pi}{7}\right)$ is

The value of \cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} is equal to

Number of solutions of cos²\left(\frac{\pi}{4}\right)(\sin x + \sqrt{2}\cos 2x) = 0 in the interval x ∈ [-2π, 2π].

Assertion (A): The minimum value of a² tan²θ + b² cot²θ is 2ab.Reason (R): For positive real numbers AM ≥ GM.

Let $P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$ and $Q = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\sin\theta\}$ be two sets. Then,

The value of $\cos A + \cos B + \cos C$ is

241. 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 $x,\, y,\, z \in R$)

The value of $2ca\sin\!\left(\dfrac{A-B+C}{2}\right)$ equals

Find the value of $\cos\dfrac{\pi}{2^2} \cdot \cos\dfrac{\pi}{2^3} \cdots \cos\dfrac{\pi}{2^{10}} \cdot \sin\dfrac{\pi}{2^{10}}$.

If $\alpha$, $\beta$ are two values of $\theta$ obtained from the equation $a\cos\theta + b\sin\theta = c$ then the value of $\tan\frac{\alpha+\beta}{2}$ is

If $2\sin\theta + 1 = 0$ and $\sqrt{3}\tan\theta = 1$ then the most general value of $\theta$ is

Let a = sin α, b = cos α and c = √(1 + sin α cos α). In a triangle with sides a, b, c, what is the angle C?

94. The diameter of the circumcircle of a triangle with sides 5, 6 and 7 is

If $\frac{1+\sin 2x}{1-\sin 2x} = \tan^2(a+x)$ for all $x$ then the numerically smallest value of $a$ is

Calculate m = \sum_{k=1}^{17} \cos\left(\frac{k\pi}{9}\right) = \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{2\pi}{9}\right) + \cos\left(\frac{3\pi}{9}\right) + \ldots + \cos\left(\frac{17\pi}{9}\right), and find the value of (m^2 + m + 2).

The value of the expression \cos^2\left(\frac{\pi}{8}\right) + \cos^2\left(\frac{3\pi}{8}\right) + \cos^2\left(\frac{5\pi}{8}\right) + \cos^2\left(\frac{7\pi}{8}\right) is

If $\sin^{-1}\frac{1}{4} + \sin^{-1}\frac{3}{5} = \sin^{-1}x$, then the value of $x$ is

The value of $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c}$ is

The minimum value of the expression $\dfrac{\sin^3\alpha + 6\sin^2\alpha + \sin\alpha + 2\cos^2\alpha - 8}{\sin\alpha - 1}$ is equal to:

If A + B = \frac{\pi}{3}, A, B > 0, then the maximum value of \tan A \cdot \tan B is

Inverse Trigonometry Questions (1043)