Trigonometry & Inverse Trigonometry Questions - JEE

Solve: tan⁻¹((x+1)/(x-1)) + tan⁻¹((x-1)/x) = tan⁻¹(-7)

Considering only the principal values of inverse functions, the set $ A = \left\{x \geq 0;\, \tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}\right\} $

$\cos[\tan^{-1}\{\sin(\cot^{-1}x)\}]=$

934. Find the number of integers not in the domain of $f(x) = \cos^{-1}\!\left(\dfrac{2-x}{2x}\right)$.

Let $S_1$ is the complete solution set of the inequality $\cos^{-1}(x) > \cos^{-1}\left(x^2\right)$ and $S_2$ is the complete solution set of the inequality $\left(\cos^{-1} x^2\right) > 0$, then $S_1 \cap S_2$ is

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x)+\tan^{-1}(2x)=\dfrac{\pi}{4}$ is:

If $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \frac{\pi}{2}$, find $x$.

If $\tan^{-1} y : \tan^{-1} x = 4:1$, express $y$ as an algebraic function of $x$. Hence or otherwise prove that $22\frac{1}{2}$ is a root of the equation $x^4 + 1 = 6x^2$.

The range of the function $ f(x) = \sin^{-1}\!\left(\log_2 \dfrac{x^2}{2}\right) $ is:

$\cos\!\left(\sin^{-1}\dfrac{3}{5}+\sin^{-1}\dfrac{5}{13}+\sin^{-1}\dfrac{33}{65}\right)$ is equal to:

Find the value of $\cos^{-1}(\cos 13)$.

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is

The value of $\tan^{-1}\left(\frac{x\cos\theta}{1-x\sin\theta}\right) - \cot^{-1}\left(\frac{\cos\theta}{x-\sin\theta}\right)$ is

Find the value of $\tan^{-1}\left(\dfrac{1}{2}\tan 2A\right) + \tan^{-1}(\cot A) + \tan^{-1}(\cot^3 A)$ for \(0

Given expression = $1 + 2^2 + 1 + 3^2 + \text{cosec}\left(\tan^{-1}\dfrac{4}{3} + \tan^{-1}\dfrac{4}{3}\right)$. Find the value of the expression.

The value of \cos^{-1}\left(\cot\left(\sin^{-1}\sqrt{\frac{1-x^2}{4}}\right)\right) + \sec^{-1}\left(\sqrt{1+x^2}\right)

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\!\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is:

The value of $\sin\left(\cos^{-1}\dfrac{1}{2} + \sin^{-1}\dfrac{\sqrt{3}}{2}\right)$ is ______.

In a triangle ABC the expression $a\cos B\cos C + b\cos C\cos A + c\cos A\cos B$ equals to:

The numerical value of $\cos[\tan^{-1}(-3) + \cot^{-1}(-3)]$ = ______.

Suppose in $\triangle ABC$ with sides a, b, c the following equation holds true \[\frac{\cos A}{a} + k_1 = \frac{\cos B}{b} + k_2 = \frac{\cos C}{c} + k_3 = \frac{a^2 + b^2 + c^2}{8}.\] If $abc = 4$, then the value of $k_1 k_2 k_3$ is:

Since f(x) is onto, the range of f(x) equals co-domain. The range of f(x) = cos−1(4x2 + 3x) is $\left[\frac{\pi}{2}, \pi - \cos^{-1}\frac{9}{16}\right]$. What is the answer? (Integer answer: 25)

Find the value of $\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ$.

In a triangle $ABC$, $a = 4$, $b = 3$, $\angle A = 60°$, then $c$ is the root of the equation

The domain of the function $ f(x) = \sin^{-1}\left[\log_3\left(\dfrac{x}{3}\right)\right] $ is:

If $\cos\theta + \sec\theta = 2$ then $\cos^n\theta + \sec^n\theta$ is equal to

If $\sin(\alpha + \beta) = 1$ and $\sin(\alpha - \beta) = \dfrac{1}{2}$, then $\tan(\alpha + 2\beta) \cdot \tan(2\alpha + \beta)$ is equal to:

From a point on the ground, the angle of elevation of the top of a tower is $ \tan^{-1}\left(\dfrac{3}{5}\right) $. The tower is 40 m away from the point. A flag is hoisted at the top of the tower and the angle of elevation of the bottom of the flag from the same point is $ \alpha $ where $ \tan\alpha = \dfrac{3}{5} $. If $ \tan(\alpha + \beta) = \dfrac{x}{40} $ (where $ \beta $ is the angle subtended by the flag at the point on the ground), find the height $ x $ of the flag (in metres).

The principal value of $\cos^{-1}\left(\cos\dfrac{7\pi}{4}\right)$ is ______.

If $\cos\theta - \sin\theta = \cos\alpha - \sin\alpha$, then the value of $|\theta + \alpha|$ 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

If $k = \displaystyle\sum_{r=0}^{10} \cos^3\dfrac{\pi r}{3}$, then the value of $\dfrac{16}{k^2}$ is

Let $f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)$ where $x \in \mathbb{R}$ and $k \geq 1$. Then $f_4(x) - f_6(x)$ equals:

989. Let $T(n) = \cos^2(30° - n°) - \cos(30° - n°)\cos(30° + n°) + \cos^2(30° + n°)$. Find the value of $4\displaystyle\sum_{n=1}^{30} nT(n)$.

The angle of elevation of the top of a vertical tower from a point $A$, due east of it is 45°. The angle of elevation of the top of the same tower from a point $B$, due south of $A$ is 30°. If the distance between $A$ and $B$ is $54\sqrt{2}$ m, then the height of the tower (in metres) is:

The general solution of $\sin x + \cos x = 1$ is given by

The given equation $3\tan^{-1}(2-\sqrt{3}) - \tan^{-1}\left(\dfrac{1}{x}\right) = \tan^{-1}\left(\dfrac{1}{2}\right)$ is solved. Find the value of $x$.

In a triangle $ABC$, with $A = \dfrac{\pi}{7}$, $B = \dfrac{2\pi}{7}$; $C = \dfrac{4\pi}{7}$, then $a^2 + b^2 + c^2$ is ($R$ = circumradius of $\triangle ABC$)

The numerical value of $\sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3)$ = ______.

If $\cot\alpha = 1$ and $\cot\alpha$, $\cot(\alpha - \beta)$ and $\cot\beta$ are in A.P., then $\tan\beta$ equals:

If $\theta + \sqrt{3}\sin\theta = 2$ and $\theta \in [0, 2\pi]$ then $\theta$ is

For statement $p$: $\theta = 240^\circ$, consider\[2\sin\left(\frac{240^\circ}{2}\right) = \sqrt{1+\sin 240^\circ} - \sqrt{1-\sin 240^\circ}\]Is statement $p$ true or false, and what about statement $q$: $\cos\left(\frac{1}{2}(A+C)\right) + \cos\left(\frac{1}{2}(B+D)\right) = 0$?

If $\left|\sin\theta + 3\sin\left(\theta - \frac{\pi}{5}\right)\right| \leq a$ for all $\theta$ then the least value of $a$ is

The value of $\sin 10°\sin 30°\sin 50°\sin 70°$ is __________ (up to four decimal places).

Given $x = \sin^{-1}(\sin 10)$ and $y = \cos^{-1}(\cos 10)$. Find the value of $y - x$.

Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is __________ .

The numerical value of $2\tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{7}$ is ______.

If $p \in (0, \pi)$ then the set of values of $p$ for which $\sin p \cdot \cos^3 p > \sin^3 p \cdot \cos p$ holds, is ______

The number of real values of $x$ for which $\sin(e^x) = 5^x + 5^{-x}$ is

Chapter Test1(b). If in a △ABC, A = p and sin B = q then cos C = ______Choose the correct answer(s):(a) If in the △ABC, cos A · cos B + sin A · sin B · sin C = 1 then the triangle is

Trigonometry & Inverse Trigonometry Questions (1013)