Inverse Trigonometry Questions - JEE

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

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

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).

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)

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

The maximum value of the function $f(x) = \dfrac{4\cot^{-1}x}{\pi} - \dfrac{\pi}{4\cot^{-1}(-x)}$ occurs at $x$ equal to

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

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 $\alpha$.

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

The period of $f(\theta) = \sin^2\theta$ is:

Find number of solutions of the equation sin-1(|log₂₆(cos x) - 1|) + cos-1(|3 log₂₆(cos x) - 7|) = π/2, if x ∈ [0, 4π].

$\cot\dfrac{a+1}{a-b} + \cot\dfrac{b+1}{b-c} + \cot\dfrac{c+1}{c-a} = \underline{\quad}$.

ABC is a triangular park with $AB = AC = 100$ metres. 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 $\text{cosec}^{-1}(2\sqrt{2})$ respectively, then the height of the tower (in metres) is __________ (up to four decimal places).

The minimum value of $\frac{r_1 r_2}{r_3}$ in a triangle is (symbols have their usual meaning)

If $x$ be real, prove that $\frac{x^2 - 2x\cos\alpha + 1}{x^2 - 2x\cos\beta + 1}$ lies between $\sin^2\frac{\alpha}{2}\cdot\csc^2\frac{\beta}{2}$ and $\cos^2\frac{\alpha}{2}\cdot\sec^2\frac{\beta}{2}$.

A continuous even periodic function $f$ with period 8 is such that $f(0)=0$, $f(1)=-2$, $f(2)=1$, $f(3)=2$, $f(4)=3$, then the value of $\tan^{-1}(\tan(f(-5)+f(20)) + \cos^{-1}(f(-10)+f(17)))$ is equal to:

In triangle $ABC$ if $\dfrac{[\Delta ABC]}{R} = 4$, then the value of $a\cos A + b\cos B + c\cos C$ is:[Note: $R$ is the circumradius of triangle $ABC$ and $[\Delta ABC]$ is the area of $\Delta ABC$]

The number of solutions of the equation $8\tan^2\theta + 9 = 6\sec\theta$ in the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ is

If in a triangle ABC, $b\cos^2\frac{A}{2} + \cos^2\frac{B}{2} = \frac{3c}{2}$, then minimum value of $\frac{1}{5}\left(\frac{a+c}{2c-a} + \frac{b+c}{2c-b}\right)$ is equal to

If $\tan\alpha^2 = \tan(\alpha - \beta)\cdot\tan(\alpha + \beta)$, then which of the following is correct?(Given: $0 , \(\tan\alpha > 0$)

In a triangle $ABC$, if $A + C = 2B$ and $A + B + C = 180^\circ$ with $\sin A + \sin C = 2\sin^2 B$, find the value of some expression (answer 30).

If $a$, $b$, $g$, and $d$ are four solutions of the equation $\tan\left(\theta + \frac{\pi}{4}\right) = 3\tan 3\theta$, then $\tan a \tan b \tan g \tan d$ equals

If $|\sin x + \cos x| = |\sin x| + |\cos x|$ ($\sin x, \cos x \neq 0$), then in which quadrant does $x$ lie?

In a triangle ABC, let $\angle C = \pi/2$. If r is the inradius and R is the circumradius of the triangle ABC, then 2(r + R) equals

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:

If $\cos(\theta - \alpha)$, $\cos\theta$, $\cos(\theta + \alpha)$ are in HP, then $\cos\theta \sec\dfrac{\alpha}{2}$ is equal to

The number of values of $\theta$ in $\left[0, \dfrac{\pi}{2}\right]$ satisfying $2\cos\theta + \sin\theta = 1$ $\left(\theta \neq \dfrac{\pi}{2}\right)$ is

Inverse Trigonometry Questions (1043)