Inverse Trigonometry Questions - JEE

Find the value of a + b + c where OA = r cot(π/4 − q/2) = 2r cot(q/2), and tan(q/2) = (−3 ± √17)/2.

Let $\cos(\alpha+\beta)=\dfrac{4}{5}$ and let $\sin(\alpha-\beta)=\dfrac{5}{13}$, where $0 \le \alpha,\ \beta \le \dfrac{\pi}{4}$, then $\tan 2\alpha =$

Minimum possible area of the triangle is :

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:

Let $0 \leq \alpha, \beta, \gamma, \delta \leq \pi$ where $\beta$ and $\gamma$ are not complementary such that$2\cos \alpha + 6\cos \beta + 7\cos \gamma + 9\cos \delta = 0$ and $2\sin \alpha - 6\sin \beta + 7\sin \gamma - 9\sin \delta = 0$If $\frac{\cos(\alpha + \delta)}{\cos(\beta + \gamma)} = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive numbers, then the value of $(m + n)$ is equal to:

The angle of elevation of a cloud from a point $250$ m above a lake is $15°$ and angle of depression of its reflection in the lake is $45°$. The height of the cloud is

If 1 + \sin 3x + \cos 3x = \frac{3}{2}\sin 2x, then x is

The value of $\cos\left(\cos^{-1}\left(\frac{1}{8}\right)\right)$ is equal to

If $\cot^{-1}\left(\frac{n}{p}\right) = \frac{p}{6}$, $n \in \mathbb{N}$, then the maximum value of $\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)$ is $'n'$ is

The number of positive integral solutions of $\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)$ is

The value of $\cos\left(\cos^{-1}\left(\cos\left(\sin^{-1}\left(\frac{63}{8}\right)\right)\right)\right)$ is

$\sin^{-1}(\sin \theta) = \theta$, for all $\theta$ belonging to

$\csc^{-1}(\csc \theta) = \theta$, for all $\theta$ belonging to

Which of the following is greatest?

If $r_1 = r + r_2 + r_3$, then the triangle is

The general value of $\theta$ satisfying $\sin^2\theta + \sin\theta = 2$ is

Given, $\cot^{-1}\left(\frac{n^2 - 10n + 21.6}{\pi}\right) > \frac{\pi}{4}$ and $\cos^{-1}(x^2) = \frac{\pi}{6}$, where $n \in \mathbb{N}$. Find the minimum value of $n$.

If $\sin(\cot^{-1}(x+1)) = \cos(\tan^{-1} x)$, then the value of x is

159. 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:

If $\sin^{-1}\left(\frac{x}{5}\right) + \text{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2}$, then $x$ is

The number of distinct solutions of the equation $\cos^2 2x + \cos 4x + \sin 4x + \cos 6x + \sin 6x = \frac{5}{4}$ in the interval $[0, 2\pi]$ is

Considering the principal values of inverse trigonometric functions, the value of $\tan\!\left(2\sin^{-1}\!\left(\dfrac{2}{\sqrt{13}}\right)-2\cos^{-1}\!\left(\dfrac{3}{\sqrt{10}}\right)\right)$ is equal to:

If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to

If the domain of the function $f(x)=\cos^{-1}\!\left(\dfrac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1)$ is the interval $[\alpha,\beta]$, then $\alpha+2\beta$ is equal to:

The number of values of $x$, for which $\tan^{-1}\!\left(\dfrac{1}{x}\right) = \pi + \tan^{-1} x$, \(0

The value of the expression $\frac{2\cos^4 + \cos^4^3 - \cos^4 \alpha + \cdots - \cos^4}{2\cos^4 + \cos^4^3 - \cos^4 - \cdots - \cos^4 + 1}$ is equal to

$\sin \alpha + \sin \beta + \sin \gamma$ can be equal to

96. In the △ABC, A > B. If the measures of A and B satisfy the equation $3\sin x - 4\sin^3 x - k = 0$, \(0

Given $\sin^4\alpha + 4\cos^4\beta + 2 = 4\sqrt{2}\sin\alpha\cos\beta$where $\alpha, \beta \in [0, \pi]$, find the value of $\cos(\alpha+\beta) - \cos(\alpha-\beta)$.

The value of $\cot^{-1}\left(\frac{xy+1}{x-y}\right) + \cot^{-1}\left(\frac{yz+1}{y-z}\right) + \cot^{-1}\left(\frac{zx+1}{z-x}\right)$ is

Value of $\tan^{-1}\left(\frac{\sin 2}{1 - \cos 2}\right)$ is

If $x^2 + y^2 + z^2 = r^2$, then $\tan^{-1}\left(\frac{xy}{zr}\right) + \tan^{-1}\left(\frac{yz}{xr}\right) + \tan^{-1}\left(\frac{xz}{yr}\right)$ is equal to

If in a $\triangle ABC$, $\angle A = \tan^{-1}2$ and $\angle B = \tan^{-1}3$, then $\angle C$ is equal to

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

The inradius of $\triangle ABC$ is $100\sqrt{3}$ and the circumradius is $200\sqrt{3}$. Consider the line perpendicular to plane $ABC$ through the circumcenter of $\triangle ABC$. Note that $P, Q, O$ must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since $P, Q, O$ are collinear, and $OP = OQ$, we must have $O$ is the midpoint of $PQ$. Now, Let $K$ be the circumcenter of $\triangle ABC$, and $L$ be the foot of the altitude from $A$ to $BC$. We must have $\tan(\angle KLP + \angle QLK) = \tan(120°)$. Setting $KP = x$ and $KQ = y$, assuming WLOG $x > y$, we must have \[\tan(120°) = -\sqrt{3} = \frac{\dfrac{x+y}{100\sqrt{3}}}{\dfrac{30000 - xy}{30000}}.\] Thus $100(x+y) = xy - 30000$. Also, $\left(\dfrac{x+y}{2}\right)^2 = \left(\dfrac{x-y}{2}\right)^2 + 120000$ by the Pythagorean theorem, so $xy = 120000$, and substituting, $90000 = 100(x+y)$, or $x + y = 900$. The desired answer is $\dfrac{x+y}{2}$.

The number of solutions of the equation $2\sin^3\alpha - 7\sin^2\alpha + 7\sin\alpha - 2 = 0$ in $[0, 2\pi]$ is

Sum of series $\displaystyle\sum_{r=1}^{n} \sin^{-1}\left[\dfrac{2r+1}{r(r+1)\left(\sqrt{r^2+2r} + \sqrt{r^2-1}\right)}\right]$ is

If $\dfrac{1}{2}\sin^{-1}\!\left(\dfrac{3\sin 2\alpha}{5 + 4\cos 2\alpha}\right) = \tan^{-1} x$, then the possible value of $x$ is:

The period of the function $f(x) = \sin^4 x + \cos^4 x$ is:

In $\triangle ABC$, circumradius is 3 and inradius is 1.5 units. If the value of $a\cot^2 A + b^2\cot^3 B + c^3\cot^4 C$ is $m\sqrt{n}$ where m and n are prime numbers, then find the value of $\left(\dfrac{m-1}{n}\right)$.

If $\dfrac{p-1}{2p+3} = \sin^2\theta + 2\cos\theta + 1$ $\forall\, \theta \in R$, then $p$ must lie in the interval:

9. Consider a triangle ABC such that $\cot A + \cot B + \cot C = \cot \theta$. The possible value of $\theta$ is:

$\sin^{-1}\sqrt{\dfrac{x}{x+y}} = \tan^{-1}(\underline{\quad})$.

The value of $\cos^2 10° - \cos 10° \cos 50° + \cos^2 50°$ is

The least value of $\csc^2 x + 25\sec^2 x$ is

If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3x^2-10x-25=0$, then the value of $3\sin^2(A+B)-10\sin(A+B)\cos(A+B)-25\cos^2(A+B)$ is

Given a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. Find the height of the lamp-post.

Given $\cos^{-1}\left(\dfrac{2}{3x}\right) + \cos^{-1}\left(\dfrac{3}{4x}\right) = \dfrac{\pi}{2}$Find the value of $x$.

The value of $\sin\sqrt{x^2 - \frac{\pi^2}{36}}$ lies in the interval

Maximum possible perimeter of the triangle is :

Inverse Trigonometry Questions (1043)