Trigonometry & Inverse Trigonometry Questions - JEE

Solution set of the inequality $(\cot^{-1}x)^2 - 5(\cot^{-1}x) + 6 > 0$ is

In triangle ABC, given a/sin A = 2√2/sin 30° = 4/sin C. Find angles C and A.

Two flagstaffs stand on a horizontal plane. A and B are two points on the line joining their feet and between them. The angles of elevation of the tops of the flagstaffs as seen from A are 30° and 60° and as seen from B are 60° and 45°. If AB is 30 m, the distance between the flagstaffs in metres is

If sin θ + cos θ = a and sin³ θ + cos³ θ = b, then the value of λ³ + μ³ + ν³ is, where λa + μb + νa = 0 and λ, μ, ν are independent of θ.

The value of expression $\frac{8}{1 + \tan(100°)}$ is equal to

If $\dfrac{\cos^248°-\sin^212°}{\sin^224°-\sin^26°}=\dfrac{\alpha+\beta\sqrt{5}}{2}$, where $\alpha,\beta\in\mathbb{N}$, then $\alpha+\beta$ is equal to _____.

If $a\sin\theta - b\cos\theta = -\sin 4\theta$ and $a\cos\theta + b\sin\theta = \frac{5}{2} - \frac{3}{2}\cos 4\theta$, then $(a+b)^{2/5} + (a-b)^{2/5}$ is _______.

If $\tan\left(142\frac{1}{2}°\right) = 2 + \sqrt{2} - \sqrt{\mu} - \sqrt{\lambda}$, then $\mu + \lambda =$

The general solution of sin 2θ sec θ + √3 tan θ = 0.

Two parallel chords are drawn on the same side of the centre of a circle of radius R. It is found that they subtend an angle of θ and 2θ at the centre of the circle. The perpendicular distance between the chords is

Which of the following is the least?(a) $\sin 3$(b) $\sin 2$(c) $\sin 1$(d) $\sin 7$

In a triangle ABC, if $\tan\frac{A}{2}\tan\frac{C}{2} = \frac{1}{3}$ and $ac = 4$, then the least value of b is:(notation have their usual meaning)

Statement I: $\tan 50° - \tan 30° - \tan 20° = \tan 50° \tan 30° \tan 20°$Statement II: If $x = y + z$, then $\tan x - \tan y - \tan z = \tan x \tan y \tan z$

In a triangle ABC, if $\tan A = 2 \sin 2C$ and $3 \cos A = 2 \sin B \sin C$ then possible values of C is/are:

The minimum value of the function f(x) = sin x/√(1 - cos² x) + cos x/√(1 - sin² x) + tan x/√(sec² x - 1) + cot x/√(cosec² x - 1) whenever it is defined is

Angles A, B and C of a △ABC are in AP. If $\frac{b}{c} = \frac{\sqrt{3}}{2}$, then ∠A is equal to

The complete set of values of x satisfying the inequality sin-1(sin 5) > x2 - 4x is ( 2 - √(1 - 2π), 2 + √(1 - 2π) ), then l =

If $\sec A \tan B + \tan A \sec B = 91$, then the value of $(\sec A \sec B + \tan A \tan B)^2$ is equal to:

Let ABC be a right triangle with $\angle BAC = \frac{\pi}{2}$, then $\left(\frac{r_2}{2R^2} + \frac{r}{R}\right)$ is equal to: (where symbols used have usual meaning in a triangle)

In triangle ABC, if $2a^2b^2 + 2b^2c^2 = a^4 + b^4 + c^4$, then angle B is equal to

All positions of point P for which triangle DEF is isosceles lie on

If $x + \frac{1}{x} = 2$, the principal value of $\sin^{-1}x$ is

If $\dfrac{1}{16}(\cos 36^\circ \sin 54^\circ)^2 - \left(\dfrac{1}{4}\sin 36^\circ \sin 36^\circ\right)^2 \equiv \dfrac{\sqrt{a}-b}{c}$, find $(a + b + c)$.

In a right angled triangle ABC, the bisector of the right angle C divides AB into segments x and y. If $\tan\frac{A-B}{2} = t$, then x : y is equal to

If in a $\triangle ABC$, $\sum\sin 3A = 0$, then at least one angle of $\triangle ABC$ is:

The number of values of $x$ in the interval $[0, 3\pi]$ satisfying the equation $2\sin^2 x + 5\sin x - 3 = 0$ is

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin(x + y)$ and $|x| + |y| = 1$ is:

Consider f, g and h be three real valued functions defined on ℝ.Let f(x) = sin 3x + cos x, g(x) = cos 3x + sin x and h(x) = f²(x) + g²(x)Number of point(s) where the graphs of the two functions, y = f(x) and y = g(x) intersects in [0, π], is:

Find the number of solutions of the equations $2\sin^2 x + \sin^2 2x = 2$ and $\sin 2x + \cos 2x = \tan x$ in $[0, 4\pi]$ satisfying the condition $2\cos 2x + \sin x \le 2$.

If triangle DEF is equilateral, then P

The value of x in \left(0, \frac{π}{2}\right) satisfying the equation \frac{\sqrt{5}-1}{4} \cdot \frac{1}{\sin x} + \frac{\sqrt{10+2\sqrt{5}}}{4} \cdot \frac{1}{\cos x} = 2 is

If $\sin x + \cos x + \tan x + \cot y = 4$, where $x, y \in [0, \frac{\pi}{2}]$, then $\tan(\frac{y}{2})$ is a root of the equation:

In triangle ABC, the ratio $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ is always equal to (All symbols used have usual meaning in a triangle.)

If \sin^{-1}: [-1,1] \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] and \cos^{-1}: [-1,1] \to [0, \pi] be two bijective functions, respectively inverses of bijective functions \sin: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to [-1,1] and \cos: [0,\pi] \to [-1,1], then \sin^{-1}x + \cos^{-1}x is

703. Let $f(x) = \cos^{-1}\!\left(\sqrt{\sin^{-1}\!\left(\sec\!\left(\ln\!\left(\dfrac{2x^2+3x-2}{x^2-3x+2}\right)\right)\right)}\right)$. Find the value of $1 + \left(\displaystyle\sum \alpha_i^2\right)$ where $\alpha_i$ represents the integers in the range of $f(x)$. If there are no integers in the range of $f(x)$, then enter your answer as zero.

If the sum of all values of $\theta$, $0 \le \theta \le 2\pi$ satisfying the equation$(8\cos^4 \theta - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12$is $k\pi$, then $k$ is equal to:

Given the equations with solutions in the interval (0, π/2):(i) sin a = 1/2, so cos a = √3/2(ii) cos β = 1/3, so sin β = √(8/9)(iii) sin γ = cos γ = 1/√2(iv) cos γ = 1, sin γ = 0Find: cos a + cos β + cos γ

In a triangle ABC, BC = 3, AC = 4 and AB = 5. The value of $\sin A + \sin 2B + \sin 3C$ equals

If the sum of all solutions of the equation $3\cot^2 \theta + 10\cot \theta + 3 = 0$ in $[0, 2\pi]$ is $k\pi$ where $k \in \mathbb{I}$, then find the value of $k$.

If $\Delta$ be area of incircle of a triangle $ABC$ and $A_1, A_2, A_3$ be the area of excircles then find the least value of $$\frac{A_1A_2A_3}{729A^3}$$

If $x\sin\theta = y\sin\left(\theta+\frac{2\pi}{3}\right) = z\sin\left(\theta+\frac{4\pi}{3}\right)$, then $\sum xy=$

Statement I: $\csc^{-1}\left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) > \sec^{-1}\left(\frac{1}{2} + \frac{1}{2}\right)$Statement II: $\csc^{-1}x > \sec^{-1}x$ if \(1

If $\cot \frac{A}{2} = \frac{a+b+c}{4\Delta}$, then △ABC is

If inside a big circle exactly n (n > 3) small circles, each of radius r, can be drawn in such a way that each small circle touches the big circle and also touches both its adjacent small circles, then the radius of big circle is

Find $\tan\frac{A}{2}$ where notations have their usual meaning.

In a triangle ABC, medians AD and BE are drawn. If $AD = 4$; $\angle DAB = \frac{\pi}{6}$ and $\angle ABE = \frac{\pi}{3}$ then the area of the triangle ABC is:

In any triangle ABC, the value of $\frac{r_1 + r_2}{1 + \cos C}$ is equal to (where notation have their usual meaning):

The radius of the circle passing through the incentre $I$ of $\triangle ABC$ and through the end points of $BC$ is given by:

In a $\triangle ABC$; inscribed circle with centre $I$ touches sides $AB, AC$ and $BC$ at $D, E, F$ respectively. Let area of quadrilateral $ADIE$ is $5$ square units and area of quadrilateral $BFID$ is $10$ square units. Find the value of $$\frac{\cos\left(\frac{C}{2}\right)}{\sin\left(\frac{A-B}{2}\right)}$$

Which one of the following function contains only one integer in its range?[Note: sgn(k) denotes the signum function of k.](a) $f(x) = |\cos\frac{1-x^2}{x}|$

Trigonometry & Inverse Trigonometry Questions (1013)