Inverse Trigonometry Questions - JEE

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) $f(x) = |\cos\frac{1-x^2}{x}|$

The ratio in which the curve $y = \left[\sin\frac{2x}{4} + \cos\frac{x}{4}\right]$, where [·] denote greatest integer function divides the curve S1 is:

If A represents the area of acute angled triangle ABC, then $\sqrt{a^2b^2 - 4A^2} + \sqrt{b^2c^2 - 4A^2} + \sqrt{c^2a^2 - 4A^2}$ is equal to

Each side of an equilateral triangle subtends an angle of 60° at the top of a tower h m high located at the centre of the triangle. If a is the length of each side of the triangle, then

The number of points in interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ where the graphs of the curves $y = \cos x$ and $y = \sin 3x$, with $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$, intersect is:

The value of $\dfrac{\sqrt{3}\,\text{cosec}\,20°-\sec20°}{\cos20°\cos40°\cos60°\cos80°}$ is equal to

In a triangle ABC; AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circle DEF is

Two men on the opposite sides of a tower measure the angles of elevation of the top of the tower as $45°$ and $30°$ respectively. If the height of the tower is $40$ m, then the distance between the men is

If 2 tan-1(1/5) - sin-1(3/5) = -cos-1(9l/65), then l =

Let a, b, c be sides of a triangle ABC and D denotes its area. If $a = 2$, $D = \sqrt{3}$, and $a\cos C + \sqrt{3}a\sin C - b - c = 0$, then find the value of $(b + c)$.

Solution of equation $\cot^{-1}x + \sin^{-1}\frac{1}{\sqrt{1+x^2}} = \frac{7\pi}{1}$ is

In $\triangle ABC$, if circumradius $'R'$ and inradius $'r'$ are connected by relation $R^2 - 4Rr + 8r^2 - 12r + 9 = 0$, then the greatest integer which is less than the semiperimeter of $\triangle ABC$ 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)General solution of the equation h(x) = 4, is:

In triangle ABC, if cos(a) = 1/2 = 1/4 from triangle OED, and θ = π - 2a, find the area of triangle ABC where BD = 2cot(θ/2) = 2cot(π/2 - a) = 2tan(a) = 2√15 and AC = 3.

In a triangle the length of two larger sides are 10 and 9 respectively. If the angles are in A.P., the length of third side can be:

If in a $\triangle ABC$, $a = 5$, $b = 4$ and $\cos(A - B) = \frac{31}{32}$, then the third side $c$ is equal to

The range of the function f(x) = tan−1x + ½ sin−1x is:

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

Inverse Trigonometry Questions (1043)