Inverse Trigonometry Questions - JEE

In a triangle, $\cos A = \dfrac{b^2+c^2-a^2}{2bc}$. If $a = 4$, $b = 3$, $\cos A = \cos 60°$, find $c$.

714. If $ \cot(\theta - \alpha),\; 3\cot\theta,\; \cot(\theta + \alpha) $ are in A.P. and $ \theta $ is not an integral multiple of $ \dfrac{\pi}{2} $, then find the value of $ \dfrac{2\sin^2\theta}{\sin^2\alpha} $.

If $\cot y = \frac{\sin x - \sin z}{\cos z - \cos x}$ then which of the following is possible?

Ex. 81: Let N denotes the number of solution of the equation f(θ) = 0 in [0, 4π] where f(θ) = sin θ - cos 2θ - 1, then the value of N + 1 is

Ex. 14: If $\csc \frac{7\pi}{32} + \csc \frac{7\pi}{16} + \csc \frac{7\pi}{8} + \csc \frac{7\pi}{4} = \csc \frac{7\pi}{2} - \cot \frac{7\pi}{k}$, then the value of k is

If $\tan \alpha, \tan \beta$ satisfy equation (i) and $\cos \gamma, \cos \delta$ satisfy equation (ii), then $\tan \alpha \cdot \tan \beta + \cos \gamma + \cos \delta$ can be equal to

A cone of base radius a has its apex at height h above the centre O of the base. If $OA = OB = AB = a$ (so triangle OAB is equilateral), and considering triangle OBH where $\tan 30^\circ = \dfrac{h}{a}$, then h equals:

The value of $x$ for which $\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}x)$ is

Given $3(\sin\theta - \cos\theta)^4 + 6(\sin\theta + \cos\theta)^2 + 4\sin^6\theta$, simplify the expression.

Find the value of $\left(1 + \cos \frac{3\pi}{8}\right)\left(1 + \cos \frac{5\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right)$.

If A and B are acute positive angles satisfying the equations $3\sin 2A + 2\sin^2 B = 1$ and $3\sin 2A - 2\sin 3B = 0$, then $A + 2B$ is equal to

In the following incomplete sentences, fill in the blanks so that the resulting sentences may become true.(a) For all real $x$, the value of $\sin^2 x \cdot \cos^2 x \leq k$, $k$ being the least possible. Then $k =$ ______.

If A is the area and 2S the sum of sides of a triangle, then

Let Statement I: The equation sin x = f(x) has no solution, where f(x) = x² + x + 1Statement II: The curve y = sin x and y = f(x) do not intersect each other when graph is observed.

The number of solutions of the equation $\tan x + \sec x = 2\cos x$ lying in the interval $[0, 2\pi]$ is

Ex. 23: Statement I If the sides of a triangle are 13, 14, 15 then the radius of incircle = 4Statement II In triangle ABC, $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$ and $r = \dfrac{A}{s}$

If in a triangle ABC, $\cos A \cdot \cos B + \sin A \cdot \sin B \cdot \sin^n C = 1$, $n \in N$, then prove that the sides are in the ratio $1:1:\sqrt{2}$.Find the value of $n$.

A tower stands at the centre of a circular park. A and C are two points on the boundary of the park such that AB subtends an angle of 45° and CB subtends an angle of 30° at the foot of the tower, where B is the foot of the tower. If the radius of the park is 18 m, then the height of the tower (in m) is:

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

Let $S_1$ and $S_2$ be the areas of inscribed and circumscribed polygons of 10 sides respectively and $S_3$ is the area of regular polygon of 20 sides inscribed in a circle, then

The product of the sines of the angles of a triangle is $p$ and the product of their cosines is $q$. Then, the tangents of the angles are the roots of the equation

The angles A, B and C of a ΔABC are in AP and a : b = 1 : $\sqrt{3}$. If c = 4 cm, then the area (in sq cm) of this triangle is

If the angles A, B and C of triangle ABC are in arithmetic progression and a, b, c represents length of sides opposite to angles A, B and C respectively, then the value of $\dfrac{a+c}{\sqrt{(a^2 - ac + c^2)}}$ is:

Let tan−1 y = tan−1 x + tan−1 \left(\frac{2x}{1-x^2}\right), where |x| . Then, a value of y is

The value of \[\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}\] is equal to

Let $0 and \(x = X \cos \theta + Y \sin \theta$, $y = X \sin \theta - Y \cos \theta$ such that $x^2 + 2xy + y^2 = aX^2 + bY^2$, where $a$ and $b$ are constants. Then

The value of $\displaystyle\sum_{r=0}^{10} \cos^3\dfrac{r\pi}{3}$ is equal to $\dfrac{-a}{b}$, then the value of $b$ is (where g.c.d of $(a,b)$ is 1).

In any triangle, if ($\sin A + \sin B + \sin C$)($\sin A + \sin B - \sin C$) = 3$\sin A\sin B$,then find the angle $\frac{C}{10}$ (in degree).

Let $A_1, A_2, A_3, \ldots, A_n$ be the vertices of an $n$-sided regular polygon such that $\dfrac{1}{A_1 A_2} = \dfrac{1}{A_1 A_3} + \dfrac{1}{A_1 A_4}$. Find the value of $n$.

If cot (α + β) = 0, then sin (α + 2β) = ?

If a, b, A are given and $b_1, b_2$ are two values of the third side b such that $b_2 = 2b_1$. Then, $\sin A$ is equal to

In triangle ABC, if $\tan\frac{A}{4} + \tan\frac{B}{4} + \tan\frac{C}{4} = 1$, then triangle ABC is

In triangle ABC, if $a^2 + c^2 = 2002b^2$, then $\frac{\cot A + \cot C}{\cot B}$ equals

If $f_4(x) - f_6(x) = \frac{1}{4}(\sin^4 x + \cos^4 x) - \frac{1}{6}(\cos^6 x + \sin^6 x)$, then the value of this expression equals:

In a right-angled isosceles triangle $\Delta ADE$ with $AE = 10$, so that $AD = DE = 5\sqrt{2}$. In right triangle $ACD$, $\tan\beta = \dfrac{CD}{AD}$. Then the area of $\Delta ABC$ is:

Given $\sin 3x = \cos 2x$, the number of solutions in $x \in \left(\dfrac{\pi}{2}, \pi\right)$ is:

If cos θ = 1/(x + 1/(2x)), then what is 2/(x² + 1/x) equal to

If cos 2θ = (√2 + 1) cos θ - 1/√2, then the value of θ is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 metre away from the tree the angle of elevation becomes 30°. The breadth of the river is:

If a cos³ α + 3a cos α sin² α = m and a sin³ α + 3a cos² α sin α = n, then (m + n)^(2/3) + (m - n)^(2/3) is equal to

If P = cos(cos x) + sin(cos x), then the least and greatest value of P respectively are:

The perimeter of a △ABC is 48 cm and one side is 20 cm. Then remaining sides of △ABC must be greater than:

In triangle ABC, if ∠A = 30°, b = 10 and a = x, then the values of x for which there are 2 possible triangles is given by (All symbols have usual meaning in a triangle)

In triangle ABC, if AC = 8, BC = 7, and D lies between A and B such that AD = 2, BD = 4, then the length CD equals

In a triangle with one angle π/3, the lengths of the sides form an A.P. If the length of the greatest side is 7 cm, the radius of the circumcircle of the triangle is

The value of the expression \[\tan\!\left(\tan^{-1}\!\left(\frac{1}{2}\right)+\tan^{-1}\!\left(\frac{2}{9}\right)+\tan^{-1}\!\left(\frac{1}{8}\right)+\tan^{-1}\!\left(\frac{2}{25}\right)+\tan^{-1}\!\left(\frac{1}{18}\right)+\cdots\cdots\infty\right)\] is:

Determine the smallest positive value of x (in degrees) for which $\tan(x + 100°) = \tan(x + 50°) \cdot \tan x \cdot \tan(x - 50°)$.

Consider an obtuse angled triangle with sides 8 cm, 15 cm and $x$ cm (largest side being 15 cm). If $x$ is an integer, then find the number of possible triangles.

If a, b, c are the sides of a triangle, then the minimum value of $\dfrac{2a}{b+c-a} + \dfrac{2b}{c+a-b} + \dfrac{2c}{a+b-c}$ is

The general solution-set of the equation $\cos x + \cos 5x = 2$ is:

Inverse Trigonometry Questions (1043)