Trigonometry & Inverse Trigonometry Questions - JEE

Given that $\cos(\alpha + \beta) = \dfrac{4}{5}$ and $\sin(\alpha - \beta) = \dfrac{5}{13}$, where $\alpha + \beta \in \left[0, \dfrac{\pi}{2}\right]$ and $\alpha - \beta \in \left[0, \dfrac{\pi}{4}\right]$, then $\tan 2\alpha$ is equal to:

The sum of solutions in $(0, 2\pi)$ of the equation $\cos x \cos\!\left(\dfrac{\pi}{3} - x\right)\cos\!\left(\dfrac{\pi}{3} + x\right) = \dfrac{1}{4}$ is:

If $\cos 2\theta = \sin\alpha$ then the most general relation between $\theta$ and $\alpha$ is (where $n \in \mathbb{Z}$)

A tower $T_1$ of height 60 m is located exactly opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_2$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is

Paragraph (Questions 7-8): Let the incircle of $\triangle ABC$ touch sides $BC, CA, AB$ at $A_1, B_1, C_1$ respectively. The incircle of $\triangle A_1B_1C_1$ touches its sides $B_1C_1, C_1A_1, A_1B_1$ at $A_2, B_2, C_2$ respectively and so on.Find $\lim_{n \to \infty} \angle A_n$

If $\cos^2 x = t$, and the equation $5\left[\frac{1-t}{t} - t\right] = 2(2t-1) + 9$ is satisfied, then find the value of $\cos 4x$.

With the usual notation, in triangle ABC, if $\angle A + \angle B = 120°$, $a = \sqrt{3} + 1$ and $b = \sqrt{3} - 1$, then the ratio $\angle A : \angle B$ is

If in the pedal triangle DEF of an acute-angled triangle ABC, the sides are denoted by l, m, n, then l/(a²) + m/(b²) + n/(c²) is equal to

If $\tan\theta_1, \tan\theta_2, \tan\theta_3$ are the real roots of the equation $x^3 - (a+1)x^2 + (b-a)x - b = 0$, where $\theta_1 + \theta_2 + \theta_3 \in (0, \pi)$, then $\theta_1 + \theta_2 + \theta_3$ is equal to

AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of point A from a certain point C on the ground is 60°. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45°. Then the height of the pole is

$3 \csc 20° - \sec 20°$ is equal to

The value of $\sqrt{3}\csc 20° - \sec 20°$ is equal to

The number of values of $x$ in $[0, 5\pi]$ satisfying the equation $3\sin^2 x - 7\sin x + 2 = 0$ is

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

Trigonometry & Inverse Trigonometry Questions (1013)