Trigonometry & Inverse Trigonometry Questions - JEE

If \(\frac{1}{3} \leq \sin x a.

Let $\sec x + \tan x = \frac{22}{7}$, where \(0

If $\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi$, then $xy+yz+zx=$

$\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\tan^{-1}\!\frac{2r+1}{r^4+2r^3+r^2+1}=$

In a triangle $ABC$, $2ca\sin\dfrac{A-B+C}{2}$ is equal to

$\sin^{-1}(\sin 5)>x^2-4x$ holds for:

If $a$ and $b$ are the roots of the equation $4x^2 - 3x + a = 0$, sin $A + \cos A + \tan A + \cot A + \sec A + \cos A = 7$ and $0 < A < \frac{\pi}{2}$, then the value of $a$ must be

The value of the expression $\dfrac{\sin^2\dfrac{2\pi}{7} + \sin^2\dfrac{4\pi}{7} + \sin^2\dfrac{\pi}{7}}{\sin^2\dfrac{\pi}{7} + \sin^2\dfrac{2\pi}{7} + \sin^2\dfrac{4\pi}{7}}$ is equal to:

Solve $\theta = \tan^{-1}(2\tan^2\theta) - \frac{1}{2}\sin^{-1}\left(\frac{3\sin 20}{5 + 4\cos 20}\right)$.

Solve for $x$: $\tan^{-1} x + \tan^{-1}(1-x) = \cot^{-1}\frac{7}{9}$, where $x \in (0,1)$.

Two poles of heights 20 m and 80 m are standing on a horizontal ground. The height (in metres) of the point of intersection of the lines joining the top of each pole to the foot of the other pole is:

If $|2x + \sin^2 a| + |2x + 3 + 2\sin a| = 0$ and $4\lambda^2 = 1$, find $4\lambda^2$.

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 min from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60°. Then, the time taken (in minutes) by him, from B to reach the pillar, is

If $\sin^{-1}\!\sqrt{x/2}+\sin^{-1}\!\sqrt{1-x/4}+\tan^{-1}y=\frac{2\pi}{3}$, then which are true?

Let a and b be the lengths of the legs of a right triangle with the following properties: (a) All 3 sides of the triangle are integers. (b) The perimeter of the triangle is numerically equal to area of the triangle, it is given that a < b. The number of ordered pairs (a, b) will be :

The value of $S = \dfrac{\sin^2\dfrac{2\pi}{7}}{\sin^2\dfrac{\pi}{7}} + \dfrac{\sin^2\dfrac{4\pi}{7}}{\sin^2\dfrac{2\pi}{7}} + \dfrac{\sin^2\dfrac{\pi}{7}}{\sin^2\dfrac{4\pi}{7}}$ is:

If ABCD is a cyclic quadrilateral then $\cos A + \cos B + \cos C + \cos D$ is equal to

A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (= a) subtends an angle of 60° at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30°. The height of the tower is

A $150\left(\sqrt{3}+1\right)$ ft.

Which are correct?(A) $\cot^{-1}x=\tan^{-1}(1/x)\ \forall x\in\mathbb{R}\setminus\{0\}$(B) $f(x)=\text{sgn}(e^x)$ is into(C) $f:\mathbb{R}^+\to\mathbb{R},\,f(x)=\sin x+x$ is odd(D) $f(x)=e^x/e^{[x]}$ is periodic

$\sin[\cot^{-1}\{\tan(\cos^{-1} x)\}]$ is equal to

The number of possible solutions of x such that $\sin^2 x + \cos^2 x = 1$ is:

Let $a \in \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$ such that $\tan^{-1}\!\left(\dfrac{\tan\alpha}{3 + 2\tan^2\alpha}\right) + \tan^{-1}\!\left(\dfrac{2\tan\alpha}{3}\right) = \dfrac{\pi}{12}$, then $\alpha$ equals:

811. In $\triangle ABC$, if $\sin A \sin B \sin C + \cos A \cos B = 1$ then the value of $\cos^2 A + \sin^2 B + 2\sin^2 \dfrac{C}{2}$ is:

If \(2\le a

If \(0 \leq x

The sum of all values of $\theta \in \left(0, \dfrac{\pi}{2}\right)$ satisfying $\sin^2 2\theta + \cos^4 2\theta = \dfrac{3}{4}$ is:

If \(0

Suppose 3\sin^{-1}(\log_2 x) + \cos^{-1}(\log_2 y) = \frac{\pi}{2} and \sin^{-1}(\log_2 x) + 2\cos^{-1}(\log_2 y) = \frac{11\pi}{6} then the value of x^2 + y^2 equals

If $\cos^{-1}(2x^2-1)=2\pi-2\cos^{-1}x$, then:

Find the range of f(x) = \sin^{-1} x + \tan^{-1} x + \sec^{-1} x

Find the number of solutions of $\cos x = |1 + \sin x|$, \(0

$\text{cosec}^{-1}(\cos x)$ exists if:

If a = \tan x, then the value of \cot\left(\frac{\pi}{4} - a\right) is

If a root of the equation $n^2\sin^2 x + 2\sin x - (2n+1) = 0$ lies in $[0, \frac{\pi}{2}]$, find the minimum positive integer value of $n$.

In $\triangle ABC$, if incircle touches the sides $AB$, $BC$ and $CA$ at $P$, $Q$ and $R$ respectively and $s - a = 3$, $s - b = 5$ and $s - c = 7$, then area of the quadrilateral $QCRI$ is, where $I$ is incentre of $\triangle ABC$:[Note: Symbols used have usual meaning in $\triangle ABC$.]

If \sin\theta + \sqrt{3}\cos\theta = 6x - x^2 - 11, 0 \leq \theta \leq 4\pi, x \in \mathbb{R}, then:

If A = \cos(\cos x) + \sin(\cos x), then the least and greatest value of A are

Find the value of $\cos 3A + \cos 3B + \cos 3C$ given that $A + B + C = 180°$ (angles of a triangle), and determine under what conditions the expression equals $1 + \cos(3A + 3B)$. Specifically, evaluate: $\cos 3A + \cos 3B = 1 - \cos(3C)$, i.e., $2\cos\dfrac{3}{2}(A+B)\cos\dfrac{3}{2}(A-B) = 2\cos^2\dfrac{3}{2}(A+B)$. If $\cos\dfrac{3}{2}(A+B) = 0$, then $\dfrac{3}{2}(A+B) = 90°$, $A + B = 60°$, so $C = 120°$. What is the answer?

The principal value of \cos^{-1}\left(\cos\left(2\cot^{-1}(\sqrt{2}-1)\right)\right) is equal to

Ex. 35. Statement I: If tan⁻¹x + tan⁻¹y + tan⁻¹z = π/4 and x + y + z = 1, then arithmetic mean of odd powers of x, y, z is equal to 1/3.Statement II: For any x, y, z we have xyz − xy − yz − zx + x + y + z = 1 + (x − 1)(y − 1)(z − 1)

Find the number of solutions of the equation in the interval [0, 2π] where the graph of y = tan x and y = \frac{71}{x} intersect.

The number of integral values of k for which the equation 7\cos x + 5\sin x = 2k + 1 has a solution is

If f(x) = \cos[p^2] x + \cos[-p^2], where [\cdot] = G.I.F., then which statement is true?

If angle \theta\ be divided into two parts such that the tangent of one part is k\ times the tangent of the other and \phi\ is their difference, then \sin\phi\ is equal to

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

In a triangle ABC, if tan B + C - A}{4} tan C + A - B}{4} tan A + B - C}{4} = 1, then find the value of cos A + cos B + cos C.

If $ \alpha = \cos^{-1}\!\left(\dfrac{3}{5}\right) $, $ \beta = \tan^{-1}\!\left(\dfrac{1}{3}\right) $, where \( 0

The value of $ \tan^{-1}\!\left[\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right] $, \( |x|

Sides of a triangle ABC are in AP. If \(a

Trigonometry & Inverse Trigonometry Questions (1013)