Trigonometry & Inverse Trigonometry Questions - JEE

Let $\theta = \sin^{-1}\left(\dfrac{3\sin 2\alpha}{5 + 4\cos 2\alpha}\right)$. Then $\tan^{-1} x = \dfrac{\theta}{2}$ where $x$ equals:

If \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) , then

We have tan(15°) + tan(30°) = –p and tan(15°) + tan(30°) = –p. Given that tan(45°) = 1, find the value of (2 + q – p) where q – p = 1.

Find the value of $\cos\left(\frac{2\pi}{7}\right)\cos\left(\frac{4\pi}{7}\right)\cos\left(\frac{8\pi}{7}\right)$

Given $f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)$, find the value of $f_4(x) - f_6(x)$.

A vertical tower subtends an angle of $60°$ at a point on the same level as the foot of the tower. On moving $100$ m further from the first point in line with the tower, it subtends an angle of $30°$ at the point. If the height of the tower is $H$ m, then the value of $\frac{H}{\sqrt{3}}$ (in meters) is

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos 2\beta$ then $\tan\alpha$, $\tan\beta$ and $\tan\gamma$ are in

The value of $\sin\left(\frac{\pi}{14}\right)\sin\left(\frac{3\pi}{14}\right)\sin\left(\frac{5\pi}{14}\right)$ is

The number of solutions of the equation $e^{\sin x} - e^{-\sin x} = 4$ is

Let a, b, c be three non-zero real numbers such that the equation acosx + 2bsinx = c, x ∈ [−π/2, π/2], has two distinct real roots α and β with α + β = π/3. Then, the value of b/a is __________________________.

Let $\alpha$ and $\beta$ respectively be the maximum and minimum values of the function $f(\theta)=4\left(\sin^4\!\left(\dfrac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\!\left(\dfrac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right)$, $\theta\in\mathbb{R}$. Then $\alpha+2\beta$ is equal to:

For $x \in (0, \pi)$, the equation $\sin x + 2 \sin 2x - \sin 3x = \frac{3}{5}$ has

If PQ be a vertical tower subtending angles α, β and γ at the points A, B and C respectively on the line in the horizontal plane through the foot D of tower and on the same side of it, then BC cot α - CA cot β + AB cot γ is equal to

A regular polygon of $n$ sides is inscribed in a circle of radius $R$ and another regular polygon of $n$ sides is circumscribed about a circle of radius $r$, where $\theta = \dfrac{\pi}{n}$. Then $r + R$ equals:

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:

If $x = y = z$, $x, y, z$ are in AP, and $\tan^{-1}x$, $\tan^{-1}y$, $\tan^{-1}z$ are also in AP, find the relationship between $x$ and $z$.

If \sin^4 x + \cos^4 x = \sin x \cos x, then x is

Let $2\sin^2 x + 3\sin x - 2 \geq 0$ and \(x^2 - x - 2

Range of $f(x)=\cot^{-1}(\log_e(1-x^2))$ is:

Find the value of $16(\sin^2 18^\circ + \sin^2 36^\circ + \sin^2 54^\circ + \sin^2 72^\circ)$.

The value of \cos 12° + \cos 84° + \cos 156° + \cos 132°\ is

173. If $\cos x + \cos^2 x = 1$. Let $E = \sin^{12} x + 3\sin^{10} x + 3\sin^8 x + \sin^6 x + 2$, then the value of $\log_{\tan\frac{\pi}{3}} E$ is:

Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. The height (in m) of the lamp-post is

$96\cos\dfrac{\pi}{33}\cos\dfrac{2\pi}{33}\cos\dfrac{4\pi}{33}\cos\dfrac{8\pi}{33}\cos\dfrac{16\pi}{33}$ is equal to

The value of $ \tan^{-1}\left(\dfrac{1}{4}\right) + \tan^{-1}\left(\dfrac{2}{9}\right) $ is:

A ladder 5 m long leans against a vertical wall. The bottom of the ladder is 3 m from the wall. If the bottom of the ladder is pulled 1 m farther from the wall, how much does the top of the ladder slide down the wall

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 $\triangle ABC$.]

The value of $\cos^2 x\left(\frac{1}{3} + x\right) - \cos x \cdot \cos\left(\frac{2}{3} + x\right)$ is

If sec x cos 5x + 1 = 0, where 0

If $P$ and $Q$ are the circumcentre and orthocentre of $\triangle ABC$, then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ is equal to

$\sec^2 \theta = \frac{4xy}{(x+y)^2}$ is true if and only if

Let the plane $x+3y-2z+6=0$ meet the coordinate axes at $A$, $B$, $C$. If the orthocentre of $\triangle ABC$ is $\left(\alpha,\beta,\dfrac{6}{7}\right)$, then $98(\alpha+\beta)^2$ is equal to __________.

$\angle PQS = 90° = \angle PRS = 90°$

A number $k$ is such that $\tan[\arctan(2) + \arctan(20k)] = k$. The sum of all possible values of $k$ is ______.

Two poles standing on a horizontal ground are of heights 5 m and 10 m, respectively. The line joining their tops makes an angle of 15° with ground. Then, the distance (in m) between the poles is

The number of points of intersection of $2y = 1$ and $y = \cos x$ in $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ is

The period of $\sin^2\theta$ is:

Let period of $f(x) = \frac{|\sin x| - |\cos x|}{|\sin x + \cos x|}$ is l, then [4l] is equal to __________ where []× denotes greatest integer function.

Let $C(\alpha,\beta)$ be the circumcentre of the triangle formed by the lines $4x+3y=69$, $4y-3x=17$, and $x+7y=61$. Then $(\alpha-\beta)^2+\alpha+\beta$ is equal to

Find $\tan 20° - 33\tan^3 20° + 27\tan^2 20° - 4 = $

A flag-staff of 5 meters high stands on a building of 25 meters height. For an observer at a height of 30 meters, the flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is

A tower subtends angles $\alpha$, $2\alpha$ and $3\alpha$, respectively, at points $A$, $B$ and $C$ all lying on a horizontal line through the foot of the tower. If $\frac{AB}{BC} = 1 + p \cos p\alpha$, then the value of $p$ is

Area of circumcircle of quadrilateral PLOM is

100. In a right-angled triangle the hypotenuse is how many times as long as the distance of the orthocentre from the opposite vertex. Its acute angles are

\sqrt{3} \cos x - 3 \sin x = \sqrt{x} + 1 is solvable only if

Let H be the orthocenter of triangle ABC, then angle subtended by side BC at the centre of incircle of △CHB is:

Let S = {θ ∈ [−2π, 2π] : 2cos²θ + 3sinθ = 0}, then the sum of the elements of S is

The value of $36(4\cos^2 9°-1)(4\cos^2 27°-1)(4\cos^2 81°-1)(4\cos^2 243°-1)$ is

The value of $\cos^2\frac{\pi}{16}+\cos^2\frac{3\pi}{16}+\cos^2\frac{5\pi}{16}+\cos^2\frac{7\pi}{16}$ is

Trigonometry & Inverse Trigonometry Questions (1013)