Trigonometry & Inverse Trigonometry Questions - JEE

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

93. If in a △ABC, c = 150, b = 50√3 and B = 30° then C has the measure

In an equilateral △ABC (where symbols used have usual meanings), then r, R and r1 form:

The maximum value of $3\cos\theta + 5\sin\left(\theta - \dfrac{\pi}{6}\right)$ for any real value of $\theta$ is __________ (up to four decimal places).

If $\tan a$, $\tan b$, and $\tan g$ are the roots of the equation $x^3 - px^2 - r = 0$, then the value of $(1 + \tan^2 a)(1 + \tan^2 b)(1 + \tan^2 g)$ is equal to

If $\tan B = \frac{\sin A \cos A}{1 - n\cos^2 A}$, then $\tan(A + B)$ equals

If $\lambda = \left(\dfrac{\cos 65^\circ + \sqrt{3}\cos 85^\circ + \sin 85^\circ}{\sin 65^\circ}\right)^2$, find the value of $\lambda$.

If the equation $\cos^4\theta + \sin^4\theta + \lambda = 0$ has real solutions for $\theta$, then $\lambda$ lies in the interval

Total number of solutions of \sin x = -\frac{1}{10} is equal to

The number of solutions of the equation \[[y + [y]] = 2\cos x\] where \[y = \frac{1}{3}[\sin x + [\sin x + [\sin x]]]\] and $[\cdot]$ denotes the greatest integer function, is

If $a \cos^2 3a + b \cos^4 a = 16 \cos^6 a + 9 \cos^2 a$ is an identity, then

The number of solutions of $|\cos x| = \sin x$ such that \(0

If $\cos x + \sin x = a$ where $-\frac{\pi}{2} , then \(\cos 2x$ is equal to

Let n be a positive integer such that n ∈ ℕ. Then, sin$\left(\frac{\pi}{2n}\right)$ + cos$\left(\frac{\pi}{2n}\right)$ = $\frac{\sqrt{2}}{2}$. Find the range of n.

Trigonometry & Inverse Trigonometry Questions (1013)