Trigonometry Questions - JEE

Let $f(x) = x^4 - 8x^3 + 18x^2 - 6x + 1 - 2\sqrt{3}$, then $f\!\left(x = \cot\dfrac{\pi}{12}\right)$ is equal to:

171. The least positive value of $x$ satisfying the equation $\dfrac{\sin x}{\cos 3x} + \dfrac{\sin 3x}{\cos 9x} + \dfrac{\sin 9x}{\cos 27x} = 0$ is:

If $x$, $y$ and $z$ are real numbers that satisfy the three equations\[\begin{cases} \tan(x)+\tan(y)+\tan(z) = 6-(\cot(x)+\cot(y)+\cot(z))\\ \tan^2(x)+\tan^2(y)+\tan^2(z) = 6-(\cot^2(x)+\cot^2(y)+\cot^2(z))\\ \tan^3(x)+\tan^3(y)+\tan^3(z) = 6-(\cot^3(x)+\cot^3(y)+\cot^3(z)) \end{cases}\]Find the value of the expression $\left(\dfrac{\tan(x)}{\tan(y)}+\dfrac{\tan(y)}{\tan(z)}+\dfrac{\tan(z)}{\tan(x)}+3\tan(x)\tan(y)\tan(z)\right)$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \{|\cos x|\}$, where $\{x\}$ represents the fractional part of x. Let S be the set containing all real values x lying in the interval $[0, 2π]$ for which $f(x) = |\cos x|$. The number of elements in the set S is

If the equation \cos 3x \cos^3 x + \sin 3x \sin^3 x = 0, then x is equal to

In a triangle $ABC$, if $\tan\frac{A}{2} = \frac{5}{6}$ and $\tan\frac{C}{2} = \frac{2}{5}$, then $a, b, c$ are in

If a, b, c are in some relation involving trigonometric identities such that $\sin^2\theta + \tan^2\theta = -b/a$ ...(1)$\sin^2\theta \cdot \tan^2\theta = c/a$ ...(2)and $\lambda = \dfrac{b^2 - c^2}{ac}$, then the value of $\lambda$ is found. Also, if $x \in (0, \pi/6) \cup (5\pi/6, \pi) \equiv (\alpha, \beta) \cup (\gamma, \delta)$ for \(\log_4(8\sin x)

If $\cot\theta = \sin 2\theta$ and $\theta \ne n\pi$, $n \in \mathbb{Z}$ then $\theta$ is equal to

A flag staff stands in the centre of a rectangular field whose diagonal is 1200 m and subtends angles 15° and 45° at the mid-points of the sides of the field. The height of the flag staff is

If x\ and y\ are acute angles, such that \cos x + \cos y = \frac{3}{2}\ and \sin x + \sin y = \frac{3}{4}\, then \sin(x + y)\ equals

In the inequality below, the value of the angle is expressed in radian measure. Which one of the inequalities below is true?(a) \(\sin 1 (b) \(\sin 3 (c) \(\sin 2 (d) \(\sin 3

If $\sin^4\alpha + 4\cos^4\beta + 2 = 4\sqrt{2}\sin\alpha\cos\beta$; $\alpha, \beta \in [0, \pi]$, then $\cos(\alpha+\beta) - \cos(\alpha-\beta) =$ __________ (up to four decimal places).

The value of $\dfrac{1 - \tan^2 15°}{1 + \tan^2 15°}$ is

If $\tan\theta + \tan\left(\dfrac{\pi}{4} + \theta\right) = 0$ then the most general value of $\theta$ is (where $n \in \mathbb{Z}$)

The value of $\cos\left(\frac{\pi}{2^2}\right)\cdot\cos\left(\frac{\pi}{2^3}\right)\cdots\cos\left(\frac{\pi}{2^{10}}\right)\cdot\sin\left(\frac{\pi}{2^{10}}\right)$ is:

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:

Trigonometry Questions (1127)