Trigonometry Questions - JEE

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

One root of $\tan^{-1}\!\cot\!\left(\dfrac{3x^2+3|x|+1}{x^2+|x|+1}\right)=\dfrac{\pi}{2}-\csc\!\cdot\!\csc^{-1}\!\left(\dfrac{3|x|+2}{|x|+1}\right)$ is $2\sin\theta$, $\theta\in\left(0,\dfrac{\pi}{2}\right)$. Value of $\tan\dfrac{7\theta}{9}\cdot\tan\dfrac{10\theta}{9}\cdot\tan\dfrac{13\theta}{9}$ is

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.

169. If $\sin\alpha + \sin\beta + \sin\gamma = -3$, $\alpha, \beta, \gamma \in (0, 2\pi)$, then $\cos 2\alpha + \cos 4\beta + \cos 6\gamma$ is equal to:

$AB$ is a vertical tower. The point $A$ is on the ground and $C$ is the middle point of $AB$. The part $CB$ subtends an angle $\alpha$ at a point $P$ on the ground. If $AP = n \cdot AC$, then the correct relation is

If $\alpha$ is an integer satisfying $|\alpha| \leq 5 - |[x]|$, where $x$ is a real number for which $2x\tan^{-1}x$ is greater than or equal to $\ln(1+x^2)$, then the number of maximum non-negative possible values of $\alpha$ is (where $[.]$ denotes the greater integer function)

If \sin 3a = 4\sin a \sin(x+a)\sin(x-a), then x is equal to

99. ABC is a right-angled isosceles triangle in which ∠A = 90°. The midpoint D of AB is joined to C. The ratio of cot ∠DCA and cot ∠DCB is

If $0 , \(0 and \(\cos x \cdot \sin y = 1$, then find the possible number of values of the ordered pair $(x, y)$.

The angles of triangle PQR are

If $\tan\theta + \tan 2\theta + \sqrt{3}\tan\theta \tan 2\theta = \sqrt{3}$, then

The set of all values of $\lambda$ for which the equation $\cos^2 2x - 2\sin^4 x - 2\cos^2 x = \lambda$ has a real solution $x$ is

In triangle ABC, $\angle B (a) \(\frac{\pi}{3}$(b) $\frac{\pi}{4}$(c) $\frac{\pi}{6}$(d) $\frac{\pi}{2}$

Let maximum value of the expression $y = |k - 3|\cos 2x + |t - 4|\sin 2x + 3$ where $0 \leq k \leq 6$ and $1 \leq t \leq 7$ is equal to 6, then find the minimum value of $(k^2 + t^2)$.

If $\sin(\alpha+\beta)=1$, $\sin(\alpha-\beta)=\dfrac{1}{2}$, then $\tan(\alpha+2\beta)\tan(2\alpha+\beta)$ is equal to

If the equation $\sum_{n=0}^{10} \operatorname{arc}\cot\left(\frac{1+2^{2n+1}}{2^n}\right) = \operatorname{arc}\cot\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. The value of $\log_2\left(\frac{b+a}{a-b}\right)$, is:

The value of \sqrt{1 + \cos\frac{\pi}{8}} \cdot \sqrt{1 + \cos\frac{3\pi}{8}}\ is equal to

If the angle of elevation of a cloud from a point P which is 25 m above a lake is 30° and the angle of depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters) from the surface of the lake is __________ .

A triangle has sides 6, 7, 8. The line through its incentre parallel to the shortest side is drawn to meet the other two sides at P and Q. The length of the segment PQ is:

If \alpha + \beta = \frac{\pi}{2}\ and \beta + \gamma = \alpha\, then the value of \tan\alpha\ is

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

974. Given that $x \in \mathbb{R}$, find the minimum value of $\left(3\sqrt{5 - 4\cos x} + \sqrt{13 - 12\sin x}\right)^2$.

If $M = (\cos^2\theta - 2\cos\theta)\sec^2\phi + 9\text{cosec}^2\phi + 5\sec^2\phi$ where $\theta \in [0,\ \pi]$ and $\phi \in \left(0,\ \dfrac{\pi}{2}\right)$, then find the least value of $M$.

The value of $4\cos\dfrac{\pi}{10} - 3\sec\dfrac{\pi}{10} - 2\tan\dfrac{\pi}{10}$ is equal to

If $4\sin^4 x + \cos^4 x = 1$, then $x$ is

The number of real solutions for x from $2\cos\dfrac{x}{2} = 2^x + 2^{-x}$ is ______.

Question 586. The value of 'm' is equal to:

The number of solutions of the equation $1 + \sin^4 x - \cos^2 3x$, $x \in \left[-\dfrac{5\pi}{2}, \dfrac{5\pi}{2}\right]$ is __________ .

If \cos(\theta - \alpha) = a\ and \cos(\theta - \beta) = b\, then \sin^2(\alpha - \beta) + 2ab\cos(\alpha - \beta)\ is equal to

Circumradius of an isosceles △ABC with ∠A = ∠B is 4 times its inradius, then cos A is root of the equation:

Let $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$, $\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}$. If $\delta>0$ and area of $\triangle ABC=5\sqrt{6}$, then $\overrightarrow{CB}\cdot\overrightarrow{CA}$ is equal to

The difference between the greatest and the least possible value of the expression $3 - \cos a + \sin^2 a$ is

If \sec x \cos 5x = -1 and 0 , then x is equal to

Trigonometry Questions (1127)