Trigonometry & Inverse Trigonometry Questions - JEE

The equation $4\cos 2x = 7 - 4x + 4x^2$ has no solution for $\theta$ if $x \in \mathbb{R}$.State whether the statement is true or false.

Number of integral solutions of the equation $\log_{\sin x} \sqrt{\sin^2 x} + \log_{\cos x} \sqrt{\cos^2 x} = 2$, where $x \in [0, 6\pi]$

Let $f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, \ldots$ Then for all $x \in R$, the value of $f_4(x) - f_6(x)$ is equal to __________ (up to four decimal places).

If a\sin^{-1}x - b\cos^{-1}x = c, then a\sin^{-1}x + b\cos^{-1}x is equal to

If $a \tan \alpha + \sqrt{a^2 - 1} \tan \beta + \sqrt{a^2 + 1} \tan \gamma = 2a$, where $a$ is constant and $\alpha, \beta, \gamma$ are variable angles, then find the least value of $2727(\tan^2 \alpha + \tan^2 \beta + \tan^2 \gamma)$.

The sides of a triangle are $\sin a$, $\cos a$, $\sqrt{1 + \sin a \cos a}$ for some $0

(h) $\frac{20}{\sqrt{3}}$ meters

If $a_1, a_2, a_3, \ldots, a_n$ are in AP with common difference $5$ and if $a_i a_j \neq -1$ for $i, j = 1, 2, \ldots, n$, then $\tan^{-1}\left(\frac{5}{1 + a_1 a_2}\right) + \tan^{-1}\left(\frac{5}{1 + a_2 a_3}\right) + \tan^{-1}\left(\frac{5}{1 + a_{n-1}a_n}\right) + \ldots + \tan^{-1}\left(\frac{5}{1 + a_n a_1}\right)$ is equal to

Note that if $\tan\theta$ is positive, then $\theta$ is in the first or third quadrant, so \(0°

In a triangle $ABC$, $\angle AGB = \dfrac{\pi}{2}$, $AD = 4$, $AG = \dfrac{2}{3} \times AD$. If $\angle BAG = \dfrac{\pi}{6}$ and $\angle ABC = \dfrac{\pi}{3}$, then the area of $\triangle ABC$ is:

With the usual notation, in $\triangle ABC$, if $\angle A + \angle B = 120°$, $a = \sqrt{3} + 1$ and $b = \sqrt{3} - 1$, then the ratio $\angle A : \angle B$ is

Ex. 34. Statement I: x=kπ2, k∈I does not represent the general solution of trigonometric equation.Statement II: Both x=nπ, n∈I and x=kπ2, k∈I satisfy the trigonometric equation sin13x−sin13xcos2x=0.

The complete set of values of $x$ satisfying \(\frac{2\sin 6x}{\sin x - 1}

Given $\dfrac{b+c}{11} = \dfrac{c+a}{12} = \dfrac{a+b}{13}$, and using the cosine formula with $\dfrac{\cos A}{\alpha} = \dfrac{\cos B}{\beta} = \dfrac{\cos C}{\gamma}$, find the value of $475\alpha = 175\beta = 133\gamma$.

The general solution of $\sin 2\theta \sec\theta + \sqrt{3}\tan\theta = 0$ is

If $\sin a\theta + \cos b\theta = 0$ then the possible values of $\theta$ form

Total reflexive relations on $A$ ($|A|=n$):

$\cot^{-1}\sqrt{\frac{1+x^2}{1-x^2}}$ is equal to

The least value of $(\cos^2\theta-6\sin\theta\cos\theta+3\sin^2\theta+2)$ is

Let a vertical tower AB have its end a on the level ground. Let C be the mid-point of AB and P be a point on the ground such that $AP = 2AB$. If $\angle BPC = \beta$, then $\tan\beta$ is equal to

Let the height of a tower be $TM = h$ and $QM = MR = x$. From point $P$ (which is 200 m above ground), angles of depression to $T$ and $R$ are $45°$ and $30°$ respectively. Find $h$.

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.

Trigonometry & Inverse Trigonometry Questions (1013)