Trigonometry & Inverse Trigonometry Questions (1013)

D, E and F are the middle points of the sides of the triangle ABC, then

x_1 and x_2 are two positive values of x for which 2 \cos x, |\cos x|, and 3\sin^2 x - 2 are in GP. The minimum value of |x_1 - x_2| is equal to

If a = \sin\frac{\pi}{18} \sin\frac{7\pi}{18} \sin\frac{13\pi}{18}, then a is equal to

The sum \frac{1}{\sin 45° \sin 46°} + \frac{1}{\sin 47° \sin 48°} + \ldots + \frac{1}{\sin 133° \sin 134°} is equal to

The value of $\text{cosec}10°-\sqrt{3}\sec10°$ is equal to:

Let \(u = \cot^{-1}\sqrt{\cos 2\theta} - \tan^{-1}\sqrt{\cos 2\theta}\), then the value of \(\sin u\) is

Statement I: Let \(f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right)\). Then \(f'(2) = -\frac{2}{5}\)Statement II: \(\sin^{-1}\left(\frac{2x}{1+x^2}\right) = \pi\)

If \(\cot^{-1}\left(\frac{n^2 - 10n + 21}{\pi}\right) > \frac{\pi}{6}\), \(n \in \mathbb{N}\), then find the maximum value of \(n\).

The set of values of x, satisfying the equation \(\tan^2(\sin^{-1} x) > \frac{1}{2}\)

Number of values of \(x\) satisfying the equation \(\cos(3\arccos(x-1)) = 0\) is equal to

Simplify: \(1 + \tan^2(\tan^{-1} x) - (\sec^2(\sec^{-1} x) - 1)\)

The value of \(\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}\) is

If the mapping f(x) = mx + c, m > 0 maps [-1, 1] onto [0, 2], then \tan\left(\tan^{-1}\frac{1}{7} + \cot^{-1}8 + \cot^{-1}18\right) is equal to

If \(\cos^{-1}x - \cos^{-1}\left(\frac{y}{2}\right) = a\), then \(4x^2 - 4xy\cos a + y^2\) equals

If the range of the function \(f(x) = \tan^{-1}(3x^2 + bx + c)\) is \(\left[0, \frac{\pi}{2}\right)\) (domain is \(\mathbb{R}\)), then:

Points $D, E$ are taken on the side $BC$ of $\triangle ABC$, such that $BD = DE = EC$ and let $\angle BAD = x, \angle DAE = y, \angle EAC = z$; then $\frac{\sin(x+y)\sin(y+z)}{\sin x \sin z} =$

Sides AB and AC in an equilateral triangle ABC with side length 3 is extended to form two rays from point A. Point P is chosen outside the triangle ABC and between the two rays such that ∠ABP + ∠BCP = 180°. If the maximum length of CP is M, then \(M^2/2\) is equal to:

If the circumradius of ∆ABC is 3 units and its area is 6 square units, and ∆DEF is formed by joining the feet of perpendiculars drawn from A, B, C on sides BC, CA, AB respectively, find the perimeter of ∆DEF.

The number of solutions of the system of equations:\(2\sin^2 x + \sin^2 2x = 2\)\(\sin 2x + \cos 2x = \tan x\)in \([0, 4\pi]\) satisfying \(2\cos 2x + \sin x \le 2\) is:

If $A, B, C, D$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$, then the value of $4\sin\frac{A}{2}+3\sin\frac{B}{2}+2\sin\frac{C}{2}+\sin\frac{D}{2}$ is equal to:

12. If the value of \(f\left(\frac{\pi}{3}\right) = a + b\sqrt{c}\) where \(a, b, c \in \mathbb{N}\), then the value of \(a + b + c\) is:

If \frac{\sin^2 2x + 4\sin 4x - 4\sin 2x \times \cos 2x}{4 - \sin^2 2x - 4\sin 2x} = \frac{1}{9} and 0 , then the value of x is:

In \(\triangle A_4B_4C_4\), find the value of \(\angle A_4\)

If \(A = \sum_{r=1}^{3}\cos\frac{2r\pi}{7}\) and \(B = \sum_{r=1}^{3}\cos\frac{2r\pi}{7}\), then:

In a right angled triangle ABC with \(A = \dfrac{\pi}{2}\), a circle is drawn touching the side AB, AC and in circle of the triangle. Its radius is equal to

Let incircle of radius $4$ units of a triangle $ABC$ touches the side $BC$ at $D$. If $BD = 6, DC = 8$ and $\Delta$ be the area of triangle, then $\sqrt[4]{\Delta - 3}$ = _______.

The total number of solutions of $\tan\{x\} = \cot\{x\}$ ; where $\{x\}$ denotes the fractional part of $x$ in $[0, 2\pi)$ is _______.

If $\sin x + \sin^2 x + \sin^3 x = 1$, then $\cos^6 x - 4\cos^4 x + 8\cos^2 x$ = _______.

If $\tan\left(\frac{2\pi}{3} - x\right) = \frac{\sin\frac{2\pi}{3} - \sin x}{\cos\frac{2\pi}{3} - \cos x}$ where $0 < x < \frac{3\pi}{2}$, and the values of $x$ are $x_1$ and $x_2$, then the value of $\frac{12}{\pi}|x_2 - x_1|$ is

If $10\sin^4 u + 15\cos^4 u = 6$ and the value of $9\cos\sec^4 u + 8\sec^4 u$ is $S$, then find the value of $\frac{S}{25}$

If $\sum_{r=1}^{q}\frac{\tan 2^{r-1}}{\cos 2^r} = \tan p^n - \tan q$, then find the value of $(p + q)$

Given that for $a, b, c, d \in \mathbb{R}$, if $a\sec(200°) - c\tan(200°) = d$ and $b\sec(200°) + d\tan(200°) = c$, then find the value of $\left(\frac{a^2 + b^2 + c^2 + d^2}{bd - ac}\right)\sin 20°$

If the value of $\cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cos\frac{7\pi}{7} = -\frac{l}{2}$. Find the value of $l$

The complete set of values of $x$ satisfying $\frac{2\sin 6x}{\sin x - 1} < 0$ and $\sec^2 x - 2\sqrt{2}\tan x \leq 0$ in $\left[0, \frac{\pi}{2}\right)$ is $[a, b) \cup (c, d]$, then find the value of $\left(\frac{cd}{ab}\right)$

If the sum of all values of $\theta$, $0 \leq \theta \leq 2\pi$ satisfying the equation $(8\cos 40 - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12$ is $k\pi$, then $k$ is equal to:

Find number of solutions of the equation $\sin^{-1}(\log_2(\cos x)) - 1) + \cos^{-1}(3\log_2^2(\cos x) - 7)) = \frac{\pi}{2}$, if $x \in [0, 4\pi]$.

The two adjacent sides of a cyclic quadrilateral are $2, 5$ and the angle between them is $60°$. If the area of the quadrilateral is $4\sqrt{3}$, then the remaining two sides are:

\(\cos^{-1} l + \cos^{-1} m + \cos^{-1} n\) is equal to

Find the value of a + b + c where OA = r cot(π/4 − q/2) = 2r cot(q/2), and tan(q/2) = (−3 ± √17)/2.

Let \(\cos(\alpha+\beta)=\dfrac{4}{5}\) and let \(\sin(\alpha-\beta)=\dfrac{5}{13}\), where \(0 \le \alpha,\ \beta \le \dfrac{\pi}{4}\), then \(\tan 2\alpha =\)

Minimum possible area of the triangle is :

If \(\alpha = \frac{1}{3}\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{1}{3}\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) where \(x \geq \frac{4}{3}\), then the value of \(\dfrac{\cos 2\alpha + \sec\alpha + 3\sqrt{3}}{\sqrt{3}}\) is equal to:

Let \(0 \leq \alpha, \beta, \gamma, \delta \leq \pi\) where \(\beta\) and \(\gamma\) are not complementary such that\(2\cos \alpha + 6\cos \beta + 7\cos \gamma + 9\cos \delta = 0\) and \(2\sin \alpha - 6\sin \beta + 7\sin \gamma - 9\sin \delta = 0\)If \(\frac{\cos(\alpha + \delta)}{\cos(\beta + \gamma)} = \frac{m}{n}\) where \(m\) and \(n\) are relatively prime positive numbers, then the value of \((m + n)\) is equal to:

The angle of elevation of a cloud from a point $250$ m above a lake is $15°$ and angle of depression of its reflection in the lake is $45°$. The height of the cloud is

If 1 + \sin 3x + \cos 3x = \frac{3}{2}\sin 2x, then x is

The value of \(\cos\left(\cos^{-1}\left(\frac{1}{8}\right)\right)\) is equal to

If \(\cot^{-1}\left(\frac{n}{p}\right) = \frac{p}{6}\), \(n \in \mathbb{N}\), then the maximum value of \(\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)\) is \('n'\) is

The number of positive integral solutions of \(\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)\) is

The value of \(\cos\left(\cos^{-1}\left(\cos\left(\sin^{-1}\left(\frac{63}{8}\right)\right)\right)\right)\) is

$\sin^{-1}(\sin \theta) = \theta$, for all $\theta$ belonging to