Trigonometry & Inverse Trigonometry Questions (1013)

If (sin A - sin C)/(cos C - cos A) = cot B, then A, B and C are in

The value of \((\cos^4 1° + \cos^4 2° + \cos^4 3° + \ldots + \cos^4 179°) - (\sin^4 1° + \sin^4 2° + \sin^4 3° + \ldots + \sin^4 179°)\) equals

If \(\sin(x\cos\theta) = \cos(x\sin\theta)\) then \(\sin 2\theta\) is equal to

Let the maximum value of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$ for $x\in\left[-\dfrac{\sqrt{3}}{2},\dfrac{1}{\sqrt{2}}\right]$ be $\dfrac{m}{n}\pi^2$, where $\gcd(m,n)=1$. Then $m+n$ is equal to _____.

If $k=\tan\!\left(\dfrac{\pi}{4}+\dfrac{1}{2}\cos^{-1}\!\dfrac{2}{3}\right)+\tan\!\left(\dfrac{1}{2}\sin^{-1}\!\dfrac{2}{3}\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x$ is _____.

\(\tan^{-1}\left(\frac{c_1 x - y}{c_1 y + x}\right) + \tan^{-1}\left(\frac{c_2 - c_1}{1 + c_2 c_1}\right) + \tan^{-1}\left(\frac{c_3 - c_2}{1 + c_3 c_2}\right) + \ldots + \tan^{-1}(1)\) is equal to

Find the value of \(\frac{\cos A + \cos B}{\sin A - \sin B} + \frac{\sin A + \sin B}{\cos A - \cos B}\) (where \(n\) is even).

The value of \(\tan^{-1} \left[ \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right]\) is

Suppose that \(a\) is a non-zero real number for which \(\sin x + \sin y = a\) and \(\cos x + \cos y = 2a\). The value of \(\cos(x - y)\) is

\(\sin(\alpha - \beta)\) is equal to

In an acute angled triangle $ABC$, $\angle A = 20°$, let $DEF$ be the feet of altitudes through $A, B, C$ respectively and $H$ is the orthocentre of $\triangle ABC$. Find $$\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF}$$

The minimum value of the function \(f(x) = (3\sin x - 4\cos x - 10)(3\sin x + 4\cos x - 10)\) is

A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 m and the distance between B and C is 100 m, then the height of the balloon is given by

A variable triangle ABC is circumscribed about a fixed circle of unit radius. Side BC always touches the circle at D and has fixed direction. If B and C vary in such a way that (BD)·(CD) = 2, then the locus of vertex A will be a

Draw the curves p|cos θ| and q|sin θ| and find the number of intersection points.

If $t = x + y + z$, then $\sin x + \sin y + \sin z - \sin t$ equals:

Let $x + \frac{1}{x} = 2, y + \frac{1}{y} = -2$ and $\sin^{-1} \left(\frac{1}{x}\right) + y = mx$, then the value of $m$ is

Let A(7,0), B(4,4) and C(0,0) and triangle DEF is isosceles with DE = DF. Then, the curve on which F may lie

\(\sin^{-1}(3x - 4x^3) = \lambda \sin^{-1} x\) then \(\lambda = \underline{\quad}\).

If inside triangle ABC, a, b, c and angle A are given and \(c\sin A

The value of \(\sqrt{\sin^2\frac{2\pi}{11} - \cos\frac{8\pi}{11}}\) is equal to

If \(P = \frac{\tan(3^n + 10) - \tan\theta}{\cos(3^n \theta)}\) and \(Q = \text{(expression)}\), then

Given, \(a^2 + 2a + \csc^2\frac{x}{2} - (a+x) = 0\), then which of the following holds good?

Let f(x) = sin²³x + cos²²x and g(x) = 1 − \(\frac{1}{2}\) tan⁻¹|x|. The number of values of x in interval [−100°, 200°] satisfying the equation f(x) = sgn(g(x)), is 5a. Then, a is equal to ……

The complete set of values of a for which the function f(x) = \tan^{-1}(x^2 - 18x + a) \geq 0, \forall x \in \mathbb{R}, is

The value of \(4\cos 20° - 3\cot 20°\) is

If in a triangle ABC, cot A}{2} + cot B}{2} + cot C}{2} = X cot A}{2} cot B}{2} cot C}{2}, then find the value of X.

The sum of all values of \(\theta \in \left[0, \frac{\pi}{2}\right)\) satisfying \(\sin 2\theta + \cos 2\theta = \frac{3}{4}\) is

The maximum value of \(4\sin^2 x + 3\cos^2 x + \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right)\) is

The sides of a triangle are \(\sin\alpha\), \(\cos\alpha\) and \(\sqrt{1 + \sin\alpha\cos\alpha}\) for some \(0

If \(4x^3 - 3x - p = 0\), where \(-1 \leq p \leq 1\) has a unique root in \([-1, 1]\), then the root is

If cot θ + cot(π/4 - θ) = 2, then the general value of θ is

Example 42: The set of values of \(X \in \mathbb{R}\) such that \(\tan^2 \theta + \sec \theta = X\) holds for some \(\theta\) is

Find the value of \(\sin 20° + \cos 40° + \sin 50° + \tan 70° + \cot 80°\).

The minimum value of \(\sin^4 a + \sin^4 b + \sin^4 g\), where \(a, b, g\) are real positive angles satisfying \(a + b + g = \pi\), is

Let \(ABC\) be a right angled triangle at \(C\). If the inscribed circle touches the side \(AB\) at \(D\) and \((AD)(BD) = 11\), then find the area of \(\triangle ABC\).

Ex. 84: If x sin³θ + y cos³θ = sin θ cos θ and x sin θ - y cos θ = 0, then (x, y) lie on

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