Inverse Trigonometry Questions (1043)

95. In the △ABC, AB = 5 cm, AC = 12 cm and BC = 13 cm then the distance of A from the side BC is (in cm)

$PQR$ is a triangular park with $PQ = PR = 200$ m. A TV tower stands at the mid-point of $QR$. If the angles of elevation of the top of the tower from $P$, $Q$ and $R$ are $45°$, $30°$ and $30°$ respectively, then the height of the tower (in meters) is

175. If \(A\) lies in the fourth quadrant and \(3\tan A + 4 = 0\), then \(5\sin 2A + 2\sin A + 4\cos A\) is equal to:

810. In \(\triangle ABC\) if inradius \(r = 1\), circumradius \(R = 3\) and semiperimeter \(s = 7\), then find the value of \((a^2 + b^2 + c^2)\), where \(a, b, c\) are the sides of triangle \(ABC\).

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\displaystyle\sum_{\theta \in S} \sin^2\left(\theta + \frac{\pi}{4}\right)$ is equal to

If $\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$, then the value of $\left(a + \frac{1}{a}\right)$ is:

158. If \((\sin^{-1} x)^2 + (\sin^{-1} y)^2 + 2\sin^{-1} x \sin^{-1} y = \pi^2\), then \(x^2 + y^2\) is equal to:

If |k| = 5 and 0°

The equation sin x + sin y + sin z = -3 for 0

Ex. 24: Statement I In a triangle ABC, \(\cos^2\dfrac{A}{2}\) has the value equal to \(\dfrac{s(s-a)}{abc}\)Statement II In a triangle ABC, \(\cos\dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{bc}}\), \(\cos\dfrac{B}{2} = \sqrt{\dfrac{(s-a)(s-c)}{ac}}\), \(\cos\dfrac{C}{2} = \sqrt{\dfrac{(s-a)(s-b)}{ab}}\)

101. Two straight roads intersect at 30°. From the junction, two persons A and B start walking at the same time, one on each road. A walks at the rate of 5 km/h. At the end of 3 hours they are 9 km apart. If B walks at uniform rate then the speed of B is

Ex. 45. If $a$, $P$, $y$ are acute angles and $\cos \theta = \frac{\sin P}{\sin a}$, $\cos \theta = \frac{\sin y}{\sin a}$ and $\cos(\theta - \phi) = \sin P \sin y$, then the value of $\tan^2 a - \tan^2 P - \tan^2 y$ 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:

152. The minimum value of the expression \(\dfrac{\sin^3\alpha + 6\sin^2\alpha + \sin\alpha + 2\cos^2\alpha - 8}{\sin\alpha - 1}\) is equal to:

276. If \(\dfrac{1}{2}\sin^{-1}\!\left(\dfrac{3\sin 2\alpha}{5+4\cos 2\alpha}\right) = \tan^{-1} x\), then the possible value of \(x\) is:

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:

703. Let \(f(x) = \cos^{-1}\!\left(\sqrt{\sin^{-1}\!\left(\sec\!\left(\ln\!\left(\dfrac{2x^2+3x-2}{x^2-3x+2}\right)\right)\right)}\right)\). Find the value of \(1 + \left(\displaystyle\sum \alpha_i^2\right)\), where \(\alpha_i\) represents the integers in the range of \(f(x)\). If there are no integers in the range of \(f(x)\), then enter your answer as zero.

175. If \(A\) lies in the fourth quadrant and \(3\tan A + 4 = 0\), then \(5\sin 2A + 2\sin A + 4\cos A\) is equal to:

The angles A, B and C of a triangle ABC are in arithmetic progression. If \(2b^2 = 3c^2\) then the angle A is:

Let $|\cos\theta\cos(60^\circ-\theta)\cos(60^\circ+\theta)|\leq\dfrac{1}{8}$, $\theta\in[0,2\pi]$. Then, the sum of all $\theta\in[0,2\pi]$, where $\cos3\theta$ attains its maximum value, is:

The angle of elevation of the top $P$ of a tower from the feet of one person standing due south of the tower is $45°$ and from the feet of another person standing due west of the tower is $30°$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

If $S=\left\{x\in\mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^2+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^2+1}}\right)=\dfrac{\pi}{4}\right\}$, then $\displaystyle\sum_{x\in S}\left(\sin\frac{(x^2+x+5)\pi}{2}-\cos\frac{(x^2+x+5)\pi}{2}\right)$ is equal to _________.

For $x\in(-1,1]$, the number of solutions of the equation $\sin^{-1}x=2\tan^{-1}x$ is equal to

In $\triangle ABC$, if $\cos A+2\cos B+\cos C=2$ and the sides opposite to $A$ and $C$ are $3$ and $7$ respectively, then $\cos A-\cos C$ is equal to

The value of $\tan 9°-\tan 27°-\tan 63°+\tan 81°$ is _____.

In △ABC, a = 4, b = 12 and B = 60°, then the value of sin A is

The principal value of \(\tan^{-1}(-\sqrt{3})\) is ______.

If \sum , then a + b is equal to : 13 1 2 2 { } = a\sqrt3 + b, a, b \in Z r=1 \pi \pi \pi r\pi sin( +(r-1) ) sin( + ) 4 6 4 6

The value of (sin 70 ) (cot 10 cot 70 - 1) is ∘ ∘ ∘

If sin x + sin x = 1, x \in (0, 2 \pi ) , then (cos 12 x + tan 12 x) + 3 (cos 10 x + tan 10 x+ 2 cos 8 x + tan 8 x) + (cos 6 x + tan 6 x) is equal to :

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum 2 2 values of 16 ((sec -1 x) + (cosec -1 x) ) is :

cos(sin -1 3 + sin -1 5 + sin -1 33 ) is equal to: 5 13 65

154. Let \(f: R \to \left(0, \dfrac{2\pi}{3}\right]\) defined as \(f(x) = \cot^{-1}(x^2 - 4x + \alpha)\). The smallest integral value of \(\alpha\) such that \(f(x)\) is into function, is equal to:

Let $x=\dfrac{m}{n}$ ($m$, $n$ are co-prime natural numbers) be a solution of the equation $\cos(2\sin^{-1}x)=\dfrac{1}{9}$ and let $\alpha,\beta$ ($\alpha>\beta$) be the roots of the equation $mx^2-nx-m+n=0$. Then the point $(\alpha,\beta)$ lies on the line

If $a=\sin^{-1}(\sin5)$ and $b=\cos^{-1}(\cos5)$, then $a^2+b^2$ is equal to

172. The value of \(\cos\left[\log_5\left(\dfrac{\sin^2 A + \cos^2 A + \tan^2 A - \sec^2 A \cdot \sin^2 A}{(1 + \tan^2 A)(1 - \sin^2 A)}\right)\right]\) is equal to:

If $\dfrac{\pi}{2} \leq x \leq \dfrac{3\pi}{4}$, then $\cos^{-1}\!\left(\dfrac{12}{13}\cos x + \dfrac{5}{13}\sin x\right)$ is equal to

If for some $\alpha,\beta$; $\alpha \leq \beta$, $\alpha+\beta = 8$ and $\sec^2(\tan^{-1}\alpha)+\operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2+\beta$ is ________.

If $\alpha>\beta>\gamma>0$, then the expression $\cot^{-1}\!\left\{\beta+\dfrac{1+\beta^2}{\alpha-\beta}\right\}+\cot^{-1}\!\left\{\gamma+\dfrac{1+\gamma^2}{\beta-\gamma}\right\}+\cot^{-1}\!\left\{\alpha+\dfrac{1+\alpha^2}{\gamma-\alpha}\right\}$ is equal to:

Let $S=\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\}$. Then $\displaystyle\sum_{x\in S}(2x-1)^2$ is equal to ________.

The integral $\displaystyle\int_{1/4}^{3/4}\cos\!\left(2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)dx$ is equal to

For $n\in\mathbb{N}$, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\dfrac{\pi}{4}$, then $n$ is equal to

Given a = 6, b = 3 and \(\cos(A - B) = -\frac{1}{4}\), find \(\sin A\).

If \(k_1 = \tan 27\theta - \tan 9\theta + \tan 9\theta - \tan 3\theta + \tan 3\theta - \tan \theta\) and \(k_2 = \frac{\sin 3\theta}{\cos 3\theta} + \frac{\sin 9\theta}{\cos 9\theta} + \frac{\sin 27\theta}{\cos 27\theta}\), then

If the value of $\dfrac{3\cos36^\circ+5\sin18^\circ}{5\cos36^\circ-3\sin18^\circ}$ is $\dfrac{a\sqrt{5}-b}{c}$, where $a,b,c$ are natural numbers and $\gcd(a,c)=1$, then $a+b+c$ is equal to:

If $\sin x=-\dfrac{3}{5}$, where $\pi<x<\dfrac{3\pi}{2}$, then $80\left(\tan^2 x-\cos x\right)$ is equal to

The set of real numbers a such that \(a^2 + 2a\), \(2a + 3\), \(a^2 + 3a + 8\) are the sides of a triangle, is:

The minimum and maximum values of \(a\sin x + b\sqrt{1 - a^2}\cos x + c\) (where \(|a| 0\)) respectively are

If \(\frac{\cos 0 \cos 2\theta}{1 - \sin \theta} + \frac{\sin \theta \sin 2\theta}{1 + \cos \theta} = 1 + \cos \theta\), then number of possible values of \(\theta\) is (where \(\theta \in [0, 2\pi]\))

Let $S = \{x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1)\}$. Then $\sum_{x \in S} (2x - 1)^2$ is equal to ______.