Trigonometry & Inverse Trigonometry Questions (1013)

\(2\sin^{-1}\sqrt{\dfrac{1-x}{2}} = \cos^{-1}(\underline{\quad})\).

For \(f(x)=e^x\), \(g(x)=\sin^{-1}x\), which are necessarily true?

Find the radius of the circle escribed to the triangle ABC on the side BC if $\angle NAB = 30°$; $\angle BAC = 30°$; $AB = AC = 5$.

In a $\triangle ABC$, $\angle A > \angle B$. Let $\angle A, \angle B$ satisfy the equation $3\sin x - 4\sin^3 x - k = 0$, where $0 < k < 1$, then $\angle C$ is equal to:

An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60° and after 10 seconds the elevation is observed to be 30°. The uniform speed of the aeroplane in km/h is

If sin x + cos x = 1 + sin x cos x, then x is

Let ABC be a triangle such that ∠A = 45°, ∠B = 75°, then \(a + c\sqrt{2}\) is equal to

If \(\cos(x-y)\), \(\cos x\) and \(\cos(x+y)\) are in HP then \(\cos x \cdot \sec\left(\frac{y}{2}\right) =\) ______

In a triangle \(ABC\), it is given that \(\dfrac{[ABC]}{R} = 4\), where \(R\) is the circumradius. Show that \(\sum_{\text{cyc}} a\cos A = 4R \prod_{\text{cyc}} \sin A\) and find \(\prod_{\text{cyc}} \sin A\).

Ex. 62: If in triangle ABC, $\tan A + \tan B + \tan C = 6$ and $\tan A \tan B = 2$, then $\sin^2 A : \sin^2 B : \sin^2 C$ is

The period of the function \( f(x) = \sin^4 x + \cos^4 x \) is:

If \(\tan(\alpha - \beta) = \dfrac{\sin(2\beta)}{3 - \cos(2\beta)}\), then \(\tan\alpha = f(\beta)\). The value of \(f\!\left(\dfrac{\pi}{3}\right)\) equals:

176. If \(\alpha = \sin\theta\,|\sin\theta|\) and \(\beta = \cos\theta\,|\cos\theta|\) where \(\theta \in \left[\dfrac{199\pi}{2},\, 100\pi\right]\), then:

The angle of elevation of tower from a point A due south of it is 30° and from a point B due west of it is 45°. If the height of the tower be 100 m, then AB =

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:

If sin x₁ + sin x₂ + sin x₃ + ... + sin x₂₀₀₈ = 2008, then find the value of sin²⁰⁰⁸ x₁ + sin²⁰⁰⁸ x₂ + sin²⁰⁰⁸ x₃ + ... + sin²⁰⁰⁸ x₂₀₀₈.

In a triangle ABC if \(3 \sin A + 4 \cos B = 6\); \(4 \sin B + 3 \cos A = 1\) then possible value(s) of ∠C be:

If √2 cos A = cos B + cos³ B, and √2 sin A = sin B - sin³ B then sin(A - B) = ?

If 2 tan-1(1/5) - sin-1(3/5) = -cos-1(63/l), then l =

If \(\cos(a+b)=\frac{3}{5}\), \(\sin(a-b)=\frac{5}{13}\) and \(0

The principal value of \(\cos^{-1}\left(\cos\dfrac{10\pi}{7}\right)\) is

If \( f(x) = 2\tan^{-1}x + \sin^{-1}\left(\dfrac{2x}{1+x^2}\right) \), \( x > 1 \), then \( f(t) \) is equal to

If \( \sin^{-1} x = 2\sin^{-1} a \) has a solution, then \( |a| \) satisfies:

Find the sum of squares of all values of x satisfying the equation \(2\tan^{-1}x = \dfrac{\pi}{2} - 2(\pi - 2\tan^{-1}x)\) (considering all cases based on the range of x).

782. Find the number of integral values of \(x\) satisfying the inequality\[\frac{\left(2^{\tan^{-1}x} - 4\right)(x-4)(x-10)}{x! - (x-1)!}

If \(\sin(2\cos^{-1}\frac{1}{\sqrt{5}})+\cos(2\tan^{-1}\frac{1}{3})=\frac{p}{q}\) (coprime), units digit of \((p-q)^{2k+1}\), \(k\in\mathbb{N}\) can be:

\(2\tan^{-1} x = \tan^{-1}\dfrac{2x}{1-x^2}\) is true if

The inequality \(\sin^{-1}(\sin 5) > x^2 - 4x\) holds if

Number of solutions of \(\sin^{-1}(1-x)-2\sin^{-1}x=\pi/2\):

If $\tan^{-1}(2n) - \tan^{-1}(n) = \tan^{-1}(2n)$, then find $n$.

\(\tan^{-1} x + \tan^{-1}\dfrac{2x}{1-3x^2} = \pi + \tan^{-1}\dfrac{3x - x^3}{1-3x^2}\) \((x > 0)\) is true if

\(\tan^{-1}\!\left(1-x^2-\dfrac{1}{x^2}\right)+\sin^{-1}\!\left(x^2+\dfrac{1}{x^2}-1\right)\), \(x\ne 0\), equals:

262. If \(\sec^{-1}(x) + \tan^{-1}\sqrt{9y^2 - 1} + \sin^{-1}(x^2 + y^2) = \lambda\) has no solution, then exhaustive set of values of \(\lambda\) is equal to:

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 an into function, is equal to:

Let \(f(x) = \sin x + \cos x + \tan x + \arcsin x + \arccos x + \arctan x\). If \(M\) and \(m\) are maximum and minimum values of \(f(x)\), then their arithmetic mean is equal to

239. Let \(a \in \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)\) such that \(\tan^{-1}\!\left(\dfrac{\tan\alpha}{3 + 2\tan^2\alpha}\right) + \tan^{-1}\!\left(\dfrac{2\tan\alpha}{3}\right) = \dfrac{\pi}{12}\), then \(\alpha\) equals:

\(4\cot^{-1} 3 + \sin^{-1}\dfrac{1}{\sqrt{5}} - \sin^{-1}\dfrac{1}{\sqrt{5}} = \underline{\quad}\).

142. If \(x=\sin^{-1}(\sin 10)\) and \(y=\cos^{-1}(\cos 10)\), then \(y-x\) is equal to:

934. Find the number of integers not in the domain of \(f(x) = \cos^{-1}\!\left(\dfrac{2 - x}{2x}\right)\).

Find the value of m where m = tan²(sec⁻¹ 2) + cot²(cosec⁻¹ 3), then find m² + m + 10

If $\frac{x + \frac{3}{x}}{} = 2$, then the value of $\sin^{-1} x$ is

Match relations on \(A=\{a,b,c\}\): (I)–(IV) with properties P,Q,R,S.

If \[2y = \cot^{-1}\left(\frac{3\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)^2\], where \(x \in \left(0, \frac{\pi}{2}\right)\), then \(\frac{dy}{dx}\) is equal to

\(\sin^{-1}(\sin 10)=\)

If \(\cos^{-1}x-\cos^{-1}(y/2)=\alpha\), then \(4x^2-4xy\cos\alpha+y^2=\)

Sum to infinite terms of the series \(\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{2}{2^3} + \tan^{-1}\frac{2}{2^5} + \cdots + \tan^{-1}\frac{1}{2^{2n-1}} + \cdots\) is

\(3\tan^{-1} x = \tan^{-1}\dfrac{3x - x^3}{1-3x^2}\) \((x > 0)\) if \(x

If the sum of all the solutions of \tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}, \quad -1 < x < 1, x \neq 0, \text{ is } \alpha - \frac{4}{\sqrt{3}}, \text{ then } \alpha \text{ is equal to}

Solve \(\sin[2\cos^{-1}\{\cot(2\tan^{-1}x)\}] = 0\); express \(x\) in the form \(a \pm b\).

For \( x \in (-\infty, -1] \), if \( f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) - 2\tan^{-1}x \) and \( g(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 4\tan^{-1}x \), then \( f(x) + g(x) \) equals: