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

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Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The principal value of

\tan^{- 1} (\cot \frac{43 π}{4})

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\sin^{- 1} \frac{3}{5} - \sin^{- 1} \frac{8}{17} =

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

A value of

x

satisfying the equation

\sin [\cot^{- 1} (1 + x)] = \cos [\tan^{- 1} x],

is:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of

\sin^{- 1} (\frac{12}{13}) - \sin^{- 1} (\frac{3}{5})

is equal to:

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of

\cot (\sum_{n = 1}^{19} \cot^{- 1} (1 + \sum_{p = 1}^{n} 2 p))

is:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

α = {c o s}^{- 1} (\frac{3}{5})

β = {t a n}^{- 1} (\frac{1}{3})

, where

0 < α, β < \frac{π}{2},

then

α - β

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of

\tan^{- 1} [\frac{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}}],

|x| < \frac{1}{2}, x \neq 0,

is equal to:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

a + \frac{π}{2} < 2 \tan^{- 1} x + 3 \cot^{- 1} x < b

then

' a'

and

' b'

are respectively.

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

f (x) = 2 \tan^{- 1} x + \sin^{- 1} (\frac{2 x}{1 + x^{2}}), x > 1

, then

f (5)

is equal to

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\sin^{- 1} (\frac{x}{5}) + {c o s e c}^{- 1} (\frac{5}{4}) = \frac{π}{2},

then the value of

x

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\cos^{- 1} x - \cos^{- 1} \frac{y}{2} = α,

where

- 1 \leq x \leq 1, - 2 \leq y \leq 2, x \leq \frac{y}{2},

then for all

x, y, 4 x^{2} - 4 x y \cos α + y^{2}

is equal to :

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Considering only the principal values of inverse functions, the set

A = \{x \geq 0 : \tan^{- 1} (2 x) + \tan^{- 1} (3 x) = \frac{π}{4}\}

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

4 {s i n}^{- 1} x + 6 {c o s}^{- 1} x = 3 π,

then

x =

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The solution of

\tan^{- 1} 2 x + \tan^{- 1} 3 x = \frac{π}{4}

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

f^{'} (x) = \tan^{- 1} (\sec x + \tan x), - \frac{π}{2} < x < \frac{π}{2}

and

f (0) = 0

, then

f (1)

is equal to:

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

S

is the sum of the first 10 terms of the series,

\tan^{- 1} (\frac{1}{3}) + \tan^{- 1} (\frac{1}{7}) + \tan^{- 1} (\frac{1}{13}) + \tan^{- 1} (\frac{1}{21}) + \dots \dots

then

\tan (S)

is equal to :

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\cos [2 \sin^{- 1} \frac{3}{4} + \cos^{- 1} \frac{3}{4}] =

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of

4 \tan^{- 1} (\frac{1}{5}) - \frac{π}{4} =

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

x = {s i n}^{- 1} (\sin 10)

and

y = {c o s}^{- 1} (\cos 10),

then

y - x

is equal to:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Let

\tan^{- 1} y = \tan^{- 1} x + \tan^{- 1} (\frac{2 x}{1 - x^{2}})

, where

|x| < \frac{1}{\sqrt{3}}

,Then a value of

y

Let x=mn(m,n are co-prime natural numbers) be a solution of the equation cos2sin-1x=19 and let α,β(α>β) be the roots of the equation mx2-nx-m+n=0. Then the point (α,β) lies on the line

Important Questions on Inverse Trigonometric Functions

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