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

EASY

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Evaluate $cosec (\cos^{- 1} \frac{3}{5})$

Important Questions on Inverse Trigonometric Functions

EASY

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

If

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

then the value of

x

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:

EASY

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

If

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

, then

f (5)

is equal to

MEDIUM

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

If

α

and

β

are roots of the equation

x^{2} + 5 |x| - 6 = 0

then the value of

|\tan^{- 1} α - \tan^{- 1} β|

is

EASY

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

Let

(a, b) \subset (0, 2 π)

be the largest interval for which

\sin^{- 1} (\sin θ) - \cos^{- 1} (\sin θ) > 0, θ \in (0, 2 π)

, holds . If

α x^{2} + β x + \sin^{- 1} (x^{2} - 6 x + 10) + \cos^{- 1} (x^{2} - 6 x + 10) = 0

and

α - β = b - a

, then

α

is equal to;

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

For any positive integer

n

, define

f_{n} : (0, \infty) \to R

as

f_{n} (x) = \sum_{i = 1}^{n} \tan^{- 1} (\frac{1}{1 + (x + j) (x + j - 1)})

for all

x \in (0, \infty)

.

$($ Here, the inverse trigonometric function $\tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2})$ $)$

Then, which of the following statement(s) is (are) TRUE?

HARD

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

If

α = 3 \sin^{- 1} (\frac{6}{11}) & β = 3 \cos^{- 1} (\frac{4}{9}),

where the inverse trigonometric functions take only the principal values, then the correct options(s) is(are)

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

is

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

The value of

\cos^{- 1} (\cot (\frac{π}{2})) + \cos^{- 1} (\sin (\frac{2 π}{3}))

is

EASY

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

Find the value of

\tan^{- 1} [2 \cos (2 \sin^{- 1} \frac{1}{2})]

.

EASY

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

The principal value of

\tan^{- 1} (\cot \frac{43 π}{4})

is

HARD

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

The value of

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

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}

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:

MEDIUM

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

If

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

If

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

and

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

then

y - x

is equal to:

EASY

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

The value of

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

is:

MEDIUM

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

If

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

then,

x =