Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

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How do you find ${sech}^{- 1} x$ value?

Important Questions on Hyperbolic Functions

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

e^{(\sec h^{- 1} \frac{1}{2} + \tan h^{- 1} \frac{1}{2} + \sin h^{- 1} \frac{1}{2})} =

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

Let

k > 0

and

t = {sech}^{- 1} (\frac{1}{2}) - {cosech}^{- 1} (\frac{3}{k}) .

If

3 e^{t} = 2 + \sqrt{3},

then

k =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

The domain of the function

f (x) = \sec^{- 1} (3 x - 4) + \tan h^{- 1} (\frac{x + 3}{5})

is

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

\tan h^{- 1} (\frac{1}{3}) + \cot h^{- 1} (2) =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

\sin h^{- 1} (- 2) + cosec h^{- 1} (- 2) + \cot h^{- 1} (- 2) =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

\cot h^{- 1} (2) + cosec h^{- 1} (- 2 \sqrt{2}) =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

x = - \frac{1}{2}, \sinh^{- 1} x + cosec h^{- 1} x =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

x = \log (\frac{1}{y} + \sqrt{1 + \frac{1}{y^{2}}}) \Rightarrow y

is equal to

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

\sinh x = 3

, then show that

x = \log_{e} (3 + \sqrt{10})

.

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If for

\sinh x = 3

, the value of

x

obtained is in the form of

\log_{e} (a + \sqrt{b})

, then find the value of

a + b

.

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

\sinh (2 \tanh^{- 1} x) = \frac{11}{60},

then

x =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

\tan h^{- 1} x = alog (\frac{1 + x}{1 - x}), |x| < 1,

then a is equal to

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

{sech}^{- 1} (\frac{1}{2}) - {cosech}^{- 1} (\frac{3}{4}) = \log_{e} k,

then

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

\sinh (x) = 3

, then show that

x = \log_{e} (3 + \sqrt{10})

.

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

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

, where

x, y \in R

, then

\tan h^{- 1} (i y) =

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

{coth}^{- 1} 3 + \tanh^{- 1} \frac{1}{3} - {cosech}^{- 1} (- \sqrt{3}) =

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

\sinh x = \frac{3}{4}

and

\cosh y = \frac{5}{3}

then

x + y =

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

\tan h^{- 1} \frac{1}{2} + \cot h^{- 1} 3 =

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Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If

2 \sin h^{- 1} (\frac{a}{\sqrt{1 - a^{2}}}) = \log (\frac{1 + x}{1 - x})

then

x =