# Hyperbolic Functions Formulas

Do you need any help in solving the Hyperbolic Functions problems? If yes, then check out our list of Hyperbolic Functions Formulas. These formulas of Hyperbolic Functions can aid you to solve all basic Hyperbolic Functions like sine, cosine, tan, cosech, sech related problems with ease & quick. So, refer to the below provided Hyperbolic Functions formula sheet and recall all formulas daily during study time. This can help you do Hyperbolic Functions calculations little faster and easier.

## Hyperbolic Functions Formulae Tables | List of Hyperbolic Functions Formulas

Solving mathematical problems in less time can be possible by using simple formulas or else using Calculators. So, memorizing all math concepts formulas like Hyperbolic Functions Formulas can make your calculations faster. Just have a look at the Hyperbolic Functions Formulae Tables, Sheet, and List from here & prepare well to become master in problem-solving.

1. Hyperbolic Functions

- sin x = \(\frac{e^{x}-e^{-x}}{2}\)
- cosh x = \(\frac{e^{x}+e^{-x}}{2}\)
- tanh x = \(\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\)
- coth x = \(\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\)
- cosech x = \(\frac{2}{e^{x}-e^{-x}}\)
- sech x = \(\frac{2}{e^{x}+e^{-x}}\)

2. Domain & Range of Hyperbolic Functions

Function | Domain | Range |

sinh x | R | R |

cosh x | R | [1, ∞ ) |

tanh x | R | (-1, 1) |

coth x | R_{0} |
R – [-1, 1] |

cosech x | R_{0} |
R_{0} |

sech x | R | (0, 1] |

3. Formulae for Hyperbolic Function

(A) Square Formulae

- cosh
^{2}x – sinh^{2}x = 1 - sech
^{2}x + tanh^{2}x = 1 - coth
^{2}x – cosech^{2}x = 1 - cosh
^{2}x + sinh^{2}x = cosh 2x

(B) Expansion Formulae

- sinh (x ± y) = sinh x cosh y ± cosh x sinh y
- cosh (x ± y) = cosh x cosh y ± sinh x sinh y
- tanh (x ± y) = \(\frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)

(C) (i) sinh x + sinh y = 2 sinh \(\frac{x+y}{2}\) cosh \(\frac{x-y}{2}\)

(ii) sinh x – sinh y = 2 cosh \(\frac{x+y}{2}\) sinh \(\frac{x-y}{2}\)

(iii) cosh x + cosh y = 2 cosh \(\frac{x+y}{2}\) cosh \(\frac{x-y}{2}\)

(iv) cosh x – cosh y = 2 sinh \(\frac{x+y}{2}\) sinh \(\frac{x-y}{2}\)

(D) (i) sinh 2x = 2 sinh x cosh x

= \(\frac{2 \tanh x}{1-\tanh ^{2} x}\)

(ii) cosh 2x = cosh^{2} x + sinh^{2} x

= 2 cosh^{2} x – 1

= 1 + 2 sinh^{2} x

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

(iii) tan 2x = \(\frac{2 \tanh x}{1+\tanh ^{2} x}\)

(E) (i) sinh 3x = 3 sinh x + 4 sinh^{3} x

(ii) cosh 3x = 4 cosh^{3} x – 3 cosh x

(iii) tan 3x = \(\frac{2 \tanh x+\tanh ^{3} x}{1+3 \tanh ^{2} x}\)

(F) (i) coshx + sinhx = e^{x}

(ii) cosh x – sinh x = e^{-x}

(iii) (cosh x + sinh x)^{n} = cosh nx + sinh nx

4. Relation between Hyperbolic and Circular Function

(i) sin (ix) = i sinh x

sinh (ix) = i sin x

sinhx = – i sin (ix)

sin x = – i sinh (ix)

(ii) cos (ix) = cosh x

cosh (ix) = cos x

cosh x = cos (ix)

cos x = cosh (ix)

(iii) tan (ix) = i tanh x

tanh (ix) = i tan x

tanh x = – i tan (ix)

tan x = – i tanh (ix)

(iv) cot (ix) = – i coth x

coth (ix) = – i cot x

coth x = i cot (ix)

cot x = i coth (ix)

(v) sec (ix) = sech x

sech (ix) = sec x

sech x = sec (ix)

sec x = sech (ix)

(vi) cosec (ix) = – i cosech x

cosech (ix) = – i cosec x

cosech x = i cosec (ix)

cosec x = i cosech (ix)

5. Period of Hyperbolic Functions

Period of sinh x = 2πi ; cosh x = 2πi and tanh x = πi

6. Relation between Inverse Hyperbolic Function and Inverse Circular Function

- sinh
^{-1}x = – i sin^{-1}(ix) - cosh
^{-1}x = – i cos^{-1}x - tan
^{-1}x = – i tan^{-1}(ix) - coth
^{-1}x = i cot^{-1}(ix) - sech
^{-1}x = – i sec^{-1}x - cosech
^{-1}x = i cosec^{-1}(ix)

7. Relation between Inverse Hyperbolic Functions and Logarithmic Functions

- sinh
^{-1}x = log(x + \(\sqrt{x^{2}+1}\))(-∞ < x < ∞) - cosh
^{-1}x = log (x + \(\sqrt{x^{2}-1}\)) (x ≥ 1) - tanh
^{-1}x = \(\frac{1}{2}\) log \(\left(\frac{1+x}{1-x}\right)\)(|x| < 1)

Note: Formulae for values of cosech^{-1} x, sech^{-1} x and coth^{-1} x may be obtained by replacing x by 1/x in the values of sinh^{-1} x , cosh^{-1} x and tanh^{-1} x respectively.