Definitions
The hyperbolic functions are defined by analogy with trigonometric functions:
cosh x = e x + e − x 2 \cosh x = \frac{e^x + e^{-x}}{2} cosh x = 2 e x + e − x
sinh x = e x − e − x 2 \sinh x = \frac{e^x - e^{-x}}{2} sinh x = 2 e x − e − x
tanh x = sinh x cosh x = e x − e − x e x + e − x \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} tanh x = c o s h x s i n h x = e x + e − x e x − e − x
Related functions:
sech x = 1 cosh x , cosech x = 1 sinh x , coth x = 1 tanh x \text{sech } x = \frac{1}{\cosh x}, \quad \text{cosech } x = \frac{1}{\sinh x}, \quad \coth x = \frac{1}{\tanh x} sech x = c o s h x 1 , cosech x = s i n h x 1 , coth x = t a n h x 1
Relation to Trigonometric Functions
The connection between hyperbolic and trig functions involves the imaginary unit:
cosh ( i x ) = cos x \cosh(ix) = \cos x cosh ( i x ) = cos x
sinh ( i x ) = i sin x \sinh(ix) = i\sin x sinh ( i x ) = i sin x
cos ( i x ) = cosh x \cos(ix) = \cosh x cos ( i x ) = cosh x
sin ( i x ) = i sinh x \sin(ix) = i\sinh x sin ( i x ) = i sinh x
Why the name “hyperbolic”? Like how cos θ \cos\theta cos θ sin θ \sin\theta sin θ x 2 + y 2 = 1 x^2 + y^2 = 1 x 2 + y 2 = 1 cosh t \cosh t cosh t sinh t \sinh t sinh t x 2 − y 2 = 1 x^2 - y^2 = 1 x 2 − y 2 = 1
Osborne’s Rule
Osborne’s rule converts trigonometric identities to hyperbolic identities:
Replace cos \cos cos cosh \cosh cosh sin \sin sin sinh \sinh sinh Change the sign wherever there is a product of two sinh \sinh sinh
Examples:
Trigonometric Hyperbolic cos 2 θ + sin 2 θ = 1 \cos^2\theta + \sin^2\theta = 1 cos 2 θ + sin 2 θ = 1 cosh 2 x − sinh 2 x = 1 \cosh^2 x - \sinh^2 x = 1 cosh 2 x − sinh 2 x = 1 cos ( A + B ) = cos A cos B − sin A sin B \cos(A+B) = \cos A\cos B - \sin A\sin B cos ( A + B ) = cos A cos B − sin A sin B cosh ( A + B ) = cosh A cosh B + sinh A sinh B \cosh(A+B) = \cosh A\cosh B + \sinh A\sinh B cosh ( A + B ) = cosh A cosh B + sinh A sinh B sin ( A + B ) = sin A cos B + cos A sin B \sin(A+B) = \sin A\cos B + \cos A\sin B sin ( A + B ) = sin A cos B + cos A sin B sinh ( A + B ) = sinh A cosh B + cosh A sinh B \sinh(A+B) = \sinh A\cosh B + \cosh A\sinh B sinh ( A + B ) = sinh A cosh B + cosh A sinh B cos 2 θ = cos 2 θ − sin 2 θ \cos 2\theta = \cos^2\theta - \sin^2\theta cos 2 θ = cos 2 θ − sin 2 θ cosh 2 x = cosh 2 x + sinh 2 x \cosh 2x = \cosh^2 x + \sinh^2 x cosh 2 x = cosh 2 x + sinh 2 x
Key Identities
Fundamental:
cosh 2 x − sinh 2 x = 1 \cosh^2 x - \sinh^2 x = 1 cosh 2 x − sinh 2 x = 1
Addition formulae:
cosh ( A ± B ) = cosh A cosh B ± sinh A sinh B \cosh(A \pm B) = \cosh A\cosh B \pm \sinh A\sinh B cosh ( A ± B ) = cosh A cosh B ± sinh A sinh B sinh ( A ± B ) = sinh A cosh B ± cosh A sinh B \sinh(A \pm B) = \sinh A\cosh B \pm \cosh A\sinh B sinh ( A ± B ) = sinh A cosh B ± cosh A sinh B tanh ( A + B ) = tanh A + tanh B 1 + tanh A tanh B \tanh(A + B) = \frac{\tanh A + \tanh B}{1 + \tanh A\tanh B} tanh ( A + B ) = 1 + t a n h A t a n h B t a n h A + t a n h B
Double angle:
cosh 2 x = cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 1 + 2 sinh 2 x \cosh 2x = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x cosh 2 x = cosh 2 x + sinh 2 x = 2 cosh 2 x − 1 = 1 + 2 sinh 2 x sinh 2 x = 2 sinh x cosh x \sinh 2x = 2\sinh x\cosh x sinh 2 x = 2 sinh x cosh x
Graph Properties
Parity:
cosh x \cosh x cosh x even : cosh ( − x ) = cosh x \cosh(-x) = \cosh x cosh ( − x ) = cosh x sinh x \sinh x sinh x odd : sinh ( − x ) = − sinh x \sinh(-x) = -\sinh x sinh ( − x ) = − sinh x tanh x \tanh x tanh x odd : tanh ( − x ) = − tanh x \tanh(-x) = -\tanh x tanh ( − x ) = − tanh x
Key values:
cosh 0 = 1 \cosh 0 = 1 cosh 0 = 1 sinh 0 = 0 \sinh 0 = 0 sinh 0 = 0 tanh 0 = 0 \tanh 0 = 0 tanh 0 = 0
Asymptotic behaviour:
cosh x ≥ 1 \cosh x \geq 1 cosh x ≥ 1 x x x x = 0 x = 0 x = 0 tanh x → ± 1 \tanh x \to \pm 1 tanh x → ± 1 x → ± ∞ x \to \pm\infty x → ± ∞ As x → + ∞ x \to +\infty x → + ∞ cosh x ≈ sinh x ≈ 1 2 e x \cosh x \approx \sinh x \approx \frac{1}{2}e^x cosh x ≈ sinh x ≈ 2 1 e x
Derivatives
d d x sinh x = cosh x \frac{d}{dx}\sinh x = \cosh x d x d sinh x = cosh x
d d x cosh x = sinh x \frac{d}{dx}\cosh x = \sinh x d x d cosh x = sinh x
d d x tanh x = sech 2 x \frac{d}{dx}\tanh x = \text{sech}^2 x d x d tanh x = sech 2 x
Compare with trig: The derivatives of sinh \sinh sinh cosh \cosh cosh sin \sin sin cos \cos cos
Inverse Hyperbolic Functions
Expressed as logarithms:
sinh − 1 x = ln ( x + x 2 + 1 ) (defined for all x ) \sinh^{-1} x = \ln(x + \sqrt{x^2 + 1}) \quad \text{(defined for all } x\text{)} sinh − 1 x = ln ( x + x 2 + 1 ) (defined for all x )
cosh − 1 x = ln ( x + x 2 − 1 ) (for x ≥ 1 ) \cosh^{-1} x = \ln(x + \sqrt{x^2 - 1}) \quad \text{(for } x \geq 1\text{)} cosh − 1 x = ln ( x + x 2 − 1 ) (for x ≥ 1 )
tanh − 1 x = 1 2 ln ( 1 + x 1 − x ) (for ∣ x ∣ < 1 ) \tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \quad \text{(for } |x| < 1\text{)} tanh − 1 x = 2 1 ln ( 1 − x 1 + x ) (for ∣ x ∣ < 1 )
Derivation of sinh − 1 \sinh^{-1} sinh − 1 If y = sinh − 1 x y = \sinh^{-1} x y = sinh − 1 x x = sinh y = e y − e − y 2 x = \sinh y = \frac{e^y - e^{-y}}{2} x = sinh y = 2 e y − e − y e y e^y e y e y e^y e y
Why Hyperbolic Functions Matter
Catenary : The shape of a hanging chain is y = a cosh ( x / a ) y = a\cosh(x/a) y = a cosh ( x / a ) Special relativity : Rapidity (the hyperbolic angle) uses tanh \tanh tanh Differential equations : Solutions to d 2 y d x 2 = y \frac{d^2y}{dx^2} = y d x 2 d 2 y = y sinh \sinh sinh cosh \cosh cosh Complex analysis : Bridge between trigonometric and exponential functions
Hyperbolic functions appear wherever phenomena have exponential growth/decay combined with symmetry, such as in thermal physics, fluid dynamics, and electrical transmission lines.
