Hyperbolic Functions
Hyperbolic functions are similar to trigonometric functions, but are defined in terms of exponentials. There are three fundamental hyperbolic functions: sinh, cosh and tanh:
![Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_e315bb00a6744cedb145d335ca639f2f~mv2.jpg/v1/fill/w_147,h_13,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_e315bb00a6744cedb145d335ca639f2f~mv2.jpg)
Similarly, the reciprocal of each function exists:
![Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_a3d5e946edf3435c90b2eea0d1c04258~mv2.jpg/v1/fill/w_147,h_13,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_a3d5e946edf3435c90b2eea0d1c04258~mv2.jpg)
Hyperbolic Graphs
![Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_da839470de2f4dad9793ce6e9604e960~mv2.jpg/v1/fill/w_147,h_47,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_da839470de2f4dad9793ce6e9604e960~mv2.jpg)
For any value of x, sinh(-x) = -sinh(x)
y = sinh(x) has no asymptotes
For any value of x, cosh(-x) = -cosh(x)
y = cosh(x) never goes below y=1
y = tanh(x) has asymptotes at y = ±1, and always stays between these
Inverse Hyperbolic Functions
Just like sin, cos and tan, the hyperbolic functions have inverses, arcsinh, arcosh and artanh:
![Inverse hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_4329042ac34f49beae9eeb57148177fb~mv2.jpg/v1/fill/w_147,h_16,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_4329042ac34f49beae9eeb57148177fb~mv2.jpg)
The graphs of these are their respective reflections in the line y=x:
![Inverse hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_38aa76cf574f45c1abdcc2325ffe58a7~mv2.jpg/v1/fill/w_147,h_49,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_38aa76cf574f45c1abdcc2325ffe58a7~mv2.jpg)
Hyperbolic Identities & Equations
The same identities exist for hyperbolic functions as they do for trigonometric functions:
sinh(A ± B) ≡ sinh(A) cosh(B) ± cosh(A) sinh(B)
cosh(A ± B) ≡ cosh(A) cosh(B) ∓ sinh(A) sinh(B)
Equations with a sinh² in them, however, are different:
cosh²(x) - sinh²(x) ≡ 1
Note that here, the sinh² is negative (the trigonometric identity is sin² + cos² ≡ 1). This is known as Osborn's rule:
According to Osborn's rule, when using trigonometric identities as hyperbolic identities, any sinh² must be multiplied by -1.
Differentiating Hyperbolic Functions
This is very similar to trigonometric functions:
![Hyperbolic differentiation. Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_391638000c7b411d9e2f713dd841ab87~mv2.jpg/v1/fill/w_147,h_11,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_391638000c7b411d9e2f713dd841ab87~mv2.jpg)
Note that the derivative of cosh(x) is positive sinh(x), not negative.
The inverse functions differentiate as such:
![Hyperbolic differentiation. Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_f610b66e06ec4ccf962770b4f9e41477~mv2.jpg/v1/fill/w_147,h_17,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_f610b66e06ec4ccf962770b4f9e41477~mv2.jpg)
Integrating Hyperbolic Functions
Simply the reverse of differentiation, but remember the " +c " and that the signs are different:
![Hyperbolic integration. Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_b4395ed5fa414ff4b07440e56112a049~mv2.jpg/v1/fill/w_147,h_10,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_b4395ed5fa414ff4b07440e56112a049~mv2.jpg)
The inverse functions can also be integrated:
![Hyperbolic integration. Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_8d7ed7d5a0ba459aa1c31a0915d136ec~mv2.jpg/v1/fill/w_156,h_22,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_8d7ed7d5a0ba459aa1c31a0915d136ec~mv2.jpg)
These standard results for when the equation you need to integrate does not have either (x²+1) or (x²-1) in the root in the denominator:
![Hyperbolic integration. Hyperbolic functions. Free online a-level further maths core pure 2 CP2 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_4ed8b0c6481244e39041112f111fa186~mv2.jpg/v1/fill/w_135,h_19,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_4ed8b0c6481244e39041112f111fa186~mv2.jpg)