Hyperbolic functions are functions built from exponential expressions and appear often in calculus, physics, engineering, and advanced modeling. This cheat sheet helps students quickly compare sinh⁡x\sinh x, cosh⁡x\cosh x, tanh⁡x\tanh x, and their reciprocal functions. It is useful for remembering definitions, identities, derivatives, integrals, and inverse relationships without searching through a textbook.

Key Facts

  • The main definitions are sinh⁡x=ex−e−x2\sinh x = \frac{e^x - e^{-x}}{2}, cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2}, and tanh⁡x=sinh⁡xcosh⁡x\tanh x = \frac{\sinh x}{\cosh x}.
  • The fundamental hyperbolic identity is cosh⁡2x−sinh⁡2x=1\cosh^2 x - \sinh^2 x = 1.
  • The reciprocal hyperbolic functions are sech⁡x=1cosh⁡x\operatorname{sech} x = \frac{1}{\cosh x}, csch⁡x=1sinh⁡x\operatorname{csch} x = \frac{1}{\sinh x}, and coth⁡x=cosh⁡xsinh⁡x\coth x = \frac{\cosh x}{\sinh x}.
  • The basic derivatives are ddxsinh⁡x=cosh⁡x\frac{d}{dx}\sinh x = \cosh x, ddxcosh⁡x=sinh⁡x\frac{d}{dx}\cosh x = \sinh x, and ddxtanh⁡x=sech⁡2x\frac{d}{dx}\tanh x = \operatorname{sech}^2 x.
  • The reciprocal derivatives include ddxsech⁡x=−sech⁡xtanh⁡x\frac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x\tanh x, ddxcsch⁡x=−csch⁡xcoth⁡x\frac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x\coth x, and ddxcoth⁡x=−csch⁡2x\frac{d}{dx}\coth x = -\operatorname{csch}^2 x.
  • Useful integrals include ∫sinh⁡x dx=cosh⁡x+C\int \sinh x\,dx = \cosh x + C, ∫cosh⁡x dx=sinh⁡x+C\int \cosh x\,dx = \sinh x + C, and ∫sech⁡2x dx=tanh⁡x+C\int \operatorname{sech}^2 x\,dx = \tanh x + C.
  • The inverse forms include arsinh⁡x=ln⁡(x+x2+1)\operatorname{arsinh} x = \ln\left(x + \sqrt{x^2 + 1}\right) and arcosh⁡x=ln⁡(x+x2−1)\operatorname{arcosh} x = \ln\left(x + \sqrt{x^2 - 1}\right) for x≥1x \ge 1.
  • Hyperbolic functions are not periodic, and cosh⁡x\cosh x is even while sinh⁡x\sinh x and tanh⁡x\tanh x are odd.

Vocabulary

Hyperbolic function
A function defined using exponential expressions that is related to the geometry of a hyperbola.
Hyperbolic sine
The function sinh⁡x=ex−e−x2\sinh x = \frac{e^x - e^{-x}}{2}, which is odd and has derivative cosh⁡x\cosh x.
Hyperbolic cosine
The function cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2}, which is even and has derivative sinh⁡x\sinh x.
Hyperbolic tangent
The function tanh⁡x=sinh⁡xcosh⁡x\tanh x = \frac{\sinh x}{\cosh x}, which has horizontal asymptotes at y=1y = 1 and y=−1y = -1.
Inverse hyperbolic function
A function such as arsinh⁡x\operatorname{arsinh} x or arcosh⁡x\operatorname{arcosh} x that reverses a hyperbolic function on an appropriate domain.
Fundamental identity
The identity cosh⁡2x−sinh⁡2x=1\cosh^2 x - \sinh^2 x = 1, which is the hyperbolic counterpart of a trigonometric Pythagorean identity.

Common Mistakes to Avoid

  • Using the circular identity cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1 for hyperbolic functions is wrong because the correct identity is cosh⁡2x−sinh⁡2x=1\cosh^2 x - \sinh^2 x = 1.
  • Forgetting the negative signs in reciprocal derivatives is wrong because ddxsech⁡x=−sech⁡xtanh⁡x\frac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x\tanh x and ddxcsch⁡x=−csch⁡xcoth⁡x\frac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x\coth x.
  • Treating sinh⁡x\sinh x and cosh⁡x\cosh x as periodic is wrong because hyperbolic functions are built from exponentials and do not repeat like sine and cosine.
  • Writing cosh⁡x=ex−e−x2\cosh x = \frac{e^x - e^{-x}}{2} is wrong because the plus sign belongs in cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2}.
  • Ignoring domains of inverse hyperbolic functions is wrong because arcosh⁡x\operatorname{arcosh} x is real only for x≥1x \ge 1.

Practice Questions

  1. 1 Evaluate sinh⁡(0)\sinh(0), cosh⁡(0)\cosh(0), and tanh⁡(0)\tanh(0) using the exponential definitions.
  2. 2 Differentiate f(x)=3cosh⁡x−2tanh⁡xf(x) = 3\cosh x - 2\tanh x.
  3. 3 Find ∫(4sinh⁡x+5sech⁡2x) dx\int \left(4\sinh x + 5\operatorname{sech}^2 x\right)\,dx.
  4. 4 Explain why cosh⁡x\cosh x has a minimum value but sinh⁡x\sinh x does not.

Understanding Hyperbolic Functions Reference

The word hyperbolic comes from the hyperbola, just as ordinary sine and cosine are linked to the circle. This geometric link explains why the two families look similar but follow different rules. Circular functions describe motion that repeats as an angle keeps turning.

Hyperbolic functions describe growth, decay, and curved shapes that keep changing instead of returning to an earlier value. Their key relationship involves a difference rather than a sum.

That small change affects signs throughout identities, derivatives, and graph shapes. It is one reason students should not assume that a familiar trigonometric rule transfers unchanged.

The graphs give useful clues before any calculation begins. Hyperbolic cosine has its lowest point at zero and rises on both sides, making a smooth U shaped curve. Hyperbolic sine passes through the origin and increases continuously from negative values to positive values.

Hyperbolic tangent has an S shape. It gets closer and closer to one on the right and negative one on the left, without reaching either value. This limiting behavior makes it useful when a model needs an output that stays within fixed bounds.

Pay attention to symmetry too. Symmetry can quickly reveal whether a proposed equation, graph, or numerical answer makes sense.

A classic real world example is a hanging cable. A flexible cable with its own weight, such as a power line between poles, forms a catenary curve. Its height can be modeled using hyperbolic cosine, not an ordinary parabola.

A parabola may look close over a short distance, but the difference matters for long bridges or cables under load. Hyperbolic functions appear in solutions to differential equations as well.

They can describe the shape of a beam, temperature patterns in certain materials, and some models of population change. In special relativity, hyperbolic functions help describe velocity changes near the speed of light, where ordinary addition of speeds no longer works.

Inverse hyperbolic functions need extra care. The word inverse means an operation that undoes a function. It does not mean reciprocal.

For example, the inverse of hyperbolic sine gives an input value from an output value, while the reciprocal function gives one divided by a function value. These are completely different ideas. Domain restrictions matter most for inverse hyperbolic cosine because not every real output can come from it.

When differentiating a hyperbolic function with an expression inside it, use the chain rule and include the derivative of the inside expression. When integrating, look for a reverse derivative pattern, then check signs carefully. A quick graph or a few test values can catch many errors.