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Hyperbolic Functions: Definitions and Graphs

Calculus 2 · Chapter 3: Hyperbolic FunctionsSection 3.1 · lesson 12 of 5411 min · free

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In this lesson

Definitions and graphs - built from exponentials, not angles.

  • Define all six hyperbolic functions from e^{x} and e^{-x}
  • Derive the fundamental identity \cosh^{2}x - \sinh^{2}x = 1
  • Read domain, range and symmetry off the definitions
  • See why the name is 'hyperbolic'

Transcript

Everything said in the video, section by section (1,603 words). Read along, or search it.

MAT-265 - Calculus 2 · Section 3.1

Hyperbolic Functions

Stretch a chain loosely between two hands and let it sag. The curve it forms looks parabolic. But it isn't. It's called a catenary, and its equation involves a function built directly from exponentials. The same curve describes power lines drooping between poles, and the cables of suspension structures. The Gateway Arch in St. Louis is shaped as an inverted, weighted catenary — and the National Park Service publishes its defining equation in terms of the hyperbolic cosine. Hyperbolic functions arise wherever exponential growth and decay combine. And despite the name, and the suggestive notation — sinch, cosh, tanh — they are not defined using triangles or angles at all. They're defined purely from e to the x and e to the minus x.

The Shape

A Hanging Cable Is Not a Parabola

So why do they look so much like trigonometry? Because they obey identities so similar to those of ordinary sine and cosine that the parallel is impossible to miss. And understanding that parallel is the goal of this section. The connection traces back to a single geometric fact. Ordinary trig functions parametrize a circle. Hyperbolic functions parametrize a hyperbola. That's it. That's the whole reason for the name, and for every identity that follows. Hold onto that, because we'll make it precise by the end.

Definitions

Six Functions From Two Building Blocks

Every hyperbolic function is assembled from just two exponential building blocks. e to the x, and e to the minus x. The two primary functions are defined as the antisymmetric and symmetric averages. Hyperbolic sine is e to the x minus e to the minus x, all over two. Hyperbolic cosine is e to the x plus e to the minus x, over two. One difference, one sum. That's the only distinction between them. The names are read cinch — or shine — for sinh, and cosh, rhyming with gosh. From these two, the remaining four follow exactly as in ordinary trigonometry. As ratios and reciprocals. Tanh is sinh over cosh. Coth is its reciprocal. Sech is one over cosh, and csch is one over sinh. Now, two features deserve immediate attention, because they determine the domains. First. Cosh x is always positive, since e to the x and e to the minus x are both positive. So sech x is defined everywhere. It never divides by zero. Second. Sinh x equals zero only when e to the x equals e to the minus x. Which happens only at x equals zero. So coth x and csch x are undefined at zero, and nowhere else.

The Fundamental Identity

One Sign Changes Everything

Now the single most important relationship in this chapter. It's the hyperbolic analogue of sine squared plus cosine squared equals one. And we can compute it directly from the definitions. No cleverness required. Take cosh squared x minus sinh squared x. Write each one out and square it. Expand. The first square gives e to the two x, plus two, plus e to the minus two x, all over four. The second gives e to the two x, minus two, plus e to the minus two x, over four. Subtract. The e to the two x terms cancel. The e to the minus two x terms cancel. What survives is two minus minus-two. Which is four. Over four. So cosh squared x minus sinh squared x equals one. For every real x. Now — look carefully at that identity, and compare it to the circular one. Where circular trigonometry has a plus, this has a minus. That minus sign is the algebraic seed of everything that distinguishes hyperbolic from circular functions. And it's precisely what links these functions to the hyperbola x squared minus y squared equals one.

Structure

Domain, Range and Symmetry

Let's collect the structural facts, because they all follow from the definitions rather than needing memorization. Each function is built from e to the x and e to the minus x, which are defined and continuous for all real x. So the domains follow directly from where a denominator might vanish. Nothing else can go wrong. The symmetry follows from something equally simple. When you replace x by minus x, e to the x and e to the minus x simply swap places. Look at sinh of minus x. The swap turns the numerator into e to the minus x, minus e to the x — which is the negative of what you started with. So sinh is odd. Symmetric through the origin. For cosh, the numerator is a sum, and swapping the two terms in a sum changes nothing. So cosh is even. Symmetric about the y-axis. Two range facts are worth verifying rather than memorizing. The minimum of cosh x occurs at x equals zero, where cosh zero is one half of one plus one, which is one. And since cosh grows without bound on either side, its range is one to infinity. For tanh x, divide numerator and denominator by e to the x. You get one minus e to the minus two x, over one plus e to the minus two x. That approaches one as x goes to infinity, and minus one as x goes to minus infinity. But it never reaches either. So the range is the open interval from minus one to one, with horizontal asymptotes at y equals plus and minus one.

Figure 3.1.1

Each Graph Inherits Its Shape From Its Exponentials

The graphs make all of that algebra concrete. And each one inherits its shape from the exponential pieces that build it. Start with sinh x. Since it's one half e to the x, minus one half e to the minus x, the graph behaves like one half e to the x for large positive x — the other term fades away. And like minus one half e to the minus x for large negative x. It passes through the origin with the gentle, ever-steepening S-shape of an odd function. Increasing everywhere. Now cosh x. As the sum of a rising and a falling exponential, the graph is their average. It's symmetric about the y-axis, dips to its minimum value of one at x equals zero, and opens upward in that characteristic catenary shape. For large absolute x, it hugs whichever exponential dominates. And tanh x. Bounded between its asymptotes at y equals one and y equals minus one. It rises smoothly through the origin with an S-shape that flattens at both ends. That bounded, gradually saturating profile is exactly why tanh is a favorite squashing function — in fields from neural networks to fluid dynamics. You feed it anything, and it hands you back something between minus one and one.

Figure 3.1.2

Circles Versus Hyperbolas - Where the Name Comes From

Now let's make good on the promise from the beginning. Where does the name hyperbolic actually come from? Recall that ordinary cosine and sine parametrize the unit circle. As the parameter t runs through the real numbers, the point cosine t, sine t moves around the curve x squared plus y squared equals one. And it does so precisely because cosine squared plus sine squared equals one. The hyperbolic functions do the same job, for a different curve. Using the identity cosh squared t minus sinh squared t equals one, the point cosh t, sinh t always satisfies x squared minus y squared equals one. That's the unit hyperbola. As t ranges over all reals, this point sweeps out the right branch — the branch with x at least one, since cosh t is always at least one. And that is the origin of the term. Just as cosine and sine are circular functions tied to the circle, cosh and sinh are tied to the hyperbola. But the parallel runs deeper still, in how the parameter t is interpreted. For the circle, t can be read as twice the area of the circular sector swept from the positive x-axis to the point. For the hyperbola, t is likewise twice the area of the corresponding hyperbolic sector. Not an angle. An area. Which is why the hyperbolic parameter is sometimes called the hyperbolic angle — even though no rotation is involved anywhere.

Key Takeaways

What to Carry Forward

Let me leave you with something to think about. The circular identity is cosine squared plus sine squared equals one. The hyperbolic identity is cosh squared minus sinh squared equals one. Only one sign differs. Yet that sign changes a bounded, repeating circle into an unbounded, non-repeating hyperbola. Why does the plus sign force the circular functions to be periodic, while the minus sign lets the hyperbolic functions grow without bound? Consider what each identity permits. On a circle, both coordinates are trapped in the interval minus one to one. There's nowhere to go but around. But on a hyperbola, the x-coordinate can be arbitrarily large — as long as y keeps pace. So cosh and sinh are free to increase forever, and never return to a starting value. So — five things to carry forward. The hyperbolic functions are defined from exponentials, not angles. The fundamental identity is cosh squared minus sinh squared equals one. The circular identity, with a minus sign — and that's the source of every structural difference. Sinh is odd with range all reals. Cosh is even with range one to infinity and minimum at zero comma one. Tanh is odd and bounded by plus and minus one. Coth and csch are undefined at zero because sinh zero is zero. Cosh and sech are defined everywhere because cosh is never zero. And the point cosh t comma sinh t parametrizes the right branch of the unit hyperbola, with the parameter equal to twice the swept sector area. That's where the name comes from. Next up is section three point two, Hyperbolic Identities — where the whole catalogue of trig identities reappears, with predictable sign changes.

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FAQ

Questions students ask us

About Hyperbolic Functions: Definitions and Graphs

What are the definitions of the six hyperbolic functions?
The hyperbolic functions are defined from exponentials, not angles: \sinh x = \frac{e^x - e^{-x}}{2} and \cosh x = \frac{e^x + e^{-x}}{2}. The other four follow as in trigonometry: \tanh x = \frac{\sinh x}{\cosh x}, \coth x = \frac{\cosh x}{\sinh x}, \operatorname{sech} x = \frac{1}{\cosh x}, and \operatorname{csch} x = \frac{1}{\sinh x}. Since \cosh x > 0 always, sech is defined everywhere. Coth and csch are undefined only at x = 0, where \sinh x = 0.
How do you prove cosh^2 x - sinh^2 x = 1?
You prove it by squaring the exponential definitions and subtracting. Expanding gives \cosh^2 x = \frac{e^{2x} + 2 + e^{-2x}}{4} and \sinh^2 x = \frac{e^{2x} - 2 + e^{-2x}}{4}. When you subtract, the e^{2x} and e^{-2x} terms cancel, leaving \frac{2 - (-2)}{4} = 1. So \cosh^2 x - \sinh^2 x = 1 for every real x. This is the hyperbolic analogue of \cos^2 x + \sin^2 x = 1.
Why are hyperbolic functions called hyperbolic?
They are called hyperbolic because the point (\cosh t, \sinh t) traces the unit hyperbola x^2 - y^2 = 1, just as (\cos t, \sin t) traces the unit circle. This follows directly from the identity \cosh^2 t - \sinh^2 t = 1. Since \cosh t \ge 1, the point sweeps out only the right branch. The parameter t is not an angle. It equals twice the area of the hyperbolic sector swept out from the positive x-axis.
Is cosh^2 x + sinh^2 x = 1 like the trig identity?
No, the correct identity is \cosh^2 x - \sinh^2 x = 1, with a minus sign. Copying the plus sign from \cos^2 x + \sin^2 x = 1 is a common mistake. At x = 1, for example, \cosh^2 1 + \sinh^2 1 = \cosh 2 \approx 3.76, not 1. That single sign difference explains why cosh and sinh are unbounded and not periodic. The x-coordinate on a hyperbola can grow without limit, while both coordinates on a circle stay between -1 and 1.

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