Hyperbolic Functions Explained

Hang a chain between two hooks and it sags into a graceful curve. It looks like a parabola, but it is not: its true shape is described by the hyperbolic cosine. Hyperbolic functions behave remarkably like the ordinary trigonometric functions, yet they are built from the exponential function. Once you see how they work, you will find them in cable design, relativity, fluid flow and even neural networks.

Definitions

The three main hyperbolic functions are defined using \( e^x \):

$$ \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2} $$

$$ \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$

They are pronounced "shine", "cosh" and "than" (or "tanch"). The reciprocal functions are \( \operatorname{sech} x = 1/\cosh x \), \( \operatorname{cosech} x = 1/\sinh x \) and \( \coth x = 1/\tanh x \).

A quick way to remember them: \( \cosh x \) is the average of \( e^x \) and \( e^{-x} \), and \( \sinh x \) is half their difference. In fact, \( e^x = \cosh x + \sinh x \), so every exponential splits neatly into an even part (cosh) and an odd part (sinh).

Graphs and Key Values

• \( \cosh x \) is even, has minimum value \( \cosh 0 = 1 \), and looks like a U-shaped curve. It is always at least 1.
• \( \sinh x \) is odd, passes through the origin, and increases everywhere, taking all real values.
• \( \tanh x \) is odd, increases from \(-1\) to \( 1 \), with horizontal asymptotes \( y = \pm 1 \). It looks like a stretched letter S.

For large positive \( x \), \( e^{-x} \) becomes tiny, so both \( \sinh x \) and \( \cosh x \) are approximately \( \tfrac{1}{2}e^x \).

Why "Hyperbolic"?

The point \( (\cos t, \sin t) \) moves around the unit circle \( x^2 + y^2 = 1 \). In exactly the same way, the point \( (\cosh t, \sinh t) \) moves along the right-hand branch of the unit hyperbola \( x^2 - y^2 = 1 \). That is where the name comes from. The functions were studied systematically by the Italian Jesuit mathematician Vincenzo Riccati in the 1760s, and Johann Heinrich Lambert developed them further around 1768.

The Fundamental Identity

Worked Example 1: Proving \( \cosh^2 x - \sinh^2 x = 1 \)

Square each definition:

$$ \cosh^2 x = \frac{(e^x + e^{-x})^2}{4} = \frac{e^{2x} + 2 + e^{-2x}}{4} $$

$$ \sinh^2 x = \frac{(e^x - e^{-x})^2}{4} = \frac{e^{2x} - 2 + e^{-2x}}{4} $$

The middle term is 2 because \( e^x \cdot e^{-x} = e^0 = 1 \). Subtract:

$$ \cosh^2 x - \sinh^2 x = \frac{(e^{2x} + 2 + e^{-2x}) - (e^{2x} - 2 + e^{-2x})}{4} = \frac{4}{4} = 1 $$

Compare this with \( \cos^2 x + \sin^2 x = 1 \). The only difference is a sign.

Osborn's Rule

This pattern holds generally. Osborn's rule says: take any trigonometric identity, replace each trig function by its hyperbolic version, and change the sign of every term containing a product (or implied product) of two sines. For example:

• \( \cos 2x = 1 - 2\sin^2 x \) becomes \( \cosh 2x = 1 + 2\sinh^2 x \)
• \( \sin 2x = 2\sin x\cos x \) becomes \( \sinh 2x = 2\sinh x\cosh x \) (no product of two sines, so no change)
• \( 1 + \tan^2 x = \sec^2 x \) becomes \( 1 - \tanh^2 x = \operatorname{sech}^2 x \)

Derivatives and Integrals

Differentiating the exponential definitions gives pleasingly simple results:

$$ \frac{d}{dx}\sinh x = \cosh x, \qquad \frac{d}{dx}\cosh x = \sinh x, \qquad \frac{d}{dx}\tanh x = \operatorname{sech}^2 x $$

Notice there is no minus sign in the derivative of cosh, unlike the derivative of cos. Integrals follow by reversing these.

Inverse Hyperbolic Functions

Because hyperbolic functions are built from exponentials, their inverses can be written with logarithms:

$$ \sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right) $$

$$ \cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right), \quad x \ge 1 $$

Worked Example 2: Solving \( \sinh x = 2 \)

Method from first principles. Write the equation using exponentials:

$$ \frac{e^x - e^{-x}}{2} = 2 \quad \Rightarrow \quad e^x - e^{-x} = 4 $$

Multiply through by \( e^x \):

$$ e^{2x} - 1 = 4e^x \quad \Rightarrow \quad e^{2x} - 4e^x - 1 = 0 $$

This is a quadratic in \( u = e^x \): \( u^2 - 4u - 1 = 0 \). By the quadratic formula,

$$ u = \frac{4 \pm \sqrt{16 + 4}}{2} = 2 \pm \sqrt{5} $$

Since \( e^x > 0 \), reject \( 2 - \sqrt{5} \approx -0.236 \). So

$$ e^x = 2 + \sqrt{5} \quad \Rightarrow \quad x = \ln(2 + \sqrt{5}) \approx \ln(4.2361) \approx 1.4436 $$

This agrees with the formula \( \sinh^{-1} 2 = \ln(2 + \sqrt{4 + 1}) \). Check: \( e^{1.4436} \approx 4.236 \) and \( e^{-1.4436} \approx 0.236 \), so \( \sinh x \approx (4.236 - 0.236)/2 = 2 \). Correct.

Worked Example 3: The Hanging Cable

A flexible cable hanging under its own weight takes the shape of a catenary:

$$ y = a\cosh\left(\frac{x}{a}\right) $$

where \( a \) depends on the tension and the cable's weight per metre. Suppose a power cable has \( a = 50 \) m and hangs between two towers 100 m apart, at \( x = -50 \) and \( x = 50 \).

Sag (drop from the towers to the lowest point):

$$ \text{sag} = a\cosh\left(\frac{50}{a}\right) - a\cosh 0 = 50(\cosh 1 - 1) $$

Using \( \cosh 1 \approx 1.5431 \):

$$ \text{sag} \approx 50 \times 0.5431 \approx 27.2 \text{ m} $$

Length of cable. The arc length of a catenary from \( -L \) to \( L \) is \( 2a\sinh(L/a) \), so

$$ \text{length} = 2 \times 50 \times \sinh 1 \approx 100 \times 1.1752 \approx 117.5 \text{ m} $$

Sanity check: the cable must be longer than the 100 m straight-line distance, and with a deep sag of about 27 m, an extra 17.5 m is reasonable. Galileo believed a hanging chain formed a parabola; in 1691 Leibniz, Christiaan Huygens and Johann Bernoulli independently showed it is a catenary.

Other Applications

• Falling with air resistance: a skydiver's speed is \( v(t) = v_T\tanh(gt/v_T) \), approaching the terminal velocity \( v_T \). With \( v_T = 55 \) m/s and \( g = 9.8 \) m/s², after 5 s the speed is \( 55\tanh(0.891) \approx 39.2 \) m/s, and after 10 s it is about 52.0 m/s, already 94% of terminal velocity. Without air resistance the speed after 10 s would be 98 m/s, so the tanh model shows how strongly drag limits the fall.
• Special relativity: velocities combine using tanh through a quantity called rapidity.
• Machine learning: \( \tanh \) is a classic activation function in neural networks, squashing any input to between \(-1\) and \( 1 \).

Did You Know?

• The Gateway Arch in St. Louis, USA, 192 m tall, is shaped as a weighted (flattened) catenary.
• An inverted catenary is the ideal shape for a free-standing arch made of stones, because it carries its weight purely in compression; the architect Antoni Gaudí used hanging-chain models to design his buildings.
• \( \cosh x \) and \( \cos x \) are linked through complex numbers: \( \cosh(ix) = \cos x \).

Key Takeaways

• \( \sinh x \) and \( \cosh x \) are the odd and even parts of \( e^x \), and \( \tanh x \) is their ratio.
• The key identity is \( \cosh^2 x - \sinh^2 x = 1 \); Osborn's rule converts trig identities to hyperbolic ones.
• Derivatives are simple: sinh and cosh differentiate to each other with no minus sign.
• Inverse hyperbolic functions are logarithms, found by solving a quadratic in \( e^x \).
• A hanging cable forms a catenary, \( y = a\cosh(x/a) \).

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