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The hyperbolic sine and cosine are combinations of exponentials: sinh x = (eˣ - e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. They satisfy cosh²x - sinh²x = 1, so the point (cosh t, sinh t) lies on the hyperbola x² - y² = 1, just as (cos t, sin t) lies on the circle x² + y² = 1. Their derivatives are simple: (sinh x)' = cosh x and (cosh x)' = sinh x. They appear in the shape of a hanging cable, in some integrals, and in solutions of differential equations.
Derive each identity and derivative directly from the exponential definitions, so nothing has to be memorized blind. Graph eˣ/2 and e⁻ˣ/2 and build cosh and sinh from them by adding and subtracting. Compare each hyperbolic identity with its trigonometric counterpart and note where the signs differ.
Some combinations of eˣ and e⁻ˣ come up so often that they have their own names. Define
sinh x = (eˣ - e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2,
read "hyperbolic sine" and "hyperbolic cosine." The other four hyperbolic functions are built from these the same way the trigonometric ones are: tanh x = sinh x / cosh x, and sech, csch and coth are the reciprocals of cosh, sinh and tanh.
Their graphs come from the exponentials. The function cosh x is the average of eˣ and e⁻ˣ. It is even, has its minimum value 1 at x = 0, and rises in both directions, so its graph looks like a U. The function sinh x is odd, passes through the origin, and increases everywhere. The function tanh x increases from -1 to 1 and has horizontal asymptotes at y = ±1.
The name comes from a key identity. Square the definitions: cosh²x = (e²ˣ + 2 + e⁻²ˣ)/4 and sinh²x = (e²ˣ - 2 + e⁻²ˣ)/4. Subtracting gives cosh²x - sinh²x = 1. So the point (cosh t, sinh t) always lies on the right branch of the hyperbola x² - y² = 1. Compare cos²t + sin²t = 1 for the unit circle. Many hyperbolic identities look like trigonometric ones with some signs changed, for example sinh(2x) = 2 sinh x cosh x and cosh(2x) = cosh²x + sinh²x.
Derivatives follow from the rule (eˣ)' = eˣ and (e⁻ˣ)' = -e⁻ˣ:
(sinh x)' = (eˣ + e⁻ˣ)/2 = cosh x, and (cosh x)' = (eˣ - e⁻ˣ)/2 = sinh x.
Notice that both signs are positive, unlike (cos x)' = -sin x. The quotient rule then gives (tanh x)' = sech²x. Each derivative formula gives an integral formula too, such as ∫ cosh x dx = sinh x + C.
The inverse functions can be written with logarithms. Solving y = sinh x for x gives sinh⁻¹ y = ln(y + √(y² + 1)). This explains why integrals such as ∫ dx/√(x² + 1) produce either an inverse hyperbolic sine or the equivalent logarithm.
Hyperbolic functions also model physical shapes. A uniform cable hanging between two poles takes the shape of a catenary, y = a cosh(x/a). The differential equation y'' = y has the general solution y = A cosh x + B sinh x, which parallels y'' = -y and its solution y = A cos x + B sin x.