Hyperbolic functions are built from exponential functions and appear throughout calculus, geometry, physics, and engineering. The main functions, sinh x, cosh x, and tanh x, behave in ways that resemble sine, cosine, and tangent, but they are based on hyperbolas rather than circles. They are especially useful when solving differential equations and evaluating integrals involving square roots such as sqrt(x^2 + a^2).
One famous real-world shape described by hyperbolic cosine is the catenary, the curve made by a hanging chain.
Key Facts
- sinh x = (e^x - e^(-x))/2
- cosh x = (e^x + e^(-x))/2
- cosh^2 x - sinh^2 x = 1
- d/dx sinh x = cosh x and d/dx cosh x = sinh x
- Integral sinh x dx = cosh x + C and integral cosh x dx = sinh x + C
- Catenary equation: y = a cosh(x/a), where a controls how steep or wide the hanging chain is.
Vocabulary
- Hyperbolic function
- A function defined using exponentials that is related to the geometry of a hyperbola.
- sinh x
- The hyperbolic sine function, defined by sinh x = (e^x - e^(-x))/2.
- cosh x
- The hyperbolic cosine function, defined by cosh x = (e^x + e^(-x))/2.
- tanh x
- The hyperbolic tangent function, defined by tanh x = sinh x / cosh x.
- Catenary
- The curve formed by an ideal flexible chain hanging under its own weight, modeled by y = a cosh(x/a).
Common Mistakes to Avoid
- Using cos^2 x + sin^2 x = 1 for hyperbolic functions is wrong because the correct identity is cosh^2 x - sinh^2 x = 1.
- Assuming d/dx cosh x = -sinh x is wrong because hyperbolic cosine differentiates to positive sinh x, unlike ordinary cosine.
- Forgetting the chain rule in expressions like cosh(3x) is wrong because d/dx cosh(3x) = 3sinh(3x), not sinh(3x).
- Treating the catenary as a parabola is wrong because a hanging chain follows y = a cosh(x/a), not a quadratic equation, although the two can look similar near the lowest point.
Practice Questions
- 1 Compute sinh 0, cosh 0, and tanh 0 using the exponential definitions.
- 2 Find the derivative of f(x) = 4cosh(2x) - 3sinh(x).
- 3 A cable hangs in the shape y = a cosh(x/a). Explain how increasing a changes the shape of the catenary and why this is not the same as changing the coefficient of a parabola.