Hyperbolic Trigonometry

Hyperbolic Trigonometry

A variation of trigonometry. Hyperbolic trig functions are defined using e^x and e^–x. The six hyperbolic trig functions relate to each other in ways that are similar to conventional trig functions. Hyperbolic trig plays an important role when trig functions have imaginary or complex arguments.

Note: Hyperbolic trigonometry has no relation whatsoever to hyperbolic geometry.

Definitions of six hyperbolic trig functions: sinh, cosh, tanh, csch, sech, coth expressed using e^x and e^(-x).

Key Formula

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

Where:

$x$ = Any real (or complex) number — the input to the hyperbolic function
$e$ = Euler's number, approximately 2.71828
$\sinh(x)$ = Hyperbolic sine of x
$\cosh(x)$ = Hyperbolic cosine of x

Worked Example

Problem: Compute sinh(2) and cosh(2) using the definitions, then verify that cosh²(2) − sinh²(2) = 1.

Step 1: Write the definitions with x = 2.

\sinh(2) = \frac{e^2 - e^{-2}}{2}, \qquad \cosh(2) = \frac{e^2 + e^{-2}}{2}

Step 2: Evaluate the exponentials. We have e² ≈ 7.3891 and e⁻² ≈ 0.1353.

e^2 \approx 7.3891, \quad e^{-2} \approx 0.1353

Step 3: Substitute to find sinh(2) and cosh(2).

\sinh(2) \approx \frac{7.3891 - 0.1353}{2} = \frac{7.2538}{2} \approx 3.6269

Step 4: Similarly compute cosh(2).

\cosh(2) \approx \frac{7.3891 + 0.1353}{2} = \frac{7.5244}{2} \approx 3.7622

Step 5: Check the fundamental identity cosh²(x) − sinh²(x) = 1.

3.7622^2 - 3.6269^2 \approx 14.1541 - 13.1544 \approx 1.000 \; \checkmark

Answer: sinh(2) ≈ 3.6269, cosh(2) ≈ 3.7622, and the identity cosh²(2) − sinh²(2) = 1 is confirmed.

Another Example

This example explores symmetry properties and a special value (x = 0) rather than numerical computation, showing that hyperbolic functions mirror the even/odd behavior of cos and sin.

Problem: Find tanh(0) and show that sinh(x) is an odd function while cosh(x) is an even function.

Step 1: Recall that tanh(x) = sinh(x)/cosh(x). Evaluate at x = 0.

\tanh(0) = \frac{\sinh(0)}{\cosh(0)} = \frac{\frac{e^0 - e^0}{2}}{\frac{e^0 + e^0}{2}} = \frac{0}{1} = 0

Step 2: Show sinh is odd by computing sinh(−x).

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

Step 3: Show cosh is even by computing cosh(−x).

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

Answer: tanh(0) = 0. sinh(−x) = −sinh(x) (odd function), and cosh(−x) = cosh(x) (even function).

Frequently Asked Questions

What is the difference between hyperbolic trig functions and regular trig functions?

Regular trig functions (sin, cos) are defined using the unit circle, while hyperbolic trig functions (sinh, cosh) are defined using the exponential function e^x. The key identity for circular trig is sin²x + cos²x = 1, but for hyperbolic trig it becomes cosh²x − sinh²x = 1 (note the minus sign). Despite different geometric origins, the two families are linked through complex numbers: cos(ix) = cosh(x) and sin(ix) = i·sinh(x).

When do you use hyperbolic trig functions?

Hyperbolic functions appear whenever you solve certain differential equations, such as y″ − y = 0 (whose solutions are sinh and cosh rather than sin and cos). They also describe the shape of a hanging cable (catenary), relativistic velocity addition in physics, and the geometry of the unit hyperbola x² − y² = 1. In calculus, they simplify many integrals and substitution techniques.

Does hyperbolic trigonometry relate to hyperbolic geometry?

Despite sharing the word "hyperbolic," these are distinct subjects. Hyperbolic trigonometry defines functions using exponentials and relates to the algebraic hyperbola x² − y² = 1. Hyperbolic geometry is a non-Euclidean geometry where parallel lines behave differently. However, hyperbolic trig functions do appear in formulas within hyperbolic geometry, so there is a useful connection at an advanced level.

Hyperbolic Trig Functions vs. Circular (Ordinary) Trig Functions

	Hyperbolic Trig Functions	Circular (Ordinary) Trig Functions
Definition	Defined via exponentials: (e^x ± e^{−x})/2	Defined via the unit circle or ratios of right-triangle sides
Fundamental identity	cosh²x − sinh²x = 1	cos²x + sin²x = 1
Curve associated	Unit hyperbola: x² − y² = 1	Unit circle: x² + y² = 1
Periodicity	Not periodic (for real inputs)	Periodic with period 2π
Range of 'cosine'	cosh(x) ≥ 1 for all real x	−1 ≤ cos(x) ≤ 1
Complex connection	cosh(x) = cos(ix)	cos(x) = cosh(ix)

Why It Matters

Hyperbolic trig functions show up repeatedly in calculus when you integrate expressions like

\frac{1}{\sqrt{x^2-1}}

or solve second-order differential equations with constant coefficients. In physics, they describe the shape of suspension bridge cables (catenaries) and appear in special relativity through rapidity. Mastering them also deepens your understanding of Euler's formula and the link between exponential and trigonometric functions via complex numbers.

Common Mistakes

Mistake: Using the identity cosh²x + sinh²x = 1 (wrong sign).

Correction: The correct fundamental identity is cosh²x − sinh²x = 1. The minus sign is the critical difference from the circular identity cos²x + sin²x = 1.

Mistake: Assuming hyperbolic functions are periodic like sin and cos.

Correction: For real inputs, sinh(x) and cosh(x) grow exponentially and are not periodic. Only when the argument is complex do hyperbolic functions exhibit periodicity (with period 2πi).

Related Terms

Trigonometry — The parent field; hyperbolic trig parallels it
Trig Functions — Circular counterparts of hyperbolic functions
e — The base of the exponential used to define sinh and cosh
Imaginary Numbers — Link circular and hyperbolic trig via Euler's formula
Complex Numbers — Domain where hyperbolic and circular trig unify
Hyperbolic Geometry — Distinct topic despite sharing the name 'hyperbolic'