2.1 Hyperbolic Functions

2026 Syllabus Objectives

By the end of these notes, you will be able to:

  1. Understand the definitions of the hyperbolic functions sinh x, cosh x, tanh x, sech x, cosech x, and coth x in terms of the exponential function.
  2. Sketch the graphs of hyperbolic functions.
  3. Prove and use identities involving hyperbolic functions (e.g. cosh²x – sinh²x ≡ 1, sinh 2x ≡ 2 sinh x cosh x).
  4. Understand and use the definitions of the inverse hyperbolic functions, and derive and use their logarithmic forms.

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Objective 1 — Definitions of Hyperbolic Functions

What are hyperbolic functions?

You already know the trigonometric functions — sin, cos, tan, and so on. Hyperbolic functions are a family of functions that look and behave very similarly to trigonometric functions, but they are built from the exponential function eˣ instead of circles.

The letter "h" at the end of each name stands for "hyperbolic". You pronounce them as: "shine" (sinh), "cosh" (cosh), "than" (tanh), and so on.

The Two Fundamental Definitions

Everything begins with just two definitions. All other hyperbolic functions come from these two.

sinh⁡x=ex−e−x2\boxed{\sinh x = \frac{e^x - e^{-x}}{2}}

cosh⁡x=ex+e−x2\boxed{\cosh x = \frac{e^x + e^{-x}}{2}}

Plain English: sinh x is half the difference between eˣ and e⁻ˣ. cosh x is half the sum of eˣ and e⁻ˣ.

Tip to remember the difference: The one with the minus sign in the middle is sinh (think: s for subtraction).

The Remaining Four Hyperbolic Functions

Just like in trigonometry where tan = sin/cos, the other hyperbolic functions are defined as ratios:

tanh⁡x=sinh⁡xcosh⁡x=ex−e−xex+e−x\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}

sech x=1cosh⁡x=2ex+e−x\text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}} cosech x=1sinh⁡x=2ex−e−x(x≠0)\text{cosech } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}} \quad (x \neq 0)

coth⁡x=cosh⁡xsinh⁡x=ex+e−xex−e−x(x≠0)\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \quad (x \neq 0)

Plain English: sech is "1 over cosh", cosech is "1 over sinh", coth is "cosh over sinh". These are the reciprocal hyperbolic functions, just like sec, cosec, and cot in trigonometry.

Even and Odd Functions — A Key Property

  • cosh x is an even function: cosh(−x) = cosh x. Its graph is symmetric about the y-axis.
  • sinh x is an odd function: sinh(−x) = −sinh x. Its graph has rotational symmetry about the origin.

Proof that cosh is even: cosh⁡(−x)=e−x+ex2=cosh⁡x✓\cosh(-x) = \frac{e^{-x} + e^{x}}{2} = \cosh x \checkmark

Proof that sinh is odd: sinh⁡(−x)=e−x−ex2=−ex−e−x2=−sinh⁡x✓\sinh(-x) = \frac{e^{-x} - e^{x}}{2} = -\frac{e^x - e^{-x}}{2} = -\sinh x \checkmark

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