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MathidiaMathematics Academy
Hyperbolic functions
Lesson 31 of 75
Introduction

Hyperbolic definitions

Objective. Define sinh, cosh and tanh exponentially.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The hyperbolic functions are defined using the exponential function: cosh x = (eˣ + e⁻ˣ)/2 and sinh x = (eˣ − e⁻ˣ)/2, with tanh x = sinh x / cosh x. These definitions mean cosh x is always even and at least 1, while sinh x is odd and can take any real value, mirroring but differing from the trigonometric cos and sin. For example, cosh 0 = (1+1)/2 = 1 and sinh 0 = (1−1)/2 = 0, matching the trigonometric values at zero even though the functions behave very differently for large x.

Worked example

Use the exponential definitions to find the exact value of cosh(ln 2).

2
Illustrated card showing 2
  1. 1cosh x = (eˣ + e⁻ˣ)/2, with x = ln 2
  2. 2e^(ln 2) = 2, and e^(−ln 2) = 1/2
  3. 3Substitute: cosh(ln 2) = (2 + 1/2)/2
  4. 4Simplify: (5/2)/2 = 5/4

Answer: 5/4

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Common mistake. Students assume cosh and sinh are bounded like cos and sin, forgetting that cosh x and sinh x grow without bound as x increases, since they are built from the unbounded exponential function. Always check behaviour using the exponential definitions, not trigonometric intuition.

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