Hyperbolic Function Definitions
Why Study Hyperbolic Functions?
When a heavy cable hangs freely between two supports, it does not form a parabola. It forms a shape called a catenary, described by the hyperbolic cosine function. This same function appears in the Gateway Arch in St. Louis, in the cables of suspension bridges, and in the analysis of ocean waves.
Hyperbolic functions arise naturally whenever exponential growth and decay combine in symmetric ways. They’re called “hyperbolic” because they relate to the hyperbola $x^2 - y^2 = 1$ the same way trigonometric functions relate to the circle $x^2 + y^2 = 1$.
Before You Start: Quick Self-Check
Can you answer these questions? If not, review the linked prerequisites first.
1. What is $e^{\ln 5}$?
Answer: $e^{\ln 5} = 5$. If this wasn’t immediate, review Exponential Functions.
2. Is $\sin(-x) = \sin x$ or $\sin(-x) = -\sin x$?
Answer: $\sin(-x) = -\sin x$ (sine is odd). If this wasn’t clear, review Trigonometric Functions.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Concept | Hyperbolic Functions |
| Chapter | 6.7 |
| Difficulty | Beginner |
| Time | ~15 minutes |
Key Concepts
The Basic Definitions
The six hyperbolic functions are defined in terms of exponentials:
$$\boxed{\sinh x = \frac{e^x - e^{-x}}{2} \qquad \cosh x = \frac{e^x + e^{-x}}{2}}$$
$$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$
The reciprocal functions are:
$$\text{csch } x = \frac{1}{\sinh x} \qquad \text{sech } x = \frac{1}{\cosh x} \qquad \coth x = \frac{\cosh x}{\sinh x}$$
Understanding the Definitions
Think of $\sinh x$ and $\cosh x$ as the “difference” and “sum” of exponential growth and decay:
- $\sinh x$ measures how much $e^x$ exceeds $e^{-x}$ (divided by 2)
- $\cosh x$ measures the average of $e^x$ and $e^{-x}$
Notice that $\cosh x + \sinh x = e^x$ and $\cosh x - \sinh x = e^{-x}$; these are useful facts for simplification.
Graphs and Behavior
y y
| / | \ /
| / | \ /
| / | \_/ (1,0)
-----+--/-------- x -------+--------- x
/ | |
/ | |
/ | y = sinh x | y = cosh xKey observations:
| Function | Domain | Range | Even/Odd | Key Feature |
|---|---|---|---|---|
| $\sinh x$ | $\mathbb{R}$ | $\mathbb{R}$ | Odd | Passes through origin |
| $\cosh x$ | $\mathbb{R}$ | $[1, \infty)$ | Even | Minimum value of 1 at $x=0$ |
| $\tanh x$ | $\mathbb{R}$ | $(-1, 1)$ | Odd | Horizontal asymptotes at $y = \pm 1$ |
Why “Hyperbolic”?
For trig functions: if $x = \cos t$ and $y = \sin t$, then $x^2 + y^2 = 1$ (the unit circle).
For hyperbolic functions: if $x = \cosh t$ and $y = \sinh t$, then $x^2 - y^2 = 1$ (the right branch of the unit hyperbola).
This geometric connection gives these functions their name.
Connection to Trig Functions
| Trig Function | Hyperbolic Analog |
|---|---|
| $\sin x$ | $\sinh x$ |
| $\cos x$ | $\cosh x$ |
| $\tan x$ | $\tanh x$ |
| $\csc x$ | $\text{csch } x$ |
| $\sec x$ | $\text{sech } x$ |
| $\cot x$ | $\coth x$ |
The parallels are striking, but there are important differences in signs (which we’ll see in identities and derivatives).
Practice Problems
Find the exact values of $\sinh(\ln 2)$ and $\cosh(\ln 2)$.
Write $3\sinh x + 4\cosh x$ in terms of $e^x$ and $e^{-x}$.
Compute $\sinh(\ln 4) + \cosh(\ln 4)$ and simplify your answer to a single number.
Express $\tanh(\ln x)$ as a rational function of $x$ (where $x > 0$).
The expression $5e^x + 3e^{-x}$ can be written in the form $\alpha \cosh(x + \beta)$ for appropriate constants $\alpha > 0$ and $\beta$.
(a) Find the exact values of $\alpha$ and $\beta$.
(b) Verify your answer by expanding $\alpha \cosh(x + \beta)$ and showing it equals $5e^x + 3e^{-x}$.
CCI-Style Conceptual Questions
Question 1: Which statement correctly describes the relationship between $\sinh x$ and $\cosh x$?
(A) $\sinh x + \cosh x = 1$ (B) $\sinh x + \cosh x = e^x$ (C) $\sinh x \cdot \cosh x = 1$ (D) $\sinh x / \cosh x = 1$
Answer
(B) Adding the definitions: $\frac{e^x - e^{-x}}{2} + \frac{e^x + e^{-x}}{2} = \frac{2e^x}{2} = e^x$
Question 2: The minimum value of $\cosh x$ is:
(A) 0 (B) 1 (C) $e$ (D) There is no minimum
Answer
(B) The function $\cosh x$ achieves its minimum at $x = 0$, where $\cosh 0 = 1$. Since $\cosh x$ is even and increases as $\vert x\vert $ increases, 1 is the global minimum.
Question 3 (Explain Why): A student claims that $\sinh x$ can never equal $\cosh x$. Is this correct? Explain your reasoning.
Answer
Incorrect. We have $\sinh x = \cosh x$ when $\frac{e^x - e^{-x}}{2} = \frac{e^x + e^{-x}}{2}$, which simplifies to $e^{-x} = 0$. Since $e^{-x} > 0$ for all real $x$, there is no solution for finite $x$.
However, as $x \to +\infty$, both functions approach $\frac{e^x}{2}$, so they become arbitrarily close. The student’s claim is technically correct for finite values, but the reasoning should reference that $e^{-x} \neq 0$ rather than making an unfounded assertion.
Common Misconceptions
hyperbolic functions are just alternative notation for the circular trigonometric functions, so their properties are identical.
This is the trig-as-algebra-symbols error. Although $\sinh$ and $\cosh$ share naming conventions with $\sin$ and $\cos$, they are defined through exponentials ($\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$) and have substantially different properties. Unlike $\cos x$, which is bounded between $-1$ and $1$, $\cosh x \geq 1$ for all real $x$ and grows without bound. The fundamental identity is $\cosh^2 x - \sinh^2 x = 1$, with a minus sign, not plus as in the circular Pythagorean identity.
Mastery Checklist
Mental Model
The Sum-Difference Pair:
Think of $\cosh x$ and $\sinh x$ as the “average” and “half-difference” of exponential growth ($e^x$) and decay ($e^{-x}$). Just as the average of two numbers is always at least as large as their difference, $\cosh x \geq \vert \sinh x\vert $ always.
When $x$ is large and positive, $e^{-x} \approx 0$, so both functions behave like $\frac{1}{2}e^x$.
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|---|---|---|
| Skills Index | Hyperbolic Identities |
Last updated: 2026-01-22