Hyperbolic Cosine (cosh)
cosh(x) = (e^x + e^(-x))/2. The hyperbolic cosine is the even analog of circular cosine, with range [1, ∞). The catenary curve: y = a·cosh(x/a).
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cosh(ix) = cos(x) — hyperbolic and circular cosine connect via complex numbers. d/dx[cosh(x)] = sinh(x) — the derivative is the companion hyperbolic sine. A hanging chain minimizes potential energy; the solution is y = a·cosh(x/a).
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Why: cosh models the catenary — the shape of a hanging chain or cable. Suspension bridges, power lines, and the Gateway Arch follow cosh curves.
How: cosh(x) = (e^x + e^(-x))/2. For small |x|, cosh(x) ≈ 1 + x²/2. The identity cosh²(x) - sinh²(x) = 1 parallels cos²+sin²=1.
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Hyperbolic Value Breakdown
All 6 Hyperbolic Functions
cosh vs |sinh|
Calculation Breakdown
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🧮 Fascinating Math Facts
Suspension bridges use catenary curves: y = a·cosh(x/a) — the shape of a hanging cable.
— Wolfram MathWorld
cosh is even: cosh(-x) = cosh(x). The graph is symmetric about the y-axis.
— Paul's Notes
Key Takeaways
- • cosh(x) = (e^x + e^(-x))/2 — the hyperbolic cosine is half the sum of exponentials
- • cosh is an even function: cosh(-x) = cosh(x). Range: [1, ∞)
- • cosh(0) = 1 is the minimum. For small x, cosh(x) ≈ 1 + x²/2
- • The catenary shape (hanging chain) is y = a·cosh(x/a). cosh² - sinh² = 1
- • Derivative: d/dx[cosh(x)] = sinh(x). Used in suspension bridges and arch structures
Did You Know?
How the Hyperbolic Cosine Works
The hyperbolic cosine is the average of e^x and e^(-x). It is always ≥ 1 and describes the shape of a hanging chain (catenary) — the curve that minimizes potential energy under uniform gravity.
Exponential Definition
cosh(x) = (e^x + e^(-x))/2. Both terms are positive, so cosh(x) ≥ 1. The minimum cosh(0) = 1 occurs when e^x = e^(-x) = 1. For large |x|, cosh grows like e^|x|/2.
Relation to Regular Trig
cosh(ix) = cos(x) and cos(ix) = cosh(x). The identity cosh² - sinh² = 1 mirrors cos² + sin² = 1 but with a minus sign — reflecting the hyperbola vs circle geometry.
Geometric Interpretation (Catenary)
A uniform chain hanging under gravity forms the catenary y = a·cosh(x/a). The Gateway Arch, suspension bridges, and power lines follow this shape. cosh describes the "even" part of e^x.
Expert Tips
Memorize Key Values
cosh(0)=1, cosh(1)≈1.543, cosh(2)≈3.762. For small x, cosh(x)≈1+x²/2. Use the Sinh Calculator for sinh values.
Even Function Property
cosh(-x) = cosh(x) means the graph has y-axis symmetry. Compare with Cosine Calculator — both cos and cosh are even.
Catenary Applications
y = a·cosh(x/a) gives sag, arc length, and tension. The Tanh Calculator gives slope: dy/dx = sinh(x/a).
The Identity cosh² - sinh² = 1
Always verify: (e^x+e^(-x))²/4 - (e^x-e^(-x))²/4 = 1. If you know cosh, then sinh = ±√(cosh² - 1), with sign from x.
Why Use This Calculator vs. Other Tools?
| Feature | This Calculator | Scientific Calculator | Manual Computation |
|---|---|---|---|
| All 6 hyperbolic functions at once | ✅ | ❌ One at a time | ❌ |
| Exponential form (e^x, e^(-x)) | ✅ | ❌ | ✅ Slow |
| Visual charts & breakdown | ✅ | ❌ | ❌ |
| Step-by-step explanation | ✅ | ❌ | ✅ |
| Identity cosh² - sinh² = 1 check | ✅ | ❌ | ⚠️ Manual |
| Copy & share results | ✅ | ❌ | ❌ |
| Screenshot-ready summary | ✅ | ❌ | ❌ |
| Preset examples | ✅ | ❌ | ❌ |
Frequently Asked Questions
Why is cosh always ≥ 1?
cosh(x) = (e^x + e^(-x))/2 is the average of two positive numbers. The minimum occurs at x=0 where e^0 + e^0 = 2, so cosh(0) = 1. For x ≠ 0, at least one exponential exceeds 1.
What is a catenary?
A catenary is the curve formed by a hanging chain or cable under uniform gravity. Its equation is y = a·cosh(x/a). Suspension bridges, power lines, and the Gateway Arch follow this shape.
What is the derivative of cosh?
d/dx[cosh(x)] = sinh(x). This mirrors d/dx[cos(x)] = -sin(x) but without the minus sign. The integral ∫cosh(x)dx = sinh(x) + C.
How does cosh differ from cos?
cos uses a circle and is bounded [-1,1] and periodic. cosh uses a hyperbola, is unbounded above with minimum 1, and is not periodic. cosh(ix) = cos(x) connects them.
What is cosh² - sinh²?
Always exactly 1. Expand (e^x+e^(-x))²/4 - (e^x-e^(-x))²/4. The cross terms cancel, leaving (e^(2x)+e^(-2x)+2 - e^(2x)-e^(-2x)+2)/4 = 1.
Where is cosh used in engineering?
Suspension bridge design, power line sag, arch structures, cable systems, and soap film surfaces (catenoid). Any hanging flexible structure under gravity.
Why is cosh called even?
cosh(-x) = cosh(x). The graph is symmetric about the y-axis. This follows from e^(-(-x)) + e^(-x) = e^x + e^(-x).
What is the inverse of cosh?
arcosh(x) = ln(x + √(x²-1)) for x ≥ 1. The inverse exists on [1, ∞) because cosh is strictly increasing there.
Hyperbolic Cosine by the Numbers
Official & Educational Sources
Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (structural engineering, aerospace), always verify with certified computational tools. Not a substitute for professional engineering analysis.
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