Hyperbolic Sine (sinh)
sinh(x) = (e^x - e^(-x))/2. The hyperbolic sine is the odd analog of circular sine, with range (-∞, ∞). Used in catenary curves, special relativity, and differential equations.
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sinh(ix) = i·sin(x) — hyperbolic and circular trig connect via complex numbers. d/dx[sinh(x)] = cosh(x) — the derivative is the companion hyperbolic cosine. Catenary: a hanging chain has shape y = a·cosh(x/a) — the derivative involves sinh.
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Why: sinh models hanging cables (catenary), appears in special relativity (Lorentz factor), and solves y'' = y. The derivative of sinh is cosh.
How: sinh(x) = (e^x - e^(-x))/2. For small |x|, sinh(x) ≈ x. The identity cosh²(x) - sinh²(x) = 1 parallels sin²+cos²=1.
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Hyperbolic Value Breakdown
All 6 Hyperbolic Functions
|sinh| vs cosh
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
A hanging chain forms a catenary: y = a·cosh(x/a). The slope involves sinh.
— Wolfram MathWorld
sinh appears in special relativity — the Lorentz factor and rapidity use hyperbolic functions.
— MIT OCW
Key Takeaways
- • sinh(x) = (e^x - e^(-x))/2 — the hyperbolic sine is half the difference of exponentials
- • sinh is an odd function: sinh(-x) = -sinh(x). Range: all reals
- • sinh(0) = 0. For small x, sinh(x) ≈ x. For large |x|, sinh grows exponentially
- • Used in catenary curves (hanging cables), special relativity, and wave equations
- • Fundamental identity: cosh²(x) - sinh²(x) = 1. Derivative: d/dx[sinh(x)] = cosh(x)
Did You Know?
How the Hyperbolic Sine Works
The hyperbolic sine is defined via exponentials, analogous to how sin(θ) relates to e^(iθ) via Euler's formula. sinh(x) captures the "odd" part of e^x.
Exponential Definition
sinh(x) = (e^x - e^(-x))/2. For x > 0, e^x dominates and sinh grows like e^x/2. For x < 0, sinh is negative. Unlike sin(x), sinh is not periodic and is unbounded.
Relation to Regular Trig
sinh(ix) = i·sin(x) and sin(ix) = i·sinh(x). Hyperbolic functions satisfy similar identities: cosh² - sinh² = 1 (vs cos² + sin² = 1).
Geometric Interpretation (Hyperbola)
On the unit hyperbola x² - y² = 1, the point (cosh(t), sinh(t)) traces the right branch. sinh(t) is the y-coordinate, analogous to sin(θ) on the unit circle.
Expert Tips
Memorize Key Values
sinh(0)=0, sinh(1)≈1.175, sinh(2)≈3.627. For small x, sinh(x)≈x. Use the Cosh Calculator for cosh values.
Odd Function Property
sinh(-x) = -sinh(x) means the graph has origin symmetry. Compare with Sine Calculator — both sin and sinh are odd.
Catenary & Special Relativity
Catenary: y = a·cosh(x/a). Arc length involves sinh. In relativity, rapidity φ satisfies v/c = tanh(φ). Try the Tanh Calculator.
The Identity cosh² - sinh² = 1
Always verify: (e^x+e^(-x))²/4 - (e^x-e^(-x))²/4 = 1. This is the hyperbolic analog of the Pythagorean identity.
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
How does sinh differ from sin?
sin uses a circle and is bounded [-1,1] and periodic. sinh uses a hyperbola, is unbounded, and grows exponentially. sinh(ix) = i·sin(x) connects them via complex numbers.
What is the derivative of sinh?
d/dx[sinh(x)] = cosh(x). This elegant relationship mirrors d/dx[sin(x)] = cos(x). The integral ∫sinh(x)dx = cosh(x) + C.
What is the range of sinh?
sinh(x) has range (-∞, ∞) — all real numbers. For any real y, there exists x such that sinh(x) = y. sinh is strictly increasing.
Where is sinh used in real life?
Catenary curves (suspension bridges, power lines), special relativity (velocity addition, rapidity), solutions to differential equations, and signal processing.
Why is sinh called odd?
sinh(-x) = -sinh(x). Graphically, the curve has rotational symmetry about the origin — reflecting through the origin gives the same graph.
What is cosh² - sinh²?
Always exactly 1. Expand (e^x+e^(-x))²/4 - (e^x-e^(-x))²/4 to verify. This is the fundamental hyperbolic identity.
How do I compute sinh without a calculator?
Use the definition: sinh(x) = (e^x - e^(-x))/2. For small x, sinh(x) ≈ x. For large x, sinh(x) ≈ e^x/2.
What is the inverse of sinh?
arsinh(x) = ln(x + √(x²+1)). It is defined for all real x. The inverse exists because sinh is strictly increasing.
Hyperbolic Sine 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 (aerospace, physics simulations), always verify with certified computational tools. Not a substitute for professional analysis.
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