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Hyperbolic functions are the exponential-based cousins of the familiar trigonometric functions — instead of being defined from a circle, they're defined from the two branches of a hyperbola, which is exactly where the name comes from. Just as (cos θ, sin θ) traces out every point on a unit circle, the pair (cosh t, sinh t) traces out every point on the hyperbola x² − y² = 1, and this parallel structure means hyperbolic functions satisfy their own set of identities that closely mirror — but aren't identical to — familiar trigonometric ones. These functions show up constantly in calculus and differential equations, and they have real, tangible physical meaning too: the shape a freely hanging cable or chain naturally forms under gravity (called a catenary) is described exactly by the cosh function, and hyperbolic functions also appear throughout special relativity and in the tanh activation function widely used in neural networks. This tool computes all six hyperbolic functions — sinh, cosh, tanh, and their three reciprocals csch, sech, and coth — directly from their exponential definitions.
How it works
The two foundational hyperbolic functions are computed directly from the exponential function: sinh(x) = (eˣ − e⁻ˣ) ÷ 2, and cosh(x) = (eˣ + e⁻ˣ) ÷ 2. From these, tanh(x) is calculated as sinh(x) ÷ cosh(x), matching the same sine-over-cosine relationship that defines the ordinary tangent function. The three reciprocal functions then follow by simple division: csch(x) = 1 ÷ sinh(x), sech(x) = 1 ÷ cosh(x), and coth(x) = 1 ÷ tanh(x) — each undefined exactly where its corresponding base function equals zero.
- Enter x.
- Click Calculate to see your results.
Examples
At x = 0
sinh(0) = 0, cosh(0) = 1, and tanh(0) = 0 — these baseline values are a quick way to sanity-check any hyperbolic function calculation.
At x = 1
sinh(1) ≈ 1.1752, cosh(1) ≈ 1.5431, and tanh(1) ≈ 0.7616, computed directly from the exponential definitions using e ≈ 2.71828.
A reciprocal function
At x = 1, sech(1) = 1 ÷ cosh(1) ≈ 1 ÷ 1.5431 ≈ 0.6481 — showing how the reciprocal functions follow immediately once the base functions are known.
Who should use it
- Calculus and differential equations coursework involving hyperbolic functions.
- Physics problems involving catenaries (hanging cable shapes) or special relativity.
- Understanding the math behind the tanh activation function in machine learning.
Industry applications
- Physics and engineering (structural analysis, relativity)
- Calculus education
- Machine learning (activation functions)
Advantages
- Computes all six hyperbolic functions in one step.
- Correctly flags the undefined reciprocal functions at x=0.
- Uses the exact exponential definitions rather than an approximation.
Limitations
- Limited to x between -50 and 50, since cosh(x) and sinh(x) grow extremely quickly beyond that.
Common mistakes to avoid
- Confusing the hyperbolic identity cosh²(x) − sinh²(x) = 1 with the trigonometric identity cos²(θ) + sin²(θ) = 1 — note the minus sign.
- Forgetting that csch and coth are undefined at x = 0, since they involve dividing by sinh(0) = 0 or tanh(0) = 0.
- Assuming hyperbolic functions are periodic like their trigonometric counterparts — they are not; sinh and cosh grow without bound as x increases.
Best practices
- Remember cosh(x) is always ≥ 1 for real x, which is a quick way to catch a computational error.
- Use the identity cosh²(x) − sinh²(x) = 1 to double-check a calculation involving both functions at the same x.
- Watch for the specific undefined points (x = 0 for csch and coth) when evaluating the reciprocal functions.
Tips
- tanh(x) is bounded between -1 and 1 for all real x, which is why it's a popular activation function in neural networks.
- Use x = 0 as a quick sanity check: sinh(0) = 0, cosh(0) = 1, tanh(0) = 0.