Evaluate \(\sinh(x)\), \(\cosh(x)\), \(\tanh(x)\), reciprocal functions, and their logarithmic inverses. Includes exponential identities and live interactive function curves.
\(\cosh^2(x) - \sinh^2(x) = \) 1.000000
Satisfies unit hyperbola equation \(u^2 - v^2 = 1\).
\(1 - \tanh^2(x) = \text{sech}^2(x) = \) 0.420000
Identical values confirm trigonometric analog.
Input any positive, negative, or zero real number, or choose exact presets like \(\ln(2)\).
View computed \(\sinh\), \(\cosh\), and \(\tanh\), along with reciprocal values and logarithmic inverses.
Confirm that \(\cosh^2(x) - \sinh^2(x) = 1\) and observe the geometric position on the curves.
A freely hanging uniform cable under gravity naturally forms a catenary described by \(y = a \cosh(x/a)\), crucial for designing bridges and transmission lines.
Rapidity \(\theta\) in Einstein's special relativity is additive: velocity \(v = c \tanh(\theta)\), and relativistic Lorentz boosts use \(\cosh(\theta)\) and \(\sinh(\theta)\).
The one-dimensional steady-state heat equation for extended surfaces and heat sink fins is governed directly by solutions involving \(\sinh(mx)\) and \(\cosh(mx)\).
In hyperbolic space, distance metrics and trigonometry on the Poincaré disk model rely on hyperbolic sines and cosines rather than circular functions.
Calculates all primary functions (\(\sinh, \cosh, \tanh\)) and their reciprocals (\(\text{csch}, \text{sech}, \coth\)) in real time.
Evaluates inverse functions with rigorous domain boundary enforcement (\(x \ge 1\) for \(\text{arcosh}\), \(|x| < 1\) for \(\text{artanh}\)).
Plots the distinctive curves of \(\sinh\), \(\cosh\), and \(\tanh\) simultaneously with live evaluation indicators.
At first glance, \(\sinh(x)\) and \(\cosh(x)\) seem like exotic creations. In reality, they are the most natural way to describe shapes in nature, from the curve of the Golden Gate Bridge cables to the geometry of spacetime in Einstein's theory of relativity.
In ordinary trigonometry, as angle \(\theta\) sweeps around the origin, the point \((\cos\theta, \sin\theta)\) traces out the unit circle \(x^2 + y^2 = 1\). Hyperbolic functions do the exact same thing for the unit hyperbola \(x^2 - y^2 = 1\)! As the hyperbolic parameter \(t\) varies, the coordinates \((\cosh t, \sinh t)\) trace out the smooth curve of the right hyperbola branch, satisfying the foundational identity:
Every hyperbolic function is built from \(e^x\) and \(e^{-x}\). \(\cosh(x) = \frac{e^x + e^{-x}}{2}\) is the average of growth and decay (an even, symmetric U-shape). \(\sinh(x) = \frac{e^x - e^{-x}}{2}\) is half the difference (an odd, origin-crossing curve).
Notice the magical sign flip: unlike circular trig where \(\frac{d}{dx}\cos x = -\sin x\), in hyperbolic calculus, there are no negative signs: \(\frac{d}{dx}\sinh x = \cosh x\) and \(\frac{d}{dx}\cosh x = +\sinh x\)!
Any flexible hanging chain or telephone wire suspended between two poles hangs in a catenary curve \(y = a\cosh(x/a)\), not a parabola! In special relativity, successive Lorentz velocity boosts add linearly via hyperbolic rapidity: \(\theta = \operatorname{artanh}(v/c)\).
Because \(\cosh(x) \ge 1\) for all real \(x\), you can never evaluate \(\operatorname{arcosh}(x)\) for any number smaller than 1! Entering \(x = 0.5\) is undefined in real numbers.
Unlike \(\tan(x)\) which sweeps from \(-\infty\) to \(+\infty\), \(\tanh(x)\) is strictly bounded between \(-1\) and \(+1\). Consequently, \(\operatorname{artanh}(x)\) only exists inside the open interval \((-1, 1)\).
Students mistakenly write \(\cosh^2(x) + \sinh^2(x) = 1\). Remember: it is \(\cosh^2(x) - \sinh^2(x) = 1\). The plus version actually evaluates to \(\cosh(2x)\)!
| Property | Circular Trigonometry | Hyperbolic Trigonometry |
|---|---|---|
| Governing Geometric Curve | Unit Circle: \(x^2 + y^2 = 1\) | Unit Hyperbola: \(x^2 - y^2 = 1\) |
| Fundamental Identity | \(\cos^2(t) + \sin^2(t) = 1\) | \(\cosh^2(t) - \sinh^2(t) = 1\) |
| Derivative of Cosine | \(\frac{d}{dx}\cos x = -\sin x\) (Negative sign) | \(\frac{d}{dx}\cosh x = +\sinh x\) (Positive sign) |
| Periodicity along Real Line | Periodic (\(2\pi\)) | Aperiodic (Periodic only along imaginary axis \(2\pi i\)) |
Step 1: Calculate \(e^{\ln 2} = 2\) and \(e^{-\ln 2} = 1/2\).
Step 2: \(\sinh(\ln 2) = \frac{2 - 1/2}{2} = \frac{3/2}{2} = \frac{3}{4} = 0.75\).
Step 3: \(\cosh(\ln 2) = \frac{2 + 1/2}{2} = \frac{5/2}{2} = \frac{5}{4} = 1.25\).
Step 4: \(\tanh(\ln 2) = \frac{0.75}{1.25} = \frac{3}{5} = 0.6\).
Verify Identity: \((5/4)^2 - (3/4)^2 = \frac{25 - 9}{16} = \frac{16}{16} = 1\).
Step 1: Recall definition \(\text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1})\).
Step 2: For \(x = 2\): \(\sqrt{2^2 + 1} = \sqrt{5} \approx 2.236068\).
Step 3: \(\text{arsinh}(2) = \ln(2 + \sqrt{5}) = \ln(4.236068) \approx 1.443635\).
Check: \(\sinh(1.443635) = \frac{e^{1.443635} - e^{-1.443635}}{2} = \frac{4.236068 - 0.236068}{2} = \frac{4}{2} = 2\).
