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Sinh (Hyperbolic Sine) — Definition, Formula & Examples

Sinh (pronounced "sinch") is the hyperbolic sine function, defined as half the difference between exe^x and e−xe^{-x}. It maps any real number to a real output and resembles a steeper version of the regular sine curve stretched along the y-axis.

For any real number xx, the hyperbolic sine is defined by sinh⁡(x)=ex−e−x2\sinh(x) = \frac{e^x - e^{-x}}{2}. It is an odd function (sinh⁡(−x)=−sinh⁡(x)\sinh(-x) = -\sinh(x)), is unbounded, and satisfies the identity cosh⁡2(x)−sinh⁡2(x)=1\cosh^2(x) - \sinh^2(x) = 1.

Key Formula

sinh⁡(x)=ex−e−x2\sinh(x) = \frac{e^x - e^{-x}}{2}
Where:
  • xx = Any real number (the input to the function)
  • ee = Euler's number, approximately 2.71828

How It Works

To evaluate sinh⁡(x)\sinh(x), compute exe^x and e−xe^{-x}, subtract the second from the first, and divide by 2. The function is odd, so sinh⁡(0)=0\sinh(0) = 0 and the graph passes through the origin with point symmetry. Its derivative is cosh⁡(x)\cosh(x), making it straightforward to differentiate and integrate. In applications, sinh⁡\sinh describes the shape of hanging cables (catenaries) and appears in solutions to certain differential equations.

Worked Example

Problem: Evaluate sinh(2).
Step 1: Compute e2e^2 and e−2e^{-2}.
e2≈7.3891,e−2≈0.1353e^2 \approx 7.3891, \quad e^{-2} \approx 0.1353
Step 2: Subtract and divide by 2.
sinh⁡(2)=7.3891−0.13532=7.25382≈3.6269\sinh(2) = \frac{7.3891 - 0.1353}{2} = \frac{7.2538}{2} \approx 3.6269
Answer: sinh⁡(2)≈3.6269\sinh(2) \approx 3.6269

Why It Matters

Sinh appears in physics and engineering whenever you solve second-order linear differential equations with constant coefficients, such as the wave equation or heat equation. In calculus courses, knowing that ddxsinh⁡(x)=cosh⁡(x)\frac{d}{dx}\sinh(x) = \cosh(x) simplifies integration problems involving exponential expressions.

Common Mistakes

Mistake: Confusing the formula with cosh by adding instead of subtracting the exponentials.
Correction: Remember: sinh uses subtraction (ex−e−xe^x - e^{-x}), while cosh uses addition (ex+e−xe^x + e^{-x}). The 's' in sinh can remind you of 'subtraction.'

Related Terms