Sinh (Hyperbolic Sine) — Definition, Formula & Examples

Sinh (pronounced "sinch") is the hyperbolic sine function, defined as half the difference between

e^x

and

e^{-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 $x$ , the hyperbolic sine is defined by $\sinh(x) = \frac{e^x - e^{-x}}{2}$ . It is an odd function ( $\sinh(-x) = -\sinh(x)$ ), is unbounded, and satisfies the identity $\cosh^2(x) - \sinh^2(x) = 1$ .

Key Formula

\sinh(x) = \frac{e^x - e^{-x}}{2}

Where:

$x$ = Any real number (the input to the function)
$e$ = Euler's number, approximately 2.71828

How It Works

To evaluate

\sinh(x)

, compute

e^x

and

e^{-x}

, subtract the second from the first, and divide by 2. The function is odd, so

\sinh(0) = 0

and the graph passes through the origin with point symmetry. Its derivative is

\cosh(x)

, making it straightforward to differentiate and integrate. In applications,

\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

e^2

and

e^{-2}

e^2 \approx 7.3891, \quad e^{-2} \approx 0.1353

Step 2: Subtract and divide by 2.

\sinh(2) = \frac{7.3891 - 0.1353}{2} = \frac{7.2538}{2} \approx 3.6269

Answer:

\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

\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 (

e^x - e^{-x}

), while cosh uses addition (

e^x + e^{-x}

). The 's' in sinh can remind you of 'subtraction.'