How is sinh(x) different from sin(x)?

The ordinary sine function sin(x) is periodic and bounded between −1 and 1, arising from the unit circle. Hyperbolic sine sinh(x) is not periodic and grows without bound, arising from the unit hyperbola. Their formulas also differ: sin(x) uses complex exponentials with imaginary arguments, while sinh(x) uses real exponentials. They are connected by the relation sinh(x) = −i sin(ix).

Hyperbolic Sine — Definition, Formula & Examples

Hyperbolic sine, written as sinh(x), is a function defined using exponential functions that mirrors some properties of the ordinary sine function but traces a hyperbola rather than a circle. It is calculated as half the difference between

e^x

and

e^{-x}

For any real number $x$ , the hyperbolic sine function is defined by $\sinh(x) = \dfrac{e^x - e^{-x}}{2}$ , where $e$ is Euler's number. The function is odd, meaning $\sinh(-x) = -\sinh(x)$ , has domain $(-\infty, \infty)$ , range $(-\infty, \infty)$ , 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 input
$e$ = Euler's number, approximately 2.71828

How It Works

To evaluate

\sinh(x)

, substitute the value of

x

into the exponential formula and compute. The function grows exponentially for large positive

x

(behaving like

\frac{1}{2}e^x

) and decreases exponentially for large negative

x

. Its derivative is

\cosh(x)

, and its integral is

\cosh(x) + C

, making it straightforward in calculus. Hyperbolic sine appears naturally when solving certain differential equations, modeling hanging cables (catenaries), and describing relativistic velocity addition in physics.

Worked Example

Problem: Evaluate sinh(2).

Step 1: Write the definition of sinh and substitute x = 2.

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

Step 2: Compute the exponential values. We have e² ≈ 7.3891 and e⁻² ≈ 0.1353.

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

Step 3: Subtract and divide by 2.

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

Answer: sinh(2) ≈ 3.6269

Another Example

Problem: Find the derivative of f(x) = sinh(3x).

Step 1: Recall that the derivative of sinh(u) with respect to x is cosh(u) · du/dx (chain rule).

\frac{d}{dx}[\sinh(u)] = \cosh(u) \cdot \frac{du}{dx}

Step 2: Here u = 3x, so du/dx = 3.

f'(x) = \cosh(3x) \cdot 3

Step 3: Write the final result.

f'(x) = 3\cosh(3x)

Answer: f'(x) = 3 cosh(3x)

Visualization

Why It Matters

Hyperbolic sine is essential in Calculus II and differential equations courses, where it appears in solutions to linear ODEs with constant coefficients. Engineers use it to model the shape of a freely hanging cable (catenary curve), and physicists encounter it in special relativity when describing rapidity. Mastering sinh and its companion functions also prepares you for integration techniques involving hyperbolic substitution.

Common Mistakes

Mistake: Using a plus sign instead of a minus sign in the formula, writing (eˣ + e⁻ˣ)/2.

Correction: That expression defines cosh(x), not sinh(x). Hyperbolic sine uses subtraction: sinh(x) = (eˣ − e⁻ˣ)/2.

Mistake: Assuming sinh(x) is periodic or bounded like sin(x).

Correction: Unlike sin(x), sinh(x) has no period and grows exponentially. Its range is all real numbers, not [−1, 1].

Related Terms

Hyperbolic Cosine — Companion even function, derivative of sinh
Hyperbolic Tangent — Ratio sinh/cosh, another hyperbolic function
Inverse Hyperbolic Functions — Includes arcsinh, the inverse of sinh
Exponential Function — Building block in the sinh definition
Catenary — Curve described by cosh, involves sinh in arc length
Sine — Circular counterpart of hyperbolic sine