How is cosh different from cos?

Cos (cosine) is a trigonometric function based on circles and oscillates between −1 and 1. Cosh (hyperbolic cosine) is based on exponentials and always outputs values greater than or equal to 1. They share similar-looking identities — for instance, $\cos^2(x) + \sin^2(x) = 1$ mirrors $\cosh^2(x) - \sinh^2(x) = 1$ — but the sign difference and the shapes of their graphs are fundamentally different.

Cosh — Definition, Formula & Examples

Cosh is the standard abbreviation for the hyperbolic cosine function, written as

\cosh(x)

. It returns the average of

e^x

and

e^{-x}

and produces a smooth, U-shaped curve called a catenary.

The hyperbolic cosine function is defined for all real numbers $x$ by $\cosh(x) = \frac{e^{x} + e^{-x}}{2}$ , where $e$ is Euler's number. It is an even function with range $[1, \infty)$ , satisfying the identity $\cosh^2(x) - \sinh^2(x) = 1$ .

Key Formula

\cosh(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

\cosh(x)

, compute

e^x

and

e^{-x}

, add them, and divide by 2. Because you are averaging two positive exponentials, the output is always at least 1. The function is symmetric about the

y

-axis, meaning

\cosh(-x) = \cosh(x)

. Engineers use cosh to describe the shape a hanging cable or chain naturally assumes under gravity — this curve is called a catenary. In calculus, cosh has the convenient property that its derivative is

\sinh(x)

, and the derivative of

\sinh(x)

\cosh(x)

, creating a clean differentiation cycle.

Worked Example

Problem: Evaluate cosh(2) to four decimal places.

Step 1: Compute

e^2

e^{2} \approx 7.3891

Step 2: Compute

e^{-2}

e^{-2} \approx 0.1353

Step 3: Add the two values and divide by 2.

\cosh(2) = \frac{7.3891 + 0.1353}{2} = \frac{7.5244}{2} = 3.7622

Answer:

\cosh(2) \approx 3.7622

Another Example

Problem: Verify that cosh(0) = 1.

Step 1: Substitute

x = 0

into the definition.

\cosh(0) = \frac{e^{0} + e^{-0}}{2}

Step 2: Since

e^0 = 1

, simplify.

\cosh(0) = \frac{1 + 1}{2} = 1

Answer:

\cosh(0) = 1

, confirming the minimum value of the function.

Visualization

Why It Matters

Cosh appears throughout college-level calculus, differential equations, and physics. Structural engineers rely on the catenary curve

y = a\cosh(x/a)

to design arches and suspension cables that distribute load evenly. In differential equations,

\cosh(x)

and

\sinh(x)

serve as the fundamental solutions to

y'' - y = 0

, making them as central to that course as sine and cosine are to oscillatory problems.

Common Mistakes

Mistake: Confusing the sign in the identity: writing

\cosh^2(x) + \sinh^2(x) = 1

instead of the correct form.

Correction: The hyperbolic Pythagorean identity uses a minus sign:

\cosh^2(x) - \sinh^2(x) = 1

. The plus sign belongs to the circular identity

\cos^2(x) + \sin^2(x) = 1

Mistake: Assuming

\cosh(x)

can output values less than 1 or negative values.

Correction: Because

e^x

and

e^{-x}

are both positive, their average is always at least 1. The minimum value

\cosh(0) = 1

is a global minimum.

Related Terms

Hyperbolic Cosine — Full-name page for the same function
Sinh (Hyperbolic Sine) — Partner function; derivative of cosh
Hyperbolic Sine — Odd counterpart defined as (eˣ − e⁻ˣ)/2
Hyperbolic Tangent — Ratio sinh/cosh, analogous to tan
Hyperbolic Secant — Reciprocal: sech(x) = 1/cosh(x)
Inverse Hyperbolic Sine — Inverse of the partner function sinh
Inverse Hyperbolic Tangent — Inverse of tanh, related via identities