Cosh and Sinh Calculator

Hyperbolic Functions:

\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \]

cosh(x):

sinh(x):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What are Hyperbolic Functions?

Hyperbolic functions (cosh and sinh) are analogs of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They appear frequently in solutions of differential equations and have applications in physics and engineering.

2. How Does the Calculator Work?

The calculator uses the exponential definitions of hyperbolic functions:

\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \]

Where:

\( x \) — Input value in radians
\( e \) — Euler's number (~2.71828)

Explanation: These functions combine exponential growth (e^x) and decay (e^-x) to produce hyperbolic analogs of cosine and sine.

3. Applications of Hyperbolic Functions

Details: Hyperbolic functions are used in:

Catenary curves (shape of hanging chains/cables)
Special relativity (Lorentz transformations)
Heat transfer calculations
Electrical engineering (transmission line theory)

4. Using the Calculator

Tips: Enter any real number (positive or negative) in radians. The calculator will compute both cosh(x) and sinh(x) simultaneously.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between cosh and regular cos?
A: While cos(x) is periodic and bounded between -1 and 1, cosh(x) grows exponentially in both positive and negative directions and is always ≥1.

Q2: Are the results dimensionless?
A: Yes, both cosh(x) and sinh(x) are unitless quantities, as they're ratios of exponential functions.

Q3: What is the range of sinh(x)?
A: sinh(x) can take any real value from -∞ to +∞, unlike sin(x) which is limited to [-1, 1].

Q4: What's special about x=0?
A: At x=0, cosh(0)=1 and sinh(0)=0, similar to cos(0)=1 and sin(0)=0 in trigonometry.

Q5: How are these related to the unit hyperbola?
A: Just as (cosθ, sinθ) parametrize the unit circle, (coshθ, sinhθ) parametrize the right branch of the unit hyperbola x²-y²=1.