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Hyperbolic Functions Calculator

Calculate sinh, cosh, tanh, csch, sech, and coth functions

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Hyperbolic Function

📐Hyperbolic Function Definitions

Hyperbolic Sine
sinh(x) = (e^x - e^(-x)) / 2
Odd function
Hyperbolic Cosine
cosh(x) = (e^x + e^(-x)) / 2
Even function
Hyperbolic Tangent
tanh(x) = sinh(x) / cosh(x)
Range: (-1, 1)
Hyperbolic Cosecant
csch(x) = 1 / sinh(x)
Undefined at x = 0
Hyperbolic Secant
sech(x) = 1 / cosh(x)
Range: (0, 1]
Hyperbolic Cotangent
coth(x) = cosh(x) / sinh(x)
Undefined at x = 0

💼Applications

Physics
• Special relativity
• Wave equations
• Catenary curves
Engineering
• Cable design
• Heat transfer
• Electric transmission
Mathematics
• Complex analysis
• Differential equations
• Geometry

Hyperbolic Functions Calculator: Compute sinh, cosh, tanh, and More

Table of Contents - Hyperbolic Functions

How to Use This Calculator - Hyperbolic Functions

Enter a value for x to calculate hyperbolic functions. For example, enter 0, 1, -2, or 0.5.

Click "Calculate" to see sinh(x), cosh(x), tanh(x), coth(x), sech(x), and csch(x). The calculator also shows inverse hyperbolic functions when applicable.

The results display exact exponential forms, decimal approximations, and key identities verified for your input value.

Understanding Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but based on hyperbolas instead of circles. They appear throughout mathematics, physics, and engineering.

The basic definitions:

sinh(x) = (e^x - e^(-x)) / 2 (hyperbolic sine) cosh(x) = (e^x + e^(-x)) / 2 (hyperbolic cosine) tanh(x) = sinh(x) / cosh(x) (hyperbolic tangent)

The other three are reciprocals: csch(x) = 1/sinh(x) (hyperbolic cosecant) sech(x) = 1/cosh(x) (hyperbolic secant) coth(x) = 1/tanh(x) (hyperbolic cotangent)

Relationship to the hyperbola:

Just as cos²(t) + sin²(t) = 1 relates to the circle x² + y² = 1, the identity cosh²(x) - sinh²(x) = 1 relates to the hyperbola x² - y² = 1.

Why "hyperbolic":

These functions parameterize the unit hyperbola the same way trig functions parameterize the unit circle. The point (cosh(t), sinh(t)) lies on the hyperbola x² - y² = 1.

Connection to exponentials:

All hyperbolic functions can be expressed using e^x and e^(-x). This makes them natural in contexts involving exponential growth, decay, and differential equations.

Even and odd functions:

cosh(x) is even: cosh(-x) = cosh(x). Its graph is symmetric about the y-axis. sinh(x) is odd: sinh(-x) = -sinh(x). Its graph has rotational symmetry about the origin.

Behavior at zero:

sinh(0) = 0 cosh(0) = 1 tanh(0) = 0

These mirror the values sin(0) = 0, cos(0) = 1, tan(0) = 0.

Asymptotic behavior:

As x approaches infinity, both sinh(x) and cosh(x) approach e^x / 2. They grow exponentially. tanh(x) approaches 1 as x approaches infinity and -1 as x approaches negative infinity.

How to Calculate Hyperbolic Functions Manually

Let me show you how to compute these functions step by step.

Example 1: sinh(0)

Calculate sinh(0)

Step 1: Use the formula sinh(x) = (e^x - e^(-x)) / 2

Step 2: Substitute x = 0 sinh(0) = (e^0 - e^0) / 2

Step 3: Evaluate e^0 = 1 sinh(0) = (1 - 1) / 2 = 0 / 2 = 0

Example 2: cosh(0)

Calculate cosh(0)

Step 1: Formula cosh(x) = (e^x + e^(-x)) / 2

Step 2: Substitute cosh(0) = (e^0 + e^0) / 2

Step 3: Evaluate cosh(0) = (1 + 1) / 2 = 2 / 2 = 1

Example 3: sinh(1)

Calculate sinh(1) approximately

Step 1: Formula sinh(1) = (e^1 - e^(-1)) / 2

Step 2: Evaluate exponentials e ≈ 2.71828 e^(-1) = 1/e ≈ 0.36788

Step 3: Compute sinh(1) = (2.71828 - 0.36788) / 2 sinh(1) = 2.3504 / 2 sinh(1) ≈ 1.1752

Example 4: cosh(ln 2)

Calculate cosh(ln 2)

Step 1: Substitute into formula cosh(ln 2) = (e^(ln 2) + e^(-ln 2)) / 2

Step 2: Simplify exponentials e^(ln 2) = 2 e^(-ln 2) = e^(ln(1/2)) = 1/2

Step 3: Calculate cosh(ln 2) = (2 + 1/2) / 2 cosh(ln 2) = (5/2) / 2 cosh(ln 2) = 5/4 = 1.25

Example 5: tanh(0)

Calculate tanh(0)

Step 1: Use definition tanh(x) = sinh(x) / cosh(x)

Step 2: We know sinh(0) = 0 cosh(0) = 1

Step 3: Divide tanh(0) = 0 / 1 = 0

Example 6: Verify identity

Verify cosh²(1) - sinh²(1) = 1

Step 1: Calculate cosh(1) cosh(1) = (e + 1/e) / 2 ≈ (2.71828 + 0.36788) / 2 ≈ 1.5431

Step 2: Calculate sinh(1) sinh(1) ≈ 1.1752 (from Example 3)

Step 3: Compute cosh²(1) - sinh²(1) 1.5431² - 1.1752² ≈ 2.3811 - 1.3811 ≈ 1.0000 ✓

The identity holds!

Example 7: sech(0)

Calculate sech(0)

Step 1: Definition sech(x) = 1 / cosh(x)

Step 2: We know cosh(0) = 1

Step 3: Calculate sech(0) = 1 / 1 = 1

Example 8: Derivatives

Find d/dx [sinh(x)]

Step 1: Use definition sinh(x) = (e^x - e^(-x)) / 2

Step 2: Differentiate d/dx [sinh(x)] = (e^x - (-1)e^(-x)) / 2 = (e^x + e^(-x)) / 2

Step 3: Recognize This equals cosh(x)

So: d/dx [sinh(x)] = cosh(x)

Example 9: Addition formula

Calculate sinh(a + b) using sinh(a) and cosh(b)

Formula: sinh(a + b) = sinh(a)cosh(b) + cosh(a)sinh(b)

Example: sinh(2) using sinh(1) and cosh(1) sinh(2) = sinh(1 + 1) = sinh(1)cosh(1) + cosh(1)sinh(1) = 2·sinh(1)cosh(1) ≈ 2(1.1752)(1.5431) ≈ 3.6269

Verify: sinh(2) = (e² - e^(-2))/2 ≈ (7.389 - 0.135)/2 ≈ 3.627 ✓

Example 10: Inverse function

Find arsinh(1) (also written sinh⁻¹(1))

Step 1: Set up equation sinh(y) = 1

Step 2: Use formula arsinh(x) = ln(x + sqrt(x² + 1))

Step 3: Calculate arsinh(1) = ln(1 + sqrt(1 + 1)) = ln(1 + sqrt(2)) = ln(1 + 1.4142...) = ln(2.4142...) ≈ 0.8814

Real-World Applications

Catenary curves:

A hanging chain or cable forms a catenary, described by y = a·cosh(x/a). This appears in suspension bridges, power lines, and architecture.

Special relativity:

Hyperbolic functions describe rapidity (relativistic velocity). The Lorentz transformation uses cosh and sinh for spacetime rotations.

Heat transfer:

Temperature distribution in fins and extended surfaces follows hyperbolic equations. Engineers use these functions to model heat dissipation.

Electrical transmission lines:

Voltage and current along transmission lines are governed by hyperbolic functions, critical for power distribution analysis.

Magnetic fields:

Field distributions around certain conductor geometries involve hyperbolic functions in electromagnetic theory.

Probability and statistics:

The hyperbolic secant distribution and certain probability density functions use sech² terms.

Fluid dynamics:

Wave patterns and flow profiles in certain conditions are described using hyperbolic functions.

Structural engineering:

Arch shapes optimized for compressive stress often follow hyperbolic curves, making structures more stable.

Common Mistakes and How to Avoid Them

Mistake 1: Confusing with trig functions

Wrong: Thinking sinh²(x) + cosh²(x) = 1

Right: cosh²(x) - sinh²(x) = 1 (note the minus sign)

Why it happens: Mixing up the hyperbolic identity with the Pythagorean identity. Remember: hyperbola has minus.

Mistake 2: Sign errors

Wrong: sinh(-x) = sinh(x)

Right: sinh(-x) = -sinh(x) (odd function)

Why it happens: Not remembering which is even and which is odd. cosh is even, sinh is odd.

Mistake 3: Wrong exponential form

Wrong: sinh(x) = (e^x + e^(-x)) / 2

Right: sinh(x) = (e^x - e^(-x)) / 2 (minus, not plus)

Why it happens: Confusing sinh and cosh formulas. sinh has minus, cosh has plus.

Mistake 4: Division by zero

Wrong: Calculating csch(0) = 1/sinh(0) = 1/0

Right: csch(0) is undefined. sinh(0) = 0, so you can't divide.

Why it happens: Not checking if the denominator is zero. Always verify.

Mistake 5: Inverse function domain

Wrong: Trying to find arcosh(-1)

Right: arcosh(x) requires x is greater than or equal to 1. arcosh(-1) is undefined.

Why it happens: Not knowing domain restrictions. cosh(x) is greater than or equal to 1 for all real x, so arcosh only defined for x is greater than or equal to 1.

Mistake 6: Derivative confusion

Wrong: d/dx[cosh(x)] = sinh(x) with a negative

Right: d/dx[cosh(x)] = sinh(x) (positive), d/dx[sinh(x)] = cosh(x)

Why it happens: Incorrectly applying chain rule thinking. These derivatives are simpler than trig.

Mistake 7: Addition formula errors

Wrong: cosh(a + b) = cosh(a) + cosh(b)

Right: cosh(a + b) = cosh(a)cosh(b) + sinh(a)sinh(b)

Why it happens: Thinking it distributes like regular addition. Use the proper addition formulas.

Related Topics

How This Calculator Works

Step 1: Input validation

Accept real number input x
Check for special values (0, 1, -1)
Verify domain for inverse functions

Step 2: Calculate basic functions

Compute e^x and e^(-x)
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)

Step 3: Calculate reciprocals

If sinh(x) ≠ 0: csch(x) = 1 / sinh(x)
If cosh(x) ≠ 0: sech(x) = 1 / cosh(x) (always defined)
If tanh(x) ≠ 0: coth(x) = 1 / tanh(x)

Step 4: Inverse functions (if requested)

arsinh(x) = ln(x + sqrt(x² + 1))
arcosh(x) = ln(x + sqrt(x² - 1)) for x ≥ 1
artanh(x) = ln((1+x)/(1-x)) / 2 for |x| is less than 1

Step 5: Verify identities

Check cosh²(x) - sinh²(x) = 1
Check tanh²(x) + sech²(x) = 1
Check coth²(x) - csch²(x) = 1

Step 6: Format output

Display exact symbolic form
Show decimal approximation
Provide exponential representation
List applicable identities

Step 7: Additional information

Show derivatives at x
Display series expansion (if small x)
Graph function behavior

FAQs

What are hyperbolic functions?

Functions defined using exponentials: sinh(x) = (e^x - e^(-x))/2, cosh(x) = (e^x + e^(-x))/2, and their ratios and reciprocals.

How are they different from trig functions?

Trig functions are based on circles; hyperbolic functions on hyperbolas. The key identity has a minus: cosh² - sinh² = 1.

What does sinh stand for?

Hyperbolic sine, pronounced "sinch" or "shine" or "s-i-n-h."

Why use hyperbolic functions?

They naturally appear in solutions to differential equations, catenary curves, relativity, and many physics and engineering problems.

What is the catenary?

The curve formed by a hanging chain, given by y = a·cosh(x/a). It's the optimal shape for arches under compression.

Are sinh and cosh even or odd?

cosh is even: cosh(-x) = cosh(x). sinh is odd: sinh(-x) = -sinh(x).

What is the fundamental identity?

cosh²(x) - sinh²(x) = 1 for all x. This mirrors cos² + sin² = 1 but with a minus sign.

How do I calculate sinh(x) without a calculator?

Use sinh(x) = (e^x - e^(-x)) / 2. Calculate e^x and e^(-x), subtract, and divide by 2.

What is tanh(x) useful for?

Activation functions in neural networks, velocity in relativity, and modeling saturation effects. It maps all reals to (-1, 1).

What are the derivatives?

d/dx[sinh(x)] = cosh(x), d/dx[cosh(x)] = sinh(x), d/dx[tanh(x)] = sech²(x).

Do addition formulas exist?

Yes, similar to trig: sinh(a+b) = sinh(a)cosh(b) + cosh(a)sinh(b), and cosh(a+b) = cosh(a)cosh(b) + sinh(a)sinh(b).

What is arsinh?

The inverse of sinh, also written sinh⁻¹. arsinh(x) = ln(x + sqrt(x² + 1)).

Can hyperbolic functions be negative?

sinh(x) can be negative (when x is less than 0). cosh(x) is always at least 1. tanh(x) ranges from -1 to 1.

What is the range of cosh(x)?

[1, ∞). The minimum value is cosh(0) = 1.

How do you pronounce these?

sinh: "sinch" or "shine", cosh: "cosh" (rhymes with "gosh"), tanh: "tanch" or "than".

What is sech used for?

The hyperbolic secant appears in soliton solutions, pulse shapes in optics, and probability distributions.

Are there double-angle formulas?

Yes. sinh(2x) = 2sinh(x)cosh(x) and cosh(2x) = cosh²(x) + sinh²(x) = 2cosh²(x) - 1.

What's the relationship to complex numbers?

sinh(ix) = i·sin(x) and cosh(ix) = cos(x). Hyperbolic functions are trig functions with imaginary arguments.

Can I integrate these?

Yes. ∫sinh(x)dx = cosh(x) + C, ∫cosh(x)dx = sinh(x) + C.

Why does cosh(0) = 1?

Because cosh(0) = (e^0 + e^0)/2 = (1 + 1)/2 = 1.