Hyperbolic Functions

🌊 The Secret World of Hyperbolic Functions

A Tale of Two Curves

Imagine you’re at a playground with two types of swings. One swing goes round and round in circles (that’s our old friend, the sine wave). But there’s another magical swing that stretches out like a rope hanging between two poles. This stretched rope shape? That’s the hyperbolic world!

🎭 Meet the Hyperbolic Family

The Three Main Characters

Think of a catenary — the curve a chain makes when you hold both ends and let it hang. This beautiful natural curve is made of hyperbolic functions!

graph TD
    A["Hyperbolic Functions"] --> B["sinh - The Stretcher"]
    A --> C["cosh - The Hanger"]
    A --> D["tanh - The Squisher"]
    B --> E["Grows forever, both ways"]
    C --> F["Always stays above 1"]
    D --> G["Stays between -1 and 1"]

📚 Chapter 1: Sinh, Cosh, and Tanh Definitions

What Are They, Really?

Remember how regular sine uses circles? Hyperbolic functions use a hyperbola instead — like the shape of two boomerangs facing away from each other.

The Magic Formulas

Sinh (pronounced “sinch”) — The Hyperbolic Sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$

Cosh (pronounced “cosh”) — The Hyperbolic Cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$

Tanh (pronounced “tanch”) — The Hyperbolic Tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

🎯 Simple Example

Let’s try x = 0:

🧠 Why “Hyperbolic”?

Picture this: On a unit circle, if you walk an arc of length t, you reach the point (cos t, sin t).

On a unit hyperbola (x² - y² = 1), if you measure area t, you reach (cosh t, sinh t)!

It’s like the hyperbola is the circle’s quirky cousin who does everything a bit differently.

📚 Chapter 2: The Hyperbolic Pythagorean Identity

The Famous Rule

For circles, we know: sin²x + cos²x = 1

For hyperbolas, there’s a twist: cosh²x - sinh²x = 1

Notice the minus sign? That’s the hyperbolic signature!

🎯 Let’s Prove It!

Starting with our definitions: $\cosh^2(x) - \sinh^2(x)$

$= \left(\frac{e^x + e^{-x}}{2}\right)^2 - \left(\frac{e^x - e^{-x}}{2}\right)^2$

$= \frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4}$

$= \frac{4}{4} = 1$ ✅

🎪 The Visual Story

Imagine cosh as a “stretchy ceiling” that’s always at least 1, and sinh as a “stretchy floor” that passes through zero. The difference between their squares? Always exactly 1!

graph TD
    A["Pythagorean Identities"] --> B["Circle: sin²x + cos²x = 1"]
    A --> C["Hyperbola: cosh²x - sinh²x = 1"]
    B --> D["Plus sign"]
    C --> E["Minus sign"]

📚 Chapter 3: Other Hyperbolic Identities

The Supporting Cast

Just like regular trig has sec, csc, and cot, we have:

🎯 More Pythagorean Friends

From the main identity, we get:

1 - tanh²x = sech²x

coth²x - 1 = csch²x

Addition Formulas (The Party Rules)

When hyperbolic functions meet:

$\sinh(a + b) = \sinh(a)\cosh(b) + \cosh(a)\sinh(b)$

$\cosh(a + b) = \cosh(a)\cosh(b) + \sinh(a)\sinh(b)$

Notice: Both terms are added for cosh (unlike regular cos which subtracts!)

🎯 Quick Example

Find sinh(2x) using sinh(x + x):

$\sinh(2x) = 2\sinh(x)\cosh(x)$

This is called the double angle formula!

Even and Odd Behavior

• cosh(-x) = cosh(x) → cosh is even (symmetric like a smile)
• sinh(-x) = -sinh(x) → sinh is odd (symmetric through origin)
• tanh(-x) = -tanh(x) → tanh is odd

📚 Chapter 4: Inverse Hyperbolic Functions

Going Backwards

What if we know sinh(x) = 2 and want to find x? We need the inverse!

The Inverse Family

🎯 The Logarithm Connection

Here’s the magic — inverses can be written as logarithms!

$\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$

$\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1}) \quad \text{for } x \geq 1$

$\tanh^{-1}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \quad \text{for } |x| < 1$

Example: Find sinh⁻¹(0)

Using the formula: $\sinh^{-1}(0) = \ln(0 + \sqrt{0 + 1}) = \ln(1) = 0$ ✅

Makes sense! sinh(0) = 0, so sinh⁻¹(0) = 0.

Domain Restrictions

graph TD
    A["Inverse Domains"] --> B["sinh⁻¹: All real numbers"]
    A --> C["cosh⁻¹: x ≥ 1 only"]
    A --> D["tanh⁻¹: -1 < x < 1 only"]
    C --> E["Because cosh ≥ 1 always"]
    D --> F["Because -1 < tanh < 1 always"]

📚 Chapter 5: Relation to Circular Functions

The Imaginary Bridge

Here’s where it gets beautiful! Hyperbolic and circular (regular) trig are connected through imaginary numbers!

The Connection Formulas

$\sinh(x) = -i \sin(ix)$ $\cosh(x) = \cos(ix)$ $\tanh(x) = -i \tan(ix)$

And going the other way:

$\sin(x) = -i \sinh(ix)$ $\cos(x) = \cosh(ix)$

🧠 What Does This Mean?

Imagine a mirror between the real and imaginary world:

• Regular trig lives on the circle (x² + y² = 1)
• Hyperbolic trig lives on the hyperbola (x² - y² = 1)
• The imaginary unit i is the bridge between them!

🎯 Example: Verify cos(ix) = cosh(x)

Using Euler’s formula: $\cos(ix) = \frac{e^{i(ix)} + e^{-i(ix)}}{2} = \frac{e^{-x} + e^{x}}{2} = \cosh(x)$ ✅

Osborn’s Rule (The Easy Swap)

To convert any circular trig identity to hyperbolic:

1. Replace sin → sinh, cos → cosh, tan → tanh
2. Whenever you see sin² (or product of two sines), flip the sign!

Example:

• Circular: cos²x + sin²x = 1
• Hyperbolic: cosh²x + (−1)sinh²x = 1 → cosh²x − sinh²x = 1 ✅

🌟 Real-World Magic

Where Do We See Hyperbolic Functions?

1. Hanging Cables: Power lines hang in a cosh curve (catenary)
2. Architecture: The Gateway Arch in St. Louis is an inverted catenary
3. Special Relativity: Rapidity in physics uses tanh
4. Engineering: Cable bridges and suspension systems

graph TD
    A["Hyperbolic Functions in Life"] --> B["🌉 Suspension Bridges"]
    A --> C["⚡ Power Lines"]
    A --> D["🚀 Relativity Physics"]
    A --> E["🏛️ Architecture"]

🎯 Quick Reference Summary

💪 You’ve Got This!

Hyperbolic functions might seem strange at first, but they’re just the hyperbola’s answer to circles. Every rule you know from regular trig has a hyperbolic cousin — sometimes with a sign flip, sometimes exactly the same.

Remember:

• sinh = the odd one, goes through zero, stretches forever
• cosh = the even one, always ≥ 1, makes that beautiful hanging rope shape
• tanh = the squished one, trapped between −1 and 1

Now go explore these curves and watch math come alive! 🚀