Welcome to the World of Hyperbolic Functions!

Welcome to Unit FP3! In this chapter, we are going to explore Hyperbolic Functions. If you’ve already studied trigonometry (sin, cos, and tan), you’re in luck—hyperbolic functions are like their "cousins." While trigonometric functions are based on circles, hyperbolic functions are based on a shape called a hyperbola.

Why do we learn this? Well, if you’ve ever seen a power line hanging between two poles, that curve is actually a hyperbolic function (a catenary)! They are also vital in engineering, physics, and even special relativity. Don't worry if this seems a bit "extra" at first; we will break it down step-by-step.

1. Defining the Hyperbolic Functions

In Further Pure 3, we define these functions using exponentials (\(e^x\)). This is because hyperbolic functions describe growth and decay patterns.

The two "parents" of all hyperbolic functions are sinh (pronounced 'shine') and cosh (pronounced 'cosh'):

  • Hyperbolic Sine: \( \sinh x = \frac{e^x - e^{-x}}{2} \)
  • Hyperbolic Cosine: \( \cosh x = \frac{e^x + e^{-x}}{2} \)

Just like in normal trig, we can build the others from these two:

  • Hyperbolic Tangent: \( \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
  • Hyperbolic Secant: \( \text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}} \)
  • Hyperbolic Cosecant: \( \text{cosech } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}} \)
  • Hyperbolic Cotangent: \( \coth x = \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \)

Quick Review Box

Remember that \(e^0 = 1\). This means:
\( \cosh 0 = \frac{1+1}{2} = 1 \)
\( \sinh 0 = \frac{1-1}{2} = 0 \)

2. Graphs and Key Properties

Understanding the shapes of these graphs helps you visualize the math!

  • \( \cosh x \): Looks like a "U" shape (similar to a parabola but steeper). It never goes below 1. It is an even function, meaning \( \cosh(-x) = \cosh(x) \).
  • \( \sinh x \): Starts low, passes through the origin (0,0), and shoots up. It is an odd function, meaning \( \sinh(-x) = -\sinh(x) \).
  • \( \tanh x \): This graph is trapped between the horizontal asymptotes \( y = 1 \) and \( y = -1 \). It looks like a stretched "S".

Did you know? Unlike \( \sin x \) and \( \cos x \), hyperbolic functions are not periodic. They don't repeat themselves; they just keep growing!

3. Hyperbolic Identities and Osborne's Rule

Just as \( \cos^2 x + \sin^2 x = 1 \), hyperbolic functions have their own set of rules. However, there is a tiny twist with the plus and minus signs.

The Fundamental Identity

\( \mathbf{\cosh^2 x - \sinh^2 x = 1} \)

(Notice the minus sign! You can prove this by plugging in the exponential definitions we learned in Section 1.)

Memory Aid: Osborne's Rule

If you know your standard trigonometric identities (from P3), you can convert them to hyperbolic ones using Osborne's Rule:
Replace every \( \cos \) with \( \cosh \) and every \( \sin \) with \( \sinh \). BUT, whenever you see a product of two sines (like \( \sin^2 x \) or \( \sin A \sin B \)), flip the sign in front of it.

Example:
Trig: \( \cos 2x = 1 - 2\sin^2 x \)
Hyperbolic: \( \cosh 2x = 1 + 2\sinh^2 x \) (The minus becomes a plus because of the \( \sinh^2 \)).

Key Takeaway Summary

Hyperbolic functions are defined by \(e^x\). Use Osborne's rule to remember identities, and remember that \( \cosh^2 x - \sinh^2 x = 1 \).

4. Solving Equations

You will often be asked to solve equations like \( a \cosh x + b \sinh x = c \). There are two main ways to tackle these:

  1. Method A: The Exponential Substitution. Replace \( \sinh x \) and \( \cosh x \) with their \(e^x\) definitions. This usually turns the equation into a quadratic in terms of \(e^x\).
  2. Method B: Using Identities. Use identities like \( \cosh^2 x - \sinh^2 x = 1 \) to get everything in terms of one function (e.g., all in \( \sinh x \)), then solve it like a standard quadratic.

Common Mistake to Avoid: When you solve for \(e^x\), remember that \(e^x\) must always be positive. If you get a solution like \(e^x = -3\), you must discard it because it is impossible for a real \(x\).

5. Inverse Hyperbolic Functions

The inverse functions are written as arsinh, arcosh, and artanh. They tell us "what value of \(x\) gives us this hyperbolic value?"

Logarithmic Equivalents

Since the original functions are based on \(e^x\), their inverses are based on natural logs (\(\ln\)). You need to know these three formulas (and how to prove them):

  • \( \mathbf{\text{arsinh } x = \ln(x + \sqrt{x^2 + 1})} \) for all \(x\)
  • \( \mathbf{\text{arcosh } x = \ln(x + \sqrt{x^2 - 1})} \) for \(x \ge 1\)
  • \( \mathbf{\text{artanh } x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)} \) for \(|x| < 1\)

Step-by-Step: How to prove \( y = \text{arsinh } x \)

  1. Start with \( x = \sinh y \).
  2. Write it as \( x = \frac{e^y - e^{-y}}{2} \).
  3. Multiply by \( 2e^y \) to get a quadratic: \( 2xe^y = (e^y)^2 - 1 \).
  4. Rearrange: \( (e^y)^2 - 2xe^y - 1 = 0 \).
  5. Use the quadratic formula to solve for \(e^y\).
  6. Take the \(\ln\) of both sides (and keep the positive root).

6. Calculus of Hyperbolic Functions

This is where things get really satisfying! The derivatives are very similar to trig, but without the annoying minus signs for \( \cosh \).

Differentiation

  • \( \frac{d}{dx}(\sinh x) = \cosh x \)
  • \( \frac{d}{dx}(\cosh x) = \sinh x \) (No minus sign here!)
  • \( \frac{d}{dx}(\tanh x) = \text{sech}^2 x \)

Integration

Integration is just the reverse process. You will also use Hyperbolic Substitution to solve tricky integrals involving square roots like \( \sqrt{x^2 + a^2} \).
Substitution Tip: If you see \( \sqrt{x^2 + a^2} \), try letting \( x = a \sinh u \). Because \( \cosh^2 u - \sinh^2 u = 1 \), the square root will simplify beautifully into \( a \cosh u \)!

Quick Review Box

Trig Derivatives: \( \sin \to \cos \), \( \cos \to -\sin \)
Hyperbolic Derivatives: \( \sinh \to \cosh \), \( \cosh \to \sinh \)
They are much friendlier!

Final Chapter Summary

Hyperbolic functions allow us to bridge the gap between exponentials and trigonometry. To succeed in this chapter:

  • Master the exponential definitions of \( \sinh \) and \( \cosh \).
  • Use Osborne’s Rule to adapt the trig identities you already know.
  • Be comfortable switching between inverse hyperbolic functions and their logarithmic forms.
  • When solving equations, always check if your value for \(e^x\) is valid (positive).

Keep practicing these substitutions and identities—they are the tools that will make the rest of FP3 much easier!