Welcome to the World of Hyperbolic Functions!
In your standard A Level Maths, you spent a lot of time with Trigonometric Functions (sin, cos, and tan), which are based on a circle. In Further Mathematics, we introduce their "cousins": Hyperbolic Functions. As the name suggests, these functions are related to the hyperbola.
Don’t worry if this seems like a lot of new terminology! You’ll soon see that many of the rules you already know for trigonometry have very similar versions here. We use these functions in real-world engineering, such as calculating how a power cable hangs between two pylons or describing the shape of a cooling tower.
1. The Basic Definitions
Unlike trig functions, which are often defined by triangles, hyperbolic functions are defined using the exponential function \( e^x \). There are three primary functions you need to know:
Hyperbolic Sine: \( \sinh x = \frac{e^x - e^{-x}}{2} \)
Hyperbolic Cosine: \( \cosh x = \frac{e^x + e^{-x}}{2} \)
Hyperbolic Tangent: \( \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
Reciprocal Hyperbolic Functions
Just like \( \sec x \), \( \text{cosec } x \), and \( \cot x \), we have reciprocal versions for hyperbolics:
- Hyperbolic Secant: \( \text{sech } x = \frac{1}{\cosh x} \)
- Hyperbolic Cosecant: \( \text{cosech } x = \frac{1}{\sinh x} \)
- Hyperbolic Cotangent: \( \text{coth } x = \frac{1}{\tanh x} \)
Quick Review: Hyperbolic functions are just combinations of \( e^x \) and \( e^{-x} \). If you ever forget an identity, you can always go back to these exponential definitions to prove it!
2. Graphs, Domains, and Ranges
Understanding the "shape" of these functions helps you visualize the math.
The Graph of \( \cosh x \)
The \( \cosh x \) graph looks like a "cup" shape or a "hanging chain." In physics, this shape is called a catenary.
- Domain: All real values of \( x \) (\( -\infty < x < \infty \)).
- Range: \( \cosh x \ge 1 \). (It never goes below 1!)
- Symmetry: It is an even function, meaning it is symmetrical about the y-axis (\( \cosh(-x) = \cosh x \)).
The Graph of \( \sinh x \)
The \( \sinh x \) graph is an "S" shape that passes through the origin.
- Domain: All real values of \( x \).
- Range: All real values of \( y \).
- Symmetry: It is an odd function (\( \sinh(-x) = -\sinh x \)).
The Graph of \( \tanh x \)
The \( \tanh x \) graph looks like a horizontal "S" trapped between two invisible lines.
- Domain: All real values of \( x \).
- Range: \( -1 < \tanh x < 1 \).
- Asymptotes: It has horizontal asymptotes at \( y = 1 \) and \( y = -1 \).
Did you know? If you hang a heavy necklace between your hands, the curve it forms is exactly the graph of \( y = \cosh x \)!
3. Hyperbolic Identities
You already know \( \cos^2 x + \sin^2 x \equiv 1 \). Hyperbolic functions have similar identities, but there is often a sign change involved.
The Fundamental Identity
\( \cosh^2 x - \sinh^2 x \equiv 1 \)
Other Key Identities
- \( \text{sech}^2 x \equiv 1 - \tanh^2 x \)
- \( \text{cosech}^2 x \equiv \text{coth}^2 x - 1 \)
- Double Angle: \( \sinh 2x \equiv 2\sinh x \cosh x \)
- Double Angle: \( \cosh 2x \equiv \cosh^2 x + \sinh^2 x \)
Memory Aid: Osborne’s Rule
To turn a trigonometric identity into a hyperbolic one, replace \( \cos \) with \( \cosh \) and \( \sin \) with \( \sinh \). BUT, if the identity involves a product of two sines (like \( \sin^2 x \) or \( \tan^2 x \) which is \( \sin^2 / \cos^2 \)), you must change the sign in front of that term.
Key Takeaway: If there is a \( \sinh^2 x \) or a \( \tanh^2 x \), the plus or minus sign in front of it usually flips compared to the trig version.
4. Differentiation and Integration
One of the best things about hyperbolic functions is that their derivatives are very "clean."
Differentiation Rules
- \( \frac{d}{dx}(\sinh x) = \cosh x \)
- \( \frac{d}{dx}(\cosh x) = \sinh x \) (Note: No minus sign here!)
- \( \frac{d}{dx}(\tanh x) = \text{sech}^2 x \)
Integration Rules
- \( \int \cosh x \, dx = \sinh x + C \)
- \( \int \sinh x \, dx = \cosh x + C \)
- \( \int \text{sech}^2 x \, dx = \tanh x + C \)
Common Mistake: Students often put a minus sign when differentiating \( \cosh x \) because they are used to trig (\( \frac{d}{dx} \cos x = -\sin x \)). In hyperbolics, both \( \sinh \) and \( \cosh \) differentiate to the positive version of each other!
5. Inverse Hyperbolic Functions
If we want to "undo" a hyperbolic function, we use \( \text{arsinh } x \), \( \text{arcosh } x \), and \( \text{artanh } x \).
Logarithmic Forms
Because the original functions are based on \( e^x \), the inverses are based on natural logs (\( \ln \)). You are required to know (and derive) these:
- \( \text{arsinh } x = \ln(x + \sqrt{x^2 + 1}) \) for all \( x \)
- \( \text{arcosh } x = \ln(x + \sqrt{x^2 - 1}) \) for \( x \ge 1 \)
- \( \text{artanh } x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right) \) for \( |x| < 1 \)
Domains and Ranges of Inverses
Remember, the domain of the inverse is the range of the original function!
- \( \text{arcosh } x \): Since \( \cosh x \) never goes below 1, \( \text{arcosh } x \) only exists for \( x \ge 1 \).
- \( \text{artanh } x \): Since \( \tanh x \) stays between -1 and 1, \( \text{artanh } x \) only exists for values between -1 and 1.
6. Integration using Substitutions
Hyperbolic functions are incredibly useful for solving integrals involving square roots of quadratics.
Case 1: Integrals involving \( \sqrt{x^2 + a^2} \)
Use the substitution \( x = a\sinh u \).
Why? Because \( a^2\sinh^2 u + a^2 = a^2(\sinh^2 u + 1) = a^2\cosh^2 u \). The square root disappears!
Case 2: Integrals involving \( \sqrt{x^2 - a^2} \)
Use the substitution \( x = a\cosh u \).
Why? Because \( a^2\cosh^2 u - a^2 = a^2(\cosh^2 u - 1) = a^2\sinh^2 u \). Again, the square root disappears!
Quick Review Box:
- \( \sqrt{x^2 + a^2} \rightarrow \) Use \( \sinh \)
- \( \sqrt{x^2 - a^2} \rightarrow \) Use \( \cosh \)
- \( \sqrt{a^2 - x^2} \rightarrow \) Use \( \sin \) (from standard A Level!)
7. Constructing Proofs
You may be asked to prove a hyperbolic identity. There are two main ways to do this:
- Method A (Exponential Definitions): Replace \( \sinh x \) and \( \cosh x \) with their \( e^x \) formulas and expand the algebra. This always works but can be messy.
- Method B (Identity Manipulation): Start with \( \cosh^2 x - \sinh^2 x \equiv 1 \) and divide by \( \cosh^2 x \) or \( \sinh^2 x \) to find the other identities, just like you do in trigonometry.
Step-by-Step Tip: When proving logarithmic forms of inverse functions, let \( y = \text{arsinh } x \), so \( x = \sinh y \). Write \( x = \frac{e^y - e^{-y}}{2} \), multiply through by \( 2e^y \) to create a quadratic equation in terms of \( e^y \), and solve using the quadratic formula!
Summary Checklist
Before the exam, make sure you can:
- State the exponential definitions for \( \sinh, \cosh, \) and \( \tanh \).
- Sketch the graphs and state their domains/ranges.
- Use Osborne's Rule to write down identities.
- Differentiate and integrate basic hyperbolic functions.
- Recall the logarithmic forms of the inverse functions.
- Choose the correct hyperbolic substitution for integration problems.
Final Encouragement: Hyperbolic functions might look intimidating because of the "h" in the names, but they behave very logically. Master the exponential definitions and Osborne's Rule, and the rest will fall into place!