Introduction to Hyperbolic Functions
Welcome to the world of Hyperbolic Functions! If you have ever looked at a power line hanging between two poles or a heavy gold chain dangling around someone’s neck, you have already seen a hyperbolic function in action. The shape that a hanging chain naturally forms is called a catenary, and it is described by the cosh function.
In this chapter, we will explore functions that look and behave very much like the trigonometry (sin, cos, and tan) you already know, but instead of being based on a circle, they are based on a hyperbola. Don't worry if this seems tricky at first—once you see the patterns, you will find they are often easier to work with than standard trig!
1. Defining the Hyperbolic Functions
The three main hyperbolic functions are sinh (pronounced "shine"), cosh (pronounced "cosh"), and tanh (pronounced "than"). Unlike standard trig, these are defined using the exponential function \( e^x \).
The Definitions:
• Hyperbolic Sine: \( \sinh x = \frac{e^x - e^{-x}}{2} \)
• Hyperbolic Cosine: \( \cosh x = \frac{e^x + e^{-x}}{2} \)
• Hyperbolic Tangent: \( \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \)
Visualising the Graphs
• Cosh x: This looks like a "U" shape (similar to a parabola but steeper). It starts at \( (0, 1) \) and goes up to infinity in both directions. Its range is \( \cosh x \ge 1 \).
• Sinh x: This looks like a "cubic" curve. It passes through the origin \( (0, 0) \). Its range is all real numbers.
• Tanh x: This looks like a "stretched S". It is trapped between two horizontal asymptotes at \( y = 1 \) and \( y = -1 \). Its range is \( -1 < \tanh x < 1 \).
Did you know?
The Gateway Arch in St. Louis, USA, is designed using an inverted cosh curve! It is one of the strongest shapes in engineering because it distributes weight perfectly.
Quick Summary: Hyperbolic functions are just combinations of \( e^x \) and \( e^{-x} \). If you can do algebra with exponentials, you can do hyperbolic functions!
2. The Fundamental Identity
In standard trig, you know that \( \cos^2 x + \sin^2 x = 1 \). Hyperbolic functions have a very similar rule, but with a tiny "twist" in the sign.
The Key Identity:
\( \cosh^2 x - \sinh^2 x = 1 \)
Common Mistake to Avoid:
Many students accidentally write a plus sign because they are so used to standard trigonometry. Memory Aid: Think of the "minus" in the equation of a hyperbola \( x^2 - y^2 = 1 \). That is why the hyperbolic identity uses a minus sign!
Key Takeaway: Always remember the minus! \( \cosh^2 x - \sinh^2 x = 1 \).
3. Calculus: Differentiation and Integration
One of the best things about hyperbolic functions is how they behave in calculus. They are much "friendlier" than trig functions because we don't have to worry about as many minus signs.
Differentiation
• \( \frac{d}{dx}(\sinh x) = \cosh x \)
• \( \frac{d}{dx}(\cosh x) = \sinh x \)
• \( \frac{d}{dx}(\tanh x) = \text{sech}^2 x \) (Note: while \( \text{sech } x \) isn't a primary focus for sketching, you should recognise it as \( \frac{1}{\cosh x} \)).
Comparison Trick:
In trig, \( \frac{d}{dx}(\cos x) = -\sin x \).
In hyperbolics, everything stays positive for sinh and cosh! There is no minus sign when you differentiate \( \cosh x \).
Integration
Since integration is just differentiation in reverse:
• \( \int \cosh x \, dx = \sinh x + c \)
• \( \int \sinh x \, dx = \cosh x + c \)
Quick Review: To differentiate or integrate sinh and cosh, you just swap them. It’s that simple!
4. Inverse Hyperbolic Functions
Just like we have \( \sin^{-1} x \), we have inverse hyperbolic functions: arsinh, arcosh, and artanh. Because hyperbolic functions are built from exponentials, their inverses can be written as logarithms.
Logarithmic Forms
You need to be able to use (and sometimes derive) these formulas:
• \( \text{arsinh } x = \ln(x + \sqrt{x^2 + 1}) \) for all \( x \)
• \( \text{arcosh } x = \ln(x + \sqrt{x^2 - 1}) \) for \( x \ge 1 \)
• \( \text{artanh } x = \frac{1}{2} \ln(\frac{1+x}{1-x}) \) for \( |x| < 1 \)
Understanding the Domains
• arcosh x: Since \( \cosh x \) never goes below 1, you can only put numbers \( \ge 1 \) into \( \text{arcosh } x \).
• artanh x: Since \( \tanh x \) stays between -1 and 1, you can only put numbers between -1 and 1 into \( \text{artanh } x \).
Step-by-Step: Deriving arsinh x
1. Start with \( y = \text{arsinh } x \), so \( x = \sinh y \).
2. Use the definition: \( x = \frac{e^y - e^{-y}}{2} \).
3. Multiply by 2: \( 2x = e^y - e^{-y} \).
4. Multiply everything by \( e^y \): \( 2xe^y = (e^y)^2 - 1 \).
5. This is a quadratic in disguise: \( (e^y)^2 - 2x(e^y) - 1 = 0 \).
6. Use the quadratic formula to solve for \( e^y \), then take the natural log!
5. Using Inverses in Integration
In your exam, you will often meet integrals that don't look like they have anything to do with hyperbolics, but the answer involves an inverse function. You should recognise these "standard forms":
Standard Integrals:
• \( \int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \text{arsinh}(\frac{x}{a}) + c \)
• \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \text{arcosh}(\frac{x}{a}) + c \)
Analogy:
Think of these as "patterns". When you see a fraction with a square root on the bottom, check if it matches the "plus" pattern (arsinh) or the "minus" pattern (arcosh).
Key Takeaway: If the denominator has \( \sqrt{x^2 + a^2} \), think arsinh. If it has \( \sqrt{x^2 - a^2} \), think arcosh.
Chapter Summary Checklist
Can you:
• State the definitions of sinh, cosh, and tanh using \( e^x \)?
• Sketch the graphs and state their domains/ranges?
• Use the identity \( \cosh^2 x - \sinh^2 x = 1 \)?
• Differentiate and integrate hyperbolic functions correctly?
• Use the logarithmic forms for inverse functions?
• Recognise the standard integrals that lead to inverse hyperbolic functions?