Hyperbolic functions

Introduction: Welcome to the World of Hyperbolic Functions!

Welcome! If you’ve already studied trigonometry, you know all about sine, cosine, and tangent. These are "circular" functions because they relate to the properties of a circle. In this chapter, we are going to meet their "cousins": the Hyperbolic Functions.

Instead of being based on a circle, these functions are based on a curve called a hyperbola (which you saw in your Coordinate Geometry studies). While they might look a bit intimidating at first with their names like sinh and cosh, don't worry! They behave very similarly to the trig functions you already know. Let's dive in and see how they work.

1. What are Hyperbolic Functions?

The three primary hyperbolic functions are sinh (pronounced "shine"), cosh (pronounced "kosh"), and tanh (pronounced "tanny" or "thank").

Unlike regular trig functions, hyperbolic functions are actually defined using the exponential function \( e^x \). This makes them very easy to calculate if you have a scientific calculator!

The Definitions:

1. Hyperbolic Sine: \(\sinh x = \frac{e^x - e^{-x}}{2}\)

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

3. Hyperbolic Tangent: \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\)

Real-World Analogy: Have you ever noticed the shape of a power line hanging between two poles? That curve isn't a parabola; it's actually a cosh curve! This shape is called a catenary.

Quick Review: Remember that \( e^0 = 1 \). If you plug \( x = 0 \) into the formulas above, you'll find that \(\sinh(0) = 0\) and \(\cosh(0) = 1\), just like \(\sin(0)\) and \(\cos(0)\)!

2. Graphs of Hyperbolic Functions

Visualizing these functions helps you understand how they behave as \( x \) gets very large or very small.

The Shape of the Curves:

• \(y = \sinh x\): This looks like a cubic curve. It starts very low, passes through the origin \((0,0)\), and goes up forever. It is an odd function (symmetrical about the origin).
• \(y = \cosh x\): This looks like a "V" or a "U" shape, similar to a parabola, but steeper. Its lowest point is at \((0,1)\). It is an even function (symmetrical about the y-axis).
• \(y = \tanh x\): This curve is "trapped" between two horizontal asymptotes at \( y = 1 \) and \( y = -1 \). It passes through the origin.

Did you know? Because \(\cosh x\) is the average of \( e^x \) and \( e^{-x} \), as \( x \) gets very large, \(\cosh x\) starts to look almost exactly like \(\frac{1}{2}e^x\). This is because \( e^{-x} \) becomes so small it's basically zero!

Key Takeaway: \(\cosh x\) is always greater than or equal to 1. \(\sinh x\) can be any value. \(\tanh x\) is always between -1 and 1.

3. Hyperbolic Identities

Just like trig functions have identities (like \(\sin^2 x + \cos^2 x = 1\)), hyperbolic functions have their own set. They look almost identical, but watch out for the minus signs!

Fundamental Identity:

\(\cosh^2 x - \sinh^2 x = 1\)

(Compare this to the trig version where it is a plus sign!)

Osborne’s Rule (A Handy Trick):

Don't worry about memorizing every single hyperbolic identity. You can convert any trigonometric identity into a hyperbolic identity using Osborne’s Rule:

1. Replace \(\sin\) with \(\sinh\) and \(\cos\) with \(\cosh\).

2. The Magic Step: Change the sign of any term that involves a product of two sines (this includes \(\sin^2\), \(\tan^2\), \(\cot^2\), and \(\csc^2\)).

Example:
Trig: \(\cos(2x) = 1 - 2\sin^2 x\)
Hyperbolic: \(\cosh(2x) = 1 + 2\sinh^2 x\) (We changed the minus to a plus because of the \(\sinh^2\) term!)

4. Inverse Hyperbolic Functions

Sometimes we know the value of the function and want to find \( x \). We use the inverse functions: \(\text{arsinh } x\), \(\text{arcosh } x\), and \(\text{artanh } x\).

Logarithmic Forms:

Because the original functions are based on \( e^x \), the inverses are based on natural logarithms (\(\ln\)). These are very important for solving equations!

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

Common Mistake: When using \(\text{arcosh } x\), remember that it only exists for \( x \geq 1 \). If you try to calculate \(\text{arcosh}(0.5)\) on your calculator, you will get an error!

5. Differentiation and Integration

Calculus with hyperbolic functions is actually easier than with trig functions because there are fewer negative signs to worry about!

Differentiation Table:

1. \(\frac{d}{dx}(\sinh x) = \cosh x\)

2. \(\frac{d}{dx}(\cosh x) = \sinh x\)

3. \(\frac{d}{dx}(\tanh x) = \text{sech}^2 x\)

Wait! Notice that \(\frac{d}{dx}(\cosh x)\) is positive \(\sinh x\). In trigonometry, the derivative of \(\cos x\) is negative \(\sin x\). This is a common place where students lose marks, so be careful!

Integration Table:

Since integration is just the reverse of differentiation:

• \(\int \cosh x \, dx = \sinh x + C\)
• \(\int \sinh x \, dx = \cosh x + C\)

Key Takeaway: In hyperbolic calculus, \(\sinh\) and \(\cosh\) just swap back and forth without changing signs! It’s like a friendly game of catch.

Summary Checklist

Before you tackle some practice questions, check if you can do the following:

- State the exponential definitions of \(\sinh x\) and \(\cosh x\).
- Sketch the graphs of \(\sinh\), \(\cosh\), and \(\tanh\).
- Use Osborne’s Rule to write down identities.
- Use the Logarithmic Forms of the inverse functions to solve for \( x \).
- Differentiate and integrate basic hyperbolic functions without mixing up the signs.

Don't worry if this seems tricky at first! Hyperbolic functions are just another tool in your mathematical toolkit. Once you get used to the exponential definitions, you'll see they are just as logical as everything else in Further Maths.