Logarithmic Forms of Inverse Hyperbolic Functions

What are the definitions of the inverse hyperbolic functions?

$arsinh x = \ln (x + \sqrt{x^{2} + 1})$ , $x \in ℝ$
$ar \cosh x = \ln (x + \sqrt{x^{2} - 1})$ , $x \geq 1$
- Since coshx is a many-to-one function, its domain is restricted to x ≥ 0 when finding the inverse
- Therefore, its range is coshx ≥ 1
- So, the domain of the inverse function is x ≥ 1
$artanh x = \frac{1}{2} \ln (\frac{1 + x}{1 - x})$ , $| x | < 1$
These three definitions are in the formula booklet

What are the graphs of the inverse hyperbolic functions and their key features?

As they are inverse functions, they are reflections of their original functions in the line y=x
$y = arsinh x$
- Domain: $x \in ℝ$
- Range: $y \in ℝ$

edexcel-al-fm-cp-4-1-2-arsinhx-graph

$y = ar \cosh x$
- Domain: $x \geq 1$
- Range: $y \geq 0$

edexcel-al-fm-cp-4-1-2-arcoshx-graph

$y = ar \tanh x$
- Domain: $- 1 < x < 1$
- Range: $y \in ℝ$

edexcel-al-fm-cp-4-1-2-artanhx-graph

How do I derive the logarithmic formulae for the inverse hyperbolic functions?

You need to be able to derive each inverse from the definition of the original
STEP 1
Write in terms of e
- $y = arsinh x$ and rearrange to $x = \sinh y$
- $x = \frac{1}{2} (e^{y} - e^{- y})$
STEP 2
Form a quadratic in terms of e^y
- Multiply by 2e^y and rearrange
- ${(e^{y})}^{2} - 2 x e^{y} - 1 = 0$
STEP 3
Solve the quadratic and find an expression for y
- $e^{y} = x \pm \sqrt{x^{2} + 1}$
  - Reject $x - \sqrt{x^{2} + 1}$ as this produces negative values as $\sqrt{x^{2} + 1} > x$ whereas $e^{y} > 0$
- $y = \ln (x + \sqrt{x^{2} + 1})$
The derivations of the other two formulae are similar
- For $arcosh x$ both $\ln (x + \sqrt{x^{2} - 1})$ and $\ln (x - \sqrt{x^{2} - 1})$ are well-defined for $x \geq 1$
  - We choose $arcosh x = \ln (x + \sqrt{x^{2} - 1})$ as $arcosh x \geq 0$ and it can be shown that $\ln (x - \sqrt{x^{2} - 1}) = \ln (\frac{1}{x + \sqrt{x^{2} - 1}}) = - \ln (x + \sqrt{x^{2} - 1})$

Examiner Tip

Be careful when working with the circular (“normal”) inverse trig functions and the inverse hyperbolic functions

Only the “ar” denotes inverse

The “c” in arcsin x, arccos x, arctan x indicates the circular functions
The hyperbolic functions have “h”, but the “h” doesn’t come immediately after the “ar”: arsinh x, arcosh x, artanh x

Be careful not to confuse these, especially when looking them up in the formulae booklet

Worked example

Starting from the definition of $\tanh x$ , show that

$artanh x = \frac{1}{2} \ln (\frac{1 + x}{1 - x})$ when $| x | < 1$ .

al-fm-4-1-2-we-solution

Logarithmic Forms of Inverse Hyperbolic Functions (Edexcel A Level Further Maths: Core Pure)

Revision Note

What are the definitions of the inverse hyperbolic functions?

How do I derive the logarithmic formulae for the inverse hyperbolic functions?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations

8.3 Simple Harmonic Motion

8.3.1 Simple Harmonic Motion

8.3.2 Damped or Forced Harmonic Motion