Hyperbolic Functions & Graphs

What are the definitions of the hyperbolic functions?

Hyperbolic sine

$\sin h x = \frac{1}{2} (e^{x} - e^{- x})$

- This can be pronounced "shine" or "sinch"
Hyperbolic cosine

$\cosh x = \frac{1}{2} (e^{x} + e^{- x})$

- This can be pronounced "cosh"
Hyperbolic tangent

$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{2 x} - 1}{e^{2 x} + 1}$

- This can be pronounced "than" or "tanch"

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

$y = \sinh x$
- Domain: $x \in ℝ$
- Range: $y \in ℝ$
- Non-stationary point of inflection at (0, 0)
- Its shape is similar to the graph of $y = x^{3}$

edexcel-al-fm-cp-4-1-1-sinhx-graph

$y = \cosh x$
- Domain: $x \in ℝ$
- Range: $y \geq 1$
- Global minimum point at (0, 1)
- Its shape is similar to the graph of $y = x^{2}$

edexcel-al-fm-cp-4-1-1-coshx-graph

$y = \tanh x$
- Domain: $x \in ℝ$
- Range: $- 1 < y < 1$
- Non-stationary point of inflection at (0, 0)
- Asymptotes at y=1 and y=-1
- Its shape is similar to the graph of $y = \arctan x$

edexcel-al-fm-cp-4-1-1-tanhx-graph

What other features of the hyperbolic functions and graphs do I need to know?

The graphs of y=sinhx and y=tanhx have rotational symmetry about the origin
- This means that
  - $\sinh (- x) = - \sinh x$
  - $\tanh (- x) = - \tanh x$
- $\sinh x$ and are therefore odd functions
The graph of y=coshx is symmetrical in the y-axis
- This means that
  - $\cosh (- x) = \cosh x$
- $\cosh x$ is therefore an even function

What may I be asked to do with hyperbolic functions and their graphs?

Sketch graphs and transformations
- e.g. $y = \sinh (2 x) - 4$
  - Write as a transformation of $y = \sinh x$ and apply the transformations in the correct order
- Where possible label the key features of the transformed graph
  - Intersections with the coordinate axes
  - Equations of any asymptotes
  - Coordinates of any turning points
Find exact values
- e.g. Find the exact value of $\sinh (\ln (5))$
  - Use the definitions to write in terms of e
  - Use $e^{\ln k} = k$ and $e^{- \ln k} = \frac{1}{k}$

Examiner Tip

When using a calculator make sure you use sinh, cosh and tanh and NOT sin, cos and tan
Questions asking for values in exact form are often easier “to see” without a calculator, using the definitions of sinh and cosh, rather than trying to type in a complicated expression with e and ln

Worked example

Find the exact values of

(i)

2 \cosh (\ln (8)) - 3 \sinh (2 \ln (2))

(ii)

3 - \tanh (2 \ln (3)) + \tanh (- 2 \ln (3))

al-fm-4-1-1-we-solution-a

Sketch the graph of

y = 3 \cosh (\frac{1}{4} x) - 1

, labelling any points where the graph crosses the coordinate axes and any turning points.

al-fm-4-1-1-we-solution-b

Hyperbolic Functions & Graphs (Edexcel A Level Further Maths: Core Pure)

Revision Note

What are the definitions of the hyperbolic functions?

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

What other features of the hyperbolic functions and graphs do I need to know?

What may I be asked to do with hyperbolic functions and their graphs?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

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