# Edexcel A Level Further Maths: Core Pure:复习笔记4.1.1 Hyperbolic Functions & Graphs

### Hyperbolic Functions & Graphs

#### What are the definitions of the hyperbolic functions?

• Hyperbolic sine

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

• This can be pronounced "cosh"
• Hyperbolic tangent

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

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

• Domain:
• Range:
• Non-stationary point of inflection at (0, 0)
• Its shape is similar to the graph of

• Domain:
• Range:
• Global minimum point at (0, 1)
• Its shape is similar to the graph of

• Domain:
• Range:
• Non-stationary point of inflection at (0, 0)
• Asymptotes at y=1 and y=-1
• Its shape is similar to the graph of

#### 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
• and  are therefore odd functions
• The graph of y=coshx is symmetrical in the y-axis
• This means that
• is therefore an even function

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

• Sketch graphs and transformations
• e.g.
• Write as a transformation of 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
• Use the definitions to write in terms of e
• Use

#### Exam 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

a) Find the exact values of
(i)
(ii)

b) Sketch the graph of , labelling any points where the graph crosses the coordinate axes and any turning points.