# Edexcel A Level Further Maths: Core Pure:复习笔记4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

### Logarithmic Forms of Inverse Hyperbolic Functions

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

• 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
• 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
• Domain:
• Range:

• Domain:
• Range:

• Domain:
• Range:

#### 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

• and rearrange to
• STEP 2
Form a quadratic in terms of ey

• Multiply by 2ey and rearrange
• STEP 3
Solve the quadratic and find an expression for y

• Reject  as this produces negative values as  whereas
• The derivations of the other two formulae are similar
• For both  and  are well-defined for
• We choose  as  and it can be shown that

#### Exam 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 , show that