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:

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

    • Domain: 
    • Range: 

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

    • Domain:
    • Range:

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

    • 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

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

 

 

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