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Edexcel A Level Further Maths: Core Pure:復(fù)習(xí)筆記4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

Category: A-level課程, 國際課程, 教材筆記, 福利干貨 Date: 2022年10月19日上午11:55

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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