Edexcel A Level Further Maths: Core Pure:復習筆記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:
- Domain:?
- Range:?
- Domain:?
- Domain:
- Range:
- Domain:
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 
- Reject
- 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 
- We choose
- For
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
- Only the “ar” denotes inverse
Worked Example
Starting from the definition of 
, show that
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