2. FP2 (MEI) Inverse hyperbolic functions
Let Maths take you Further…

3. Inverse hyperbolic functions
Before you start:
You need to be confident in manipulating exponential and logarithmic functions.
You need to have covered the work on Maclaurin series from chapter 4.
You need to have covered Calculus from chapter 1 (integration using inverse trig functions)
When you have finished… You should:
Understand and be able to use the definitions of the inverse hyperbolic functions.
Be able to use the logarithmic forms of the inverse hyperbolic functions.
Be able to integrate
and and related functions.

4. Notation sin x arcsin x sinh x arsinh x cos x arccos x cosh x arcosh x
trig. functions
inverse trig. functions
hyperbolic
inverse hyperbolic
sin x
arcsin x
sinh x
arsinh x
cos x
arccos x
cosh x
arcosh x
tan x
arctan x
tanh x
artanh x
cosec x
arccosec x
cosech x
arcosech x
sec x
arcsec x
sech x
arsech x
cot x
arccot x
coth x
arcoth x

5. Latin for arc

6. Graphs Use the graph of sinhx to sketch the graph of arsinhx
Hint: use the line y=x to help!
Remember for a function to have an inverse it has to be a one-to-one function

7. Sketch the graph of arcoshx and state its domain and range
The domain needs to be refined to ensure the function is one to one

8. Logarithmic form of the inverse hyperbolic functions
y=arsinh x so x=sinh y

9. Summary

11. Differentiating inverse hyperbolic trig. functions
Note: this can be done using the same technique that was used for differentiating inverse trig. functions
y=arcosh x
x= cosh y

12. Results We can now integrate expressions of these forms!
We can also differentiate composite functions involving inverse hyperbolic functions using the chain rule e.g.

13. Using the previous results, together with the results we established by considering inverse trig. Functions, we should now be able to integrate functions of the form:

24. Inverse hyperbolic functions
When you have finished… You should:
Understand and be able to use the definitions of the inverse hyperbolic functions.
Be able to use the logarithmic forms of the inverse hyperbolic functions.
Be able to integrate
and and related functions.

25. Independent study:
Using the MEI online resources complete the study plan for Hyperbolic functions 2
Do the online multiple choice test for this and submit your answers online.