Edexcel A Level Further Maths: Core Pure:复习笔记4.1.3 Hyperbolic Identities & Equations
Hyperbolic Identities & Equations
Are there identities linking the hyperbolic functions to the circular trig functions?
- Yes - these can be seen using de Moivre's Theorem to write
- Compare these with the definitions for the hyperbolic functions
- Therefore they can be related using the identities
What are the hyperbolic identities?
- In general, the hyperbolic identities are the same as the circular trigonometric identities except where there is a product of an even number of sinh terms, in which case the term changes sign
- e.g.
- This is referred to as Osborn’s Rule
- This occurs because of the connection with the imaginary number i
- e.g.
- All the circular trigonometric identities can be used with hyperbolic functions
- The main hyperbolic identities you are likely to need are
- These are listed in the formulae booklet
- Other identities include
- The harmonic identities can also be used with hyperbolic functions
- Hyperbolic identities involving tanhx exist
- They are not normally used as it is easier to use sinhx, coshx and their definitions
- If you do use tanhx identities, be careful with implied or ‘hidden’ products of sinhx (e.g. tanh2x)
- You can prove these identities by using the definitions of the hyperbolic functions in terms of e
Do reciprocal hyperbolic functions and identities exist?
- Yes! However, It is usually easier to deal with identities and equations involving these in terms of sinhx, coshx and their definitions
(Pronounced “coshec”)
(Pronounced “shec”)
(Pronounced “cough”)
How do I use hyperbolic identities to prove other identities?
- Start with the LHS and use the hyperbolic identities to rearrange into the RHS
- This approach can lead to what seems like a dead-end
- In such cases simplify the LHS as far as possible, ideally so that it is in terms of sinhx and/or coshx only
- Then use the sinhx and coshx definitions to write the LHS in terms of e
- Repeat this for the RHS so that the LHS and RHS ‘meet in the middle’
How do I solve equations involving hyperbolic functions?
- Use identities to create an equation in terms of sinhx or coshx only
- This should be a familiar equation to solve – linear, quadratic, etc
- Find exact answers in terms of natural logarithms
- Using the inverse hyperbolic functions definitions
- Use your calculator if exact answers are not required
- As with circular trigonometric equations, do not cancel hyperbolic terms, rearrange so the equation equals zero and factorise
- When solving equations be careful when solving coshx = k (for constant k)
- cosh-1x is not necessarily the same as arcoshx
- arcoshx is, strictly speaking, referring to the inverse function of coshx such that coshx is a one-to-one function
- Using the graph you can see that the for k>1 there are two solutions to
- This can be written in logarithmic form as
- This can be shown to be equivalent to
- If k=1 then the only solution to
is x=0 - If k<1 then there are no real solutions to
- cosh-1x is not necessarily the same as arcoshx
Exam Tip
- You can use the A Level Maths section of the formula booklet to remind you of trigonometric identities (such as sin(A±B)) which you can then adapt for the hyperbolic trig functions – don’t limit yourself to just the Further Maths section
Worked Example
a) Using the definitions of 
prove the identity 
.
prove the identity .
b) Find the real solutions, as exact values, to the equation 
.
.
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