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
  • 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

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 .


b) Find the real solutions, as exact values, to the equation .