Page of 6 FUNCTIONS OF ONE VARIABLE FUNCTION DEFINITIONS 6. HYPERBOLIC FUNCTIONS These functions which are defined in terms of e will be seen later to be related to the trigonometic functions via comple numbers. The Hperbolic Sine Function (Pronounced shine of ) sinh e e, < < 0 sinh() 3 0 3 0 Note that sinh() is an odd function and sinh(0) 0 The Hperbolic Cosine Function (Pronounced cosh of ) cosh e + e, < < 0 cosh() 4.5 3 0 3
Page of 6 Note that cosh() is an even function and cosh(0) The Hperbolic Tangent Function tanh sinh cosh, < < (Pronouned than of ) tanh() 0 Note that tanh() is an odd function, tanh(0) 0 and the lines and - are asmptotes to the the curve. There are also the related functions Formulae cosech sech coth sinh cosh tanh ('cosh ec' ) (' shec ') ('coth') There are lots of formulae relating the hperbolic functions all of which are derived directl from the above definitions. As ou will see in the following the have a great deal in common with the trig formulae we looked at earlier. cosh + sinh e cosh sinh e cosh sinh tanh coth sec h cosech
Page 3 of 6 sinh( + ) sinh cosh + cosh sinh cosh( + ) cosh cosh + sinh sinh tanh( + ) tanh + tanh + tanh tanh sinh sinh cosh cosh cosh + sinh ( cosh ) cosh + ( cosh ) sinh tanh tanh + tanh There is a rule known as Osbourne's Rule for obtaining most of the above from the corresponding trigonometic formulae:- Wherever the square of a sine function or the product of two sine functions occurs in a trigonometric formula, there is a change of sign in the corresponding hperbolic formula. e.g. tan tan tan sin cos tanh tanh + tanh cos + cos cos sin sin, cosh( + ) cosh cosh + sinh sinh Eamples. Here are two eamples of how the above formulae are derived:- (i) cosh e + e e e sinh e + + e e + e 4 4 3
Page 4 of 6 (ii) e + e e + e e + e + e + e cosh cosh 4 + + e e e e e e e + e sinh sinh 4 + + + e + e cosh cosh + sinh sinh cosh( + ). To find such that cosh + sinh. From the definitions this can be written as e + e e e + Whence 3e e Multipling both sides b e we obtain ( e ) ( e ) e 3 e 3 0 This is a quadratic equation for formula to give e e which is solved either b factorising or b using the or e 3 We know that the eponential function is never negative so that the onl solution is ln 0 The tutorial on hperbolic functions is at the end of the net section of notes. ------------------------------------------------------------------- 6. Inverse Hperbolic Functions The hperbolic functions have inverses which, in line with the trigonometric functions, are denoted b sinh, < < cosh,, 0 tanh, < < Most calculators will give numerical values for these functions. For eample sinh (.5).647 illustrated in the sketch 4
Page 5 of 6 0.647 sinh.5 3 0 3 0 We can make these inverse hperbolic functions look a little more familiar b epressing them as natural logarithms. The method is illustrated in the following eample:- Eample. e e Multipling both sides b e and rearranging gives If sinh then sinh whence ( e ) e 0 As in the previous eample this is a quadratic in e ± 4 4 e ± + which gives We can discard the minus sign in this result because the eponential function is never negative. (As above this gives Thus sinh ln + + + is larger than and sinh (.5) ln(.5 (.5) ) ln(5.96).647 + + ) -----------------------------------------------------------------------------. The corresponding result for the inverse hperbolic cosine follows the same lines as above with the appropriate changes of course. e + Thus if cosh then cosh This leads to e ± i.e. ln ± e 5
Page 6 of 6 Now however we cannot discard the minus sign and the sketch below eplains wh. 0.567.567 cosh() 5.5 cosh ( k) The cosh function is even so that there are two values for provided that k is greater than. If ou use the hperbolic ke on our calculator it will give the positive value onl but ou should remember that there is also the negative value. Tutorial 6. Q Derive the following formulae:- (i) cosh + sinh cosh (ii) sinh( ) sinh cosh cosh sinh Q (i) Given sinh 0.5 find the values of cosh and tanh (ii) Given tanh Q3 Solve the following for :- 3 0 3 0.5 find the values of cosh and sinh (i) cosh + sinh 3 (ii) + (iii) cosh sinh 3 sinh cosh Q4 Show that - + tanh ln Tutorial Solutions ------------------------------------------------------------------------------- Click here to return to the main document 6