Similarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola

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1 Universit of Hperbolic Functions The trigonometric functions cos α an cos α are efine using the unit circle + b measuring the istance α in the counter-clockwise irection along the circumference of the circle. The area of the sector so etermine is α, so we can equivalentl sa that cos α an cos α are erive from the unit circle + b measuring off a sector (shae re )of area α. The other four trigonometric functions can then be efine in terms of cos an sin. DEO ET PAT- RIE Saskatchewan Similarl, we ma efine hperbolic functions cosh α an sinh α from the unit hperbola b measuring off a sector (shae re )of area α to obtain a point P whose - an - coorinates are efine to be cosh α an sinh α. (cosh α, sinh α) Since at this point we o not et know how to compute the areas of most curve regions, we must take it on faith that the si hperbolic functions ma be epresse simpl in terms of the eponential function: sinh α eα e α cosh α eα + e α tanh α sinh α cosh α eα e α e α + e α cotanh α cosh α sinh α eα + e α e α e α sech α cosh α e α + e α cosech α sinh α e α e α Note that the omains of sinh, cosh, tanh, an sech are (, ) an the omains of cotanh an cosech are (, 0) (0, ).

2 ( e α + e α We can check that the point, eα e α ) lies on the unit hperbola: ( e α + e α ) ( e α e α ) eα + + e α eα + e α Pthagorean Ientities This gives us the first important hperbolic function ientit: cosh α sinh α This ma be use to erive two other ientities relating the two other pairs of hperbolic functions: tanh α sech α an cotanh α cosech α O an Even Ientities It is clear that sinh, tanh, cotanh an cosech are o functions, while cosh, cotanh, an sech are even, so we have the corresponing ientities: sinh( ) sinh, tanh( ) tanh, cotanh ( ) cotanh, cosech ( ) cosech cosh( ) cosh, sech ( ) sech. Sum an Difference Ientities We can use the above formulas for the hperbolic functions in terms of e to erive analogs of the ientities for the trigonometric functions: sinh α cosh β eα e α e β + e β e α+β + e α β e α+β e α β sinh β cosh α eβ e β e α + e α (eα e α )(e β + e β ) (eβ e β )(e α + e α )

3 e β+α + e β α e β+α e β α Aing these two proucts gives: sinh α cosh β + sinh β cosh α e α+β + e α β e α+β e α β + eβ+α + e β α + e β+α e β α e α+β e α β eα+β e α β e(α+β) e (α+β) sinh(α + β) an subtracting these two proucts gives: sinh α cosh β sinh β cosh α e α+β + e α β e α+β e α β eβ+α + e β α + e β+α e β α e α β e (α β) eα β e (α β) sinh(α β) Similarl, cosh α cosh β eα + e α e β + e β e α+β + e α β + e β α + e α β sinh α sinh β eα e α e β e β e α+β e α β e β α + e α β (eα + e α )(e β + e β ) (eα e α )(e β e β ) Aing these two proucts gives cosh α cosh β + sinh α sinh β e α+β + e α β + e β α + e α β + eα+β e α β e β α + e α β 3

4 e α+β + e α β eα+β + e (α+β) cosh(α + β) an subtracting them gives: cosh α cosh β sinh α sinh β e α+β + e α β + e β α + e α β eα+β e α β e β α + e α β e α β + e α+β eα β + e (α β) cosh(α β) Summarizing, we have four ientities: sinh(α + β) sinh α cosh β + sinh β cosh α sinh(α β) sinh α cosh β sinh β cosh α cosh(α + β) cosh α cosh β + sinh α sinh β cosh(α β) cosh α cosh β sinh α sinh β which are almost eactl parallel to those for the trigonometric functions an ma be use to erive sum an ifference formulas for the other four hperbolic functions. Letting β α, we get: Double an Half- Angle Ientities sinh α sinh α cosh α, cosh α cosh α + sinh α + sinh α cosh α, so cosh cosh α + α an sinh cosh α α, an thus: cosh α + cosh α cosh α an sinh α

5 cosh α cosh α + an sinh α cosh α (sinh ) (cosh ) Derivatives ( e e ) ( e + e ) e ( e ) e + ( e ) e + e e e cosh sinh (tanh ) ( ) sinh cosh cosh (sinh ) sinh (cosh ) cosh cosh cosh sinh sinh cosh cosh sinh cosh cosh sech (cotanh ) ( ) cosh sinh sinh (cosh ) cosh (sinh ) sinh sinh sinh cosh cosh sinh sinh cosh sinh sinh cosech (sech ) (cosh ) ( ) (cosh ) (cosh ) ( ) (cosh ) sinh sech tanh (cosech ) (sinh ) 5

6 ( ) (sinh ) (sinh ) ( ) (sinh ) cosh cosech cotanh Summar: (sinh ) cosh (tanh ) sech (sech ) sech tanh (cosh ) sinh (cotanh ) cosech (cosech ) cosech cotanh Graphs ofthe Hperbolic Functions sinh cosech cosh sech tanh cotanh The omains an ranges are summarize in the net table: 6

7 function omain Range sinh (, ) (, ) cosh (, ) [, ) tanh (, ) (, ) cotanh (, 0) (0, ) (, ) (, ) sech (, ) (0, ]) cosech (, 0) (0, ) (, 0) (0, ) Inverse Hperbolic Functions sinh, tanh, cotanh an cosech are one-to-one, but cosh an sech are not. For the purpose of efining the inverse of cosh an sech we will restrict their omains to [0, ). We will enote the inverse hperbolic functions b sinh, cosh, tanh, cotanh, sech, an cosech or: sinh inv, cosh inv, tanh inv, cotanh inv, sech inv, an cosech inv or even: arcsinh, arccosch, arctanh, arccothh, arcsech, an arccosech. The usual Cancellation Laws hol in the appropriate omains: sinh(sinh ) sinh (sinh ) cosh(cosh ) cosh (cosh ) tanh(tanh ) tanh (tanh ) cotanh (cotanh ) cotanh (cotanh ) sech (sech ) sech (sech ) cosech (cosech ) cosech (cosech ) The erivatives of the inverse hperbolic functions ma be foun the same wa the erivatives of the inverse trigonometric functions were foun: b ifferentiating the left-han Cancellation Laws above: Eample: Differentiating sinh(sinh ) we get 7

8 ( cosh(sinh ) sinh ), so ( ) sinh cosh(sinh ). Using the ientit cosh sinh weget cosh + sinh,so cosh + sinh an therefore cosh(sinh ) ( ) + sinh (sinh ) + sinh(sinh ) + Thus we have ( ) sinh + One ma similarl erive the erivatives of the other hperbolic functions: ( ) cosh ( ) ( ) tanh cotanh ( ) sech ( ) cosech + Eplicit Computation ofinverse Hperbolics The inverse hperbolic functions have the unusual propert that the can be eplicit compute: Eample: Solve the equation sinh for in terms of. (The solution will be sinh!) We have sinh e e, so e e or e e 0. Multipling both sies of this equation b e we get: 8

9 (e ) e 0, a quaratic equation in e which has solution e ( ) ± ( ) ()( ) ± + Since + < 0 an we must have e > 0, we get e + +. ± + Taking logarithms of both sies of this equation, we get ln( + + ), so we have sinh ln( + + ) Similarl, cosh ln( + ) an tanh ( ) + ln We then have ( cosech sinh ln ) + + ln + + ln ( + ) + Similarl cotanh ( ) + ln an ( ) + sech ln 9