. An elementary course of infinitesimal calculus . t—y 19] CONTINUITY. 43 Since sinh x and cosh x are continuous, whilst cosh xnever vanishes, it follows that tauha; is continuous for allvalues of x. Fig. 22 shews the curve y = tanh has the lines y = + 1 as asymptotes *,. It is evident from the definitions (1) that cosh *• + sinh a; = e*, cosh x — sinh x = e~^ ... (3),whence, by multiplication, cosh^ X — sinh^ x = \ (4). From this we derive, dividing by cosh a; and sinh a;,respectively, sech x=\— tanh a;, 1 ,^. cosech a; = coth a; — 1 )The forraulse (4) and (5) corresponcl to the trisjo
. An elementary course of infinitesimal calculus . t—y 19] CONTINUITY. 43 Since sinh x and cosh x are continuous, whilst cosh xnever vanishes, it follows that tauha; is continuous for allvalues of x. Fig. 22 shews the curve y = tanh has the lines y = + 1 as asymptotes *,. It is evident from the definitions (1) that cosh *• + sinh a; = e*, cosh x — sinh x = e~^ ... (3),whence, by multiplication, cosh^ X — sinh^ x = \ (4). From this we derive, dividing by cosh a; and sinh a;,respectively, sech x=\— tanh a;, 1 ,^. cosech a; = coth a; — 1 )The forraulse (4) and (5) corresponcl to the trisjoiiometncalformuliB cosa3 + siaa;= 1 (6), seca: = l +taiia;,1cosec X = cot X iiia;,| !+lJ .(7). * The numerical values (to three places) of the functions cosh x, sinh x,tanh a;, for values of x ranging from 0 to 2-5 at intervals of 0-1, are given inthe Appendix, Table E. 44 INFINITESIMAI, CALCULUS. [CH. I Again, from (1) and (3) we easily find sinh {x±y) = sinh x cosh y + cosh x sinh y, Icosh (a; + 1/) = cosh a; cosh y + sinh a; sinh y, Jwhence, as particular cases, sinh 2i» = 2 sinh a; cosh a;, 1 , , cosh 2a! = cosh a; + sinha;, J These are the analogues of the trigonometrical formulse sin (a; + y) = sin a; cos 2/+ cos a; sin y, I ..^. cos (a; + y) = cos a; c
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