chapter  1
Mathematics
Pages 30

Hyperbolic functions involve the exponential functions, ex and e−x, where e is the base of the Napierian logs (ln). (e = 2.7182828…).

Definitions:

Hyperbolic sine : sinh x e e2 x x

= −

(1.26)

Hyperbolic cosine : cosh x e e2 x x

= + −

(1.27)

Hyperbolic tangent : tanh x e ee e sinh x cosh x

x x= −

+ =

(1.28)

cosh x sinh x 1 : 1 tanh x 1cosh x sech x (hyperbolic secant) 2 2 2

− = − = = (1.29)

1.2.1 Inverse Hyperbolic Functions

Note:

ln = loge

y sinh : ‘y’equals the inverse hyperbolic sinhof x1= − (1.30)

sinh x ln(x x 1) forall valuesof x1 2= + +− (1.31)

cosh x ln(x x 1) x 11 2= ± + − ≥− (1.32)

tanh x 12 ln 1 x 1 x x 1

1 =

+

>− (1.33)

ax bx c 0 a 0, anda, bandcare real2 + + = ≠ (1.34) The roots are

x 12a ( b b 4ac) 2

= − ± − (1.35)

ax2 + bx + c = 0, a ≠ 0 and a, b and c are real.