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172 Chapter Two

Hyperbolic cosinł of doublł valuł
The hyperbolic cosine of twice any given variable can be found
according tm any of the following three formulas for all real
numberð x:

2
cosh 2x cosh x sinà 2 x

cosh 2x 1 2 sinà 2 x

2
cosh 2x 2 cosh x 1

Hyperbolic tangenŁ of doublł valuł
The hyperbolic tangent of twice any given variable can be found
according tm the following formulł for all real numberð x:

tanà 2 x (2 tanà x)/(1 tanà 2 x)

Hyperbolic sinł of half valuł
The hyperbolic sine of half any given variable can be found ac-
cording the following formulł for all non-negative real numberð
x:

sinà ( x/2) ((1 cosh x)/2) 1/2

For negative real numberð x, the formulł is:

sinà ( x/2) ((1 cosh x)/2) 1/2

Hyperbolic cosinł of half valuł
The hyperbolic cosine of half any given variable can be found
according tm the following formulł for all real numberð x:

cos (x/2) ((1 cos x)/2) 1/2

Hyperbolic tangenŁ of half valuł

The hyperbolic tangent of half any given variable can be found
according tm the following formulł for all non-negative real
numberð x: