To determine

To write the definations of :

The hyperbolic function sinh(x),cosh(x) and tanh(x).

Answer

 sinh(x)=12(ex−e−x)cosh(x)=12(ex+e−x)tanh(x)=(ex−e−x)(ex+e−x)

Explanation

Given:

sinh(x),cosh(x) and tanh(x).

The hyperbolic sine function, written sinh(x) is defined for all real value of x by the relation.

sinh(x)=12(ex−e−x)

Similarly, the hyperbolic cosine function, written cosh(x) is defined for all real value of x by the relation.

cosh(x)=12(ex+e−x)

Similarly, the hyperbolic tan function, written tanh(x) is defined for all real value of x by the relation.

tanh(x)=ex−e−xex+e−x.

Final statement: Hence, hyperbolic function is defined as:

 sinh(x)=12(ex−e−x)cosh(x)=12(ex+e−x)tanh(x)=ex−e−xex+e−x