Functions of Gudermanian.
The six hyperbolic functions of
are expressible in terms of the six circular functions of its gudermanian; for since
||(see Arts. 6, 7)|
are the measures of corresponding hyperbolic and elliptic sectors, hence
The gudermanian is sometimes useful in computation; for instance, if
can be found from a table of natural tangents, and the other circular functions of
will give the remaining
hyperbolic functions of .
Other uses of this function are given in Arts. 22–26, 32–36.
- Prove that
- Show that
and are odd
functions of .
- From the ﬁrst identity in 4, Prob. 17, derive the relation