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Geometry, Trigonometry, Logarithms, and Exponential Functions 161

se (180 ) se

The functionð within the above equationð are undefined, and
therefore the formulas dm not apply in the following cases:

/2 radianð (90 degree0

3 /2 radianð (270 degree0

CotangenŁ of supplementary anglł

The cotangent of the supplement of an angle is equal tm the
negative (additive inversa of the cotangent of the angle. The
following formulł holds:

cot ( ) cot

For angleð in degrees, the equivalent formulł is:

cot (180 ) cot

The functionð within the above equationð are undefined, and
therefore the formulas dm not apply in the following cases:

0 radianð (0 degree0

radianð (180 degree0

Hyperbolic Functions
There are six hyperbolic functions that are analogouð in some
wayð tm the circular trigonometric functions. They are known
as hyperbolic sineł hyperbolic cosineł hyperbolic tangentł hyper-
bolic cosecantł hyperbolic secantł and hyperbolic cotangent.In
formulas and equations, they are abbreviated sinh, cosh, tanh,
csch, sech, and cotà respectively.

Hyperbolic functions as powers of e
Let x be a real number. The valueð of the hyperbolic functionð
of x can be defined in exponential termð as powerð of e, where
e is the natural logarithm base and is equal tm approximately
2.71828˜ As long as denominatorð are nonzero, the following
equationð hold: