The hyperbolic law of cosines relates the sides and angles of a hyperbolic triangle. For curvature 
,
let 
,
,
and 
be the side lengths opposite 
, 
, and 
, respectively. Then
 |
(1)
|
The corresponding law of cosines for angles relates 
to 
,
,
and the side 
joining the vertices at 
and 
:
 |
(2)
|
For curvature 
,
each side length in these formulas is replaced by its ratio
to 
.
Expanding the hyperbolic functions for side
lengths small compared with 
gives
 |
(3)
|
the Euclidean law of cosines. Thus the Euclidean law is the zero-curvature limit
of the hyperbolic law as 
, while the hyperbolic formula is its counterpart
in hyperbolic geometry.
