Angle trisection is the division of an arbitraryangle into three equal angles. It was one of the three geometric
problems of antiquity for which solutions using only compass
and straightedge were sought. The problem was algebraically
proved impossible by Wantzel (1836).
Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and radians ( and , respectively), which can be trisected. Furthermore,
some angles are geometrically trisectable, but cannot be
constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary
angle can be accomplished using a markedruler
(a Neusis construction) as illustrated above
(Courant and Robbins 1996).
An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983; Steinhaus 1999, p. 7). To construct this approximation of an angle having measure , first bisect and then trisect chord (left figure above). The desired approximation is then angle
having measure (right figure above). To connect with , use the law of sines
on triangles
and
gives
(1)
so .
Since we also have ,
this can be written
(2)
Solving for
then gives
(3)
This approximation is with of even for angles as large as , as illustrated above and summarized in the following
table (Petersen 1983), where angles are measured in degrees.
and 
radians (
and 
, respectively), which can be trisected. Furthermore,
some angles are geometrically trisectable, but cannot be
constructed in the first place, such as 
(Honsberger 1991). In addition, trisection of an arbitrary
angle can be accomplished using a marked ruler
(a Neusis construction) as illustrated above
(Courant and Robbins 1996).
having measure 
, first bisect 
and then trisect chord 
(left figure above). The desired approximation is then angle
having measure 
(right figure above). To connect 
with 
, use the law of sines
on triangles 
and 
gives
.
Since we also have 
,
this can be written
![sint=2[sin(1/2alpha)cost-sintcos(1/2alpha)].](/images/equations/AngleTrisection/NumberedEquation2.svg)
then gives
of 
even for angles 
as large as 
, as illustrated above and summarized in the following
table (Petersen 1983), where angles are measured in degrees.
(
)
(
)
(
)
(
)
has Maclaurin
series
.
