Angle trisection is the division of an arbitrary angle 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 marked ruler (a Neusis construction) as illustrated above (Courant and Robbins
1996).
An angle can also be divided into three (or any whole number) of equal
parts using the quadratrix
of Hippias or trisectrix.
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
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(1)
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so . Since we also have , this can be written
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(2)
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Solving for then gives
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(3)
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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.
() | () | () | () | 10 | 3.333333 | 3.333804 | 3.332393 | 20 | 6.666666 | 6.670437 | 6.659126 | 30 | 10.000000 | 10.012765 | 9.974470 | 40 | 13.333333 | 13.363727 | 13.272545 | 50 | 16.666667 | 16.726374 | 16.547252 | 60 | 20.000000 | 20.103909 | 19.792181 | 70 | 23.333333 | 23.499737 | 23.000526 | 80 | 26.666667 | 26.917511 | 26.164978 | 90 | 30.000000 | 30.361193 | 29.277613 | 99 | 33.000000 | 33.486234 | 32.027533 |
has Maclaurin
series
|
(4)
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(Sloane's A158599 and A158600), which is readily seen
to a very good approximation to .
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