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
(1)
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so . Since we also have , this can be written
(2)
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Solving for then gives
(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 .
Wolfram
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