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Integral Proof (Posted on 2015-01-26) Difficulty: 3 of 5
Prove that

tan6(pi/9) + tan6(2pi/9) + tan6(4pi/9)

is an integer, and find the value of this integer.

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts possible approach. | Comment 2 of 3 |

More of an observation really.

WolframAlpha gives the result as exactly 33273.

tan6(pi/9) is the root of  x^3-33273x^2+11691x-27  near  x = 0.00232485 etc.

tan6(2pi/9) is the root of  x^3-33273x^2+11691x-27  near  x = 0.349045 etc.

tan6(4pi/9) is the root of  x^3-33273x^2+11691x-27  near  x = 33272.6 etc.

We can sum the roots of ax^3 + bx^2 + cx + d = 0 using this:

giving -b/a or 33273.

Edited on January 26, 2015, 10:00 pm
  Posted by broll on 2015-01-26 21:42:55

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