Prove that

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

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

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: http://mathforum.org/library/drmath/view/61024.html

giving -b/a or 33273.

Edited on January 26, 2015, 10:00 pm

Posted by broll on 2015-01-26 21:42:55