數學解方程

With reference to the information in:
http://en.wikipedia.org/wiki/Cubic_equation#Monic_formula_of_roots
We can set up the cubic stuff as follows first:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/May09/Crazyeqn3.jpg

Then the roots should be expressed in trigonometric form which are:


圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/May09/Crazyeqn4.jpg


2009-05-20 22:04:14 補充：
感謝兩位高手的高見, 原本想以 Newton's approximation 求根, 惟一時醒不起 cos 3x = 4 cos^3 x - 3 cos x 的公式.

設 x = cosθ 無疑更簡潔，但必須先證明 IxI <= 1。

可如下：

設 f(x) = 8x^3 - 6x - 1

f(-1) = -3
f(-1/2) = 1
f(0) = -1
f(1) = 1

f(x) 連續且最多只有三根，所以 若 8x^3 - 6x - 1 = 0，IxI <= 1。


當然，亦可取巧地將所得三個答案分別代入方程驗證，從而解方程。

以 cos(3t)= 4(cost)^3 - 3(cost), cos(3t) = 1/2解之, 對本題比較容易些!
3t=π/3, 5π/3, 7π/3 => x= cost= cos(π/9), cos(5π/9), cos(7π/9)

As follows:

圖片參考：http://g.imagehost.org/0051/ScreenHunter_03_May_19_22_25.gif