中五數學題(附加數)

2√3cos2x + cosx -2√3 = 0 ???

圖片參考：http://f.imagehost.org/0162/ScreenHunter_03_Jan_23_12_02.gif

2sqrt(3)cos2(x)+cos(x)-2sqrt(x)=0
is an equation that requires numerical methods to obtain a solution.
the function looks like this:
\

圖片參考：http://i263.photobucket.com/albums/ii157/mathmate/cos0.png

There are no real roots for x<0 for obvious reasons.
The following plot indicates a real root between 0.8 and 1.0.

圖片參考：http://i263.photobucket.com/albums/ii157/mathmate/cos1-1.png

A F5 solution can be found quite easily by trial and error followed by linear interpolation. In university, Newton's method could be easily applied.
For example:
let
f(x)=2*sqrt(3)*cos(x)^2+cos(x)+(-2)*sqrt(x)
f(0.8)=0.589328078
f(1.0)=-0.4484343508
from which the next estimate is x=0.91357668, and
f(0.91357668)=-0.00783
A refined estimate for x=0.912087
for which
f(0.912087)=-0.000099595
Further refinement can be easily made if required.

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