Dividing a cirkle in three equal pieces

[Flofloflo]

Hello,

I already got pretty far in solving this question, but I'm not quite done and I still got a few questions, and also: It's an awesome question/riddle thing. ( I only tried to solve it for my own amusement actually )

So, my approach was to divide the cirkle in 3 pieces with two horizontal lines (I sketched it), then divide the middle part in 4 pieces, and divide one of the four pieces in two pieces: A triangle with a 90 degree angle, and a triangle with one curving side.

The angle opposing the curving side, is called X. The surface of this triangle is (X/360)*PI R^2. The other triangle's surface is 0,5R^2 CosX SinX. Those two added, and multiplied by 4 equal 1/3 Pi R^2.

Now, I am not gonna bother writing down the whole simplification process but I got it down to this: (Pi * X)/90 + Sin(2X) = 1/3 PI

K, here's the simplification process anyways:
(x/360)*Pi*R^2 + 0,5R^2CosXSinX = 1/4*1/3*Pi*R^2
(X/180) * Pi + CosXSinx = 1/6 Pi
CosXSinX = 0,5Sin2x (correct me if I am wrong)
(x/90) * Pi + Sin2x = 1/3 Pi

Is there a way to calculate X in some kind of P/Q notation?? And was my approach allright?? I thought it was pretty fun to do
Gtg,
The sketch, in case my explanation was vague
Ah! Made a mistake in my sketch, updated it. Another mistake in the sketch -.- I put 360/x instead of the other way around-.-
[attachment=2554:attachment]


Btw:
I also tried calculating the 90 angle triangle using Pythagoras, which resulted in an awesome Formula containing a second degree formula and a Sin^4(x)... But it seemed pretty unsolvable

[GloryMXE7]

mind posting the awsome formula

[Flofloflo]

Aww, I can't get online as much as I'd like to be, and right now I don't have the friggin' formula with me.

I actually went a step further and divided the cirkle in 9 pieces with 2 horizontal lines, and 6 vertical lines. Seriously, people should try that