Fancy Four Equation

Geometry Level 4

One of the roots of the equation is :

4 + 4 + 4 x = x \sqrt{4 + \sqrt{4 + \sqrt{4 - x}}} = x

sin 4 π 19 + sin 6 π 19 + sin 10 π 19 \sin\frac{4\pi}{19}+\sin\frac{6\pi}{19}+\sin\frac{10\pi}{19} cos 4 π 19 + cos 6 π 19 + cos 10 π 19 \cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19} 2 ( cos 4 π 19 + cos 6 π 19 + cos 10 π 19 ) 2(\cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19}) 2 ( sin 4 π 19 + sin 6 π 19 + sin 10 π 19 ) 2(\sin\frac{4\pi}{19}+\sin\frac{6\pi}{19}+\sin\frac{10\pi}{19})

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1 solution

I s e t F = 4 + 4 + 4 m S n C m S n C w i t h m a n d n a s v a r i a b l e s . W h e r e S = ( sin ( 4 π 19 ) + sin ( 6 π 19 ) + sin ( 10 π 19 ) ) . C = ( cos ( 4 π 19 ) + cos ( 6 π 19 ) + cos ( 10 π 19 ) ) . I f e e d ( m , n ) t o t a l f o u r t i m e s w i t h ( 1 , 0 ) ; ( 2 , 0 ) ; ( 0 , 1 ) ; a n d ( 0 , 2 ) . I g o t F = 0 f o r n = 0 a n d m = 2. T h i s i s 2 ( C o s f u n c t i o n ) , t h e t h i r d o n e . I~ set~~~ F~=~\sqrt{4+ \sqrt{4+ \sqrt{4-m*S-n*C}}}-m*S-n*C~~~ with~ m ~and~ n ~as~ variables. \\ Where~S= \left (\sin(\dfrac{4*\pi}{19})+\sin(\dfrac{6*\pi}{19})+\sin(\dfrac{10*\pi}{19}) \right ).\\ C= \left (\cos(\dfrac{4*\pi}{19})+\cos(\dfrac{6*\pi}{19})+\cos(\dfrac{10*\pi}{19}) \right ).\\ I~ feed~ (m,n)~total~four~ times~~with~ (1,0);~ (2,0);~ (0,1);~and~ (0,2). \\ \therefore~I~got~F=0~~for~n=0~and~m=2.\\ This ~is~~~2(Cos function),~the~third~one.\\ ~~~~\\
In the calculator, in place of S and C, I had inserted the full terms.

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