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sage: a=-1/2 sage: b=sqrt(3)/2 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3) sage: N(s,digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi),digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi)-s,digits=200) -8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???)

sage: a=-1/2 sage: b=sqrt(3)/2 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3) sage: N(s,digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi),digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi)-s,digits=200) -8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???)

sage: a=-1/2 sage: b=sqrt(3)/2 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3) sage: N(s,digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi),digits=200) 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi)-s,digits=200) -8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???)(???) The text formatting does not look working well here.

sage: a=-1/2 sage: b=sqrt(3)/2 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3) sage: N(s,digits=200) N(s,digits=200)
0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi),digits=200) pi),digits=200)
0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi)-s,digits=200) pi)-s,digits=200)
-8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???) The text formatting does not look working well here.

sage: a=-1/2 sage: b=sqrt(3)/2 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3) sage: N(s,digits=200)
0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi),digits=200)
0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703 sage: N(sin(50/180pi)-s,digits=200)
-8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???) The text formatting does not look working well here.

.

sage: a=-1/2
 sage: b=sqrt(3)/2
 sage: s=(3/8(a^3+sqrt(-a^4b^2-2a^2b^4-b^6)+ab^2)^(1/3)-(-5184a^2-5184b^2)/(13824(a^3+sqrt(-a^4b^2-2a^2b^4-b^6+ab^2)^(1/3))+a/4)^(1/3)
    s=(3/8*(a^3+sqrt(-a^4*b^2-2*a^2*b^4-b^6)+a*b^2)^(1/3)-(-5184*a^2-5184*b^2)/(13824*(a^3+sqrt(-a^4*b^2-2*a^2*b^4-b^6+a*b^2)^(1/3))+a/4)^(1/3)
sage: N(s,digits=200) 
    0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703
    sage: N(sin(50/180pi),digits=200) 
 0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703
sage: N(sin(50/180*pi),digits=200)    
    0.76604444311897803520239265055541667393583245708039524585404528464215538885687472352822927668054849344996248839987004553419542483163334907118418648890238602810825947820108992619243612738937829040787703
sage: N(sin(50/180*pi)-s,digits=200)     sage: N(sin(50/180pi)-s,digits=200) 
     -8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202
-8.1651261039391273250903677461338074230805364330263310047708623423822773669493747438272352951232512935097984755386384444942102172477771762845811026642442173532418045576803620147104496959316258647247981e-202

The last calculation shows strange result. The difference should be exactly zero... (???) The text formatting does not look working well here.