續(xù)上篇。

二．求解sin(2*n*pi/37)

與cos(2*n*pi/37)的求解非常類似，但結(jié)果顯著變得復(fù)雜！

1.?Determine K, L, M of the equation x^3-k*x^2+l*x-m=0

Step1-1 k

k12=2*(sin(14*pi/37)+sin(66*pi/37)+sin(68*pi/37));

k13=2*(sin(24*pi/37)+sin(18*pi/37)+sin(32*pi/37));

k21=2*(sin(12*pi/37)+sin(16*pi/37)+sin(46*pi/37));

k22=2*(sin(10*pi/37)+sin(26*pi/37)+sin(38*pi/37));

k23=2*(sin(70*pi/37)+sin(34*pi/37)+sin(44*pi/37));

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Step1-2 l

l11=4*(sin(2*pi/37)*sin(20*pi/37)+sin(20*pi/37)*sin(52*pi/37)+sin(52*pi/37)*sin(2*pi/37));

l12=4*(sin(14*pi/37)*sin(66*pi/37)+sin(66*pi/37)*sin(68*pi/37)+sin(68*pi/37)*sin(14*pi/37));

l13=4*(sin(24*pi/37)*sin(18*pi/37)+sin(18*pi/37)*sin(32*pi/37)+sin(32*pi/37)*sin(24*pi/37));

l21=4*(sin(12*pi/37)*sin(16*pi/37)+sin(16*pi/37)*sin(46*pi/37)+sin(46*pi/37)*sin(12*pi/37));

l22=4*(sin(10*pi/37)*sin(26*pi/37)+sin(26*pi/37)*sin(38*pi/37)+sin(38*pi/37)*sin(10*pi/37));

l23=4*(sin(70*pi/37)*sin(34*pi/37)+sin(34*pi/37)*sin(44*pi/37)+sin(44*pi/37)*sin(70*pi/37));

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Step1-3 m

m11=8*sin(2*pi/37)*sin(20*pi/37)*sin(52*pi/37);

m12=8*sin(14*pi/37)*sin(66*pi/37)*sin(68*pi/37);

m13=8*sin(24*pi/37)*sin(18*pi/37)*sin(32*pi/37);

m21=8*sin(12*pi/37)*sin(16*pi/37)*sin(46*pi/37);

m22=8*sin(10*pi/37)*sin(26*pi/37)*sin(38*pi/37);

m23=8*sin(70*pi/37)*sin(34*pi/37)*sin(44*pi/37);

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Then

k11=(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

k12=(sqrt(74+2*sqrt(37))+w1*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

k13=(sqrt(74+2*sqrt(37))+(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

k21=(sqrt(74-2*sqrt(37))-w2*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

k22=(sqrt(74-2*sqrt(37))-w1*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

k23=(sqrt(74-2*sqrt(37))-(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

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l11=(w1*(3996+648*sqrt(37)+j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3)+w2*(3996+648*sqrt(37)-j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3))/6;

l12=(w2*(3996+648*sqrt(37)+j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3)+w1*(3996+648*sqrt(37)-j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3))/6;

l13=((3996+648*sqrt(37)+j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3)+(3996+648*sqrt(37)-j*12*sqrt(3)*(37+8*sqrt(37)))^(1/3))/6;

l21=(w2*(3996-648*sqrt(37)+j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3)+w1*(3996-648*sqrt(37)-j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3))/6;

l22=((3996-648*sqrt(37)+j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3)+(3996-648*sqrt(37)-j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3))/6;

l23=(w1*(3996-648*sqrt(37)+j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3)+w2*(3996-648*sqrt(37)-j*12*sqrt(3)*(37-8*sqrt(37)))^(1/3))/6;

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m11=(sqrt(74-2*sqrt(37))-(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

m12=(sqrt(74-2*sqrt(37))-w2*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

m13=(sqrt(74-2*sqrt(37))-w1*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3))/6;

m21=-(sqrt(74+2*sqrt(37))+(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

m22=-(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

m23=-(sqrt(74+2*sqrt(37))+w1*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3))/6;

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2.Solve the equation x^3-K*x^2+L*x-M=0 whose solutions must have the form that: ?R=(k+z1*(0.5*x+sqrt(-6.75)*y)^(1/3)+z2*(0.5*x+sqrt(-6.75)*y)^(1/3))/6=2*sin(2*k*pi/37). z1*z2=1; z1 can either be 1 or w1 or w2, and X, Y, K, z1 all have to do with k.

Step2-1 x, y

x11=2*k11^3-9*k11*l11+27*m11;

x12=2*k12^3-9*k12*l12+27*m12;

x13=2*k13^3-9*k13*l13+27*m13;

x21=2*k21^3-9*k21*l21+27*m21;

x22=2*k22^3-9*k22*l22+27*m22;

x23=2*k23^3-9*k23*l23+27*m23;

x11=(8*sqrt(185-22*sqrt(37))+w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))/6;

x12=(8*sqrt(185-22*sqrt(37))+(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))/6;

x13=(8*sqrt(185-22*sqrt(37))+w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))/6;

x21=-(8*sqrt(185+22*sqrt(37))+(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))/6;

x22=-(8*sqrt(185+22*sqrt(37))+w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))/6;

x23=-(8*sqrt(185+22*sqrt(37))+w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))/6;

y11=sqrt(k11^2*l11^2-27*m11^2+18*k11*l11*m11-4*k11^3*m11-4*l11^3);

y12=sqrt(k12^2*l12^2-27*m12^2+18*k12*l12*m12-4*k12^3*m12-4*l12^3);

y13=sqrt(k13^2*l13^2-27*m13^2+18*k13*l13*m13-4*k13^3*m13-4*l13^3);

y21=-sqrt(k21^2*l21^2-27*m21^2+18*k21*l21*m21-4*k21^3*m21-4*l21^3);

y22=sqrt(k22^2*l22^2-27*m22^2+18*k22*l22*m22-4*k22^3*m22-4*l22^3);

y23=sqrt(k23^2*l23^2-27*m23^2+18*k23*l23*m23-4*k23^3*m23-4*l23^3);

y11=(4*sqrt(37+6*sqrt(37))+(4*sqrt(8707765+1430946*sqrt(37))+12*sqrt(-215895-35478*sqrt(37)))^(1/3)+(4*sqrt(8707765+1430946*sqrt(37))-12*sqrt(-215895-35478*sqrt(37)))^(1/3))/6;

y12=(4*sqrt(37+6*sqrt(37))+w2*(4*sqrt(8707765+1430946*sqrt(37))+12*sqrt(-215895-35478*sqrt(37)))^(1/3)+w1*(4*sqrt(8707765+1430946*sqrt(37))-12*sqrt(-215895-35478*sqrt(37)))^(1/3))/6;

y13=(4*sqrt(37+6*sqrt(37))+w1*(4*sqrt(8707765+1430946*sqrt(37))+12*sqrt(-215895-35478*sqrt(37)))^(1/3)+w2*(4*sqrt(8707765+1430946*sqrt(37))-12*sqrt(-215895-35478*sqrt(37)))^(1/3))/6;

y21=-(4*sqrt(37-6*sqrt(37))+(4*sqrt(8707765-1430946*sqrt(37))+12*sqrt(-215895+35478*sqrt(37)))^(1/3)+(4*sqrt(8707765-1430946*sqrt(37))-12*sqrt(-215895+35478*sqrt(37)))^(1/3))/6;

y22=-(4*sqrt(37-6*sqrt(37))+w2*(4*sqrt(8707765-1430946*sqrt(37))+12*sqrt(-215895+35478*sqrt(37)))^(1/3)+w1*(4*sqrt(8707765-1430946*sqrt(37))-12*sqrt(-215895+35478*sqrt(37)))^(1/3))/6;

y23=-(4*sqrt(37-6*sqrt(37))+w1*(4*sqrt(8707765-1430946*sqrt(37))+12*sqrt(-215895+35478*sqrt(37)))^(1/3)+w2*(4*sqrt(8707765-1430946*sqrt(37))-12*sqrt(-215895+35478*sqrt(37)))^(1/3))/6;

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Step2-2 Calculate sin(2*k*pi/37) with the switch of either w1, w2, 1

sin(2*pi/37)=(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(20*pi/37)=(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(52*pi/37)=(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(14*pi/37)=(sqrt(74+2*sqrt(37))+w1*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(66*pi/37)=(sqrt(74+2*sqrt(37))+w1*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(68*pi/37)=(sqrt(74+2*sqrt(37))+w1*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*((4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w2*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(24*pi/37)=(sqrt(74+2*sqrt(37))+(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(18*pi/37)=(sqrt(74+2*sqrt(37))+(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(32*pi/37)=(sqrt(74+2*sqrt(37))+(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*(w2*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+w1*(12*sqrt(26123295+4292838*sqrt(37))+108*sqrt(-71965-11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295+4292838*sqrt(37))-108*sqrt(-71965-11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(12*pi/37)=(sqrt(74-2*sqrt(37))-w2*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(46*pi/37)=(sqrt(74-2*sqrt(37))-w2*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(16*pi/37)=(sqrt(74-2*sqrt(37))-w2*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*((4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(10*pi/37)=(sqrt(74-2*sqrt(37))-w1*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(26*pi/37)=(sqrt(74-2*sqrt(37))-w1*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(38*pi/37)=(sqrt(74-2*sqrt(37))-w1*(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*(w2*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w1*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w2*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w1*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(70*pi/37)=(sqrt(74-2*sqrt(37))-(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(34*pi/37)=(sqrt(74-2*sqrt(37))-(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-w2*(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-w1*(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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sin(44*pi/37)=(sqrt(74-2*sqrt(37))-(44*sqrt(185-14*sqrt(37))+12*sqrt(-555+42*sqrt(37)))^(1/3)-(44*sqrt(185-14*sqrt(37))-12*sqrt(-555+42*sqrt(37)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3)-(144*sqrt(185+22*sqrt(37))+18*(w1*(4*sqrt(172179096293+1900617694*sqrt(37))+12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293+1900617694*sqrt(37))-12*sqrt(-12107447655+494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111-18*sqrt(37))+w1*(12*sqrt(26123295-4292838*sqrt(37))+108*sqrt(-71965+11826*sqrt(37)))^(1/3)+w2*(12*sqrt(26123295-4292838*sqrt(37))-108*sqrt(-71965+11826*sqrt(37)))^(1/3)))^(1/3))/36;

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三．求解x^37=1的復(fù)數(shù)解xk=cos(2*k*pi/37)+j*sin(2*k*pi/37).

For example, if k=1,

xk=(-1+sqrt(37)-w2*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w1*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+(2664-288*sqrt(37)+18*((64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*((444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+(2664-288*sqrt(37)+18*((64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*((444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+j*(sqrt(74+2*sqrt(37))+w2*(44*sqrt(185+14*sqrt(37))+12*sqrt(-555-42*sqrt(37)))^(1/3)+w1*(44*sqrt(185+14*sqrt(37))-12*sqrt(-555-42*sqrt(37)))^(1/3)+w2*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))+54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(215895+35478*sqrt(37))*(11+3*sqrt(-3)))^(1/3)+(12*sqrt(215895+35478*sqrt(37))*(11-3*sqrt(-3)))^(1/3)))^(1/3)+w1*(144*sqrt(185-22*sqrt(37))+18*(w1*(4*sqrt(172179096293-1900617694*sqrt(37))+12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3)+w2*(4*sqrt(172179096293-1900617694*sqrt(37))-12*sqrt(-12107447655-494874366*sqrt(37)))^(1/3))-54*j*(4*sqrt(111+18*sqrt(37))+(12*sqrt(215895+35478*sqrt(37))*(11+3*sqrt(-3)))^(1/3)+(12*sqrt(215895+35478*sqrt(37))*(11-3*sqrt(-3)))^(1/3)))/36≈0.98561591 + 0.16900082*j.

We inform that we can approximately construct a regular 37-gon using one right triangle with a hypotenuse 1000 and another side 169.

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四．花絮——正37角星