可用的單位根：w1=(-1+sqrt(-3))/2; w2=(-1-sqrt(-3))/2; j=sqrt(-1)

核心思想：韋達(dá)定理與卡爾丹公式

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一．求解cos(2*n*pi/37)

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

Step1-1

K11=2*(cos(2*pi/37)+cos(20*pi/37)+cos(52*pi/37));

K12=2*(cos(14*pi/37)+cos(66*pi/37)+cos(68*pi/37));

K13=2*(cos(24*pi/37)+cos(18*pi/37)+cos(32*pi/37));

K21=2*(cos(12*pi/37)+cos(16*pi/37)+cos(46*pi/37));

K22=2*(cos(10*pi/37)+cos(26*pi/37)+cos(38*pi/37));

K23=2*(cos(70*pi/37)+cos(34*pi/37)+cos(44*pi/37));

Step1-2

L11=4*(cos(2*pi/37)*cos(20*pi/37)+cos(20*pi/37)*cos(52*pi/37)+cos(52*pi/37)*cos(2*pi/37));

L12=4*(cos(14*pi/37)*cos(66*pi/37)+cos(66*pi/37)*cos(68*pi/37)+cos(68*pi/37)*cos(14*pi/37)); L13=4*(cos(24*pi/37)*cos(18*pi/37)+cos(18*pi/37)*cos(32*pi/37)+cos(32*pi/37)*cos(24*pi/37));

L21=4*(cos(12*pi/37)*cos(16*pi/37)+cos(16*pi/37)*cos(46*pi/37)+cos(46*pi/37)*cos(12*pi/37));

L22=4*(cos(10*pi/37)*cos(26*pi/37)+cos(26*pi/37)*cos(38*pi/37)+cos(38*pi/37)*cos(10*pi/37));

L23=4*(cos(70*pi/37)*cos(34*pi/37)+cos(34*pi/37)*cos(44*pi/37)+cos(44*pi/37)*cos(70*pi/37));

Step1-3

M11=8*cos(2*pi/37)*cos(20*pi/37)*cos(52*pi/37);

M12=8*cos(14*pi/37)*cos(66*pi/37)*cos(68*pi/37);

M13=8*cos(24*pi/37)*cos(18*pi/37)*cos(32*pi/37);

M21=8*cos(12*pi/37)*cos(16*pi/37)*cos(46*pi/37);

M22=8*cos(10*pi/37)*cos(26*pi/37)*cos(38*pi/37);

M23=8*cos(70*pi/37)*cos(34*pi/37)*cos(44*pi/37);

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Then

K11=(-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))/6;

K12=(-1+sqrt(37)-w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6;

K13=(-1+sqrt(37)-(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6;

K21=(-1-sqrt(37)-w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

K22=(-1-sqrt(37)-(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

K23=(-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))/6;

L11=(-2+2*sqrt(37)+w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6; L12=(-2+2*sqrt(37)+(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6;

L13=(-2+2*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))/6;

L21=(-2-2*sqrt(37)+(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

L22=(-2-2*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))/6;

L23=(-2-2*sqrt(37)+w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

M11=(11-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))/6;

M12=(11-sqrt(37)-w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

M13=(11-sqrt(37)-(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3))/6;

M21=(11+sqrt(37)-(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6;

M22=(11+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))/6;

M23=(11+sqrt(37)-w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3))/6;

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

Step2-1 Determine 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;

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);

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Consequently

X11=(148-16*sqrt(37)+(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))/6;

X12=(148-16*sqrt(37)+w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))/6;

X13=(148-16*sqrt(37)+w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))/6;

X21=(148+16*sqrt(37)+w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))/6;

X22=(148+16*sqrt(37)+w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))/6;

X23=(148+16*sqrt(37)+(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))/6;

Y11=-((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;

Y12=-(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;

Y13=-(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;

Y21=-(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;

Y22=-((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;

Y23=-(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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Step2-2 Calculate cos(2*k*pi/37) based on the choice of either w1, w2, 1

cos(2*pi/37)=(-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))/36;

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cos(20*pi/37)=(-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)+w2*(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)+w1*(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))/36;

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cos(52*pi/37)=(-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)+w1*(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)+w2*(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))/36;

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cos(14*pi/37)=(-1+sqrt(37)-w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+w2*(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+w1*(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(66*pi/37)=(-1+sqrt(37)-w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+w1*(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+w2*(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(68*pi/37)=(-1+sqrt(37)-w1*(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-w2*(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+(2664-288*sqrt(37)+18*(w1*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w2*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w1*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(24*pi/37)=(-1+sqrt(37)-(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+w1*(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+w2*(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(18*pi/37)=(-1+sqrt(37)-(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(32*pi/37)=(-1+sqrt(37)-(148+32*sqrt(37)+j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)-(148+32*sqrt(37)-j*12*sqrt(3)*(37+6*sqrt(37)))^(1/3)+w2*(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3)+w1*(2664-288*sqrt(37)+18*(w2*(64084+13856*sqrt(37)+j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3)+w1*(64084+13856*sqrt(37)-j*12*sqrt(3)*(192659-31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)-72*sqrt(111)+j*4*(37-8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)-72*sqrt(111)-j*4*(37-8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(12*pi/37)=(-1-sqrt(37)-w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3)+(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(16*pi/37)=(-1-sqrt(37)-w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+w1*(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3)+w2*(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(46*pi/37)=(-1-sqrt(37)-w1*(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-w2*(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+w2*(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))+j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3)+w1*(2664+288*sqrt(37)+18*(w1*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w2*(64084-13856*sqrt(37)-j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3))-j*162*(w2*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w1*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(10*pi/37)=(-1-sqrt(37)-(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+(2664+288*sqrt(37)+18*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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))/36;

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cos(26*pi/37)=(-1-sqrt(37)-(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+w2*(2664+288*sqrt(37)+18*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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)+w1*(2664+288*sqrt(37)+18*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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))/36;

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cos(38*pi/37)=(-1-sqrt(37)-(148-32*sqrt(37)+j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)-(148-32*sqrt(37)-j*12*sqrt(3)*(37-6*sqrt(37)))^(1/3)+w1*(2664+288*sqrt(37)+18*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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)+w2*(2664+288*sqrt(37)+18*(w2*(64084-13856*sqrt(37)+j*12*sqrt(3)*(192659+31458*sqrt(37)))^(1/3)+w1*(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))/36;

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cos(70*pi/37)=(-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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(34*pi/37)=(-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)+w2*(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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3)+w1*(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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36;

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cos(44*pi/37)=(-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)+w1*(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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3)+w2*(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*(w1*(444*sqrt(3)+72*sqrt(111)+j*4*(37+8*sqrt(37)))^(1/3)+w2*(444*sqrt(3)+72*sqrt(111)-j*4*(37+8*sqrt(37)))^(1/3)))^(1/3))/36.

未完待續(xù)，下一期：求sin(2*n*pi/37)！