41边形(二)-每日热门
三.求sin(2*k*pi/41)
I.求实数项B。以下,B110,B120,B210,B220都是实数,Bmnx(x不等于0)都是纯的五次根式。前一章已经声明五次单位根f1~f4的值。
B110=2*(sin(2*pi/41)+sin(20*pi/41)+sin(36*pi/41)+sin(32*pi/41)+sin(74*pi/41))=sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))/2;
(资料图)
B111=2*(sin(2*pi/41)+f1*sin(20*pi/41)+f2*sin(36*pi/41)+f3*sin(32*pi/41)+f4*sin(74*pi/41));
B112=2*(sin(2*pi/41)+f2*sin(20*pi/41)+f4*sin(36*pi/41)+f1*sin(32*pi/41)+f3*sin(74*pi/41));
B113=2*(sin(2*pi/41)+f3*sin(20*pi/41)+f1*sin(36*pi/41)+f4*sin(32*pi/41)+f2*sin(74*pi/41));
B114=2*(sin(2*pi/41)+f4*sin(20*pi/41)+f3*sin(36*pi/41)+f2*sin(32*pi/41)+f1*sin(74*pi/41));
B120=2*(sin(18*pi/41)+sin(16*pi/41)+sin(78*pi/41)+sin(42*pi/41)+sin(10*pi/41))=sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))/2;
B121=2*(sin(18*pi/41)+f1*sin(16*pi/41)+f2*sin(78*pi/41)+f3*sin(42*pi/41)+f4*sin(10*pi/41));
B122=2*(sin(18*pi/41)+f2*sin(16*pi/41)+f4*sin(78*pi/41)+f1*sin(42*pi/41)+f3*sin(10*pi/41));
B123=2*(sin(18*pi/41)+f3*sin(16*pi/41)+f1*sin(78*pi/41)+f4*sin(42*pi/41)+f2*sin(10*pi/41));
B124=2*(sin(18*pi/41)+f4*sin(16*pi/41)+f3*sin(78*pi/41)+f2*sin(42*pi/41)+f1*sin(10*pi/41));
B210=2*(sin(6*pi/41)+sin(60*pi/41)+sin(26*pi/41)+sin(14*pi/41)+sin(58*pi/41))=sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))/2;
B211=2*(sin(6*pi/41)+f1*sin(60*pi/41)+f2*sin(26*pi/41)+f3*sin(14*pi/41)+f4*sin(58*pi/41));
B212=2*(sin(6*pi/41)+f2*sin(60*pi/41)+f4*sin(26*pi/41)+f1*sin(14*pi/41)+f3*sin(58*pi/41));
B213=2*(sin(6*pi/41)+f3*sin(60*pi/41)+f1*sin(26*pi/41)+f4*sin(14*pi/41)+f2*sin(58*pi/41));
B214=2*(sin(6*pi/41)+f4*sin(60*pi/41)+f3*sin(26*pi/41)+f2*sin(14*pi/41)+f1*sin(58*pi/41));
B220=2*(sin(28*pi/41)+sin(34*pi/41)+sin(12*pi/41)+sin(38*pi/41)+sin(52*pi/41))=sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))/2;
B221=2*(sin(28*pi/41)+f1*sin(34*pi/41)+f2*sin(12*pi/41)+f3*sin(38*pi/41)+f4*sin(52*pi/41));
B222=2*(sin(28*pi/41)+f2*sin(34*pi/41)+f4*sin(12*pi/41)+f1*sin(38*pi/41)+f3*sin(52*pi/41));
B223=2*(sin(28*pi/41)+f3*sin(34*pi/41)+f1*sin(12*pi/41)+f4*sin(38*pi/41)+f2*sin(52*pi/41));
B224=2*(sin(28*pi/41)+f4*sin(34*pi/41)+f3*sin(12*pi/41)+f2*sin(38*pi/41)+f1*sin(52*pi/41));
II.求五次根式内的表达式H。
H111=B111^5; H112=B112^5; H113=B113^5; H114=B114^5; H110=B110^5;
H121=B121^5; H122=B122^5; H123=B123^5; H124=B124^5; H120=B120^5;
H211=B211^5; H212=B212^5; H213=B213^5; H214=B214^5; H210=B210^5;
H221=B221^5; H222=B222^5; H223=B223^5; H224=B224^5; H220=B220^5;
为了方便计算,引入中间变量I:Ixyz=Hxyz/Bxy0.
I111=H111/B110; I112=H112/B110; I113=H113/B110; I114=H114/B110; I110=H110/B110;
I121=H121/B120; I122=H122/B120; I123=H123/B120; I124=H124/B120; I120=H120/B120;
I211=H211/B210; I212=H212/B210; I213=H213/B210; I214=H214/B210; I210=H210/B210;
I221=H221/B220; I222=H222/B220; I223=H223/B220; I224=H224/B220; I220=H220/B220;
再引入O、P、Q、R、S五个中间变量:
O11=(I111+I112+I113+I114+I110)/5=0.125*(5095-713*sqrt(41)-(1061+93*sqrt(41))*sqrt(205-32*sqrt(41)));
O12=(I121+I122+I123+I124+I120)/5=0.125*(5095-713*sqrt(41)+(1061+93*sqrt(41))*sqrt(205-32*sqrt(41)));
O21=(I211+I212+I213+I214+I210)/5=0.125*(5095+713*sqrt(41)+(1061-93*sqrt(41))*sqrt(205+32*sqrt(41)));
O22=(I221+I222+I223+I224+I220)/5=0.125*(5095+713*sqrt(41)-(1061-93*sqrt(41))*sqrt(205+32*sqrt(41)));
P11=(I111/f1+I112/f2+I113/f3+I114/f4+I110)/5=0.125*(-4400+760*sqrt(41)-(420+140*sqrt(41))*sqrt(205-32*sqrt(41)));
P12=(I121/f1+I122/f2+I123/f3+I124/f4+I120)/5=0.125*(-4400+760*sqrt(41)+(420+140*sqrt(41))*sqrt(205-32*sqrt(41)));
P21=(I211/f1+I212/f2+I213/f3+I214/f4+I210)/5=0.125*(-4400-760*sqrt(41)+(420-140*sqrt(41))*sqrt(205+32*sqrt(41)));
P22=(I221/f1+I222/f2+I223/f3+I224/f4+I220)/5=0.125*(-4400-760*sqrt(41)-(420-140*sqrt(41))*sqrt(205+32*sqrt(41)));
Q11=(I111/f2+I112/f4+I113/f1+I114/f3+I110)/5=0.125*(2740-380*sqrt(41)-240*sqrt(205-32*sqrt(41)));
Q12=(I121/f2+I122/f4+I123/f1+I124/f3+I120)/5=0.125*(2740-380*sqrt(41)+240*sqrt(205-32*sqrt(41)));
Q21=(I211/f2+I212/f4+I213/f1+I214/f3+I210)/5=0.125*(2740+380*sqrt(41)+240*sqrt(205+32*sqrt(41)));
Q22=(I221/f3+I222/f1+I223/f4+I224/f2+I220)/5=0.125*(2740+380*sqrt(41)-240*sqrt(205+32*sqrt(41)));
R11=(I111/f3+I112/f1+I113/f4+I114/f2+I110)/5=0.125*(1775-225*sqrt(41)+(75+35*sqrt(41))*sqrt(205-32*sqrt(41)));
R12=(I121/f3+I122/f1+I123/f4+I124/f2+I120)/5=0.125*(1775-225*sqrt(41)-(75+35*sqrt(41))*sqrt(205-32*sqrt(41)));
R21=(I211/f3+I212/f1+I213/f4+I214/f2+I210)/5=0.125*(1775+225*sqrt(41)-(75-35*sqrt(41))*sqrt(205+32*sqrt(41)));
R22=(I221/f3+I222/f1+I223/f4+I224/f2+I220)/5=0.125*(1775+225*sqrt(41)+(75-35*sqrt(41))*sqrt(205+32*sqrt(41)));
S11=(I111/f4+I112/f3+I113/f2+I114/f1+I110)/5=0.125*(-3980+680*sqrt(41)-(240+100*sqrt(41))*sqrt(205-32*sqrt(41)));
S12=(I121/f4+I122/f3+I123/f2+I124/f1+I120)/5=0.125*(-3980+680*sqrt(41)+(240+100*sqrt(41))*sqrt(205-32*sqrt(41)));
S21=(I211/f4+I212/f3+I213/f2+I214/f1+I210)/5=0.125*(-3980-680*sqrt(41)+(240-100*sqrt(41))*sqrt(205+32*sqrt(41)));
S22=(I221/f4+I222/f3+I223/f2+I224/f1+I220)/5=0.125*(-3980-680*sqrt(41)-(240-100*sqrt(41))*sqrt(205+32*sqrt(41)));
根据1+f1+f2+f3+f4=0整理得到:
H111=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f2+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);
H112=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f4+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);
H113=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f1+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);
H114=0.0625*sqrt(41+3*sqrt(41)-sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)-(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)-(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(6720-1060*sqrt(41)+100*sqrt(8405-1312*sqrt(41)))*f3+(5755-905*sqrt(41)+(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);
H121=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f1+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f2+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f3);
H122=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f2+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f4+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f1);
H123=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f3+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f1+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f4);
H124=0.0625*sqrt(41+3*sqrt(41)+sqrt(410-2*sqrt(41)))*(9075-1393*sqrt(41)+(821-7*sqrt(41))*sqrt(205-32*sqrt(41))+(-420+80*sqrt(41)+(180+40*sqrt(41))*sqrt(205-32*sqrt(41)))*f4+(6720-1060*sqrt(41)-100*sqrt(8405-1312*sqrt(41)))*f3+(5755-905*sqrt(41)-(315+135*sqrt(41))*sqrt(205-32*sqrt(41)))*f2);
H211=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f2+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);
H212=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f4+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);
H213=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f1+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);
H214=0.0625*sqrt(41-3*sqrt(41)-sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)+(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)+(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(6720+1060*sqrt(41)+100*sqrt(8405+1312*sqrt(41)))*f3+(5755+905*sqrt(41)-(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);
H221=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f1+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f2+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f3);
H222=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f2+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f4+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f1);
H223=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f3+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f1+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f4);
H224=0.0625*sqrt(41-3*sqrt(41)+sqrt(410+2*sqrt(41)))*(9075+1393*sqrt(41)-(821+7*sqrt(41))*sqrt(205+32*sqrt(41))+(-420-80*sqrt(41)-(180-40*sqrt(41))*sqrt(205+32*sqrt(41)))*f4+(6720+1060*sqrt(41)-100*sqrt(8405+1312*sqrt(41)))*f3+(5755+905*sqrt(41)+(315-135*sqrt(41))*sqrt(205+32*sqrt(41)))*f2);
2.比对五次根式的辐角,并用B、H^(1/5)线性表示sin(2*k*pi/41)。这些数值比cos(2*k*pi/41)复杂一些。
代入已知数值得到:
I.k为2次剩余
sin(2*pi/41)=(B110+f2*H111^(1/5)+f3*H114^(1/5)+f1*H112^(1/5)+f4*H113^(1/5))/10;
sin(20*pi/41)=(B110+f1*H111^(1/5)+f4*H114^(1/5)+f4*H112^(1/5)+f1*H113^(1/5))/10;
sin(36*pi/41)=(B110+H111^(1/5)+H114^(1/5)+f2*H112^(1/5)+f3*H113^(1/5))/10;
sin(32*pi/41)=(B110+f4*H111^(1/5)+f1*H114^(1/5)+H112^(1/5)+H113^(1/5))/10;
sin(74*pi/41)=(B110+f3*H111^(1/5)+f2*H114^(1/5)+f3*H112^(1/5)+f2*H113^(1/5))/10;
sin(18*pi/41)=(B120+H121^(1/5)+H124^(1/5)+f2*H122^(1/5)+f3*H123^(1/5))/10;
sin(16*pi/41)=(B120+f4*H121^(1/5)+f1*H124^(1/5)+H122^(1/5)+H123^(1/5))/10;
sin(78*pi/41)=(B120+f3*H121^(1/5)+f2*H124^(1/5)+f3*H122^(1/5)+f2*H123^(1/5))/10;
sin(42*pi/41)=(B120+f2*H121^(1/5)+f3*H124^(1/5)+f1*H122^(1/5)+f4*H123^(1/5))/10;
sin(10*pi/41)=(B120+f1*H121^(1/5)+f4*H124^(1/5)+f4*H122^(1/5)+f1*H123^(1/5))/10;
II.k不为2次剩余
sin(6*pi/41)=(B210+f3*H211^(1/5)+f2*H214^(1/5)+H212^(1/5)+H213^(1/5))/10;
sin(60*pi/41)=(B210+f2*H211^(1/5)+f3*H214^(1/5)+f3*H212^(1/5)+f2*H213^(1/5))/10;
sin(26*pi/41)=(B210+f1*H211^(1/5)+f4*H214^(1/5)+f1*H212^(1/5)+f4*H213^(1/5))/10;
sin(14*pi/41)=(B210+H211^(1/5)+H214^(1/5)+f4*H212^(1/5)+f1*H213^(1/5))/10;
sin(58*pi/41)=(B210+f4*H211^(1/5)+f1*H214^(1/5)+f2*H212^(1/5)+f3*H213^(1/5))/10;
sin(28*pi/41)=(B220+f1*H221^(1/5)+f4*H224^(1/5)+H222^(1/5)+H223^(1/5))/10;
sin(34*pi/41)=(B220+H221^(1/5)+H224^(1/5)+f3*H222^(1/5)+f2*H223^(1/5))/10;
sin(12*pi/41)=(B220+f4*H221^(1/5)+f1*H224^(1/5)+f1*H222^(1/5)+f4*H223^(1/5))/10;
sin(38*pi/41)=(B220+f3*H221^(1/5)+f2*H224^(1/5)+f4*H222^(1/5)+f1*H223^(1/5))/10;
sin(52*pi/41)=(B220+f2*H221^(1/5)+f3*H224^(1/5)+f2*H222^(1/5)+f3*H223^(1/5))/10;
四.41次单位根举例
前一章声明了数列A、G的表达式。根据exp(j*x)=cos(x)+j*sin(x)得到:
exp(2*pi*j/41)=(A110+G111^(1/5)+G114^(1/5)+f3*G112^(1/5)+f2*G113^(1/5)+j*(B110+f2*H111^(1/5)+f3*H114^(1/5)+f1*H112^(1/5)+f4*H113^(1/5)))/10;
exp(6*pi*j/41)=(A210+G211^(1/5)+G214^(1/5)+f1*G212^(1/5)+f4*G213^(1/5)+j*(B210+f3*H211^(1/5)+f2*H214^(1/5)+H212^(1/5)+H213^(1/5)))/10;
exp(18*pi*j/41)=(A120+f1*G121^(1/5)+f4*G124^(1/5)+f3*G122^(1/5)+f2*G123^(1/5)+j*(B120+H121^(1/5)+H124^(1/5)+f2*H122^(1/5)+f3*H123^(1/5)))/10;
exp(54*pi*j/41)=(A220+f2*G221^(1/5)+f3*G224^(1/5)+f4*G222^(1/5)+f1*G223^(1/5)-j*(B220+f1*H221^(1/5)+f4*H224^(1/5)+H222^(1/5)+H223^(1/5)))/10.
其他的单位根也很容易得到。
五.写在最后
这些计算结果表明正41边形可通过至少两次5等分角作出。可对up主x_4597的x^41-1=0求解视频内容,作为有益补充。但难免有缺漏不足之处,敬请诸公斧正。
谢谢大家!