sin18°=sin36°/2
sin(x/2)= √(1-cosx)/2
sin(36°/2)= √(1-cos36°)/2
2sinxcosy= sin(x+y)-sin(x-y)
2cos36°sin18°=sin54°-sin18°
sin(2x)=2sinxcosx
sin72°= 2sin36°cos36°
sin36°= 2sin18°cos18°
---------------------------(*)
sin72°sin36°=4sin36°sin18°cos36°cos18° / :sin36°
sin72°=4sin18°cos36°cos18°
sinx= cos(90°-x)
sin72°=cos(90°-72°)
cos(90°-72°)=4sin18°cos36°cos18°
cos18°=4sin18°cos36°cos18° / :cos18°
1=4sin18°cos36° / :2
1/2= 2sin18°cos36°
Inlocuim in 2cos36°sin18°=sin54°-sin18°
1/2= sin54°-sin18°
sin54°=cos(90°-54°)
sin54°=cos36°
1/2= cos36°-sin18°
a^2-b^2= (a+b)(a-b)
a=cos36°+sin18°
b=cos36°-sin18°
(cos36°+sin18°)^2 - (cos36°-sin18°)^2=
(cos36°+sin18°+ cos36°-sin18°)(cos36°+sin18°- cos36°+sin18°)=
2cos36°*2sin18°=
2(cos36°+sin18°)
sin(2x)= 2sinxcosx
sin72°=2sin36°cos36°
sin36°=2sin18°cos18°
---------------------------(*)
sin72°sin36°=4sin36°sin18°cos36°cos18° / :sin36°
sin72°=4sin18°cos36°cos18°
sinx=cos(90°-x)
sin72°=cos(90°-72)
sin72°=cos18°
cos18°=4sin18°cos36°cos18° / :cos18°
1=4sin18°cos36° / :2
1/2=2sin18°cos36°
Inlocuim in (cos36°+sin18°)^2 - (cos36°-sin18°)^2= 2cos36°*2sin18°
2cos36°*2sin18°= 2*2cos36°sin18°= 2*2*1/2= 1
(cos36°+sin18°)^2 - (cos36°-sin18°)^2= 1
Dar am aflat ca 1/2= cos36°-sin18°
(cos36°+sin18°)^2 - (1/2)^2 = 1
(cos36°+sin18°)^2 = 1/4 + 1 = 5/4
cos36°+sin18= √5/2 (cos36° si sin18 nu pot fi negativi)
(cos36°+sin18) + (cos36°-sin18°)= √5/2 + 1/2
2cos36°= √5/2 + 1/2
cos36°= (√5/2 + 1/2)/2= (√5+1)/4
Dar sin18°=sin36°/2, sin(36°/2)= √(1-cos36°)/2
sin18°= √[(1-(√5+1)/4)/2]
sin18°= (√5-1)/4
