Can you solve this?

[Milly48]

1.Show that if x = 18degrees then cos 2x = sin 3x 2.Hence find the exact value sin 18degrees 3.Then prove cos 36degrees - sin 18degrees = 1/2

[pinus]

Yes I can

[football]

Thats not a quize nor a puzle thats my homework.

[bernardo]

cos 36 = sin 54 = 0.809; 0.309; qed

[Jabba]

For the first part do what bernardo did but don't use your calculator, as tat proves nothing. You should know an equation relating cos x and sin (90 - x). For 2 and 3, what level is this? A-Level?

[Brugel]

1. cos2x=cos(2x18)=cos36=sin(90-36)=sin54=sin3x 2.sin54=cos36 so sin(36+18)=cos36 so sin36cos18+cos36sin18=cos36 so(2sin18cos18)cos18+(1-2(sin18)squared)sin18=1-2(sin18)squared so 2sin18(cos18)squared+sin18-2(sin18)cubed=1-2(sin18)squared so 2sin18(1-(sin18)squared)+sin18-2(sin18)cubed=1-2(sin18)squared so 2sin18-2(sin18)cubed+sin18-2(sin18)cubed=1-2(sin18)squared so 3sin18-4(sin18)cubed=1-2(sin18)squared so 4(sin18)cubed-2(sin18)squared-3sin18+1=0 so (sin18-1)(4(sin18)squared+2sin18-1)=0 so 4(sin18)squared+2sin18-1=0 since sin18>0 solving this using the quadratic formula and taking the positive square root gives sin18=(sqr(5)-1)/4, the exact value of sin18. 3. cos36-sin18 =1-2(sin18)squared-sin18 =1-2x(sqr(5)-1)/4x(sqr(5)-1)/4-(sqr(5)-1)/4 =1-2x(6-2sqr(5))/16-(sqr(5)-1)/4 =1-3/4+sqr(5)/4-sqr(5)/4+1/4 =1/2 as required. Sorry about the bad notation-I don't know how to do things like powers on the computer.