Film, Media & TV4 mins ago
Can you solve this?
5 Answers
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
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For more on marking an answer as the "Best Answer", please visit our FAQ.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.