What is the value of sin 18°

 

Sin18°

A=18

5A=90

2A+3A=90

2A=90-3A

Multiplying by Sin on both sides

Sin2A=Sin(90-3A).                 {Sin(90-3A)=Cos3A}

Sin2A=Cos3A

2SinACosA=4Cos^3A-3CosA

2SinACosA-4Cos^3A+3CosA=0

CosA(2SinA-4Cos^2A+3)=0

2SinA-4Cos^2A+3=0

2SinA-4(1-Sin^2A)+3=0

2SinA-4+4Sin^2A+3=0

4Sin^2A+2SinA-1=0

Than we put the formula of quadratic equations

X=-b±√b^2-4ac/2a

a=4,b=2,c=-1

-2±√2^2-4(4)(-1)/2(4)

-2±√4-(-16)/8

-2±√20/8

-2±√5×4/8

-2±2√5/8

2(-1±1√5)/8

-1±√5/4 .    Ans.......