derive the sin(18) degrees

Let x = 18, so 2x = 36 and 3x = 54. Hence 2x = 90- 3x.
Therefore, sin2x = sin(90-3x) = cos3x
Or 2sinx cosx = 4cos³x – 3cosx.
Hence either cosx = 0 which gives x = 90,
or 2sinx = 4cos²x-3 = 1-4sin²x
Therefore 4sin²x + 2sinx =1.
By Solving the quadratic equation, We have sin (18) = (sqrt(5)-1)/ 4. Thanks.

Approved

Take A=18 so that 5A=90. 5A can be written as. 2A +3A=90 2A=90-3AApply sin on both sides and solve Solve

Approved