What is the least degree of a polynomial with integer coefficient which has as a root?
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12° * 5 = 60°
let 12° = θ ̇
thus, 5θ = 60°
3θ + 2θ = 60°
cos(3θ+ 2θ) = 1/2 (since cos60° = 1/2)
cos3θcos2θ - sin3θsin2θ=1/2
(4cos³θ-3cosθ)(2cos²θ-1)-(3sinθ-4sin³θ)2sinθcosθ=1/2
let cosθ=x (4x³-3x)(2x²-1) - 2x(3-4(1-x²))(1-x²)=1/2
on solving we’ll have
32x^5- 40x³+10x -1 =0
(x-1/2)(32x^4 + 16x³- 32x²-16x +2)=0
but x≠1/2
16x^4 + 8x³ -16x² -8x + 1=0 thus degree is 4