Trigonometric solution

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

Elimination solution:

Since $\cos 12^{\circ}$ can be constructed, it is an algebraic number and thus "undefined" is not the answer.

If the degree of the polynomial was 0 or 1, then $\cos 12^{\circ}$ is a rational number. However, since it is not a rational number, the answer must be $\boxed{4}$ .