Trigonometric functions of rational numbers are algebraic numbers!

Geometry Level 5

What is the degree \(n\) of the minimal polynomial for \( \tan(1^\circ) \)?

Definition: The minimal polynomial for an algebraic number \(x\) is the monic polynomial \(f\) of degree \(n\) such that \(f(x) = 0\), \(f\) has rational coefficients, and for any \(f_1\) such that \(f_1(x) = 0\) and \(f_1\) has rational coefficients, \(\deg(f_1) \geq \deg(f)\).

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