Explicit Trigonometry: Day 6–
Back to Day 1–5
Hereafter, we put i = √-1 and promise (re^{θi})^{1/n} := r^{1/n}e^{θi/n} for r > 0, -π < θ ≤ π .
Note that ℚ[cos π/n] is an abelian extension of ℚ, whose extension degree is Euler's totient function φ(n) if n is even and φ(n)/2 if n is odd. So, our rule is to express cos π/n only using +, -, ×, ÷, and dth roots with d no greater than the largest prime divisor of φ(n), starting from rational numbers. This is possible thanks to a bit of Galois theory and excludes some boring expressions like cos π/p = ( (-1)^{1/p} + (-1)^{-1/p} )/2 for a prime p. However, for powers of a prime, our rule allows things like what we did for cos π/9.