The five fifth roots of unity add to zero

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Foundations of mathematical language

Mathematical Methods for Physics, Ch Complex numbers, the complex exponential and phasors

Problem

The five fifth roots of unity add to zero. Use that fact to find the exact values of cos⁡(2π/5)\cos(2\pi/5) and cos⁡(4π/5)\cos(4\pi/5). (a) Show that 1+2cos⁡(2π/5)+2cos⁡(4π/5)=01+2\cos(2\pi/5)+2\cos(4\pi/5)=0. (b) Find a second relation between the same two cosines, and solve for both exactly. Check your values numerically.

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