Suppose cos(theta) + i*sin(theta) is a root of the equation xⁿ + a₁*x^(n-1) + a₂*x^(n-2) +... + a_(n-1)*x + aₙ = 0, where all coefficients a_r are real. Find the value of the sum a₁*cos(theta) + a₂*cos(2*theta) +... + aₙ*cos(n*theta), i.e. the sum from r = 1 to n of a_r*cos(r*theta).

Question

Accepted Answer

-1. Putting x = e^(i*theta), the polynomial equals zero; taking the real part of e^(-in*theta) times the equation isolates 1 + sum a_r cos(r*theta) = 0, so the required sum is -1.

Suppose cos(theta) + isin(theta) is a root of the equation xⁿ + a₁x^(n-1) + a₂x^(n-2) +... + a_(n-1)x + aₙ = 0, where all coefficients a_r are real. Find the value of the sum a₁cos(theta) + a₂cos(2theta) +... + aₙcos(ntheta), i.e. the sum from r = 1 to n of a_rcos(r*theta).

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