= In this question you may use the following fact: for a polynomial p(z) +. +a1z+ao...

 

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= In this question you may use the following fact: for a polynomial p(z) +. +a1z+ao with complex coefficients ao,..., an-1, the roots of p(z) sum to 2n+an-12-1 -an-1. (A root of a polynomial p(2) is a number a such that p(a) = 0.) (a) Show that the five fifth roots of 1 are precisely the roots of the polynomial f(z) = 25-1. (b) Find the five fifth roots of 1 in polar form, and show that they sum to zero. (In fact, it is true that for any n, the n'th roots of 1 sum to zero! But you need not show this.) (c) Using the above, or otherwise, show that I 2T 4m 67 8T cos(0) + cos + cos +cos +cos = 0 5 5 5 5 and 4 6 8 sin(0) + sin + sin + sin + sin = 0. 5 5 5 5 (You must show that these equations hold exactly. Showing that they hold to a number of decimal places is not sufficient.) = In this question you may use the following fact: for a polynomial p(z) +. +a1z+ao with complex coefficients ao,..., an-1, the roots of p(z) sum to 2n+an-12-1 -an-1. (A root of a polynomial p(2) is a number a such that p(a) = 0.) (a) Show that the five fifth roots of 1 are precisely the roots of the polynomial f(z) = 25-1. (b) Find the five fifth roots of 1 in polar form, and show that they sum to zero. (In fact, it is true that for any n, the n'th roots of 1 sum to zero! But you need not show this.) (c) Using the above, or otherwise, show that I 2T 4m 67 8T cos(0) + cos + cos +cos +cos = 0 5 5 5 5 and 4 6 8 sin(0) + sin + sin + sin + sin = 0. 5 5 5 5 (You must show that these equations hold exactly. Showing that they hold to a number of decimal places is not sufficient.)