461exam2sol

# 461exam2sol - Math 461 F Exam 2 - Fri Mar 25 Spring 2011...

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Math 461 F Spring 2011 Exam 2 — Fri Mar 25 Drew Armstrong Problem 1. [6 points] (a) Accurately state Gauss and Wantzel’s theorem on the constructibility of regular poly- gons with straightedge-and-compass. A regular n -gon is constructible if and only if n is equal to a power of 2 multiplied by distinct Fermat primes. (b) Yes or no. For the following values of n , state whether the regular n -gon is constructible. n regular n -gon constructible? 5 Yes 7 No 15 Yes 17 Yes Problem 2. [6 points] In this problem we want to compute cos ( 4 π 5 ) . (a) Let ω = cos ( 4 π 5 ) + i sin ( 4 π 5 ) . Label the vertices of the given regular pentagon (in the complex plane) by powers of ω . (b) Find a formula for u = ω + ω - 1 and solve it to ﬁnd cos ( 4 π 5 ) . (Hint: The sum of the ﬁve vertices is zero.) Since u = ω + ω - 1 = ω + ω = 2 cos ( 4 π 5 ) , we wish to solve for u . We know that the sum of all of the ﬁfth roots of unity is zero. That is, ω 2 + ω + 1 + ω - 1 + ω - 2 = 0 . So we wish to express the sum of these roots in terms of

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## This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.

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461exam2sol - Math 461 F Exam 2 - Fri Mar 25 Spring 2011...

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