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(1) Recall on February 6 in class we discussed e 0 + e 2?i/n +...

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Advance Math

(1) Recall on February 6 in class we discussed e 0 + e 2?i/n + e4?i/n + · · · + e 2(n?1)?i/n = 0 and in order to explain why it wastrue we needed to show that the sum of the real parts equals 0 andthe sum of the imaginary parts is equal to 0.

(a) In class I showed the following identity for n even usingthe fact that sin(2? ? x) = ? sin(x): sin(0) + sin(2?/n) +sin(4?/n) + · · · + sin(2(n ? 1)?/n) = 0 Do the same thing for nodd (make sure it is clear, at least to yourself, why the argumentis slightly different for n even and n odd).

(b) Using the identity cos(x) = ? cos(x + ?), show that cos(0) +cos(2?/n) + cos(4?/n) + · · · + cos(2(n ? 1)?/n) = 0 for neven.

(c) Why does the same proof not work for n odd ? Show andexplain what goes wrong for the example of n = 3.

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