Question

(1) Recall on February 6 in class we discussed e 0 + e 2πi/n + e 4πi/n + · · · + e 2(n−1)πi/n = 0 and in order to explain why it was true we needed to show that the sum of the real parts equals 0 and the sum of the imaginary parts is equal to 0.

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

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

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

Answer 1