Complex Numbers

Question

Write down the 5th roots of unity.

Show that (i) w^4 + w^3 + w^2 + w + 1 = 0 , where w=e^((2*pi*i)/5)

(ii) a = w + w^4 and b = w^2 + w^3 are the roots of t^2 + t - 1 =0.

Hence, find the value of ab.

Deduce that cos(2pi/5) + cos(4pi/5) + cos(6pi/5) + cos(8pi/5) = -1

Answer 1

I won't work all of the parts of this question for you, but I will provide some hints.

The 5th roots of unity (one) can be obtained from De Moivre's Theorem; they are

e^2πk/5 for k = 0 to 4.

If you plot those points on the complex plane, you will see that they are symmetric about the x-axis.

In the sum of those roots, the imaginary parts add to 0 and the real parts of the 4 complex roots will add to -1.

With that information you should be able to answer the question.