provide a full handwritten solution to all parts

3. Roots of unity (chapter 9) ) Show, as a consequence of De Moivre's Theorem, that all roots of the equation z" = 1 are located on the unit circle, and the roots come in conjugate pairs (except for real roots which are their own conjugates). Finally conclude that the roots form the vertices of a regular polygon with n sides (or n vertices) with one of the vertices located at the complex number 1. Throughout this question, let zo be the first root in counterclockwise direction after 1. b) Suppose you have Re(zo). Explain, hypothetically, how you use value of Re(zo) to construct all the roots of unity using compass and straight edge and using techniques of Greek Constructions as in 12.1. (You will return to this method in the following items.) ) Read the proof of Theorem 12.4.12, and explain i) Why does the expression zo have to satisfy (28 + 28 + . . . zo + 1) = 0, and why must the expression be defined? i) Why could we make the assumption _ = Zo? ii) How did we conclude that zo + 2 = 2Re(zo)? 1) For n = 3, use the method of 12.4.12 to calculate Re(zo) where zo the second cube-root of unity. Use this information to draw/construct cube roots of unity. (Please use proper steps of Greek Construction) ) We want to repeat this process for n = 6, but you need to do it in two ways: First, find the geometric relationship between third and sixth roots of unity, and without any computa- tions, construct all the sixth roots of unity from the previous question. TIma i) This time, only follow the method of Theorem 12.4.12 to compute the Re(zo) where zo is the first of the sixth roots of unity after 1 in counterclockwise direction. Imz ) Finally repeat the method of the proof of Theorem 12.4.12 to compute Re(zo) where zo is the first of the fifth roots of unity (in the counterclockwise direction, after 1.) (No drawing in this part.) g) Use your value of Re(zo) from part f), and in the following diagram, use methods of Greek Construction to draw the all the fifth roots of unity using compass and straight edge. You may use the semicircle and the rectangle given. Rez