The golden ratio 1.61803 . . . is well known for having many relationships with natural phenomenon. "Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities" - see https://en.wikipedia.org/wiki/Golden_ratio. The following exercise illustrates how the golden ratio is related to the 5th roots of 1. (a) Find all solutions z C to the equation z 5 = 1. (1 mark) (b) Locate all of the solutions found in Part (a) on the Argand plane. (1/2 mark) (c) Show that the points representing the roots are the vertices of a regular pentagon. (1/2 mark) (d) Use the sum of a geometric series to find an expression for 1 + z + z 2 + z 3 + z 4 and hence show that z 5 1 = (z 1) z 4 + z 3 + z 2 + z + 1 . (1 mark) (e) f(z) = z 4 + z 3 + z 2 + z + 1 may be written as a product f(z) = g1(z)g2(z) where g1(z) = z 2 + 1z + 1, g2(z) = z 2 + 2z + 1. Show that 1 and 2 are roots of z 2 z 1. One of these roots is the famous golden ratio. (1 mark) (f) Use Parts (a) and (e) to determine cos 2 5 and cos 4 5 in exact form. (1 ma