(a) Let u 2p/7. Use the reference angle concept to explain why cos 3u cos 4u, then use

Chapter 12, Problem 53

(choose chapter or problem)

(a) Let u 2p/7. Use the reference angle concept to explain why cos 3u cos 4u, then use your calculator to confirm the result. (b) For this portion of the exercise, assume as given the following two trigonometric identities: cos 3u 4 cos3 u 3 cos u cos 4u 8 cos4 u 8 cos2 u 1 Use these identities and the result in part (a) to show that cos(2p/7) is a root of the equation 8x4 4x3 8x2 3x 1 0 (1) (c) List the prossibilities for the rational roots of equation (1). Then use synthetic division and the remainder theorem to show that there is only one rational root. Check that the reduced equation in this case is 8x3 4x2 4x 1 0 (2) (d) The work in parts (a) through (c) shows that the number cos(2p/7) is a root of equation (2). By following the same technique, it can be shown that the numbers cos(4p/7) and cos(6p/7) also are roots of equation (2)