Unformatted text preview: Let Î¶ = e 2 Ï€i/ 5 . (i) SHOW that 1 + Î¶ + Î¶ 2 + Î¶ 3 + Î¶ 4 = 0. HINT. Expand (1Î¶ )(1 + Î¶ + Î¶ 2 + Î¶ 3 + Î¶ 4 ), and use the fact that Î¶ 5 = 1. (ii) SHOW that Î¶ + Î¶ 4 = 2cos 2 Ï€ 5 . HINT. Remember that cos(2 Ï€Î¸ ) = cos( Î¸ ) and sin(2 Ï€Î¸ ) =sin( Î¸ ). (iii) SHOW that Î¶ 2 + Î¶ 3 = ( Î¶ + Î¶ 4 ) 22. (iv) Hence SHOW that x = 2cos 2 Ï€ 5 is a root of the polynomial x 2 + x1 = 0 , and that cos 2 Ï€ 5 = âˆš 51 4 . 1...
View
Full Document
 Spring '08
 Staff
 Cos, Equivalence relation, Complex number, Z1

Click to edit the document details