User Name Remember Me? Password

 Algebra Pre-Algebra and Basic Algebra Math Forum

 February 3rd, 2015, 06:23 AM #1 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Roots of unity. Find the fifth roots of unity. If $\displaystyle \omega$ is the root with smallest positive argument if $\displaystyle u=\omega +\omega^4$ and $\displaystyle v=\omega^2+\omega^3$ Show that $\displaystyle u+v=-1$ and that $\displaystyle u-v=sqrt5$ Hence find $\displaystyle \cos72^{\circ}$ February 3rd, 2015, 06:54 AM #2 Member   Joined: Jan 2015 From: Orlando, Florida Posts: 92 Thanks: 10 here's the first part: w=e^(i2pi/5) u+v is the sum of the 5th primitive roots of unity, which is (w^5-1)/(w-1)-1. e^(i2pi/5)^5=e^(i2pi)=0 so we are left with -1 Thanks from topsquark February 3rd, 2015, 07:20 AM #3 Member   Joined: Jan 2015 From: Orlando, Florida Posts: 92 Thanks: 10 To find cos(72), note that cos(72)+cos(144)+cos(216)+cos(288 )=-1, and then spam the double angle identity until you're left with only cos(72), then you can solve the polynomial for cos(72). After that the second part (u-v) should be easy Thanks from topsquark February 3rd, 2015, 07:36 AM   #4
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
 Originally Posted by USAMO Reaper To find cos(72), note that cos(72)+cos(144)+cos(216)+cos(288 )=-1, and then spam the double angle identity until you're left with only cos(72), then you can solve the polynomial for cos(72).
Cheaty answer:
Code:
algdep(cos(72*Pi/180),2) Tags roots, unity Search tags for this page

,

,

,

### buktikan bahwa 1 cos 72 cos 144 cos 216 cos 288=0

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post not3bad Complex Analysis 2 December 20th, 2014 01:35 AM Jacob0793 Complex Analysis 3 March 7th, 2014 04:58 AM mathbalarka Number Theory 37 January 14th, 2014 03:57 AM lahuxixi Number Theory 3 November 21st, 2013 07:56 PM Elladeas Abstract Algebra 2 February 19th, 2011 01:20 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.      