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February 3rd, 2015, 06:23 AM   #1
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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 $\displaystyle \cos72^{\circ}$
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February 3rd, 2015, 06:54 AM   #2
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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
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February 3rd, 2015, 07:20 AM   #3
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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
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February 3rd, 2015, 07:36 AM   #4
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Quote:
Originally Posted by USAMO Reaper View Post
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)
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