Roots of Unity question (1 Viewer)

[James Smith The Third]

Sup lads i'm back, I've worked out most of this question except for part iii of b and part c. I wrote out the whole question just for context. If anyone could give me advice on how to do part iii of b and part c it would be appreciated!

a. (i) Find the five fifth roots of unity, writing the complex roots in mod-arg form.
(ii) Show that the points in the complex plane representing these roots form a regular pentagon.
(iii) By considering the sum of these five roots, show that cos 2π 5 + cos 4π 5 = −1 2 .

b. (i) Show that z^5 − 1 = (z − 1)(z 4 + z 3 + z 2 + z + 1).
(ii) Hence show that z^4 + z^3 + z^2 + z + 1 = (z^2 − 2 cos 2π 5 z + 1)(z^2 − 2 cos 4π 5 z + 1).
(iii) By equating the coefficients of z in this identity, show that cos(π/5) = 1+ √ 5 / 4 .

c. (i) Use the substitution x = u+ 1 u to show that the equation x 2 +x−1 = 0 has roots 2 cos 2π 5 and 2 cos 4π 5 .
(ii) Deduce that cos π 5 cos 2π 5 = 1/ 4 .

 

[Drdusk]

Sup lads i'm back, I've worked out most of this question except for part iii of b and part c. I wrote out the whole question just for context. If anyone could give me advice on how to do part iii of b and part c it would be appreciated!

a. (i) Find the five fifth roots of unity, writing the complex roots in mod-arg form.
(ii) Show that the points in the complex plane representing these roots form a regular pentagon.
(iii) By considering the sum of these five roots, show that cos 2π 5 + cos 4π 5 = −1 2 .

b. (i) Show that z^5 − 1 = (z − 1)(z 4 + z 3 + z 2 + z + 1).
(ii) Hence show that z^4 + z^3 + z^2 + z + 1 = (z^2 − 2 cos 2π 5 z + 1)(z^2 − 2 cos 4π 5 z + 1).
(iii) By equating the coefficients of z in this identity, show that cos(π5) = 1+ √ 5 / 4 .

c. (i) Use the substitution x = u+ 1 u to show that the equation x 2 +x−1 = 0 has roots 2 cos 2π 5 and 2 cos 4π 5 .
(ii) Deduce that cos π 5 cos 2π 5 = 1/ 4 .

Just plugging iii into my calculator it does not seem to be true.

As for part c sub x = u + 1/u and expanding all that gives you



which is the same equation as the one in part b, so it obviously has the same roots. You just need to argue as to why there's only the two roots.

For part c-ii use the product of the roots to get you that identity

 

[James Smith The Third]

Just plugging iii into my calculator it does not seem to be true.

As for part c sub x = u + 1/u and expanding all that gives you



which is the same equation as the one in part b, so it obviously has the same roots. You just need to argue as to why there's only the two roots.

For part c-ii use the product of the roots to get you that identity

sorry i made a typo it should be π/5 not π5. thanks for the help on the substitution part though, that was really useful!

 

[Drdusk]

sorry i made a typo it should be π/5 not π5. thanks for the help on the substitution part though, that was really useful!

I know I did cos(pi/5). Seriously just type it in the calculator, it does not equal 1 + sqrt(5)/4

 

[Trebla]

When you equate the coefficients of z, you get the same result as (a)(iii) that is


Using double angles and supplementary angles this is equivalent to


and the result follows...

 

[Drdusk]

When you equate the coefficients of z, you get the same result as (a)(iii) that is


Using double angles and supplementary angles this is equivalent to


and the result follows...

The hell is my calculator doing then...

 

[Trebla]

The hell is my calculator doing then...

I think it's meant to say

 

[James Smith The Third]

When you equate the coefficients of z, you get the same result as (a)(iii) that is


Using double angles and supplementary angles this is equivalent to


and the result follows...

ah ok thanks! i ended up just using algebra by that point and got the answer. appreciate the help!