Question 1209455
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Answer: <font color=red>-1/2</font>


Explanation


Each term of
cos(2pi/7)+cos(4pi/7)+cos(6pi/7)
is of the form
cos(2k*pi/7)
where k is from the set {1,2,3}


It might not be entirely obvious, but the expression cos(2k*pi/7) suggests we'll be using the <a href="https://en.wikipedia.org/wiki/Root_of_unity">nth roots of unity</a>. 
In this case n = 7.


The roots of z^n = 1 are of the form cis(2k*pi/n) 
cis is shorthand for "cosine i sine"
for example, cis(2pi) = cos(2pi) + i*sin(2pi)
k is from the set {0,1,2,...,n-1}
Since the expression we're calculating involves cosines only, we'll focus on the real parts of cis(2k*pi/7) and ignore the imaginary parts.


I'll be using the trig identity cos(x) = cos(2pi - x) to be able to say something like cos(8pi/7) = cos(6pi/7)
Here's a bit of scratch work to show what I mean
cos(8pi/7) = cos(2pi - 8pi/7) 
= cos(14pi/7 - 8pi/7)
= cos( (14pi-8pi)/7 )
= cos(6pi/7)
This will be useful to show that we have symmetry going on.
Similar steps are applied to demonstrate cos(10pi/7) = cos(4pi/7) and cos(12pi/7) = cos(2pi/7)


Here are the real parts of the 7 roots for z^7 = 1.
cos(0) = 1
cos(2pi/7) = a
cos(4pi/7) = b
cos(6pi/7) = c
cos(8pi/7) = cos(6pi/7) = c
cos(10pi/7) = cos(4pi/7) = b
cos(12pi/7) = cos(2pi/7) = a
Once again note the symmetry.


Those 7 values add to
1+(a+b+c)+(c+b+a) = 1+2*(a+b+c)
The goal is to find a+b+c.


The roots of z^n = 1 add to 0 which is a result of <a href="https://en.wikipedia.org/wiki/Vieta%27s_formulas">Vieta's formulas</a>
This means the real parts of the roots must add to 0.
1+2*(a+b+c) = 0
a+b+c = <font color=red>-1/2</font> is the final answer.


Verification using <a href="https://www.wolframalpha.com/input?i=cos%282pi%2F7%29%2Bcos%284pi%2F7%29%2Bcos%286pi%2F7%29">WolframAlpha</a>
You can also use <a href="https://www.desmos.com/calculator/jnqugdsgy2">Desmos</a> and <a href="https://www.geogebra.org/calculator/kvtfewjr">GeoGebra</a> among many other online tools.
Keep in mind that -1/2 = -0.5 of course.


ikleyn your "solution" was complete garbage which I cleaned up. You're welcome. You need to stop being so arrogant with such a head of hot air. And you do NOT own Vieta's formulas nor anything you presented in your solution. You are a complete joke. Stop wasting everyone's time. And I just found out that you use AI in nearly all of your solutions. That explains a lot.
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