Can we show a sum of symmetrical cosine values is zero by using roots of unity?

Can we show that

$$\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{5\pi}{7}+\cos\frac{6\pi}{7}=0$$

by considering the seventh roots of unity? If so how could we do it?

Also I have observed that

$$\cos\frac{\pi}{5}+\cos\frac{2\pi}{5}+\cos\frac{3\pi}{5}+\cos\frac{4\pi}{5}=0$$

as well, so just out of curiosity, is it true that $$\sum_{k=1}^{n-1} \cos\frac{k\pi}{n} = 0$$

for all $$n$$ odd?

• Use complex numbers, something like this Commented May 21, 2019 at 21:05
• You could use that $\cos (\pi-\theta)=-\cos \theta$ instead, and just pair each cosine with its negative. Commented May 21, 2019 at 21:10
• Also, it works for all $n$'s, not just odd ones. Commented May 21, 2019 at 21:19
• because the unpaired one is $\cos (\pi /2)$ Commented May 21, 2019 at 21:41
• @Acccumulation Even for $n=1$, one has $\sum_{k=1}^{n-1}\cos(k\pi/n)=0$. Commented May 22, 2019 at 5:39

Note that $$\cos(\pi - \alpha)= - \cos(\alpha)$$ Therefore $$\cos(\frac{\pi}{7})+\cos(\frac{2\pi}{7})+\cos(\frac{3\pi}{7})+\cos(\frac{4\pi}{7})+\cos(\frac{5\pi}{7})+\cos(\frac{6\pi}{7})=$$

$$\cos(\frac{\pi}{7})+\cos(\frac{2\pi}{7})+\cos(\frac{3\pi}{7})-\cos(\frac{3\pi}{7})-\cos(\frac{2\pi}{7})-\cos(\frac{\pi}{7})=0$$

The same goes for other natural numbers $$n$$ instead of $$7$$.

• and even number also Commented May 21, 2019 at 21:41

I think you can use Euler's Formula.

The Nth roots of unity = $$e^{2\pi k i/N}$$ for values of k between $$0$$ and $$N-1$$ inclusive.

There sum from k to $$N-1$$ is a geometric series.

$$S= \sum_{k=0}^{N-1} \ e^{2\pi i k/N}=\frac{1\cdot e^{(2\pi i /N)N}-1}{e^{2\pi i /N}-1}$$

The numerator is zero for any N.

But the real part of $$S$$ is the real part of the individual terms of the sum, i.e. the cosines. The real part of the sum is zero so the sum of the real parts of the roots of unity is 0.

• <pedant>This works only for integer N greater than 1</pedant> Commented May 22, 2019 at 5:16

$$1 + \cos\theta + \cos2\theta +... + \cos n\theta = \frac{1}{2} + \frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\frac{\theta}{2})}$$
Then for $$\forall n\in\mathbb{N}, n>0$$ $$\cos\frac{\pi}{n+1}+ \cos\frac{2\pi}{n+1} +... + \cos \frac{n\pi}{n+1} = \frac{\sin\left[(n+\frac{1}{2})\frac{\pi}{n+1}\right]}{2\sin\left(\frac{\pi}{2(n+1)}\right)}-\frac{1}{2}=\\ \frac{\sin\left[\frac{2n+1}{2(n+1)}\pi\right]}{2\sin\left(\frac{\pi}{2(n+1)}\right)}-\frac{1}{2}= \frac{\sin\left[\pi-\frac{\pi}{2(n+1)}\right]}{2\sin\left(\frac{\pi}{2(n+1)}\right)}-\frac{1}{2}=\frac{1}{2}-\frac{1}{2}=0$$