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Reference: http://xkcd.com/1047/

We tried various different trigonometric identities. Still no luck.

Geometric interpretation would be also welcome.

EDIT: Very good answers, I'm clearly impressed. I followed all the answers and they work! I can only accept one answer, the others got my upvote.

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This seems helpful: in.answers.yahoo.com/question/index?qid=20110731075450AAX6Prz –  Rasmus May 3 '12 at 14:01
    
Drat. I was trying to find a way to prove $\sum_{k=1}^7\cos\frac{2k\pi-1}7=0$ which could be simplified to double the left hand side plus $\cos \pi$ equals 0. –  Mike May 3 '12 at 14:20
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+1 for the fantastic reference! –  Georges Elencwajg May 3 '12 at 16:32
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I noticed interesting reference about the subject. $\cos(\pi/7), \cos(3 \pi/7) , \cos(5 \pi/7)$ are root of the polinom $8x^3-4x^2-4x+1=0$. It is not proof but I can see easily the sum of roots are $-(-4)/8=1/2$.en.wikipedia.org/wiki/Algebraic_number –  Mathlover May 4 '12 at 0:47
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9 Answers

up vote 20 down vote accepted

Hint: start with $e^{i\pi/7} = \cos(\pi/7) + i\sin(\pi/7)$ and the fact that the lhs is a 7th root of -1.

Let $u = e^{i\pi/7}$, then we want to find $\Re(u + u^3 + u^5)$.

Then we have $u^7 = -1$ so $u^6 - u^5 + u^4 - u^3 + u^2 -u + 1 = 0$.

Re-arranging this we get: $u^6 + u^4 + u^2 + 1 = u^5 + u^3 + u$.

If $a = u + u^3 + u^5$ then this becomes $u a + 1 = a$, and rearranging this gives $a(1 - u) = 1$, or $a = 1 / (1 - u)$.

So all we have to do is find $\Re(1 / (1 - u))$.

$1 / (1 - u) = 1 / (1 - \cos(\pi/7) - i \sin(\pi/7)) = (1 - \cos(\pi/7) + i \sin(\pi/7)) / (2 - 2 \cos(\pi/7))$

so

$\Re(1/(1-u)) = (1 - \cos(\pi/7)) / (2 - 2\cos(\pi/7)) = 1/2 $

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Very elegant as the calculations are kept to the minimum! –  S4M Sep 19 '12 at 19:47
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You could obtain this by first multiplying $\cos(\pi/7)+\cos(3\pi/7)+\cos(5\pi/7)$ by $2\sin(\pi/7)$ and then applying the double angle formula for $\sin$ and sum to product formula for $\cos x \sin y$.

Let $a=\pi/7$, $b=3\pi/7$, and $c=5\pi/7$.

Using $$ 2\cos x\sin x =\sin 2x $$ and $$ \cos x\sin y= {\sin(x+y)-\sin(x-y)\over2 } $$ we have $$ \eqalign{ (\cos a+\cos b+\cos c)\cdot 2\sin a &=\color{maroon}{2\cos a\sin a}+\color{darkgreen}{2\cos b\sin a} +\color{darkblue}{2\cos c\sin a}\cr &=\color{maroon}{\sin 2a }+\color{darkgreen}{\sin(b+a)-\sin(b-a)}+\color{darkblue}{ \sin(c+a)-\sin(c-a)}\cr &=\color{teal}{\sin(2\pi/7) }+ {\color{purple}{\sin(4\pi/7)}\color{teal}{-\sin(2\pi/7)}}+ { \sin(6\pi/7)-\color{purple}{\sin(4\pi/7)}}\cr &=\sin(6\pi/7)\cr &=\sin (\pi/7)\cr &=\sin a. } $$ Now divide both sides of $$ (\cos a+\cos b+\cos c)\cdot 2\sin a =\sin a $$ by $2\sin a$.

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To elaborate on Mathlover's comment, the three numbers $\cos\frac{\pi}{7}$, $\cos\frac{3\pi}{7}$, and $\cos\frac{5\pi}{7}$ are the three roots of the monic Chebyshev polynomial of the third kind

$$\hat{V}_n(x)=\frac{\cos\left(\left(n+\frac12\right)\arccos\,x\right)}{2^n\cos\frac{\arccos\,x}{2}}=\frac1{2^n}\left(U_n(x)-U_{n-1}(x)\right)$$

where $U_n(x)=\frac{\sin((n+1)\arccos\,x)}{\sqrt{1-x^2}}$ is the usual Chebyshev polynomial of the second kind, and the last relationship is derived through the trigonometric identity

$$2\sin\frac{\theta}{2}\cos\left(\left(n+\frac12\right)\theta\right)=\sin((n+1)\theta)-\sin\,n\theta$$

The question then is essentially asking for the negative of the coefficient of the $x^2$ term of $\frac18(U_3(x)-U_2(x))$ (Vieta); we can generate the two polynomials using the definition given above, or through an appropriate recursion relation. We then have

$$\begin{align*}U_2(x)&=4x^2-1\\U_3(x)&=8x^3-4x\end{align*}$$

and we thus have

$$\hat{V}_3(x)=\frac18((8x^3-4x)-(4x^2-1))=x^3-\frac{x^2}{2}-\frac{x}{2}+\frac18$$

which yields the identities

$$\begin{align*} \cos\frac{\pi}{7}+\cos\frac{3\pi}{7}+\cos\frac{5\pi}{7}&=\frac12\\ \cos\frac{\pi}{7}\cos\frac{3\pi}{7}+\cos\frac{3\pi}{7}\cos\frac{5\pi}{7}+\cos\frac{\pi}{7}\cos\frac{5\pi}{7}&=-\frac12\\ \cos\frac{\pi}{7}\cos\frac{3\pi}{7}\cos\frac{5\pi}{7}&=-\frac18 \end{align*}$$

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Thanks a lot for clearance. you shew beauty of Mathematics –  Mathlover May 4 '12 at 9:07
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Geometric picture: if $N$ is an integer greater than 1, the points $\left(\cos\frac{2\pi j}{N},\sin\frac{2\pi j}{N}\right)$, for $0\le j\lt N$, are equally distributed on the unit circle. More precisely, this set of points is unchanged if you rotate the plane by $\frac{2\pi}{N}$ about the origin. If we think of the points as a set of equal masses, then the center of mass has to be a point that is unchanged under rotation. There is only one such point.

With a little bit of work, the sum you have can be connected with this center of mass calculation.

To elaborate on the above: The matrix that rotates a vector by $\frac{2\pi}{N}$ is $$ R=\begin{bmatrix} \cos\frac{2\pi}{N} & -\sin\frac{2\pi}{N}\\\sin\frac{2\pi}{N} & \cos\frac{2\pi}{N} \end{bmatrix}. $$ Try multiplying $R$ by any of the vectors $\left(\cos\frac{2\pi j}{N},\sin\frac{2\pi j}{N}\right)$ and then applying addition formulas. You should get $\left(\cos\frac{2\pi (j+1)}{N},\sin\frac{2\pi (j+1)}{N}\right)$. But this means that the sum $$ S=\sum_{j=0}^{N-1}\left(\cos\frac{2\pi j}{N},\sin\frac{2\pi j}{N}\right) $$ is unchanged under multiplication by $R$. Can you prove that this implies $S=(0,0)$?

You are trying to prove $$ \frac{1}{2}-\cos\frac{\pi}{7}-\cos\frac{3\pi}{7}-\cos\frac{5\pi}{7}=0. $$ To relate it to the above, multiply both sides by 2, and use $\cos\theta=-\cos(\pi+\theta)=-\cos(\pi-\theta)$ to rewrite the left hand side.

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Let $y = \cos\theta + i \cdot \sin\theta$, where $\theta$ has either of the values $$\frac{\pi}{7}, \frac{3\pi}{7}, \frac{5\pi}{7}, \pi, \frac{9\pi}{7}, \frac{11\pi}{7} \ \text{and} \ \frac{13\pi}{7}$$ Then $y^{7} = \cos{7\theta} + i \:\sin{7\theta} =-1$. Then you have $$y^{7}+1=0 \Rightarrow (y+1)(y^{6}-y^{5}+y^{4}-y^{3}+y^{2}-y +1)=0$$ Now the root $y=-1$ corresponds to the value $\theta=\pi$. The roots of the equation $$y^{6}-y^{5}+y^{4}-y^{3}+y^{2}-y +1 =0 \qquad \cdots (1)$$ are therefore $\cos\theta + i\: \sin\theta$ where $\theta$ takes other values than $\pi$. Put $2x = y + \frac{1}{y} = 2 \cos\theta$ and note that:

  • $\displaystyle y^{2}+\frac{1}{y^2} = 4x^{2}-2$

  • $\displaystyle y^{3} + \frac{1}{y^3} = 8x^{3}-6x$.

Now divide $(1)$ by $y^{3}$ and get the answer.

Another method to look at this:

$7 \theta = \text{an odd multiple of} \: \pi$, therefore $\cos{4\theta} = -\cos{3\theta}$. Take $c=\cos{\theta}$. From this you have \begin{align*} 2\cdot (2c^{2}-1)^{2} -1 &= -(4c^{2}-3c) \\\ 8c^{4} + 4c^{2}-8c^{2} -3c + 1 &=0 \\\ (c+1) \cdot (8c^{3} - 4c^{2} -4c+1) &=0 \end{align*}

The following page also may be helpful: http://www.isibang.ac.in/~sury/luckyoct10.pdf

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The three angles in the formula $\lbrace \pi/7, 3\pi/7, 5\pi/7 \rbrace $, thought of as points on the unit circle, can be completed to a set of seven angles of vertices of a regular heptagon.

These heptagon angles are, up to an integer multiple of $2 \pi$, equal to $\lbrace \pi, \pm \pi/7, \pm 3\pi/7, \pm 5\pi/7 \rbrace $.

The average $x$ coordinate of vertices of the polygon is zero, from the rotational symmetry of the heptagon.

There is also symmetry with respect to reflection in the $x$ axis: $\cos \pm x = \cos x$.

Combining these facts one gets $2S_7 - 1 = 0$ where $S_7$ is the sum of three cosines. Or $S_n = 1/2$ for odd $n \geq 3$.

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The correct denomination is heptagon. –  Did May 8 '12 at 9:48
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As a variation of my previous answer, it can be shown (see e.g. this paper) that the eigenvalues of the tridiagonal matrix

$$\begin{pmatrix}1&1&0\\1&0&1\\0&1&0\end{pmatrix}$$

are $2\cos\dfrac{\pi}{7}$, $2\cos\dfrac{3\pi}{7}$, and $2\cos\dfrac{5\pi}{7}$. Since the trace of a matrix is the same as the sum of its eigenvalues,

$$2\cos\dfrac{\pi}{7}+2\cos\dfrac{3\pi}{7}+2\cos\dfrac{5\pi}{7}=1$$

and the identity shown in the OP easily follows.

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To compute $s=\cos(\pi/7) + \cos(3 \pi/7) + \cos(5 \pi/7)$, one can continue the sum and consider $$t=\sum_{k=1}^7\cos\left(\frac{2k-1}7\pi\right). $$ Since $\cos(2\pi-\alpha)=\cos(\alpha)$ for every $\alpha$ and $\cos(\pi)=-1$, $t=2s-1$. Now, $t$ is the real part of $$ u=\mathrm e^{\mathrm i\pi/7} + \mathrm e^{3\mathrm i\pi/7} + \mathrm e^{5\mathrm i\pi/7} + \mathrm e^{7\mathrm i\pi/7} + \mathrm e^{9\mathrm i\pi/7} + \mathrm e^{11\mathrm i\pi/7} + \mathrm e^{13\mathrm i\pi/7}, $$ and $u=z+z^3+z^5+z^7+z^9+z^{11}+z^{13}$ with $z=\mathrm e^{\mathrm i\pi/7}$. This is a geometric sum with ratio $z^2\ne1$ hence $u=z(1-z^{14})/(1-z^2)$.

Finally, $z^{14}=\mathrm e^{14\mathrm i\pi/7}=1$ hence $u=0$. Thus, $t=0$, and $s=1/2$.

This is the $n=3$ case of the identity, valid for every $n\geqslant1$, $$ \color{red}{\sum\limits_{k=1}^n\cos\left(\frac{2k-1}{2n+1}\pi\right)=\frac12}. $$

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Here's yet another approach, which has very great overlap with many of the previous answers. Let $\zeta=\cos(\pi/7)+i\sin(\pi/7)$. It’s a primitive fourteenth root of unity, thus satisfies the polynomial relation (A): $\zeta^6-\zeta^5+\zeta^4-\zeta^3+\zeta^2-\zeta+1=0$. Since $\zeta+\zeta^{-1}=2\cos(\pi/7)$, and more generally $\zeta^n+\zeta^{-n}=2\cos(n\pi/7)$ for any $n$, you want to prove that $\zeta^5+\zeta^3+\zeta+\zeta^{-1}+\zeta^{-3}+\zeta^{-5}=1$. All that remains is to note that since $\zeta^7=-1$, $\zeta^5+\zeta^{-5}=-\zeta^2-\zeta^{-2}$, the desired fact falls out from relation (A).

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