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Difference between revisions of "1963 IMO Problems/Problem 5"

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Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>.
 
Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>.
  
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==Solutions==
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==Solution==
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===Solution 1===
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 +
Because the sum of the <math>x</math>-coordinates of the seventh roots of unity is <math>0</math>, we have
 +
 
 +
<cmath>\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\frac{8\pi}{7}} + \cos{\frac{10\pi}{7}} + \cos{\frac{12\pi}{7}} + \cos{\frac{14\pi}{7}} = 0</cmath>
 +
<cmath>\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\frac{8\pi}{7}} + \cos{\frac{10\pi}{7}} + \cos{\frac{12\pi}{7}} = -1</cmath>
 +
<cmath>\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\left(-\frac{6\pi}{7}\right)} + \cos{\left(-\frac{4\pi}{7}\right)} + \cos{\left(-\frac{2\pi}{7}\right)} = -1.</cmath>
 +
 
 +
Now, we can apply <math>\cos{x} = \cos{\left(-x\right)}</math> to obtain
 +
 
 +
<cmath>2\left(\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}}\right) = -1</cmath>
 +
<cmath>\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} = -\frac{1}{2}.</cmath>
 +
 
 +
Finally, since <math>\cos{x} = -\cos{(\pi-x)}</math>,
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 +
<cmath>\cos{\frac{2\pi}{7}} - \cos{\frac{3\pi}{7}} - \cos{\frac{\pi}{7}} = -\frac{1}{2}</cmath>
 +
<cmath>\cos{\frac{\pi}{7}} - \cos{\frac{2\pi}{7}} + \cos{\frac{3\pi}{7}} = \frac{1}{2}\textrm{. }\square</cmath>
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 +
~mathboy100
 +
 
 +
== Solution 2 ==
 
Let <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=S</math>. We have
 
Let <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=S</math>. We have
  
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Then, by product-sum formulae, we have
 
Then, by product-sum formulae, we have
  
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<cmath>S * 2* \sin{\frac{\pi}{7}} = \sin{\frac{2\pi}{7}}+\sin{\frac{4\pi}{7}}-\sin{\frac{2\pi}{7}}+\sin{\frac{6\pi}{7}}-\sin{\frac{4\pi}{7}}=\sin{\frac{6\pi}{7}}=\sin{\frac{\pi}{7}}</cmath>
+
<cmath>S \cdot 2 \sin{\frac{\pi}{7}} = \sin{\frac{2\pi}{7}}+\sin{\frac{4\pi}{7}}-\sin{\frac{2\pi}{7}}+\sin{\frac{6\pi}{7}}-\sin{\frac{4\pi}{7}}=\sin{\frac{6\pi}{7}}=\sin{\frac{\pi}{7}}</cmath>
  
 
Thus <math>S = 1/2</math>. <math>\blacksquare</math>
 
Thus <math>S = 1/2</math>. <math>\blacksquare</math>
  
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===Solution 2===
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== Solution 3 ==
 
Let <math>a=\sin{\frac{\pi}{7}}</math> and <math>b=\cos{\frac{\pi}{7}}</math>. From the addition formulae, we have
 
Let <math>a=\sin{\frac{\pi}{7}}</math> and <math>b=\cos{\frac{\pi}{7}}</math>. From the addition formulae, we have
  
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Clearly <math>b\neq -1</math>, so <math>8b^3-4b^2-4b+1=0</math>. This proves the result. <math>\blacksquare</math>
 
Clearly <math>b\neq -1</math>, so <math>8b^3-4b^2-4b+1=0</math>. This proves the result. <math>\blacksquare</math>
 +
 +
== Solution 4 ==
 +
Let <math>\omega=\mathrm{cis}\left(\frac{\pi}{14}\right)</math>. Thus it suffices to show that <math>\omega+\omega^{-1}-\omega^2-\omega^{-2}+\omega^3+\omega^{-3}=1</math>. Now using the fact that <math>\omega^k=\omega^{14+k}</math> and <math>-\omega^2=\omega^9</math>, this is equivalent to
 +
<cmath>\omega+\omega^3+\omega^5+\omega^7+\omega^9+\omega^{11}+\omega^{13}-\omega^7</cmath>
 +
<cmath>\omega\left(\frac{\omega^{14}-1}{\omega^2-1}\right)-\omega^7</cmath>
 +
But since <math>\omega</math> is a <math>14</math>th root of unity, <math>\omega^{14}=1</math>. The answer is then <math>-\omega^{7}=1</math>, as desired.
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 +
~yofro
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 +
 +
== Solution 5 ==
 +
We let <math>\omega = \mathrm{cis} \ \left(\dfrac{2\pi}{7} \right)</math>. We therefore have <math>w^i</math>, where <math>0 \leq i \leq 6</math>, are the <math>7^{\text{th}}</math> roots of unity. Since <math>\sum_{i = 0}^6 \omega^i = 0</math>, then <math>\sum_{i = 1}^6 \omega^i = -1</math>, so <math>\sum_{i = 1}^3 \mathrm{Re}(\omega^i) = -\dfrac{1}{2}</math>. Therefore, because <math>\mathrm{cis} \ \alpha = \cos \alpha + i \sin \alpha, \mathrm{Re}(\mathrm{cis} \ \alpha) = \cos \alpha</math>, so <cmath>\cos \left(\dfrac{2\pi}{7} \right) + \cos \left( \dfrac{4\pi}{7} \right) + \cos \left(\dfrac{6\pi}{7} \right) = -\dfrac{1}{2}</cmath>
 +
<cmath>\implies -\cos \left(\dfrac{2\pi}{7} \right) - \cos \left(\dfrac{4\pi}{7} \right) - \cos \left(\dfrac{6\pi}{7} \right) = \dfrac{1}{2}</cmath>
 +
 +
Since <math>\cos \alpha = - \cos(\pi - \alpha)</math>, we have <math>\cos \left(\dfrac{\pi}{7} \right) - \cos \left(\dfrac{2\pi}{7} \right) + \cos \left(\dfrac{3\pi}{7} \right) = \dfrac{1}{2}</math> and we are done <math>\blacksquare</math>
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 +
~[[User: Yiyj1|Yiyj1]]
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=1963|num-b=4|num-a=6}}
 
{{IMO box|year=1963|num-b=4|num-a=6}}

Latest revision as of 08:05, 1 October 2024

Problem

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$.

Solution

Because the sum of the $x$-coordinates of the seventh roots of unity is $0$, we have

\[\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\frac{8\pi}{7}} + \cos{\frac{10\pi}{7}} + \cos{\frac{12\pi}{7}} + \cos{\frac{14\pi}{7}} = 0\] \[\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\frac{8\pi}{7}} + \cos{\frac{10\pi}{7}} + \cos{\frac{12\pi}{7}} = -1\] \[\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} + \cos{\left(-\frac{6\pi}{7}\right)} + \cos{\left(-\frac{4\pi}{7}\right)} + \cos{\left(-\frac{2\pi}{7}\right)} = -1.\]

Now, we can apply $\cos{x} = \cos{\left(-x\right)}$ to obtain

\[2\left(\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}}\right) = -1\] \[\cos{\frac{2\pi}{7}} + \cos{\frac{4\pi}{7}} + \cos{\frac{6\pi}{7}} = -\frac{1}{2}.\]

Finally, since $\cos{x} = -\cos{(\pi-x)}$,

\[\cos{\frac{2\pi}{7}} - \cos{\frac{3\pi}{7}} - \cos{\frac{\pi}{7}} = -\frac{1}{2}\] \[\cos{\frac{\pi}{7}} - \cos{\frac{2\pi}{7}} + \cos{\frac{3\pi}{7}} = \frac{1}{2}\textrm{. }\square\]

~mathboy100

Solution 2

Let $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=S$. We have

\[S=\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\cos{\frac{\pi}{7}}+\cos{\frac{3\pi}{7}}+\cos{\frac{5\pi}{7}}\]

Then, by product-sum formulae, we have

\[S \cdot 2 \sin{\frac{\pi}{7}} = \sin{\frac{2\pi}{7}}+\sin{\frac{4\pi}{7}}-\sin{\frac{2\pi}{7}}+\sin{\frac{6\pi}{7}}-\sin{\frac{4\pi}{7}}=\sin{\frac{6\pi}{7}}=\sin{\frac{\pi}{7}}\]

Thus $S = 1/2$. $\blacksquare$

Solution 3

Let $a=\sin{\frac{\pi}{7}}$ and $b=\cos{\frac{\pi}{7}}$. From the addition formulae, we have

\[S=b-[b^2-a^2]+[ b(b^2-a^2)-a(2ab) ]=b-b^2+b^3+a^2(1-3b)\]

From the Trigonometric Identity, $a^2=1-b^2$, so

\[S=b-b^2+b^3+(1-b^2)(1-3b)=4b^3-2b^2-2b+1\]

We must prove that $S=1/2$. It suffices to show that $8b^3-4b^2-4b+1=0$.

Now note that $\cos{\frac{4\pi}{7}}=-\cos{\frac{3\pi}{7}}$. We can find these in terms of $a$ and $b$:

\[\cos{\frac{4\pi}{7}}=[b^2-a^2]^2-[2ab]^2=b^4-6a^2b^2+b^4=b^4-6(1-b^2)b^2+b^4=8b^4-8b^2+1\]

\[\cos{\frac{3\pi}{7}}=b[b^2-a^2]-a[2ab]=b^3-3a^2b=b^3-3(1-b^2)b=3b-4b^3\]

Therefore $8b^4-8b^2+1=-(3b-4b^3)\Rightarrow 8b^4+4b^3-8b^2-3b+1=0$. Note that this can be factored:

\[8b^4+4b^3-8b^2-3b+1=(8b^3-4b^2-4b+1)(b+1)=0\]

Clearly $b\neq -1$, so $8b^3-4b^2-4b+1=0$. This proves the result. $\blacksquare$

Solution 4

Let $\omega=\mathrm{cis}\left(\frac{\pi}{14}\right)$. Thus it suffices to show that $\omega+\omega^{-1}-\omega^2-\omega^{-2}+\omega^3+\omega^{-3}=1$. Now using the fact that $\omega^k=\omega^{14+k}$ and $-\omega^2=\omega^9$, this is equivalent to \[\omega+\omega^3+\omega^5+\omega^7+\omega^9+\omega^{11}+\omega^{13}-\omega^7\] \[\omega\left(\frac{\omega^{14}-1}{\omega^2-1}\right)-\omega^7\] But since $\omega$ is a $14$th root of unity, $\omega^{14}=1$. The answer is then $-\omega^{7}=1$, as desired.

~yofro


Solution 5

We let $\omega = \mathrm{cis} \ \left(\dfrac{2\pi}{7} \right)$. We therefore have $w^i$, where $0 \leq i \leq 6$, are the $7^{\text{th}}$ roots of unity. Since $\sum_{i = 0}^6 \omega^i = 0$, then $\sum_{i = 1}^6 \omega^i = -1$, so $\sum_{i = 1}^3 \mathrm{Re}(\omega^i) = -\dfrac{1}{2}$. Therefore, because $\mathrm{cis} \ \alpha = \cos \alpha + i \sin \alpha, \mathrm{Re}(\mathrm{cis} \ \alpha) = \cos \alpha$, so \[\cos \left(\dfrac{2\pi}{7} \right) + \cos \left( \dfrac{4\pi}{7} \right) + \cos \left(\dfrac{6\pi}{7} \right) = -\dfrac{1}{2}\] \[\implies -\cos \left(\dfrac{2\pi}{7} \right) - \cos \left(\dfrac{4\pi}{7} \right) - \cos \left(\dfrac{6\pi}{7} \right) = \dfrac{1}{2}\]

Since $\cos \alpha = - \cos(\pi - \alpha)$, we have $\cos \left(\dfrac{\pi}{7} \right) - \cos \left(\dfrac{2\pi}{7} \right) + \cos \left(\dfrac{3\pi}{7} \right) = \dfrac{1}{2}$ and we are done $\blacksquare$

~Yiyj1

See Also

1963 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
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