Difference between revisions of "1999 AIME Problems/Problem 11"
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\begin{align*} | \begin{align*} | ||
s \cdot \sin 5 = \sum_{k=1}^{35} \sin 5k \sin 5 &= \sum_{k=1}^{35} \frac{1}{2}(\cos (5k - 5)- \cos (5k + 5))\\ | s \cdot \sin 5 = \sum_{k=1}^{35} \sin 5k \sin 5 &= \sum_{k=1}^{35} \frac{1}{2}(\cos (5k - 5)- \cos (5k + 5))\\ | ||
− | &= | + | &= 0.5(\cos 0 - \cos 10 + \cos 5 - \cos 15 + \cos 10 \ldots + \cos 165 - \cos 175+ \cos 170 - \cos 180) |
+ | \end{align*}</cmath> | ||
This telescopes to <cmath>s = \frac{\cos 0 + \cos 5 - \cos 175 - \cos 180}{2 \sin 5} = \frac{1 + \cos 5}{\sin 5}.</cmath> Manipulating this to use the identity <math>\tan x = \frac{1 - \cos 2x}{\sin 2x}</math>, we get <cmath>s = \frac{1 - \cos 175}{\sin 175} \Longrightarrow s = \tan \frac{175}{2},</cmath> and our answer is <math>\boxed{177}</math>. | This telescopes to <cmath>s = \frac{\cos 0 + \cos 5 - \cos 175 - \cos 180}{2 \sin 5} = \frac{1 + \cos 5}{\sin 5}.</cmath> Manipulating this to use the identity <math>\tan x = \frac{1 - \cos 2x}{\sin 2x}</math>, we get <cmath>s = \frac{1 - \cos 175}{\sin 175} \Longrightarrow s = \tan \frac{175}{2},</cmath> and our answer is <math>\boxed{177}</math>. | ||
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==Alternate Solution== | ==Alternate Solution== | ||
− | We note that <math>\sin x = \mbox{Im } e^{ix}</math>. We thus have that | + | We note that <math>\sin x = \mbox{Im } e^{ix}\text{*}</math>. We thus have that |
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
\sum_{k = 1}^{35} \sin 5k &= \sum_{k = 1}^{35} \mbox{Im } e^{5ki}\\ | \sum_{k = 1}^{35} \sin 5k &= \sum_{k = 1}^{35} \mbox{Im } e^{5ki}\\ | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
The desired answer is thus <math>175 + 2 = \boxed{177}</math>. | The desired answer is thus <math>175 + 2 = \boxed{177}</math>. | ||
+ | |||
+ | *Only if <math>x</math> is in radians, which it is not. However, the solution is still viable, so keep reading. | ||
+ | ==Solution 3== | ||
+ | Let <math>x=e^{\frac{i\pi}{36}}</math>. By Euler's Formula, <math>\sin{5k^\circ}=\frac{x^k-\frac{1}{x^{k}}}{2i}</math>. | ||
+ | |||
+ | The sum we want is thus <math>\frac{x-\frac{1}{x}}{2i}+\frac{x^2-\frac{1}{x^{2}}}{2i}+\cdots+\frac{x^{35}-\frac{1}{x^{35}}}{2i}</math> | ||
+ | |||
+ | We factor the <math>\frac{1}{2i}</math> and split into two geometric series to get <math>\frac{1}{2i}\left(\frac{-\frac{1}{x^{35}}(x^{35}-1)}{x-1}+\frac{x(x^{35}-1)}{x-1}\right)</math> | ||
+ | |||
+ | However, we note that <math>x^{36}=-1</math>, so <math>-\frac{1}{x^{35}}=x</math>, so our two geometric series are actually the same. We combine the terms and simplify to get <math>\frac{1}{i}\left(\frac{x^{36}-x}{x-1}\right)</math> | ||
+ | |||
+ | Apply Euler's identity and simplify again to get <math>\frac{1}{i}\left(\frac{-x-1}{x-1}\right)</math> | ||
+ | |||
+ | Now, we need to figure out how to express this as the tangent of something. We note that <math>\tan(5k^\circ)=\frac{\sin(5k^\circ)}{\cos(5k^\circ)}=\frac{\frac{x^k-\frac{1}{x^k}}{2i}}{\frac{x^k+\frac{1}{x^k}}{2}}=\frac{1}{i}\frac{x^{2k}-1}{x^{2k}+1}</math>. | ||
+ | |||
+ | So, we set the two equal to each other to solve for <math>k</math>. Cross multiplying gets <math>(-x-1)(x^{2k}+1)=(x-1)(x^{2k}-1)</math>. Expanding yields <math>-x^{2k+1}-x-x^{2k}-1=x^{2k+1}-x-x^{2k}+1</math>. Simplifying yields <math>x^{2k+1}=-1</math>. Since <math>2k+1=36</math> is the smallest solution, we have <math>k=\frac{35}{2}</math>, and the argument of tangent is <math>5k=\frac{175}{2}</math>. The requested sum is <math>175+2=\boxed{177}</math>. | ||
+ | ==Solution 4== | ||
+ | This solution is inspired by [[1997 AIME Problems/Problem 11]]. | ||
+ | First of all, <math>\sum_{k=1}^{35}\sin 5k=\sum_{k=0}^{35}\sin 5k</math> because the difference is <math>\sin 0 = 0</math>. Now consider the sum <math>\sum_{k=0}^{35}\cos 5k</math>. Since <math>\cos 5=-\cos 175</math>, <math>\cos 10=-\cos 170</math> and so on, this simplifies to <math>\cos 0=1</math>. | ||
+ | Now, consider the ratio | ||
+ | <cmath>\dfrac{\sum\limits_{n=0}^{35}\sin 5k}{\sum\limits_{n=0}^{35}\cos 5k.}</cmath> | ||
+ | Using the sum-to-product identities, the numerator can be written as | ||
+ | <cmath>\sin \frac{175}{2} (\cos \frac{175}{2} + \cos \frac {165}{2}+\cdots + \cos \frac{5}{2}.)</cmath> | ||
+ | Similarly (via sum-to-product identities), the denominator is equivalent to | ||
+ | <cmath>\cos \frac{175}{2} (\cos \frac{175}{2} + \cos \frac {165}{2}+\cdots + \cos \frac{5}{2}.)</cmath> | ||
+ | If we substitute these equivalent expressions into the original ratio, many parts cancel. We are left with | ||
+ | <cmath>\frac{\sin \frac{175}{2}}{\cos \frac{175}{2}} = \tan{\frac{175}{2}}.</cmath> | ||
+ | Aha! Remembering that the original denominator <math>\sum_{k=0}^{35}\cos 5k</math> was equal to 1, we realize that the numerator, <math>\sum_{k=0}^{35}\sin 5k=\tan{\frac{175}{2}}</math>. Since <math>\sum_{k=0}^{35}\sin 5k=\sum_{k=1}^{35}\sin 5k</math>, we get <math>\tan \frac{175}{2}</math>, or <math>m+n=\boxed{177}</math>, as our final answer. <math>\blacksquare</math> | ||
+ | ~ewei12 | ||
== See also == | == See also == |
Latest revision as of 12:39, 29 June 2024
Problem
Given that where angles are measured in degrees, and and are relatively prime positive integers that satisfy find
Solution
Let . We could try to manipulate this sum by wrapping the terms around (since the first half is equal to the second half), but it quickly becomes apparent that this way is difficult to pull off. Instead, we look to telescope the sum. Using the identity , we can rewrite as
This telescopes to Manipulating this to use the identity , we get and our answer is .
Alternate Solution
We note that . We thus have that The desired answer is thus .
- Only if is in radians, which it is not. However, the solution is still viable, so keep reading.
Solution 3
Let . By Euler's Formula, .
The sum we want is thus
We factor the and split into two geometric series to get
However, we note that , so , so our two geometric series are actually the same. We combine the terms and simplify to get
Apply Euler's identity and simplify again to get
Now, we need to figure out how to express this as the tangent of something. We note that .
So, we set the two equal to each other to solve for . Cross multiplying gets . Expanding yields . Simplifying yields . Since is the smallest solution, we have , and the argument of tangent is . The requested sum is .
Solution 4
This solution is inspired by 1997 AIME Problems/Problem 11. First of all, because the difference is . Now consider the sum . Since , and so on, this simplifies to . Now, consider the ratio Using the sum-to-product identities, the numerator can be written as Similarly (via sum-to-product identities), the denominator is equivalent to If we substitute these equivalent expressions into the original ratio, many parts cancel. We are left with Aha! Remembering that the original denominator was equal to 1, we realize that the numerator, . Since , we get , or , as our final answer. ~ewei12
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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