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Difference between revisions of "2013 AIME I Problems/Problem 14"
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== Problem 14 ==
+
== Problem ==
 
For <math>\pi \le \theta < 2\pi</math>, let
 
For <math>\pi \le \theta < 2\pi</math>, let
  
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so that <math>\frac{P}{Q} = \frac{2\sqrt2}{7}</math>. Then <math>\sin\theta = -\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
so that <math>\frac{P}{Q} = \frac{2\sqrt2}{7}</math>. Then <math>\sin\theta = -\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
−
== Solution ==
+
==Solution 1==
−
===Solution 1===
 
  
 
Noticing the <math>\sin</math> and <math>\cos</math> in both <math>P</math> and <math>Q,</math> we think of the angle addition identities: <cmath>\sin(a + b) = \sin a \cos b + \cos a \sin b, \cos(a + b) = \cos a \cos b + \sin a \sin b.</cmath> With this in mind, we multiply <math>P</math> by <math>\sin \theta</math> and <math>Q</math> by <math>\cos \theta</math> to try and use some angle addition identities. Indeed, we get  
 
Noticing the <math>\sin</math> and <math>\cos</math> in both <math>P</math> and <math>Q,</math> we think of the angle addition identities: <cmath>\sin(a + b) = \sin a \cos b + \cos a \sin b, \cos(a + b) = \cos a \cos b + \sin a \sin b.</cmath> With this in mind, we multiply <math>P</math> by <math>\sin \theta</math> and <math>Q</math> by <math>\cos \theta</math> to try and use some angle addition identities. Indeed, we get  
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This is a system of equations; rearrange and rewrite to get <cmath>P(1 + 2 \sin \theta) + 2Q \cos \theta = 2 \cos \theta</cmath> and <cmath>P \cos^2 \theta + Q \cos \theta(2 + \sin \theta) = 2 \cos \theta.</cmath> Subtract the two and rearrange to get <cmath>\dfrac{P}{Q} = \dfrac{\cos \theta}{2 + \sin \theta} = \dfrac{2 \sqrt{2}}{7}.</cmath> Then, square both sides and use Pythagorean Identity to get a quadratic in <math>\sin \theta.</math> Factor that quadratic and solve for <math>\sin \theta = -17/19, 1/3.</math> The answer format tells us it's the negative solution, and our desired answer is <math>17 + 19 = \boxed{036}.</math>
 
This is a system of equations; rearrange and rewrite to get <cmath>P(1 + 2 \sin \theta) + 2Q \cos \theta = 2 \cos \theta</cmath> and <cmath>P \cos^2 \theta + Q \cos \theta(2 + \sin \theta) = 2 \cos \theta.</cmath> Subtract the two and rearrange to get <cmath>\dfrac{P}{Q} = \dfrac{\cos \theta}{2 + \sin \theta} = \dfrac{2 \sqrt{2}}{7}.</cmath> Then, square both sides and use Pythagorean Identity to get a quadratic in <math>\sin \theta.</math> Factor that quadratic and solve for <math>\sin \theta = -17/19, 1/3.</math> The answer format tells us it's the negative solution, and our desired answer is <math>17 + 19 = \boxed{036}.</math>
  
−
===Solution 2===
+
==Solution 2==
  
 
Use sum to product formulas to rewrite <math>P</math> and <math>Q</math>
 
Use sum to product formulas to rewrite <math>P</math> and <math>Q</math>
Line 41: Line 40:
 
Then use the pythagorean identity to solve for <math>\sin\theta</math>, <math>\sin\theta = -\frac{17}{19} \implies \boxed{036}</math>
 
Then use the pythagorean identity to solve for <math>\sin\theta</math>, <math>\sin\theta = -\frac{17}{19} \implies \boxed{036}</math>
  
−
===Solution 3===
+
==Solution 3==
  
 
Note that <cmath>e^{i\theta}=\cos(\theta)+i\sin(\theta)</cmath>
 
Note that <cmath>e^{i\theta}=\cos(\theta)+i\sin(\theta)</cmath>
Line 69: Line 68:
 
Clearing denominators and solving for <math>x</math> gives sine as <math>x=-\frac{17}{19}</math>.
 
Clearing denominators and solving for <math>x</math> gives sine as <math>x=-\frac{17}{19}</math>.
  
−
<math>017+019=\boxed{036}</math>
+
<math>017+019=\boxed{036}
  
−
===Solution 4===
+
==Solution 4==
−
A bit similar to Solution 3. We use <math>\phi = \theta+90^\circ</math> because the progression cycles in <math>P\in (\sin 0\theta,\cos 1\theta,-\sin 2\theta,-\cos 3\theta\dots)</math>. So we could rewrite that as <math>P\in(\sin 0\phi,\sin 1\phi,\sin 2\phi,\sin 3\phi\dots)</math>.
+
A bit similar to Solution 3. We use </math>\phi = \theta+90^\circ<math> because the progression cycles in </math>P\in (\sin 0\theta,\cos 1\theta,-\sin 2\theta,-\cos 3\theta\dots)<math>. So we could rewrite that as </math>P\in(\sin 0\phi,\sin 1\phi,\sin 2\phi,\sin 3\phi\dots)<math>.
  
−
Similarly, <math>Q\in (\cos 0\theta,-\sin 1\theta,-\cos 2\theta,\sin 3\theta\dots)\implies Q\in(\cos 0\phi,\cos 1\phi, \cos 2\phi, \cos 3\phi\dots)</math>.
+
Similarly, </math>Q\in (\cos 0\theta,-\sin 1\theta,-\cos 2\theta,\sin 3\theta\dots)\implies Q\in(\cos 0\phi,\cos 1\phi, \cos 2\phi, \cos 3\phi\dots)<math>.
  
−
Setting complex <math>z=q_1+p_1i</math>, we get <math>z=\frac{1}{2}\left(\cos\phi+\sin\phi i\right)</math>
+
Setting complex </math>z=q_1+p_1i<math>, we get </math>z=\frac{1}{2}\left(\cos\phi+\sin\phi i\right)<math>
  
−
<math>(Q,P)=\sum_{n=0}^\infty z^n=\frac{1}{1-z}=\frac{1}{1-\frac{1}{2}\cos\phi-\frac{i}{2}\sin\phi}=\frac{1-0.5\cos\phi+0.5i\sin\phi}{\dots}</math>.
+
</math>(Q,P)=\sum_{n=0}^\infty z^n=\frac{1}{1-z}=\frac{1}{1-\frac{1}{2}\cos\phi-\frac{i}{2}\sin\phi}=\frac{1-0.5\cos\phi+0.5i\sin\phi}{\dots}<math>.
  
−
The important part is the ratio of the imaginary part <math>i</math> to the real part. To cancel out the imaginary part from the denominator, we must add <math>0.5i\sin\phi</math> to the numerator to make the denominator a difference (or rather a sum) of squares. The denominator does not matter. Only the numerator, because we are trying to find <math>\frac{P}{Q}=\tan\text{arg}(\Sigma)</math> a PROPORTION of values. So denominators would cancel out.
+
The important part is the ratio of the imaginary part </math>i<math> to the real part. To cancel out the imaginary part from the denominator, we must add </math>0.5i\sin\phi<math> to the numerator to make the denominator a difference (or rather a sum) of squares. The denominator does not matter. Only the numerator, because we are trying to find </math>\frac{P}{Q}=\tan\text{arg}(\Sigma)<math> a PROPORTION of values. So denominators would cancel out.
  
−
<math>\frac{P}{Q}=\frac{\text{Re}\Sigma}{\text{Im}\Sigma}=\frac{0.5\sin\phi}{1-0.5\cos\phi}=\frac{\sin\phi}{2-\cos\phi}=\frac{2\sqrt{2}}{7}</math>.
+
</math>\frac{P}{Q}=\frac{\text{Re}\Sigma}{\text{Im}\Sigma}=\frac{0.5\sin\phi}{1-0.5\cos\phi}=\frac{\sin\phi}{2-\cos\phi}=\frac{2\sqrt{2}}{7}<math>.
  
−
Setting <math>\sin\theta=y</math>, we obtain
+
Setting </math>\sin\theta=y<math>, we obtain
 
<cmath>\frac{\sqrt{1-y^2}}{2+y}=\frac{2\sqrt{2}}{7}</cmath>
 
<cmath>\frac{\sqrt{1-y^2}}{2+y}=\frac{2\sqrt{2}}{7}</cmath>
 
<cmath>7\sqrt{1-y^2}=2\sqrt{2}(2+y)</cmath>
 
<cmath>7\sqrt{1-y^2}=2\sqrt{2}(2+y)</cmath>
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<cmath>y=\frac{-32\pm\sqrt{4900}}{114}=\frac{-16\pm 35}{57}</cmath>.
 
<cmath>y=\frac{-32\pm\sqrt{4900}}{114}=\frac{-16\pm 35}{57}</cmath>.
  
−
Since <math>y<0</math> because <math>\pi<\theta<2\pi</math>, <math>y=\sin\theta=-\frac{51}{57}=-\frac{17}{19}</math>. Adding up, <math>17+19=\boxed{036}</math>.
+
Since </math>y<0<math> because </math>\pi<\theta<2\pi<math>, </math>y=\sin\theta=-\frac{51}{57}=-\frac{17}{19}<math>. Adding up, </math>17+19=\boxed{036}<math>.
  
−
===Solution 5 (lots of room for sillies, I wouldn't recommend it)===
+
==Solution 5 (lots of room for sillies, I wouldn't recommend it)==
  
−
We notice <math>\sin\theta=\frac{-i}{2}(e^{i\theta}-e^{-i\theta})</math> and <math>\cos\theta=\frac{1}{2}(e^{i\theta}+e^{-i\theta})</math>
+
We notice </math>\sin\theta=\frac{-i}{2}(e^{i\theta}-e^{-i\theta})<math> and </math>\cos\theta=\frac{1}{2}(e^{i\theta}+e^{-i\theta})<math>
  
−
With these, we just quickly find the sum of the infinite geometric series' in <math>P</math> and <math>Q</math>. <math>P</math> has 2 parts, the <math>\cos</math> and the <math>\sin</math> parts. The <math>\cos</math> part is: <math>\frac12\cos\theta-\frac18\cos3\theta+\cdots</math>, which can be turned into: <math>\frac14(e^{i\theta}(1-\frac{e^{i2\theta}}{4}+\cdots)+e^{-i\theta}(1-\frac{e^{-i2\theta}}{4}+\cdots))</math>, which is <math>\frac{1}{4}(\frac{e^{i\theta}}{1+\frac{1}{4}e^{i2\theta}}+\frac{e^{-i\theta}}{1+\frac{1}{4}e^{-i2\theta}})</math>. This turns into <math>\frac{5(e^{i\theta}+e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}</math>.
+
With these, we just quickly find the sum of the infinite geometric series' in </math>P<math> and </math>Q<math>. </math>P<math> has 2 parts, the </math>\cos<math> and the </math>\sin<math> parts. The </math>\cos<math> part is: </math>\frac12\cos\theta-\frac18\cos3\theta+\cdots<math>, which can be turned into: </math>\frac14(e^{i\theta}(1-\frac{e^{i2\theta}}{4}+\cdots)+e^{-i\theta}(1-\frac{e^{-i2\theta}}{4}+\cdots))<math>, which is </math>\frac{1}{4}(\frac{e^{i\theta}}{1+\frac{1}{4}e^{i2\theta}}+\frac{e^{-i\theta}}{1+\frac{1}{4}e^{-i2\theta}})<math>. This turns into </math>\frac{5(e^{i\theta}+e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}<math>.
  
−
Following the same process as above, we find that the <math>\sin</math> part of <math>P</math> is <math>\frac{2i(e^{i2\theta}-e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}</math>, the <math>\cos</math> part of <math>Q</math> is <math>\frac{16+2(e^{i2\theta}+e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}</math>, and finally, the <math>\sin</math> part of <math>Q</math> is <math>\frac{3i(e^{i\theta}-e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}</math>.
+
Following the same process as above, we find that the </math>\sin<math> part of </math>P<math> is </math>\frac{2i(e^{i2\theta}-e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}<math>, the </math>\cos<math> part of </math>Q<math> is </math>\frac{16+2(e^{i2\theta}+e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}<math>, and finally, the </math>\sin<math> part of </math>Q<math> is </math>\frac{3i(e^{i\theta}-e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}<math>.
  
−
We convert all 4 of these equations into trig, and we end up getting <math>\frac{2\sqrt{2}}{7}=\frac{10\cos{\theta}-4\sin{2\theta}}{16+4\cos{2\theta}-6\sin{\theta}}</math>, we divide by <math>2</math> on both numerator and denominator, and we get <math>\frac{2\sqrt{2}}{7}=\frac{5\cos{\theta}-2\sin{2\theta}}{8+2\cos{2\theta}-3\sin{\theta}}</math>. We use some trig identities and we get <math>\frac{\cos{\theta}(5-4\sin{\theta})}{10-4\sin^2{\theta}-3\sin{\theta}}=\frac{2\sqrt2}{7}</math>. We factor the denominator into <math>(5-4\sin\theta)(2+\sin\theta)</math>. We cancel out <math>5-4\sin\theta</math> on both numerator and denominator to get <math>\frac{\cos\theta}{2+\sin\theta}=\frac{2\sqrt2}{7}</math>. We set <math>\sin\theta</math> as <math>x</math>, and we just solve a quadratic in terms of <math>x</math>, <math>\frac{1-x^2}{x^2+4x+4}=\frac{8}{49}</math>, cross multiply and simplify, and we get <math>57x^2+32x-17=0</math>. We can actually factor this to get <math>(19x+17)(3x-1)=0</math>, which yields the 2 solutions <math>x=-\frac{17}{19}, x=\frac{1}{3}</math>. Since <math>\pi\leq\theta<2\pi</math>, the latter solution is deemed invalid, and we are left with <math>\sin\theta=-\frac{17}{19}</math>. Our final answer is <math>17+19=\boxed{036}</math>.
+
We convert all 4 of these equations into trig, and we end up getting </math>\frac{2\sqrt{2}}{7}=\frac{10\cos{\theta}-4\sin{2\theta}}{16+4\cos{2\theta}-6\sin{\theta}}<math>, we divide by </math>2<math> on both numerator and denominator, and we get </math>\frac{2\sqrt{2}}{7}=\frac{5\cos{\theta}-2\sin{2\theta}}{8+2\cos{2\theta}-3\sin{\theta}}<math>. We use some trig identities and we get </math>\frac{\cos{\theta}(5-4\sin{\theta})}{10-4\sin^2{\theta}-3\sin{\theta}}=\frac{2\sqrt2}{7}<math>. We factor the denominator into </math>(5-4\sin\theta)(2+\sin\theta)<math>. We cancel out </math>5-4\sin\theta<math> on both numerator and denominator to get </math>\frac{\cos\theta}{2+\sin\theta}=\frac{2\sqrt2}{7}<math>. We set </math>\sin\theta<math> as </math>x<math>, and we just solve a quadratic in terms of </math>x<math>, </math>\frac{1-x^2}{x^2+4x+4}=\frac{8}{49}<math>, cross multiply and simplify, and we get </math>57x^2+32x-17=0<math>. We can actually factor this to get </math>(19x+17)(3x-1)=0<math>, which yields the 2 solutions </math>x=-\frac{17}{19}, x=\frac{1}{3}<math>. Since </math>\pi\leq\theta<2\pi<math>, the latter solution is deemed invalid, and we are left with </math>\sin\theta=-\frac{17}{19}<math>. Our final answer is </math>17+19=\boxed{036}<math>.
  
 
~ASAB
 
~ASAB
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omg this was definitely the hardest problem on the AIME I
 
omg this was definitely the hardest problem on the AIME I
  
−
===Solution 6===
+
==Solution 6==
 
Follow solution 3, up to the point of using the geometric series formula
 
Follow solution 3, up to the point of using the geometric series formula
 
<cmath>Q+iP=\frac{1}{1+\frac{\sin(\theta)}{2}-\frac{Qi\cos(\theta)}{2}}</cmath>
 
<cmath>Q+iP=\frac{1}{1+\frac{\sin(\theta)}{2}-\frac{Qi\cos(\theta)}{2}}</cmath>
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<cmath>Pi+\frac{Pi}{2}\sin\theta-\frac{Qi}{2}\cos\theta = 0</cmath>
 
<cmath>Pi+\frac{Pi}{2}\sin\theta-\frac{Qi}{2}\cos\theta = 0</cmath>
  
−
We can then write <math>P = 2 \sqrt{2} k</math>, and <math>Q = 7k</math>, (<math>k \neq 0</math>). Thus, we can substitute and divide out by k.
+
We can then write </math>P = 2 \sqrt{2} k<math>, and </math>Q = 7k<math>, (</math>k \neq 0<math>). Thus, we can substitute and divide out by k.
 
<cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\cos\theta\ =\ 0</cmath>
 
<cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\cos\theta\ =\ 0</cmath>
 
<cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\sqrt{1-\sin^{2}\theta}=\ 0</cmath>
 
<cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\sqrt{1-\sin^{2}\theta}=\ 0</cmath>
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<cmath>\left(3\sin\theta-1\right)\left(19\sin\theta+17\right) = 0</cmath>
 
<cmath>\left(3\sin\theta-1\right)\left(19\sin\theta+17\right) = 0</cmath>
  
−
Since <math>\pi \le \theta < 2\pi</math>, we get <math>\sin \theta < 0</math>, and thus, <math>\sin\theta = \frac{-19}{17} \implies \boxed{036}</math>
+
Since </math>\pi \le \theta < 2\pi<math>, we get </math>\sin \theta < 0<math>, and thus, </math>\sin\theta = \frac{-19}{17} \implies \boxed{036}$
  
 
-Alexlikemath
 
-Alexlikemath

Revision as of 19:13, 24 January 2021

Problem

For $\pi \le \theta < 2\pi$, let

\begin{align*} P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots \end{align*}

and

\begin{align*} Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots \end{align*}

so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$. Then $\sin\theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution 1

Noticing the $\sin$ and $\cos$ in both $P$ and $Q,$ we think of the angle addition identities: \[\sin(a + b) = \sin a \cos b + \cos a \sin b, \cos(a + b) = \cos a \cos b + \sin a \sin b.\] With this in mind, we multiply $P$ by $\sin \theta$ and $Q$ by $\cos \theta$ to try and use some angle addition identities. Indeed, we get \begin{align*} P \sin \theta + Q \cos \theta &= \cos \theta + \dfrac{1}{2}(\cos \theta \sin \theta - \sin \theta \cos \theta) - \dfrac{1}{4}(\sin{2 \theta} \sin \theta + \cos{2 \theta} \cos{\theta}) - \cdots \\ &= \cos \theta - \dfrac{1}{4} \cos \theta - \dfrac{1}{8} \sin{2 \theta} - \dfrac{1}{16} \cos{3 \theta} + \cdots \\ &= \cos \theta - \dfrac{1}{2}P \end{align*} after adding term-by-term. Similar term-by-term adding yields \[P \cos \theta + Q \sin \theta = -2(Q - 1).\] This is a system of equations; rearrange and rewrite to get \[P(1 + 2 \sin \theta) + 2Q \cos \theta = 2 \cos \theta\] and \[P \cos^2 \theta + Q \cos \theta(2 + \sin \theta) = 2 \cos \theta.\] Subtract the two and rearrange to get \[\dfrac{P}{Q} = \dfrac{\cos \theta}{2 + \sin \theta} = \dfrac{2 \sqrt{2}}{7}.\] Then, square both sides and use Pythagorean Identity to get a quadratic in $\sin \theta.$ Factor that quadratic and solve for $\sin \theta = -17/19, 1/3.$ The answer format tells us it's the negative solution, and our desired answer is $17 + 19 = \boxed{036}.$

Solution 2

Use sum to product formulas to rewrite $P$ and $Q$


$P \sin\theta\ + Q \cos\theta\ = \cos \theta\ - \frac{1}{4}\cos \theta + \frac{1}{8}\sin 2\theta + \frac{1}{16}\cos 3\theta - \frac{1}{32}\sin 4\theta + ...$

Therefore, $P \sin \theta - Q \cos \theta = -2P$

Using $\frac{P}{Q} = \frac{2\sqrt2}{7}, Q = \frac{7}{2\sqrt2} P$

Plug in to the previous equation and cancel out the "P" terms to get: $\sin\theta - \frac{7}{2\sqrt2} \cos\theta = -2$.

Then use the pythagorean identity to solve for $\sin\theta$, $\sin\theta = -\frac{17}{19} \implies \boxed{036}$

Solution 3

Note that \[e^{i\theta}=\cos(\theta)+i\sin(\theta)\]

Thus, the following identities follow immediately: \[ie^{i\theta}=i(\cos(\theta)+i\sin(\theta))=-\sin(\theta)+i\cos(\theta)\] \[i^2 e^{i\theta}=-e^{i\theta}=-\cos(\theta)-i\sin(\theta)\] \[i^3 e^{i\theta}=\sin(\theta)-i\cos(\theta)\]

Consider, now, the sum $Q+iP$. It follows fairly immediately that:

\[Q+iP=1+\left(\frac{i}{2}\right)^1e^{i\theta}+\left(\frac{i}{2}\right)^2e^{2i\theta}+\ldots=\frac{1}{1-\frac{i}{2}e^{i\theta}}=\frac{2}{2-ie^{i\theta}}\] \[Q+iP=\frac{2}{2-ie^{i\theta}}=\frac{2}{2-(-\sin(\theta)+i\cos(\theta))}=\frac{2}{(2+\sin(\theta))-i\cos(\theta)}\]

This follows straight from the geometric series formula and simple simplification. We can now multiply the denominator by it's complex conjugate to find:

\[Q+iP=\frac{2}{(2+\sin(\theta))-i\cos(\theta)}\left(\frac{(2+\sin(\theta))+i\cos(\theta)}{(2+\sin(\theta))+i\cos(\theta)}\right)\] \[Q+iP=\frac{2((2+\sin(\theta))+i\cos(\theta))}{(2+\sin(\theta))^2+\cos^2(\theta)}\]

Comparing real and imaginary parts, we find: \[\frac{P}{Q}=\frac{\cos(\theta)}{2+\sin(\theta)}=\frac{2\sqrt{2}}{7}\]

Squaring this equation and letting $\sin^2(\theta)=x$:

$\frac{P^2}{Q^2}=\frac{\cos^2(\theta)}{4+4\sin(\theta)+\sin^2(\theta)}=\frac{1-x^2}{4+4x+x^2}=\frac{8}{49}$

Clearing denominators and solving for $x$ gives sine as $x=-\frac{17}{19}$.

$017+019=\boxed{036}

==Solution 4== A bit similar to Solution 3. We use$ (Error compiling LaTeX. Unknown error_msg)\phi = \theta+90^\circ$because the progression cycles in$P\in (\sin 0\theta,\cos 1\theta,-\sin 2\theta,-\cos 3\theta\dots)$. So we could rewrite that as$P\in(\sin 0\phi,\sin 1\phi,\sin 2\phi,\sin 3\phi\dots)$.

Similarly,$ (Error compiling LaTeX. Unknown error_msg)Q\in (\cos 0\theta,-\sin 1\theta,-\cos 2\theta,\sin 3\theta\dots)\implies Q\in(\cos 0\phi,\cos 1\phi, \cos 2\phi, \cos 3\phi\dots)$.

Setting complex$ (Error compiling LaTeX. Unknown error_msg)z=q_1+p_1i$, we get$z=\frac{1}{2}\left(\cos\phi+\sin\phi i\right)$$ (Error compiling LaTeX. Unknown error_msg)(Q,P)=\sum_{n=0}^\infty z^n=\frac{1}{1-z}=\frac{1}{1-\frac{1}{2}\cos\phi-\frac{i}{2}\sin\phi}=\frac{1-0.5\cos\phi+0.5i\sin\phi}{\dots}$.

The important part is the ratio of the imaginary part$ (Error compiling LaTeX. Unknown error_msg)i$to the real part. To cancel out the imaginary part from the denominator, we must add$0.5i\sin\phi$to the numerator to make the denominator a difference (or rather a sum) of squares. The denominator does not matter. Only the numerator, because we are trying to find$\frac{P}{Q}=\tan\text{arg}(\Sigma)$a PROPORTION of values. So denominators would cancel out.$\frac{P}{Q}=\frac{\text{Re}\Sigma}{\text{Im}\Sigma}=\frac{0.5\sin\phi}{1-0.5\cos\phi}=\frac{\sin\phi}{2-\cos\phi}=\frac{2\sqrt{2}}{7}$.

Setting$ (Error compiling LaTeX. Unknown error_msg)\sin\theta=y$, we obtain <cmath>\frac{\sqrt{1-y^2}}{2+y}=\frac{2\sqrt{2}}{7}</cmath> <cmath>7\sqrt{1-y^2}=2\sqrt{2}(2+y)</cmath> <cmath>49-49y^2=8y^2+32y+32</cmath> <cmath>57y^2+32y-17=0\rightarrow y=\frac{-32\pm\sqrt{1024+4\cdot 969}}{114}</cmath> <cmath>y=\frac{-32\pm\sqrt{4900}}{114}=\frac{-16\pm 35}{57}</cmath>.

Since$ (Error compiling LaTeX. Unknown error_msg)y<0$because$\pi<\theta<2\pi$,$y=\sin\theta=-\frac{51}{57}=-\frac{17}{19}$. Adding up,$17+19=\boxed{036}$.

==Solution 5 (lots of room for sillies, I wouldn't recommend it)==

We notice$ (Error compiling LaTeX. Unknown error_msg)\sin\theta=\frac{-i}{2}(e^{i\theta}-e^{-i\theta})$and$\cos\theta=\frac{1}{2}(e^{i\theta}+e^{-i\theta})$With these, we just quickly find the sum of the infinite geometric series' in$P$and$Q$.$P$has 2 parts, the$\cos$and the$\sin$parts. The$\cos$part is:$\frac12\cos\theta-\frac18\cos3\theta+\cdots$, which can be turned into:$\frac14(e^{i\theta}(1-\frac{e^{i2\theta}}{4}+\cdots)+e^{-i\theta}(1-\frac{e^{-i2\theta}}{4}+\cdots))$, which is$\frac{1}{4}(\frac{e^{i\theta}}{1+\frac{1}{4}e^{i2\theta}}+\frac{e^{-i\theta}}{1+\frac{1}{4}e^{-i2\theta}})$. This turns into$\frac{5(e^{i\theta}+e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}$.

Following the same process as above, we find that the$ (Error compiling LaTeX. Unknown error_msg)\sin$part of$P$is$\frac{2i(e^{i2\theta}-e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}$, the$\cos$part of$Q$is$\frac{16+2(e^{i2\theta}+e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}$, and finally, the$\sin$part of$Q$is$\frac{3i(e^{i\theta}-e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}$.

We convert all 4 of these equations into trig, and we end up getting$ (Error compiling LaTeX. Unknown error_msg)\frac{2\sqrt{2}}{7}=\frac{10\cos{\theta}-4\sin{2\theta}}{16+4\cos{2\theta}-6\sin{\theta}}$, we divide by$2$on both numerator and denominator, and we get$\frac{2\sqrt{2}}{7}=\frac{5\cos{\theta}-2\sin{2\theta}}{8+2\cos{2\theta}-3\sin{\theta}}$. We use some trig identities and we get$\frac{\cos{\theta}(5-4\sin{\theta})}{10-4\sin^2{\theta}-3\sin{\theta}}=\frac{2\sqrt2}{7}$. We factor the denominator into$(5-4\sin\theta)(2+\sin\theta)$. We cancel out$5-4\sin\theta$on both numerator and denominator to get$\frac{\cos\theta}{2+\sin\theta}=\frac{2\sqrt2}{7}$. We set$\sin\theta$as$x$, and we just solve a quadratic in terms of$x$,$\frac{1-x^2}{x^2+4x+4}=\frac{8}{49}$, cross multiply and simplify, and we get$57x^2+32x-17=0$. We can actually factor this to get$(19x+17)(3x-1)=0$, which yields the 2 solutions$x=-\frac{17}{19}, x=\frac{1}{3}$. Since$\pi\leq\theta<2\pi$, the latter solution is deemed invalid, and we are left with$\sin\theta=-\frac{17}{19}$. Our final answer is$17+19=\boxed{036}$.

~ASAB

omg this was definitely the hardest problem on the AIME I

==Solution 6== Follow solution 3, up to the point of using the geometric series formula <cmath>Q+iP=\frac{1}{1+\frac{\sin(\theta)}{2}-\frac{Qi\cos(\theta)}{2}}</cmath>

Moving everything to the other side, and considering only the imaginary part, we get <cmath>Pi+\frac{Pi}{2}\sin\theta-\frac{Qi}{2}\cos\theta = 0</cmath>

We can then write$ (Error compiling LaTeX. Unknown error_msg)P = 2 \sqrt{2} k$, and$Q = 7k$, ($k \neq 0$). Thus, we can substitute and divide out by k. <cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\cos\theta\ =\ 0</cmath> <cmath>2\sqrt{2}+\sqrt{2}\sin\theta-\frac{7}{2}\sqrt{1-\sin^{2}\theta}=\ 0</cmath> <cmath>2\sqrt{2}+\sqrt{2}\sin\theta\ =\frac{7}{2}\left(\sqrt{1-\sin^{2}\theta}\right)</cmath> <cmath>8+8\sin\theta+2\sin^{2}\theta=\frac{49}{4}-\frac{49}{7}\sin^{2}\theta</cmath> <cmath>\frac{57}{4}\sin^{2}\theta+8\sin\theta-\frac{17}{4} = 0</cmath> <cmath>57\sin^{2}\theta+32\sin\theta-17 = 0</cmath> <cmath>\left(3\sin\theta-1\right)\left(19\sin\theta+17\right) = 0</cmath>

Since$ (Error compiling LaTeX. Unknown error_msg)\pi \le \theta < 2\pi$, we get$\sin \theta < 0$, and thus,$\sin\theta = \frac{-19}{17} \implies \boxed{036}$

-Alexlikemath

See also

2013 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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All AIME Problems and Solutions

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