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
 
+
<cmath>\begin{align*}
<math>\begin{align*}</math>
+
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
<math>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</math>
+
\end{align*}</cmath>
<math>\end{align*}</math>
 
 
 
 
and
 
and
 
+
<cmath>\begin{align*}
<math>\begin{align*}</math>
+
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
<math>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</math>
+
\end{align*}</cmath>
<math>\end{align*}</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>.
 
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===
 
<math>\begin{align*}</math>
 
<math>P \sin\theta\ + Q \cos\theta\ = \cos\theta\ - \frac{1}{2}\ P</math>
 
<math>\end{align*}</math>
 
and
 
<math>\begin{align*}</math>
 
<math>P \cos\theta\ + Q \sin\theta\ = -2(Q-1)</math>
 
<math>\end{align*}</math>
 
 
 
Solving for P, Q we have
 
  
 +
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:
  
<math>\frac{P}{Q} = \frac{\cos\theta }{2 + \sin\theta } = \frac{2\sqrt2}{7}</math>
+
<cmath>\sin(a + b) = \sin a \cos b + \cos a \sin b, \cos(a + b) = \cos a \cos b - \sin a \sin b</cmath>
  
Square both side, and use polynomial rational root theorem to solve <math>sin\theta</math>
+
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
 +
<cmath>\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*}</cmath>
 +
after adding term-by-term. Similar term-by-term adding yields <cmath>P \cos \theta + Q \sin \theta = -2(Q - 1).</cmath>
 +
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> Since we're given <math>\pi\leq\theta<2\pi,</math> <math>\sin\theta</math> is nonpositive. We therefore use the negative solution, and our desired answer is <math>17 + 19 = \boxed{036}.</math>
  
<math>\sin\theta = -\frac{17}{19} </math>
+
==Solution 2==
  
The answer is <math>\boxed{036}</math>
+
Use sum to product formulas to rewrite <math>P</math> and <math>Q</math>
  
===Solution 2===
 
<math>\begin{align*}</math>
 
Use sum to product formulas to rewrite <math>P</math> and <math>Q</math>
 
<math>\end{align*}</math>
 
  
<math>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 + ... </math>
+
<cmath>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 + ... </cmath>
  
<math>\begin{align*}</math>
+
Therefore, <cmath>P \sin \theta - Q \cos \theta = -2P</cmath>
Therefore, <math>P \sin \theta - Q \cos \theta = -2P</math>
 
<math>\end{align*}</math>
 
  
<math>\begin{align*}</math>
+
Using <cmath>\frac{P}{Q} =  \frac{2\sqrt2}{7}, Q = \frac{7}{2\sqrt2} P</cmath>
Using <math>\frac{P}{Q} =  \frac{2\sqrt2}{7}</math>, <math>Q = \frac{7}{2\sqrt2} P</math>
 
<math>\end{align*}</math>
 
  
<math>\begin{align*}</math>
+
Plug in to the previous equation and cancel out the "P" terms to get: <cmath>\sin\theta - \frac{7}{2\sqrt2} \cos\theta = -2</cmath>
Plug in to the previous equation and cancel out the "P" terms to get: <math>\sin\theta - \frac{7}{2\sqrt2} \cos\theta = -2</math>
 
<math>\end{align*}</math>
 
  
Then use the pythagorean identity to solve for <math>\sin\theta</math>, <math>\sin\theta = \boxed{036}</math>
+
Then use the pythagorean identity to solve for <math>\sin\theta</math>, <cmath>\sin\theta = -\frac{17}{19} \implies \boxed{036}</cmath>
  
===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 85: Line 69:
  
 
<math>017+019=\boxed{036}</math>
 
<math>017+019=\boxed{036}</math>
 +
 +
==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>.
 +
 +
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>
 +
 +
<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.
 +
 +
<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
 +
<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 <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 (utterly disgusting)==
 +
 +
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 observe that both <math>P</math> and <math>Q</math> can be split into <math>2</math> parts, namely the terms which contain the <math>\cos</math> and the terms which contain the <math>\sin .</math>
 +
 +
The <math>\cos</math> part of <math>P</math> can be expressed as:
 +
<cmath>\begin{align*}\frac12\cos\theta-\frac18\cos3\theta+\cdots&=\frac14\left(e^{i\theta}\left(1-\frac{e^{i2\theta}}{4}+\cdots\right)+e^{-i\theta}\left(1-\frac{e^{-i2\theta}}{4}+\cdots\right)\right) \
 +
&= \frac{1}{4}\left(\frac{e^{i\theta}}{1+\frac{1}{4}e^{i2\theta}}+\frac{e^{-i\theta}}{1+\frac{1}{4}e^{-i2\theta}}\right)\
 +
&= \frac{5(e^{i\theta}+e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}.\end{align*}</cmath>
 +
 +
Repeating the above process, we find that the <math>\sin</math> part of <math>P</math> is <cmath>\frac{2i(e^{i2\theta}-e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}},</cmath> the <math>\cos</math> part of <math>Q</math> is <cmath>\frac{16+2(e^{i2\theta}+e^{-i2\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}},</cmath> and finally, the <math>\sin</math> part of <math>Q</math> is <cmath>\frac{3i(e^{i\theta}-e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}.</cmath>
 +
 +
Converting back to trigonometric form, we have <cmath>\begin{align*}\frac{2\sqrt{2}}{7}&=\frac{10\cos{\theta}-4\sin{2\theta}}{16+4\cos{2\theta}-6\sin{\theta}}\
 +
&=\frac{5\cos{\theta}-2\sin{2\theta}}{8+2\cos{2\theta}-3\sin{\theta}}.\end{align*}</cmath> Using the <math>\sin</math> double identity and simplifying, we have <cmath>\frac{2\sqrt2}{7}=\frac{\cos{\theta}(5-4\sin{\theta})}{10-4\sin^2{\theta}-3\sin{\theta}}.</cmath> Factoring the denominator, we have <cmath>10-4\sin^2{\theta}-3\sin{\theta}=(5-4\sin\theta)(2+\sin\theta).</cmath> Simplifying <cmath>\begin{align*}\frac{2\sqrt2}{7}&= \frac{\cos{\theta}(5-4\sin{\theta})}{(5-4\sin\theta)(2+\sin\theta)}\
 +
&=\frac{\cos\theta}{2+\sin\theta}.\end{align*}</cmath> We set <math>\sin\theta</math> as <math>x</math>, and by the Pythagorean Identity, we have <math>57x^2+32x-17=0</math>. This factors into <math>(19x+17)(3x-1)=0</math>, which yields the 2 solutions <math>x=-\frac{17}{19}, x=\frac{1}{3}</math>. As <math>\pi\leq\theta<2\pi</math>, the latter root is erroneous, and we are left with <math>\sin\theta=-\frac{17}{19}</math>. Thus, our final answer is <math>17+19=\boxed{036}</math>.
 +
 +
~ASAB
 +
 +
==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 <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}\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 <math>\pi \le \theta < 2\pi</math>, we get <math>\sin \theta < 0</math>, and thus, <math>\sin\theta = \frac{-17}{19} \implies \boxed{036}</math>
 +
 +
-Alexlikemath
 +
 +
 +
==Video Solution==
 +
https://youtu.be/036u51CF-EQ?si=SHTrTwSg3LMnE_yH
 +
 +
~MathProblemSolvingSkills.com
 +
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2013|n=I|num-b=13|num-a=15}}
 
{{AIME box|year=2013|n=I|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:18, 21 August 2024

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 $\sin \theta = -17/19, 1/3.$ Since we're given $\pi\leq\theta<2\pi,$ $\sin\theta$ is nonpositive. We therefore use 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 $\phi = \theta+90^\circ$ 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, $Q\in (\cos 0\theta,-\sin 1\theta,-\cos 2\theta,\sin 3\theta\dots)\implies Q\in(\cos 0\phi,\cos 1\phi, \cos 2\phi, \cos 3\phi\dots)$.

Setting complex $z=q_1+p_1i$, we get $z=\frac{1}{2}\left(\cos\phi+\sin\phi i\right)$

$(Q,P)=\sum_{n=0}^\infty z^n=\frac{1}{1-z}=\frac{1}{1-\frac{1}{2}\cos\phi-\frac{i}{2}\sin\phi}=\frac{1-0.5\cos\phi+0.5i\sin\phi}{\dots}$.

The important part is the ratio of the imaginary part $i$ 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 $\sin\theta=y$, we obtain \[\frac{\sqrt{1-y^2}}{2+y}=\frac{2\sqrt{2}}{7}\] \[7\sqrt{1-y^2}=2\sqrt{2}(2+y)\] \[49-49y^2=8y^2+32y+32\] \[57y^2+32y-17=0\rightarrow y=\frac{-32\pm\sqrt{1024+4\cdot 969}}{114}\] \[y=\frac{-32\pm\sqrt{4900}}{114}=\frac{-16\pm 35}{57}\].

Since $y<0$ because $\pi<\theta<2\pi$, $y=\sin\theta=-\frac{51}{57}=-\frac{17}{19}$. Adding up, $17+19=\boxed{036}$.

Solution 5 (utterly disgusting)

We notice $\sin\theta=-\frac{i}{2}(e^{i\theta}-e^{-i\theta})$ and $\cos\theta=\frac{1}{2}(e^{i\theta}+e^{-i\theta})$

We observe that both $P$ and $Q$ can be split into $2$ parts, namely the terms which contain the $\cos$ and the terms which contain the $\sin .$

The $\cos$ part of $P$ can be expressed as: \begin{align*}\frac12\cos\theta-\frac18\cos3\theta+\cdots&=\frac14\left(e^{i\theta}\left(1-\frac{e^{i2\theta}}{4}+\cdots\right)+e^{-i\theta}\left(1-\frac{e^{-i2\theta}}{4}+\cdots\right)\right) \\ &= \frac{1}{4}\left(\frac{e^{i\theta}}{1+\frac{1}{4}e^{i2\theta}}+\frac{e^{-i\theta}}{1+\frac{1}{4}e^{-i2\theta}}\right)\\  &= \frac{5(e^{i\theta}+e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}.\end{align*}

Repeating the above process, we find that the $\sin$ 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 \[\frac{3i(e^{i\theta}-e^{-i\theta})}{17+4e^{i2\theta}+4e^{-i2\theta}}.\]

Converting back to trigonometric form, we have \begin{align*}\frac{2\sqrt{2}}{7}&=\frac{10\cos{\theta}-4\sin{2\theta}}{16+4\cos{2\theta}-6\sin{\theta}}\\ &=\frac{5\cos{\theta}-2\sin{2\theta}}{8+2\cos{2\theta}-3\sin{\theta}}.\end{align*} Using the $\sin$ double identity and simplifying, we have \[\frac{2\sqrt2}{7}=\frac{\cos{\theta}(5-4\sin{\theta})}{10-4\sin^2{\theta}-3\sin{\theta}}.\] Factoring the denominator, we have \[10-4\sin^2{\theta}-3\sin{\theta}=(5-4\sin\theta)(2+\sin\theta).\] Simplifying \begin{align*}\frac{2\sqrt2}{7}&= \frac{\cos{\theta}(5-4\sin{\theta})}{(5-4\sin\theta)(2+\sin\theta)}\\ &=\frac{\cos\theta}{2+\sin\theta}.\end{align*} We set $\sin\theta$ as $x$, and by the Pythagorean Identity, we have $57x^2+32x-17=0$. This factors into $(19x+17)(3x-1)=0$, which yields the 2 solutions $x=-\frac{17}{19}, x=\frac{1}{3}$. As $\pi\leq\theta<2\pi$, the latter root is erroneous, and we are left with $\sin\theta=-\frac{17}{19}$. Thus, our final answer is $17+19=\boxed{036}$.

~ASAB

Solution 6

Follow solution 3, up to the point of using the geometric series formula \[Q+iP=\frac{1}{1+\frac{\sin(\theta)}{2}-\frac{Qi\cos(\theta)}{2}}\]

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

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

Since $\pi \le \theta < 2\pi$, we get $\sin \theta < 0$, and thus, $\sin\theta = \frac{-17}{19} \implies \boxed{036}$

-Alexlikemath


Video Solution

https://youtu.be/036u51CF-EQ?si=SHTrTwSg3LMnE_yH

~MathProblemSolvingSkills.com


See also

2013 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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