Difference between revisions of "2013 AIME I Problems/Problem 14"

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<math>\frac{P}{Q} = \frac{cos\theta\ ( sin\theta + 2)}{8 + 8sin\theta + 2sin^2\theta }&#036;
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<math>\frac{P}{Q} = \frac{cos\theta\ ( sin\theta + 2)}{8 + 8sin\theta + 2sin^2\theta }</math>
 
 
</math>
 
  
 
Square both side, and use polynomial rational root theorem to solve <math>sin\theta</math>
 
Square both side, and use polynomial rational root theorem to solve <math>sin\theta</math>

Revision as of 16:00, 21 March 2013

Problem 14

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

$\begin{align*}$ (Error compiling LaTeX. Unknown error_msg) $P &= \frac12\cos\theta - \frac14\sin 2\theta - \frac18\cos 3\theta + \frac{1}{16}\sin 4\theta + \frac{1}{32} \cos 5\theta - \frac{1}{64} \sin 6\theta - \frac{1}{128} \cos 7\theta + \cdots$ (Error compiling LaTeX. Unknown error_msg) $\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

and

$\begin{align*}$ (Error compiling LaTeX. Unknown error_msg) $Q &= 1 - \frac12\sin\theta -\frac14\cos 2\theta + \frac18 \sin 3\theta + \frac{1}{16}\cos 4\theta - \frac{1}{32}\sin 5\theta - \frac{1}{64}\cos 6\theta +\frac{1}{128}\sin 7\theta + \cdots$ (Error compiling LaTeX. Unknown error_msg) $\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

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

(solution) $\begin{align*}$ (Error compiling LaTeX. Unknown error_msg) $P sin\theta\ + Q cos\theta\ = cos\theta\ - \frac{1}{2}\ P$ $\end{align*}$ (Error compiling LaTeX. Unknown error_msg) and $\begin{align*}$ (Error compiling LaTeX. Unknown error_msg) $P cos\theta\ + Q sin\theta\ = -2(Q-1)$ $\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Solve for P, Q we have


$\frac{P}{Q} = \frac{cos\theta\ ( sin\theta + 2)}{8 + 8sin\theta + 2sin^2\theta }$

Square both side, and use polynomial rational root theorem to solve $sin\theta$

$sin\theta = -\frac{17}{19}$

The answer is 036

See also

2013 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions