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If $t \in (0,\pi)$, how can we find the integer solutions to the system in $x,y,z$ with $y,z\neq0$

\begin{align} (y+z)\cos(3t) &= xyz \sin(3t) \\ x \sin(3t) &= 2 \frac{\cos(3t)}{y} +2 \frac{\sin(3t)}{z} \\ xyz \sin(3t) &= (y+2z)\cos(3t)+y \sin(3t) \end{align}

Thank you for any hints/responces!

Cameron Williams
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AgnostMystic
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  • You should get in the habit of putting parentheses around the argument of functions. $f3t+y$ is very ambiguous (though probably clear for this particular example) and can lead to confusion. I edited your post to hopefully better reflect the actual problem at hand. Plus, function notation necessitates parentheses. We don't write $fx =$, we write $f(x) = $. – Cameron Williams Sep 21 '20 at 12:32
  • The first step is to let $s = \cos(3t),$ and then calculate $\sin(3t)$ in terms of $s.$ Then, rewrite the equations, removing the trig functions. Then determine which theorems or previously solved problems you think might be pertinent here. Regard them as your tools (i.e. the problem's background). Next, spend 30 minutes to an hour trying to use your tools to solve the problem. Finally, regardless of how far you get, edit your query to detail the problem's background (from your perspective) and to show your work. – user2661923 Sep 21 '20 at 12:36
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    If you multiply the middle equation by $yz$ to clear the fractions, then the LHS of the bottom two equations are the same. Equating the right hand sides of those two equations then shows that you must have $\sin(3t)=\cos(3t)$. Once you have that things get considerably simpler. – Jaap Scherphuis Sep 21 '20 at 12:42

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