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Sine Degree Radian Equality

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $ x $ for which the sine of $ x $ degrees is the same as the sine of $ x $ radians are $ \frac{m\pi}{n-\pi} $ and $ \frac{p\pi}{q+\pi} $, where $ m $, $ n $, $ p $, and $ q $ are positive integers. Find $ m+n+p+q $.

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$\frac{a}{b}$
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0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$

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