Question 1104138
{{{ sin( 2x ) = -1/2 }}}
Let {{{ 2x = z }}}
{{{ sin( z ) = -1/2 }}}
The sin function is negative in 
the 3rd and 4th quadrants
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{{{ z = pi + pi/6 + 2n*pi }}}
{{{ z = 2pi - pi/6 + 2n*pi }}}
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{{{ 2x = pi + pi/6 + 2n*pi }}}
{{{ x = pi/2 + pi/12 + n*pi }}}
{{{ x = ( 7pi )/12 + n*pi }}}
and
{{{ 2x = 2pi - pi/6 + 2n*pi }}}
{{{ x = pi - pi/12 + n*pi }}}
{{{ x = ( 11pi )/12 + n*pi }}}
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check:
suppose {{{ n = 3 }}}
{{{ x = ( 7pi )/12 + 3pi }}}
{{{ 2x = (7pi)/6  + 6pi }}}
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{{{ 6pi }}} takes you back to {{{ 0pi }}}
and
{{{ (7pi)/6 = pi + pi/6 }}}
and
{{{ sin( (7pi)/6 ) = -1/2  }}}
OK