Question 1156517
{{{ 5 + 2cos(x) - 8*sin^2(x) = 0 }}}
Use identity:
{{{ sin^2(x) + cos^2(x) = 1 }}}
{{{ sin^2(x) = 1 - cos^2(x) }}}
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{{{ 5 + 2cos(x) - 8*( 1 - cos^2(x)) = 0 }}}
{{{ 5 + 2cos(x) - 8 + 8cos^2(x) = 0 }}}
{{{ 8cos^2(x) + 2 cos(x) - 3 = 0 }}}
Let {{{ cos(x) = z }}}
{{{ 8z^2 + 2z - 3 = 0 }}}
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Use quadratic formula:
{{{ z = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ a = 8 }}}
{{{ b = 2 }}}
{{{ c = -3 }}}
{{{ z = (-2 +- sqrt( 2^2-4*8*(-3) ))/(2*8) }}}
{{{ z = (-2 +- sqrt( 4 + 96 )) / 16 }}}
{{{ z = (-2 +- sqrt( 100 )) / 16 }}}
{{{ z = ( -2 + 10 ) / 16 }}}
{{{ z = 1/2 }}}
and
{{{ z = ( -2 - 10 ) / 16 }}}
{{{ z = -12/16 }}}
{{{ z = -3/4 }}}
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{{{ cos(x) = z }}}
{{{ cos(x) = 1/2 }}}
{{{ x = arc cos( 1/2 ) }}}
{{{ x = pi / 3 }}}
and
{{{ x = 2pi - pi/3 }}}
{{{ x = ( 5pi )/3 }}}
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{{{ cos(x) = z }}}
{{{ cos(x) = -3/4 }}}
{{{ x = arc cos( -3/4 ) }}}
{{{ x = 138.59 *( pi / 180 ) }}}
{{{ x = .77pi }}}
and
{{{ x = ( 360 - 138.59 )*( pi / 180 ) }}}
{{{ x = 221.41*( pi / 180 ) }}}
{{{ x = 1.23pi }}}
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My answers are:
{{{ x = pi/3 }}}
{{{ x = ( 5pi )/3 }}}
{{{ x = .77pi }}}
{{{ x = 1.23pi }}}
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I'll check one of the answers.  You can check the rest
{{{ 5 + 2cos(x) - 8*sin^2(x) = 0 }}}
{{{ 5 + 2*cos( .77pi ) - 8*sin^2( .77pi )= 0 }}}
{{{ 5 + (-1.5) - 3.5 = 0 }}}
{{{ 0 = 0 }}}
Looks OK
To convert back to degrees, multiply answers by {{{ 180/pi }}}
Definitely get a 2nd opinion if needed
and check my math