Question 1071226
{{{ a = 12 }}} ( vertical component )
{{{ b = -5 }}} ( horizontal component )
{{{ c^2 = 12^2 + (-5)^2 }}}
{{{ c^2 = 144 + 25 }}}
{{{ c^2 = 169 }}}
{{{ c = 13 }}}
{{{ sin( theta ) = 12/13 }}}
{{{ cos( theta ) = -5/13 }}}
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{{{ cos( 2*theta ) = 1 - 2*( 12/13 )^2 }}}
{{{ cos( 2*theta ) = 1 - 2*( 144/169 ) }}}
{{{ cos( 2*theta ) = 169/169 - 288/169 }}}
{{{ cos( 2*theta ) = -119/169 }}}
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{{{ tan( 2*theta ) = ( 2*tan( theta ) ) / ( 1 - ( tan(theta))^2 ) }}}
{{{ tan( 2*theta ) = ( 2*( -12/5 ) ) / ( 1 - ( -12/5 )^2 ) }}}
{{{ tan( 2*theta ) = ( -24/5 ) / (  25/25 - 144/25 ) }}}
{{{ tan( 2*theta ) = ( -120/25 ) / (  25/25 - 144/25 ) }}}
{{{ tan( 2*theta ) = ( -120 ) / ( 25 - 144 ) }}}
{{{ tan( 2*theta ) = 120/119 }}}
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On my calculator I get:
{{{ theta = 112.6199 }}}
{{{ 2*theta = 225.2397 }}}
My answers check on calculator
{{{ cos( 2*theta ) = -.70414 }}}
{{{ tan( 2*theta ) = 1.0084 }}}
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Note that {{{ 2*theta }}} is in the 3rd quadrant
check all the math!