Question

angle x lies in the third quadrant and tanx=7/24

determiner an exact value for cos2x
determiner an exact value for sin2x

Answer 1

sin ( 2 x ) = 2 tan x / ( 1 + tan² x )

cos ( 2 x ) = ( 1 - tan² x ) / ( 1 + tan² x )


tan x = 7 / 24 so:

cos ( 2 x ) = ( 1 - tan² x ) / ( 1 + tan² x )

cos ( 2 x ) = [ 1 - ( 7 / 24 )² ] / [ 1 + ( 7 / 24 )² ]

cos ( 2 x ) = ( 1 - 49 / 576) + ( 1 + 49 / 576 )

cos ( 2 x ) = ( 576 / 576 - 49 / 576 ) / ( 576 / 576 + 49 / 576 )

cos ( 2 x ) = ( 527 / 576 ) / (625 / 576 )

cos ( 2 x ) = 576 ∙ 527 / 576 ∙ 625

cos ( 2 x ) = 527 / 625


sin ( 2 x ) = 2 tan x / ( 1 + tan² x )

sin ( 2 x ) = 2 ∙ ( 7 / 24 ) / [ 1 + ( 7 / 24 )² ]

sin ( 2 x ) = ( 14 / 24 ) / ( 1 + 49 / 576 )

sin ( 2 x ) = ( 14 / 24 ) / ( 576 / 576 + 49 / 576 )

sin ( 2 x ) = ( 14 / 24 ) / ( 625 / 576 )

sin ( 2 x ) = 14 ∙ 576 / 24 ∙ 625

sin ( 2 x ) = 8064 / 15000

sin ( 2 x ) = 24 ∙ 336 / 24 ∙ 625

sin ( 2 x ) = 336 / 625