If tan ` x=(-3)/(4) " and" (3pi)/(2) lt x lt 2 pi ` find the value of
`"(i) " sin 2x , "(ii) " cos 2x , "(iii) " tan 2x`

Explanation

Since x lies in Quadrant IV we have
cos x `gt ` 0 sin , x `lt` 0 and tan x `lt` 0
`:. " sec " x = sqrt(1+ tan^(2) x) = sqrt((1 +(9)/(16)) = sqrt((25)/(16)) = (5)/(4) rArr " cos " x = (4)/(5) `
And sin `x= -sqrt(1- "cos"^(2) x ) = - sqrt((1-(16)/(25)) =- sqrt((9)/(25)) = (-3)/(5)`
`:. " tan " x = ("sin x")/(" cos x ") = ((-3)/(5) xx (5)/(4)) = (-3)/(4)`
`(i) " sin " 2x =2"sin x cos x "`
`={2 xx ((-3))/(5) xx (4)/(5)} = (-24)/(25)`
`(ii) " cos " 2x =(2 "cos"^(2) x -1)`
` =(2 xx (16)/(25) -1) =(7)/(25)`
`(iii) " tan " 2x = ("sin 2x ")/(" cos 2x ") =((-24)/(25) xx (25)/(7)) =(-24)/(7) `