Question

, x in quadrant II

Answer 1

As, x is in quadrant II,

π 2 <x<π π 4 < x 2 < π 2

Therefore, sin x 2 , cos x 2 and tan x 2 are all positive.

Form the question,

tanx=− 4 3

So,

sec 2 x=1+ tan 2 x =1+ ( − 4 3 ) 2 =1+ 16 9 = 25 9

Therefore,

cos 2 x= 9 25 cosx=± 3 5

As x is in quadrant II, cosx is negative.

cosx=− 3 5

Now, for cos x 2

cosx=2 cos 2 x 2 −1 − 3 5 = cos 2 x 2 −1 cos 2 x 2 = 1 5 cos x 2 = 1 5         ( ∵cos x 2  is positive )

For sin x 2 ,

sin 2 x 2 + cos 2 x 2 =1 sin 2 x 2 + ( 1 5 ) 2 =1 sin 2 x 2 = 4 5          ( ∵sin x 2  is positive ) sin x 2 = 2 5

For tan x 2 ,

tan x 2 = sin x 2 cos x 2 = 2 5 1 5 =2

Hence, the respective values of sin x 2 , cos x 2 and tan x 2 are 2 5 ,  1 5 and 2.