Question

Find for , x in quadrant III

Answer 1

As, x is in quadrant III,

π<x< 3π 2 π 2 < x 2 < 3π 4

Therefore, cos x 2 and tan x 2 are negative, whereas sin x 2 is positive.

It is given that,

cosx=− 1 3

Here,

cosx=1−2 sin 2 x 2 sin 2 x 2 = 1−cosx 2 = 1−( − 1 3 ) 2 = 2 3

Therefore,

sin x 2 = 2 3         ( sin x 2  is positive ) sin x 2 = 6 3

Now,

cosx=2 cos 2 x 2 −1 cos 2 x 2 = 1+cosx 2 = 1+( − 1 3 ) 2 = 1 3

Now, for cos x 2

cos x 2 =− 1 3

For tan x 2 ,

tan x 2 = sin x 2 cos x 2 = 2 3     − 1 3     =− 2

Simplify the above equation,

tan x 2 =− 2

Hence, the respective values of sin x 2 , cos x 2 and tan x 2 are 6 3 ,  − 3 3 and − 2 .