Question

Find sin x/2, Cos x/2 and tqn x/2 for sin x = -1/4, in quadrant 2nd​

Answer 1

Answer :

x is in 2nd quadrant then x is between 90 and 180 then cos x is negative and x/2 is between 0 and 90 which means that sin(x/2) , cos (x/2) and tan(x/2) are all positive.

cos x = - sqrt( 1 / (1 + (tan x)^2)) =

= - sqrt( 1/ (1 + 16/9))

= - sqrt( 9 / 25 ) = - 3/5.

(sin x/2)^2 = (1 - cos x)/2 = ( 1 + 3/5)/2

= 4/5

sin x/2 = 2/sqrt(5).

(cos x/2)^2 = (1 + cos x)/2 = ( 1 - 3/5)/2

= 1/5

cos x/2 = 1/sqrt(5).

tan x/2 = (sin x/2)/(cos x/2) = 2.

Answer 2

ANSWER

Given that x is in second quadrant

90<x<180

45<2x<90

2x lies in first quadrant

So,

sinx,cosx,tanx are positive in  first quadrant

tanx=1−tan22x2tan(2x)

3−4=1−tan22x2tan(2x)

Solving the above quadratic equation, we get,

tan(2x)=2−1    or  tan(2x)=2

∴tan(2x)=2(∵2x) lies in first quadrant

1+tan2(2x)=sec2(2