Find sinx2 cosx2 and tanx2 from the given information secx

Find sin(x/2), cos(x/2), and tan(x/2) from the given information. sec(x) = 7/6, 270 degree

Solution

sec x =7/6 x is in IV quadrant ie 270 < x < 360⁰

cos x = 1/ sec x

cos x = 6/7   x is in IV quadrant

from cos x = 2 cos²(x/2) - 1

2 cos²(x/2) = cos x + 1

cos (x/2) = +_ ( cos x + 1 / 2)^1/2

cos (x/2) = +_ ( [6/7] + 1 / 2)^1/2

cos (x/2) = +_ 0.963

[ since 270 < x< 360 , So

( 270/2) < (x/2) < (360 /2) = 135 < (x/2)< 180 ie in II quadrant.

in II quadrant cos (theta) is -Ve.

cos(x/2) = - 0.963

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from sin²(x/2) + cos²(x/2) = 1

sin (x/2) = +_ ( 1- cos²(x/2) )^1/2

sin (x/2) = +_ ( 1- ( - 0.963)² )^1/2   135 < (x/2)< 180

sin (x/2) = + 0.2695   

[ in II quadrant sin(theta) is +Ve]

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

= 0.2695 / - 0.963 = - 0.2798