Find sin x /2 , cos x /2 , and tan x /2 from the given information. cos(x) = − 24 25 , 180° < x < 270°

Question

Find sin(x/2), cos(x/2), and tan (x/2) from the given information.

cos(x)=-24/25, 180°<x<270°

Answer 1

If 180° < x < 270°, 90° <x/2 < 135°, so x/2 is in the second quadrant, where sin is positive, cos is negative, and tan is negative.

cos x/2 = -√((1+cosx)/2) = -√((1-24/25)/2) = -√(1/50) = -√2/10

sin x/2 = √((1-cosx)/2) = √((1+24/25)/2) = √(49/50) = 7√2/10

tan x/2 =sin x/2/(cos x/2) = 7√2/10/(-√2/10) = -7

If you prefer, you can calculate tan x/2 = - √((1-cos x)/(1+cosx)) = -√((1+24/25)/(1-24/25)) = -√49 = -7

tan x/2 may also be written as sin x/(1+cos x)

Then, you calculate sin(x) as -7/25 since it is in the 3rd quadrant.

-7/25/(1-24/25) = -7