Question

Find sin x/2, cos x/2, and tan x/2 from the given information. Please Help!!!
tan x = 2, 0° < x < 90°
sin x/2=
cos x/2=
tan x/2=

Answer 1

To find tan x/2, use the trig identity tan 2x = (2tan x/2)/(1 - tan^2 x/2)
Call t = tan x/2 -> tan x = 2 = 2t/(1 - t^2)->
2t^2 + 2t - 2 = 0 -> t^2 + t - 1 = 0. Solve this quadratic equation.
D = 1 + 4 = 5-> VD = V5
Two real roots: t = (-1 + V5)/2 and (-1 - V5)/2 (rejected because tan x > 0
tan x/2 = t = 0.62 = (sin x/2)/(cos x/2)-> sin x/2 = 0.62*cos x/2 (1)
sin^2 x/2 + cos^2 x/2 = 1 (2)
0.38*cos^2 x/2 + cos^2 x/2 = 1
cos^2 x/2 (1 +0.38) = 1
cos^2 x/2 = 1/1.38 = 0.72 -> cos x/2 = 0.85
sin x/2 = 0.62*(0.85) ->sin x/2 = 0.53
Check with equation (2): (0.85)^2 + (0.53)^2 = 1. Correct.
Check with equation (1): tan x = 0.53/0.85 = 0.62. Correct