SOLUTION: I was given the following problem but I am having difficulty with it: compute the exact values of sin x/2, cos x/2, tan x/2 tan x=3/4, -pi<x<-pi/2 Can you show me the pro

Algebra ->  Trigonometry-basics -> SOLUTION: I was given the following problem but I am having difficulty with it: compute the exact values of sin x/2, cos x/2, tan x/2 tan x=3/4, -pi<x<-pi/2 Can you show me the pro      Log On


   

Question 818300: I was given the following problem but I am having difficulty with it:
compute the exact values of sin x/2, cos x/2, tan x/2
tan x=3/4, -pi Can you show me the process of how to solve this?
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
I was given the following problem but I am having difficulty with it:
compute the exact values of sin x/2, cos x/2, tan x/2
tan x=3/4, -pi
***
Identities:
sin%28x%2F2%29=sqrt%28%281-cos%28x%29%29%2F2%29
cos%28x%2F2%29=sqrt%28%281%2Bcos%28x%29%29%2F2%29
tan%28x%2F2%29=sin%28x%29%2F%281%2Bcos%28x%29%29
..
Assume reference angle x is quadrant I
You are working with a 3-4-5 right triangle in quadrant I
sin(x)=3/5
cos(x)=4/5
..
sin%28x%2F2%29=sqrt%28%281-%284%2F5%29%29%2F2%29=sqrt%28%281%2F5%29%2F2%29=1/√10=√10/10
cos%28x%2F2%29=sqrt%28%281%2Bcos%28x%29%29%2F2%29=sqrt%28%281%2B%284%2F5%29%29%2F2%29=√(9/10)=3/√10=3√10/10

..
calculator check:
tan(x)=3/4
x≈36.8698˚
x/2≈18.4349˚
..
sin(x/2)≈sin(18.4349)≈0.3162..
exact value as calculated=√10/10≈0.3162..
..
cos(x/2)≈cos(18.4349)≈0.9486..
exact value as calculated=3√10/10≈0.9486..
..
tan(x/2)=tan(18.4349)≈0.3333..
exact value as calculated=1/3