SOLUTION: Use the given information to find cos(x/2), sin(x/2), and tan(x/2): Cos (x)=-4/5 180 degrees< x < 270 degrees

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Question 810472: Use the given information to find cos(x/2), sin(x/2), and tan(x/2):
Cos (x)=-4/5
180 degrees< x < 270 degrees
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Use the given information to find exact answers for cos(x/2), sin(x/2), and tan(x/2):
Cos (x)=-4/5
180 degrees< x < 270 degrees
***
you are working with a 3-4-5 reference right triangle in quadrant III where sin>0, cos<0, tan>0.
..
cos(x)=-4/5 (given)
sin(x)=-3/5
tan(x)=sin/cos=3/5
..
cos%28x%2F2%29=-sqrt%28%281%2Bcos%28x%29%29%2F2%29%29
cos%28x%2F2%29=-sqrt%28%281-4%2F5%29%2F2%29=-sqrt%28%281%2F5%29%2F2%29=-sqrt%281%2F10%29
..
sin%28x%2F2%29=sqrt%28%281-cos%28x%29%29%2F2%29%29
sin%28x%2F2%29=sqrt%28%281%2B4%2F5%29%2F2%29=sqrt%28%289%2F5%29%2F2%29=sqrt%289%2F10%29
..
tan%28x%2F2%29=sin%28x%29%2F%281%2Bcos%28x%29%29
tan%28x%2F2%29=%28-3%2F5%29%2F%281-4%2F5%29=%28-3%2F5%29%2F%281%2F5%29=-3
..
Check:(w/calculator)
cos(x)=-4/5(in quadrant III)
x≈216.87˚
x/2≈108.43˚
..
cos(x/2)=cos(108.43˚)≈-0.316
exact value=-√(1/10)≈-0.316
..
sin(x/2)=sin(108.43˚)≈0.9487
exact value=√(9/10)≈0.9487
..
tan(x/2)=tan(108.43˚)≈-3.0000
exact value=-3.0000