Question 810472
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
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you are working with a 3-4-5 reference right triangle in quadrant III where sin>0, cos<0, tan>0.
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cos(x)=-4/5 (given)
sin(x)=-3/5
tan(x)=sin/cos=3/5
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{{{cos(x/2)=-sqrt((1+cos(x))/2))}}}
{{{cos(x/2)=-sqrt((1-4/5)/2)=-sqrt((1/5)/2)=-sqrt(1/10)}}}
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{{{sin(x/2)=sqrt((1-cos(x))/2))}}}
{{{sin(x/2)=sqrt((1+4/5)/2)=sqrt((9/5)/2)=sqrt(9/10)}}}
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{{{tan(x/2)=sin(x)/(1+cos(x))}}}
{{{tan(x/2)=(-3/5)/(1-4/5)=(-3/5)/(1/5)=-3}}}
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Check:(w/calculator)
cos(x)=-4/5(in quadrant III)
x&#8776;216.87&#730;
x/2&#8776;108.43&#730;
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cos(x/2)=cos(108.43&#730;)&#8776;-0.316
exact value=-&#8730;(1/10)&#8776;-0.316
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sin(x/2)=sin(108.43&#730;)&#8776;0.9487
exact value=&#8730;(9/10)&#8776;0.9487
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tan(x/2)=tan(108.43&#730;)&#8776;-3.0000
exact value=-3.0000