Question 831733
Use a double- or half-angle identity to find the exact value of each trigonometric expression below if csc x = 4 and 90 degrees < x < 180 degrees
a) sin x/2
b) cos x/2
c) tan x/2
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cscx=4
sinx=1/4
cosx=-&#8730;(1-sin^2x)=-&#8730;(1-(1/16))=-&#8730;(15/16)=-&#8730;15/4
note: cos<0,sin>0 in quadrant II
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sin(x/2)={{{sqrt((1+cos(x))/2)=sqrt((1+sqrt(15)/4)/2)=sqrt((4+sqrt(15))/8))}}}
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cos(x/2)={{{sqrt((1-cos(x))/2)=sqrt((1-sqrt(15)/4)/2)=sqrt((4-sqrt(15))/8))}}}
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tan(x/2)={{{(sin(x))/(1-cos(x))=(1/4)/((1-sqrt(15)/4))=(1/4)/((4-sqrt(15))/4)=1/(4-sqrt(15))}}}
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Calculator check:
sinx=1/4
x&#8776;165.52&#730; in quadrant II
x/2&#8776;82.76&#730;
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sin(x/2)&#8776;sin(82.76)&#8776;0.9920..
exact value=&#8730;(4+&#8730;15)/8&#8776;0.9920..
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cos(x/2)&#8776;cos(82.76)&#8776;0.1260..
exact value=&#8730;(4-&#8730;15)/8&#8776;0.1260..
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tan(x/2)=tan(82.76)&#8776;7.872..
exact value=1/(4-&#8730;15)&#8776;7.872..