Question 869336
Find sin(x/2), cos(x/2), and tan(x/2) from the given information.
tan(x)= 2sqrt6 and 0^degrees < x < 90^degrees
{{{tan(x)=2sqrt(6)}}}
hypotenuse of reference right triangle in quadrant I={{{sqrt((2sqrt(6))^2+1)=sqrt(24+1)=sqrt(25)=5)}}}
{{{sin(x)=2sqrt(6)/5}}}
{{{cos(x)=1/5}}}
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{{{sin(x/2)=sqrt((1-cos(x))/2)=sqrt((1-1/5)/2)=sqrt((4/5)/2)=sqrt(4/10)=2/sqrt(10)}}}=2&#8730;10/10=&#8730;10/5
{{{cos(x/2)=sqrt((1+cos(x))/2)=sqrt((1+1/5)/2)=sqrt((6/5)/2)=sqrt((3/5))}}}=&#8730;(3/5)
tan(x/2)=sin/cos=(&#8730;10/5)/&#8730;(3/5)=&#8730;50/5&#8730;3
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Calculator check:
tan(x)=2&#8730;6
x&#8776;78.46
x/2&#8776;39.23&#730;
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sin(x/2)&#8776;sin 39.23&#8776;0.6324
exact value as calculated=&#8730;10/5&#8776;0.6324
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cos(x/2)&#8776;cos 39.23&#8776;0.7746
exact value as calculated=&#8730;(3/5)&#8776;0.7746
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tan(x/2)=tan 39.23&#8776;0.8164
exact value as calculated=&#8730;50/(5&#8730;3)&#8776;0.8164