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
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Identities:
{{{sin(x/2)=sqrt((1-cos(x))/2)}}}
{{{cos(x/2)=sqrt((1+cos(x))/2)}}}
{{{tan(x/2)=sin(x)/(1+cos(x))}}}
..
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
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{{{sin(x/2)=sqrt((1-(4/5))/2)=sqrt((1/5)/2)}}}=1/√10=√10/10
{{{cos(x/2)=sqrt((1+cos(x))/2)=sqrt((1+(4/5))/2)}}}=√(9/10)=3/√10=3√10/10
{{{tan(x/2)=sin(x)/(1+cos(x))=(3/5)/(1+(4/5))=(3/5)/(9/5)=1/3}}}
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calculator check:
tan(x)=3/4
x≈36.8698˚
x/2≈18.4349˚
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sin(x/2)≈sin(18.4349)≈0.3162..
exact value as calculated=√10/10≈0.3162..
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cos(x/2)≈cos(18.4349)≈0.9486..
exact value as calculated=3√10/10≈0.9486..
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tan(x/2)=tan(18.4349)≈0.3333..
exact value as calculated=1/3