Question 742055
The question is if tan=3/2 and pie is less than or equal to theta, which is less than three pi over two, i have to find the exact value for sin(two theta), cos (two theta), sin(theta/two), and cos (theta/two)
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tan=3/2, (&#960; &#8804; x < 3&#960;/2)(Q3)
let O=opposite side
let A=adjacent side
let H=hypotenuse
..
tanx=3/2=O/A
O=-3, A=-2
H=&#8730;(O^2+A^2)=&#8730;(9+4)=&#8730;13
sinx=O/H=-3/&#8730;13
cosx=A/H=-2/&#8730;13
..
sin(2x)=2sinxcosx=2*(-2/&#8730;13)(-3/&#8730;13)=12/13
cos(2x)=1-2sin^2x=1-2(-3/&#8730;13)^2=1-18/13=-5/13
{{{sin(x/2)=sqrt((1-cosx)/2)}}}
&#8730;((1-(-2/&#8730;13)/2))=&#8730;((1+.5547)/2)=&#8730;(1.5547/2)
{{{cos(x/2)=-sqrt((1+cosx)/2)}}}
-&#8730;((1+(-2/&#8730;13)/2))=-&#8730;((1-.5547/2)
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Check with calculator:
tanx=3/2(in Q3)
x&#8776;56.31º+180&#8776;236.31º
2x&#8776;472.62-360&#8776;112.62º
x/2=118.15º
..
sin2x=sin(112.62º)&#8776;0.923..
as computed,12/13&#8776;0.923..
..
cos2x=cos(112.62)=-0.3846..
as computed,-5/13&#8776;-0.3846..
..
sin(x/2)=sin(118.15)&#8776;0.8817..
as computed,&#8776;0.8817..
..
cos(x/2=cos(118.15)=-0.4718..
as computed,&#8776;-0.4718..