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)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! 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, (π ≤ x < 3π/2)(Q3)
let O=opposite side
let A=adjacent side
let H=hypotenuse
..
tanx=3/2=O/A
O=-3, A=-2
H=√(O^2+A^2)=√(9+4)=√13
sinx=O/H=-3/√13
cosx=A/H=-2/√13
..
sin(2x)=2sinxcosx=2*(-2/√13)(-3/√13)=12/13
cos(2x)=1-2sin^2x=1-2(-3/√13)^2=1-18/13=-5/13
√((1-(-2/√13)/2))=√((1+.5547)/2)=√(1.5547/2)
-√((1+(-2/√13)/2))=-√((1-.5547/2)
..
Check with calculator:
tanx=3/2(in Q3)
x≈56.31º+180≈236.31º
2x≈472.62-360≈112.62º
x/2=118.15º
..
sin2x=sin(112.62º)≈0.923..
as computed,12/13≈0.923..
..
cos2x=cos(112.62)=-0.3846..
as computed,-5/13≈-0.3846..
..
sin(x/2)=sin(118.15)≈0.8817..
as computed,≈0.8817..
..
cos(x/2=cos(118.15)=-0.4718..
as computed,≈-0.4718..
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