Question 676063: Given that tan(theta)=5/4 , pi< theta < 3pi/2, find sin(2theta), cos(2theta), sin(theta/2), cos(theta/2), tan(2theta), and tan(theta/2)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Given that tan(theta)=5/4 , pi< theta < 3pi/2, find sin(2theta), cos(2theta), sin(theta/2), cos(theta/2), tan(2theta), and tan(theta/2)
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use x in place of theta
interval: π ≤ x ≤ 3π/2
You are working with a reference angle in quadrant III where tan>0, sin & cos<0
given tan x=5/4=opp side/adj side
hypotenuse=√(5^2+4^2)=√(25+16)=√41
sinx=5/√41
cosx=4/√41
..
sin(2x)=2sinxcosx=2*5/√41*4/√41=40/41
check: (with calculator set to radians)
tan^-1(5/4)≈.896
sin(2x)=sin(1.792)≈.9756
40/41≈.9756
..
do the remaining problems by the same procedure.
substitute, solve and check: note: (x=0.896 radians)
..
cos(2x)=cos^2x-sin^2x
sin(x/2)=-[√(1-cosx)/2)]
cos(x/2)=-[√(1+cosx)/2)]x
tan(2x)=(2tanx)/(1-tan^2x)
tan(x/2)=sinx/(1+cosx)
..
This is a lot of work. good luck!
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