SOLUTION: exact value of tan3x when cos x=3/5

Question 519012: exact value of tan3x when cos x=3/5
Answer by mamiya(56) (Show Source): You can put this solution on YOUR website!
we know that tan 3x= sin3x/cos3x, let call this formula (a)
formulas to keep in mind before you start reading
sin(a+b)= sin(a)cos(b)+ sin(b)cos(a), let call it (1)
cos(a+b)= cos(a)cos(b)- sin(a)sin(b), let call it (2)
cos(2x)= cos^2(x)- sin^2(x)
sin(2x)= 2cos(x)sin(x)
sin^2(x)+ cos^2(x)= 1

base on the (1), sin3x= sin(2x+x)
= sin(2x)cos(x)+ sin(x)cos(2x)
= 2 cos^2(x)sin(x)+ sin(x)[cos^2(x)-sin^2(x)]
= sin(x)[ 2cos^2(x)+ cos^2(x)- ( 1-cos^2(x) ]
= sin(x)[ 4cos^2(x)-1 ]
= [sqrt of (1-cos^2(x)][4cos^2(x)-1]
= [ sqrt of (1-(3/5)^2) ][ 4*(3/5)^2 -1 ]
= 4/5 X 11/25
= 44/125
the (2) , cos3x = cos(2x+x)
= cos(2x)cos(x)- sin(x)sin(2x)
= [ cos^2(x)-sin^2(x)]cos(x) - 2sin^2(x)cos(x)
= cos(x)[ cos^2(x)-(1- cos^2(x)) - 2(1-cos^2(x)) ]
= cos(x)[ 4cos^2(x) - 3]
= 3/5[ 4*(3/5)^2 - 3]
= 3/5 X -39/25
= -117/125
so, tan 3x= 44/125:(-117/125)
= - 44/117

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