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Find the exact value of tan(x-y) if sin x=8/17 cosy = 3/5

Find the exact value of tan(x-y) if sin x=8/17 cosy = 3/5-example-1

1 Answer

11 votes

To solve the the question we proceed as follows:

From trigonometric laws


(cos x)^2+(sin x)^2=1


(cos x)^2+(sin x)^2=1


sin (x-y)=sin
x
sin
y-sin
y
cos
x


cos (x-y)=cos
x
cos
y+sin
x
xin
y

si
x=(8)/(17)


cos
x=sqrt(1-(sin x)^2)=sqrt(1-64/289)=sqrt((225)/(289) )=(15)/(17)


cos
y=(3)/(5)


sin
x= sqrt(1- (cos x)^2)= sqrt(1-(9)/(25) )=sqrt((16)/(25) )=(4)/(5)

thus


tan (x-y)=[sin (x-y)]/[cos (x-y)]

=[sin x cos y-sin y cos x]/[cos x cos y+sin x sin y]

plugging in the values we obtain:


[8/17 *3/5-4/5*15/7]/[15/17*3/5+8/17*4/5]

simplifying


[24/85-60/85]/[45/85+32/85]


=-(36)/(77)

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