Question

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

Answer 1

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)`