Question

How do you find the exact value `cos(x+y)` if `tanx=5/3,siny=1/3`?

Answer 1

`cos(x+y)=(6sqrt2-5)/(3sqrt34)`

Explanation:

Use the cosine angle addition formula:

`cos(x+y)=cosxcosy-sinxsiny`

We need to determine `sinx`, `cosx`, and `cosy` from whatever we have.

Determining `mathbf(sinx,cosx: )`

`tan^2x+1=sec^2x" "" "" "`use `tanx=5/3`

`25/9+1=sec^2x`

`secx=sqrt(34/9)=sqrt34/3`

Then:

`color(blue)(cosx=1/secx=3/sqrt34`

Now using:

`sin^2x+cos^2x=1" "" "" "`where `cosx=3/sqrt34`

`sin^2x+9/34=1`

`color(blue)(sinx=sqrt(25/34)=5/sqrt34`

Determining `mathbf(cosy: )`

`sin^2y+cos^2y=1" "" "" "`use `siny=1/3`

`1/9+cos^2y=1`

`color(blue)(cosy=sqrt(8/9)=(2sqrt2)/3`

Returning to `mathbf(cos(x+y): )`

`cos(x+y)=cosxcosy-sinxsiny`

`color(white)(cos(x+y))=3/sqrt34((2sqrt2)/3)-5/sqrt34(1/3)`

`color(white)(cos(x+y))=(6sqrt2-5)/(3sqrt34)`

Note these are all assuming that `x` and `y` are in the first quadrant (everything is positive).