Question

How do you find the exact value of `tan(u+v)` given that `sinu=5/13` and `cosv=-3/5`?

Answer 1

`tan(u+v)=63/16`

Explanation:

Given that `sinu=5/13`, then `tanu=5/12`.

In a right-angled triangle, the sine of an angle is the ratio between the opposite side and the hypotenuse. Given both of these, we can work out the remaining side, and also the tangent of that angle.

`"adj"=sqrt(13^2-5^2)=12`

`tanu="opp"/"adj"=5/12`

Given that `cosv=-3/5`, `tanv=4/3`

`tanv = sqrt(sec^2v-1)` given `secv=1/cosv`

`cosv=-3/5 rarr secv=-5/3`

`tanv=sqrt((-5/3)^2-1)=4/3`

`tan(u+v)=(tanu+tanv)/(1-tanutanv)`

`tan(u+v)=(5/12+4/3)/(1-(5/12)(4/3))=63/16`