Find the exact value of sin(u+v) given that sin u=3/5 and cos v= -24/25. (Both u and v are in quadrant 2.)

Hello!

We know the formula `sin(u + v) = sin(u)cos(v) + cos(u)sin(v),` but it is not enough because we a given only `sin(u)` and `cos(v),` but neither `cos(u)` nor `sin(v).`

But we can find them using the identity `sin^2(x) + cos^2(x) = 1` and the information about the quadrant. Indeed, in the quadrant 2 sine is positive while cosine is negative, i.e.

`cos(u) = - sqrt(1 - sin^2(u)) = - sqrt(1 - 3^2/5^2) = - 4/5`

and

`sin(v)...

Hello!

We know the formula `sin(u + v) = sin(u)cos(v) + cos(u)sin(v),` but it is not enough because we a given only `sin(u)` and `cos(v),` but neither `cos(u)` nor `sin(v).`

But we can find them using the identity `sin^2(x) + cos^2(x) = 1` and the information about the quadrant. Indeed, in the quadrant 2 sine is positive while cosine is negative, i.e.

`cos(u) = - sqrt(1 - sin^2(u)) = - sqrt(1 - 3^2/5^2) = - 4/5`

and

`sin(v) = + sqrt(1 - cos^2(v)) = sqrt(1 - 24^2/25^2) = 7/25.`

This way we obtain

`sin(u + v) =sin(u)cos(v) + cos(u)sin(v) = `

`= (3/5)*(-24/25) + (-4/5)(7/25) = (-72 - 28)/125 = -100/125 = -4/5.`

It is not hard to find `cos(u + v),` too, using the similar formula

`cos(u + v) = cos(u) cos(v) - sin(u) sin(v). `

The answer for your question is  -4/5.