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) = + 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.
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