Hello!
We know the formula %7D%7D%7D%3D%7B%5Csin%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Ccos%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D%2B%7B%5Ccos%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Csin%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D%2C "sin(u + v) = sin(u)cos(v) + cos(u)sin(v),")
but it is not enough because we a given only %7D%7D%7D "sin(u)")
and %7D%7D%7D%2C "cos(v),")
but neither %7D%7D%7D "cos(u)")
nor %7D%7D%7D. "sin(v).")
But we can find them using the identity %7D%2B%7B%7B%5Ccos%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7Bx%7D%5Cright)%7D%3D%7B1%7D "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.
%7D%7D%7D%3D-%5Csqrt%7B%7B%7B1%7D-%7B%7B%5Csin%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%3D-%5Csqrt%7B%7B%7B1%7D-%5Cfrac%7B%7B%7B3%7D%7D%5E%7B%7B2%7D%7D%7D%7B%7B%7B5%7D%7D%5E%7B%7B2%7D%7D%7D%7D%7D%3D-%5Cfrac%7B%7B4%7D%7D%7B%7B5%7D%7D "cos(u) = - sqrt(1 - sin^2(u)) = - sqrt(1 - 3^2/5^2) = - 4/5")
and
%7D%7D%7D%5Cldots "sin(v)...")
Hello!
We know the formula but it is not enough because we a given only and but neither nor
But we can find them using the identity and the information about the quadrant. Indeed, in the quadrant 2 sine is positive while cosine is negative, i.e.
and
%7D%7D%7D%3D%2B%5Csqrt%7B%7B%7B1%7D-%7B%7B%5Ccos%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D%3D%5Csqrt%7B%7B%7B1%7D-%5Cfrac%7B%7B%7B24%7D%7D%5E%7B%7B2%7D%7D%7D%7B%7B%7B25%7D%7D%5E%7B%7B2%7D%7D%7D%7D%7D%3D%5Cfrac%7B%7B7%7D%7D%7B%7B25%7D%7D. "sin(v) = + sqrt(1 - cos^2(v)) = sqrt(1 - 24^2/25^2) = 7/25.")
This way we obtain
%7D%7D%7D%3D%7B%5Csin%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Ccos%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D%2B%7B%5Ccos%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Csin%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D%3D "sin(u + v) =sin(u)cos(v) + cos(u)sin(v) =")
%7D%5Ccdot%7B%5Cleft(-%5Cfrac%7B%7B24%7D%7D%7B%7B25%7D%7D%5Cright)%7D%2B%7B%5Cleft(-%5Cfrac%7B%7B4%7D%7D%7B%7B5%7D%7D%5Cright)%7D%7B%5Cleft(%5Cfrac%7B%7B7%7D%7D%7B%7B25%7D%7D%5Cright)%7D%3D%5Cfrac%7B%7B-%7B72%7D-%7B28%7D%7D%7D%7B%7B125%7D%7D%3D-%5Cfrac%7B%7B100%7D%7D%7B%7B125%7D%7D%3D-%5Cfrac%7B%7B4%7D%7D%7B%7B5%7D%7D. "= (3/5)*(-24/25) + (-4/5)(7/25) = (-72 - 28)/125 = -100/125 = -4/5.")
It is not hard to find %7D%7D%7D%2C "cos(u + v),")
too, using the similar formula
%7D%7D%7D%3D%7B%5Ccos%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Ccos%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D-%7B%5Csin%7B%7B%5Cleft(%7Bu%7D%5Cright)%7D%7D%7D%7B%5Csin%7B%7B%5Cleft(%7Bv%7D%5Cright)%7D%7D%7D. "cos(u + v) = cos(u) cos(v) - sin(u) sin(v).")
The answer for your question is -4/5.
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