Question

How do you evaluate the expression `sin(u+v)` given `cosu=4/7` with `0<u<pi/2` and `sinv=-9/10` with `pi<v<(3pi)/2`?

Answer 1

`- (36 + sqrt627)/70`

Explanation:

Use trig identity:
sin (u + v) = sin u.cos v + sin v cos u
First, find sin u and cos v
`cos u = 4/7` --> `sin^2 u = 1 - 16/49 = 33/49`
`sin u = sqrt33/7` (take the positive answer because u is in Q. 1)
`sin v = - 9/10` --> `cos^2 v = 1 - 81/100 = 19/100`
`cos v = - sqrt19/10` (cos v is negative, v is in Q. 3)
We have:
`sin (u + v) = (sqrt33/7)(- sqrt19/10) + (-9/10)(4/7) =`
`= - (36 + sqrt627)/70`