Question

How do you evaluate the expression `sin(u+v)` given `sinu=3/5` with `pi/2<u<p` and `cosv=-5/6` with `pi<v<(3pi)/2`?

Answer 1

`sin(u+v)=(-15+4sqrt11)/36=-0.0481`

Explanation:

As `sinu=3/5` and domain of `u` is given by `pi/2 < u < pi` i.e. `u` is in second quadrant and `cosu` is negative and

`cosu=-sqrt(1-(3/5)^2)=-sqrt(1-9/25)=-sqrt(16/25)=-4/5`

Further as `cosv=-5/6` with `pi < v < (3pi)/2`, hence `v` is in third quadrant and `sinv` is negative and

`sinv=-sqrt(1-(-5/6)^2)=-sqrt(1-25/36)=-sqrt(11/36)=-sqrt11/6`

Now `sin(u+v)=sinucosv+cosusinv`

= `3/5xx(-5/6)+(-4/5)xx(-sqrt11/6)`

= `(-15)/30+(4sqrt11)/30`

= `(-15+4sqrt11)/36`

= `(-15+4xx3.317)/36=-1.732/36=-0.0481`