Question

Question #0bee5

Answer 1

`sin(tan^-1 (3/4)+cos^-1 (5/13)) = (3/5)(5/13)+(4/5)(12/13) = 63/65`

Explanation:

`tan^-1(3/4)` is a `theta` between `-pi/2` and `pi/2` with `tan theta = 3/4`

`cos^-1(5/13)` is a `phi` between `0` and `pi` with `cos phi = 5/13`

We have been asked to find `sin(theta + phi)`.

We know that `sin(theta + phi) = sin theta cos phi + cos theta sin phi`

We know that `tan theta = 3/4` and `0 < theta < pi/2`, we can find that `sin theta = 3/5` and `cos theta = 4/5`

We know that `cos phi = 5/13` and `0 < phi < pi/2` so we find `sin phi = 12/13`.

Therefore

`sin(tan^-1 (3/4)+cos^-1 (5/13)) = (3/5)(5/13)+(4/5)(12/13) = 63/65` .