Question

How do you evaluate `sin(arctan(3/4))` without a calculator?

Answer 1

`sin(arctan(3/4))=+-3/5`

Explanation:

`sin(arctan(3/4))` literally means sine of an angle whose tangent is `3/4`. Let the angle be `theta=arctan(3/4)`

As `tantheta=3/4`, `cottheta=4/3` (as it is reciprocal)

Hence `csc*2theta=1+cot^2theta=1+16/9=25/9`

Hence `csctheta=+-5/3` and `sintheta=+-3/5`

Note that as `tantheta` is positive, it is in firs or third quadrant and hence, we can have `sintheta` positive as well as negative.