Question

How do you evaluate `arctan [tan ((-2pi)/3)]`?

Answer 1

`arctan[tan((-2pi)/3)] = pi/3`

Explanation:

For a ratio `r`, `arctan(r) = theta`
`color(white)("XXXX")`where `theta epsilon [-pi/2, pi/2]`
and
`color(white)("XXXX")` `tan(theta) = r`
`color(white)("XXXX")` `color(white)("XXXX")` `color(white)("XXXX")`(definition)

The question therefore becomes:
`color(white)("XXXX")`For what angle `theta epsilon [-pi/2, pi/2]` is
`color(white)("XXXX")` `color(white)("XXXX")` `color(white)("XXXX")` `tan(theta) = tan((-2pi)/3)` ?

Note that the angle `((-2pi)/3)` occurs in the third quadrant and is a positive value (based on CAST or recognition that both the "rise" and the "run" are negative, so the ratio `tan = rise/run` is positive).

The required angle `theta` must occur in the first quadrant (since `theta epsilon [-pi/2, pi,2]` and `tan(theta) >= 0`)

Note that the reference angle (see diagram above) for `(-(2pi)/3)` is `pi/3`.