Question

What is/ how do I find the least positive coterminal arc and the reference arc of -10pi/3?

Answer 1

First add multiples of `2pi`.

Explanation:

Since `-(10pi)/3` is negative, we add `2pi` repeatedly until the angle becomes positive.

`-(10pi)/3 + 2pi = -(10pi)/3 + 6pi/3 = -(4pi)/3`
This is still negative. Add another `2pi`.

`-(4pi)/3 + 2pi = -(4pi)/3 + 6pi/3 = (2pi)/3`

The coterminal angle is `(2pi)/3`.

[ASIDE: If you had noticed that `2pi` was not sufficient, you could have added `4pi` to begin with -- instead of adding `2pi` twice.]

For every angle that is an integer multiple of `pi/3` in reduced form, the reference angle in Quadrant I is `pi/3`.