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

Jun 24, 2017

First add multiples of $2 \pi$.

Explanation:

Since $- \frac{10 \pi}{3}$ is negative, we add $2 \pi$ repeatedly until the angle becomes positive.

$- \frac{10 \pi}{3} + 2 \pi = - \frac{10 \pi}{3} + 6 \frac{\pi}{3} = - \frac{4 \pi}{3}$
This is still negative. Add another $2 \pi$.

$- \frac{4 \pi}{3} + 2 \pi = - \frac{4 \pi}{3} + 6 \frac{\pi}{3} = \frac{2 \pi}{3}$

The coterminal angle is $\frac{2 \pi}{3}$.

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

For every angle that is an integer multiple of $\frac{\pi}{3}$ in reduced form, the reference angle in Quadrant I is $\frac{\pi}{3}$.