### Video Transcript

Find one angle with positive
measure and one angle with negative measure which are coterminal to an angle with
measure two π by three.

Letβs take our angle with measure
two π by three radians. Now, if youβre not usually
confident with radians, donβt worry too much. π radians is 180 degrees; itβs
half a turn. So, two π by three is two-thirds
of this. Letβs draw the initial side of this
angle. And since itβs positive, weβre
going to measure in a counterclockwise direction. And so, the terminal side is likely
to be somewhere around here. Next, we recall what it means for
two angles to be coterminal. Coterminal angles are angles that
share the same initial and terminal sides.

So, essentially, we want to find
alternative ways to express the exact same angle. If we want to find, then, an angle
with positive measure, weβre going to need to keep going in the counterclockwise
direction. In fact, weβre going to need to
complete a full turn to get back to this terminal side, where a full turn is equal
to two π radians. So, the positive coterminal angle
will be the equivalent to adding two π by three radians and two π radians. We can rewrite, of course, two π
as six π over three. And the purpose of doing this is to
ensure we have a pair of equivalent denominators. And once we do, we can add the
numerators. Two plus six is eight. So, two π by three plus six π by
three is eight π by three radians. And so, we have our angle with
positive measure.

What about the angle with negative
measure? Well, an angle with negative
measure is measured in a clockwise direction. We start at the same initial side,
but we travel in the opposite direction to get to the terminal side. Since the full turn is two π
radians, to find the size of this angle, weβre going to subtract two π by three
from two π. Once again, if we rewrite two π as
six π by three, we can then subtract the numerators. Six π minus two π is four π. And so, the magnitude of the
negative angle will be four π by three. And we, therefore, say that our two
angles are eight π by three and negative four π by three.