Question Video: Finding Positive and Negative Measures of Angles Coterminal to a given Angle

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.

Let’s take our angle with measure two 𝜋 by three radians. Now, if you’re not usually confident with radians, don’t worry too much. 𝜋 radians is 180 degrees; it’s half a turn. So, two 𝜋 by three is two-thirds of this. Let’s draw the initial side of this angle. And since it’s positive, we’re 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, we’re going to need to keep going in the counterclockwise direction. In fact, we’re 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, we’re 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.