Question Video: Identifying Multiple Equivalent Coterminal Angles Mathematics

Find a positive and a negative coterminal angle for 340°.

02:54

Video Transcript

Find a positive and a negative coterminal angle for 340 degrees.

Remember, when we think about angles in a standard position, the angle is measured by the amount of rotation from the initial side to the terminal side. If we measure our angle in a counterclockwise direction, we consider the measurement to be positive. And if we measure in a clockwise direction, the measurement is negative. Now we also say that if two angles in standard position have the same terminal side, then they are called coterminal angles. So, let’s draw the angle 340 degrees out first.

The vertex of our angle needs to lie at the origin, and its initial side must lie on the positive 𝑥-axis. So, that’s the yellow line shown. It’s positive 340 degrees, so we’re going to measure 340 degrees in a counterclockwise direction. A full turn, of course, is 360 degrees. So it’s a little bit less than a full turn. And so our angle in standard position will look a little something like this. We need to find another positive coterminal angle and a negative coterminal angle. We’ll start with a negative coterminal angle since it’s a little easier to work out.

Since we’re measuring from the initial side to the terminal side but we want the measurement to be negative, we’re going to go in a clockwise direction. That’s simply here to here. And so we’re going to subtract the angle we were given, 340 degrees, from the amount of degrees in a full turn, that’s 360. 360 minus 340 is 20 or 20 degrees. So the angle is 20 degrees. But because we’re measuring it in a clockwise direction, we say that it’s negative 20 degrees.

So that’s our negative coterminal angle. But how do we find the positive one? Well, in fact, what we’re going to do is complete the 340-degree turn, and then we’re going to complete another full turn. In doing so, we end up back at the same terminal angle. And so to find the value of this positive coterminal angle, we’re going to add a full turn onto the initial turn. So 340 plus that full turn, 360 degrees. 340 plus 360 is 700. And of course, we measure this in a counterclockwise direction, so it’s positive 700 degrees. And so the positive and negative coterminal angle for 340 degrees are 700 degrees and negative 20 degrees.

And, of course, it follows that we could continue to complete turns and still end up on that terminal side. This means that there are, in fact, an infinite number of variations for both the positive and negative coterminal angle.

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