Lesson Video: Angles in Standard Position Mathematics

In this video, we will learn how to identify angles in standard position and find the positive and negative measures of their equivalent angles.

13:08

Video Transcript

In this lesson, we’ll learn how to identify angles in standard position and find the positive and negative measures of their equivalent angles. We’re going to be referring to the four quadrants throughout this lesson, so we’ll remind ourselves what we mean by these. We split the coordinate plane into four quarters called quadrants. We label them in a counterclockwise direction and use Roman numerals to do so, with this quadrant being the first, this being the second, this being the third, and this being the fourth.

We might also be used to measuring angles using degrees, such that a full turn is 360 degrees. Now, whilst not a prerequisite to accessing the video, it’s worth noting that we can also measure angles using radians where a full turn, 360 degrees, is equal to two 𝜋 radians. So, this lesson is about angles in standard position. But what does that actually mean? An angle is said to be in standard position if its vertex is located at the origin and one ray lies on the positive 𝑥-axis. The ray that lies on the 𝑥-axis is called the initial side, and the other ray is called the terminal side.

We also say that if the terminal side of an angle lies on one of the axes, such as at 90 degrees or 180 degrees, it’s called a quadrantal angle. The angle is then measured by the amount of rotation from the initial side to the terminal side. If the angle is measured in a counterclockwise direction, we say that the measurement is positive. But if we measure the angle in a clockwise direction, we say that the measurement is negative.

Our final definition refers to two angles which are in standard position and have the same terminal side, and those are called coterminal, so we can see from the diagram here that 60 degrees and negative 300 degrees are coterminal angles. So, let’s practice identifying whether angles are actually in standard position.

Is the angle in standard position?

An angle is said to be in standard position if its vertex is located at the origin and one ray, which we call the initial side, lies on the positive 𝑥-axis. Now, our angle is bound by the rays highlighted in yellow. We do indeed see that one of the sides lies on the positive 𝑥-axis and the vertex does appear to be at the origin. We also know that it is indeed the initial side of our angle that lies on the positive 𝑥-axis rather than the terminal side. And this is because we’re given an arrow that tells us the direction.

If the arrow was moving in the opposite direction, we couldn’t actually say that the angle is in standard position because the initial side would not lie on the positive 𝑥-axis. And so, we see that the angle given satisfies all of our criteria to be in standard position. And the answer is yes.

Is the angle in standard position?

Remember, we say that an angle is in standard position if its vertex is located at the origin and one ray, which is the initial side, lies on the positive 𝑥-axis. Our angle is the angle measured in a counterclockwise direction between the two lines highlighted in yellow. One of these rays does indeed lie on the positive 𝑥-axis. And in fact, we can see it’s the initial side because we’re given the direction in which the angle is measured. We can also see that the vertex, the point where our initial side and terminal side meet, is at the origin. And so our angle satisfies all the relevant criteria required to be considered in standard position. And the answer is yes.

Let’s now look an example defined by an ordered pair.

Does the ordered pair given by the ray joining 𝐶𝐴 and the ray joining 𝐶 to 𝐷 express an angle in standard position?

We know that for an angle to be in standard position, its vertex must be centered at the origin and the initial side must lie on the positive 𝑥-axis. Now, our angle is defined by this ordered pair. So we have the ray joining 𝐶 and 𝐴 and the ray joining 𝐶 and 𝐷. The ray joining 𝐶 to 𝐴 is this one, and that does indeed lie on the positive 𝑥-axis. We know it’s the initial side of our angle because it’s mentioned first. Then the ray that joins 𝐶 to 𝐷 is this one here, meaning the angle we’re interested in is this one. But does the vertex of this lie at the origin? Well, no, the vertex is over here somewhere. It’s someway along the positive 𝑥-axis. And so the ordered pair defined by the ray joining 𝐶 to 𝐴 and the ray joining 𝐶 to 𝐷 does not express an angle in standard position. And the answer is no.

There are, however, a couple of angles that are in standard position here. The first would be given by the ordered pair the ray joining 𝑂 to 𝐶 and the ray joining 𝑂 to 𝐸. The ray 𝑂𝐶 lies on the positive 𝑥-axis, and then the vertex is centered at the origin. Similarly, we could begin with the same initial side, and that’s the ray joining 𝑂 to 𝐶, and measure through to the ray joining 𝑂 to 𝐺. The vertex for this angle is still located at the origin, and so the angle is also in standard position.

In our next example, we’ll look at how we can find the value of coterminal angles.

State the positive related angle for the angle shown.

Remember, when thinking about angles in 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, we say that the measurement is negative. We also say that if two angles in standard position have the same terminal side, they’re called coterminal angles. And so here we have negative 340 degrees. And of course, since this is negative, it’s measured in a clockwise direction.

To find the size of the coterminal angle, we need to identify where the terminal side is. The terminal side is this one. And so to find the positive related angle, we need to find the measure of this angle here, this acute angle. And so let’s use one of the key angle facts. We know that angles around a point sum to 360 degrees. And so we take the measure of the reflex angle to be 340 degrees — remember, the negative just tells us the direction — and we’re going to subtract its value from 360. 360 minus 340 is 20 degrees. Since this angle is indeed being measured in a counterclockwise direction from its initial side, we say that it’s positive. And so the positive related angle for the angle we were given is 20 degrees.

Now that we’ve seen how to find the value of coterminal angles, let’s look at finding multiple coterminal angles.

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.

In our final example, we’ll look at how to identify the quadrant in which an angle in standard position lies.

In which quadrant does the angle negative 242 degrees lie?

Remember, if we take the coordinate plane, we can split it into four quarters which are called quadrants. We label these in a counterclockwise direction, and we use Roman numerals to do so. We say that this quadrant is the first quadrant. We have the second over here. The third then lies down here. And this is the fourth quadrant. So in which quadrant does the angle negative 242 degrees lie?

Well, to find out, we’re going to begin by adding the initial side of our angle onto the diagram. The vertex of our angle has to lie at the origin and its initial side must be located on the positive 𝑥-axis as shown. Since the angle is negative, we need to travel in a clockwise direction. We’ll do this in intervals of 90 degrees. 90 degrees takes us to this line here. Then another 90 degrees takes us half a turn. It takes us 180 degrees. And our terminal side would lie on the negative 𝑥-axis. If we were to continue another 90 degrees, that would take us all the way to 270 degrees.

And in fact, of course, since we’re traveling in a clockwise direction, they should all be negative. But we only wanted 242 degrees. And so we’re not going to quite complete that last quarter of a turn. We can therefore say that the angle negative 242 degrees lies in the second quadrant.

Let’s recap the key points from this lesson. An angle is said to be in standard position if its vertex is located at the origin and one ray is on the positive 𝑥-axis. The ray that lies on the 𝑥-axis is called the initial side. And the other ray that forms the angle is called the terminal side. We also define a quadrantal angle to be made up of one in which the terminal side lies on one of the axes, such as at 270 degrees or 360 degrees. Then, the angle is measured by the amount of rotation from the initial side to the terminal side. And if we measure in a counterclockwise direction, that measurement is positive. If we measure in a clockwise direction, the measurement is negative.

Finally, we said that if two angles in standard position have the same terminal side, then we call those coterminal angles.

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