Lesson Explainer: Angles in Standard Position

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

Each of the angles that we will look at will be graphed on the coordinate plane. The coordinate plane is formed by the intersection of a horizontal line, which we refer to as the -axis, and a vertical line, which we refer to as the -axis. The point of intersection of the two lines is called the origin. The two axes divide the coordinate plane into four regions, which we call quadrants. We label the quadrants in a counterclockwise direction with roman numerals as follows.

Whether or not an angle is in standard position on the coordinate plane depends upon the position of one of the rays that make up the angle and on the position of the angle’s vertex.

Definition: Angle in Standard Position

An angle is in standard position if its vertex is at the origin on the coordinate plane and if one ray is coincident with the positive -axis. The ray that is coincident with the positive -axis is called the initial side of the angle, and it is where the angle is measured from. The other ray is called the terminal side.

An example of an angle in standard position is in the figure below.

We can see that its initial side, , is coincident with the positive -axis or the part of the -axis to the right of the origin. We can also see that its vertex, point , is at the origin.

The measure of an angle is the amount of rotation from its initial side to its terminal side and can be given in degrees or radians. One full revolution of the terminal side is equivalent to or radians. If an angle is measured in a counterclockwise direction, we classify it as positive, and if it is measured in a clockwise direction, we classify it as negative.

We can use the fact that the measures of angles in the same direction around a point sum to to find a negative angle with the same terminal side as a given positive angle and a positive angle with the same terminal side as a given negative angle. These pairs of angles are called coterminal angles.

Definition: Coterminal Angles

Coterminal angles are angles in standard position on the coordinate plane that have the same terminal side.

Every positive angle has a negative coterminal angle, and every negative angle has a positive coterminal angle. The angles measuring and in standard position are examples of coterminal angles, because each of their terminal sides lies on the positive -axis as follows.

These angles are also referred to as quadrantal angles, as is any angle in standard position that has its terminal side on one of the axes of the coordinate plane.

The angles measuring and in standard position are other examples of coterminal angles, because their terminal sides are in the same position relative to the positive -axis. In other words, the angles have the same terminal side.

Notice that the measure of the angle is more than the measure of the angle. We can add any multiple of to an angle to produce the measure of one of its coterminal angles.

Let’s begin our examination of angles with an example in which we must decide if a given angle is in standard position.

Example 1: Identifying Angles in Standard Position

Is the angle in standard position?

Answer

Recall that an angle is in standard position if its vertex is at the origin on the coordinate plane and one ray is coincident with the positive -axis. The ray that lies on the positive -axis is called the initial side of the angle, and the other ray is called the terminal side.

Notice that the angle in the figure is being measured in a counterclockwise direction. We can see that one of the rays that makes up the angle, or the angle’s initial side, does indeed lie on the positive -axis. We can also see that the other ray that makes up the angle, or the angle’s terminal side, is located in quadrant I of the coordinate plane. The point where the two rays meet, or the vertex of the angle, lies at the origin.

Thus, we can say that, yes, the angle in the figure is in standard position.

In the last example, we looked at an angle in standard position. Let’s compare the angle in that example with the one in the figure below.

Here, we can see that although the vertex of the angle is at the origin on the coordinate plane, the angle’s initial side lies not on the positive -axis, but on the negative -axis. Therefore, had this been the figure we were given in the example, we would have said that, no, the angle is not in standard position.

The following angle is another angle that is not in standard position.

Here, although the vertex of the angle is at the origin on the coordinate plane, the angle is being measured from a side that is not on the positive -axis. This is indicated by the direction of the arrow in the figure. Had the arrow pointed in the opposite direction, the angle would have been in standard position.

Now let’s look at another angle and determine if it is in standard position.

Example 2: Determining if an Angle is in Standard Position

Is the angle in standard position?

Answer

Remember that if the vertex of an angle is at the origin of the coordinate plane, and if one ray, referred to as the initial side, is coincident with the positive -axis, then the angle is said to be in standard position. The position of the other ray, referred to as the terminal side, does not matter.

In the figure we have been given, the angle is being measured in a counterclockwise direction. The angle’s initial side lies on the positive -axis of the coordinate plane, while the other ray, or the angle’s terminal side, lies in quadrant IV. Also, the vertex, or the point where the two rays meet, lies at the origin.

Therefore, we can say that, yes, the angle in the figure is in standard position.

Note

Suppose that, instead, we had been given the following figure.

In this figure, our angle is being measured in a clockwise direction, which means that its measure is not positive, but negative. However, the initial side of the angle is again on the positive -axis of the coordinate plane, and the vertex is again at the origin, so had the figure appeared this way, we would still have said that, yes, the angle is in standard position.

In our next example, we will determine if an angle expressed as an ordered pair is in standard position.

Example 3: Understanding if an Angle is in Standard Position

Does the ordered pair express an angle in standard position?

Answer

We know that for the ordered pair to express an angle in standard position, the angle the ordered pair represents would have to have its initial side coincident with the positive -axis of the coordinate plane and its vertex at the origin. Since is given first in the ordered pair, it is the initial side of our angle. We can see that this ray lies entirely on the positive -axis.

It meets the terminal side, , of our angle at point , the angle’s vertex. This point is also on the positive -axis of the coordinate plane but not at the origin.

Since the vertex is not at the origin, we can say that, no, the ordered pair does not express an angle in standard position.

Note

Although the angle expressed by the ordered pair is not in standard position, we can identify multiple angles in the figure that are in standard position. One is given by the ordered pair , another is given by the ordered pair , and a third is given by the ordered pair . For each of these angles, the initial side is on the positive -axis, and the vertex, , is at the origin.

We should also consider the angle given by the ordered pair . The vertex, , of this angle is at the origin, but the angle’s initial side, , is not on the positive -axis. Thus, we know that this angle is not in standard position.

The example that follows asks us to find a positive coterminal angle for a given angle with a negative measure. Recall that coterminal angles in standard position on the coordinate plane share the same terminal side.

Example 4: Finding Coterminal Angles

State the smallest positive angle that is equivalent to the angle shown.

Answer

We know that the angle shown is in standard position because its initial side is coincident with the positive -axis and its vertex is at the origin. We can also see that this angle is being measured in a clockwise direction, which is why the measure of the angle is a negative number of degrees.

The smallest positive angle that is equivalent to the angle shown is being measured in a counterclockwise direction. We know that it is also in standard position because it has the same initial side and vertex as the angle. It also has the same terminal side as the angle, which means that this angle and the angle are coterminal angles.

Since the negative sign in the measure of the angle just tells us the direction in which the angle is begin measured, let’s ignore it for the time being and take the measure of this angle to be . Remember that the measures of angles in the same direction around a point sum to , so by subtracting from , we can find the measure of the smallest positive angle that is equivalent to the angle shown to be

Next, not only will we find a negative coterminal angle for a given angle with a positive measure, but we will find a positive coterminal angle as well.

Example 5: Identifying Multiple Equivalent Coterminal Angles

Find a positive and a negative coterminal angle for .

Answer

Recall that coterminal angles are angles in standard position on the coordinate plane that have the same terminal side. Let’s begin by finding a negative coterminal angle for .

Remember that positive angles in standard position on the coordinate plane are measured in a counterclockwise direction, while negative angles are measured in a clockwise direction. We can use this fact, along with the fact that coterminal angles share the same terminal side, to draw the angle and a negative coterminal angle as shown below.

Notice that both angles share an initial side on the positive -axis and a vertex at the origin. Recall that the measures of angles circling a point sum to , so to find the measure of a negative coterminal angle for , we can begin by subtracting from to get

Now, since the negative coterminal angle is being measured in a clockwise direction, we can say that its measure is .

Next, let’s find the measure of a positive coterminal angle for . To do so, we will consider the fact that one full turn about a point measures . If, after rotating the terminal side of an angle in standard position on a coordinate plane counterclockwise about the origin, we were to then rotate it more degrees about the origin in the counterclockwise direction, our terminal side would be in the same position it was before the additional rotation. In other words, the resulting angle would be coterminal with the angle. This is shown in the following two figures.

Thus, to find the measure of a positive coterminal angle for , we can add to to get

In summary, we can say that a positive coterminal angle for is , and a negative coterminal angle for is .

In the last example, we found a positive and a negative coterminal angle for . It is worth noting that since we can fully rotate the terminal side of an angle in standard position on the coordinate plane about the origin any number of times, and are not the only positive and a negative coterminal angles for . To find some additional positive coterminal angles, we could continue to add to get and so on. To find some additional negative coterminal angles, we could continue to subtract to get and so on. There are actually an infinite number of positive and a negative coterminal angles for , or indeed, any angle.

As a final example, we will look at a problem in which we are given an angle and are asked to determine the quadrant of the coordinate plane in which it lies.

Example 6: Determining the Quadrant in Which a Given Angle Lies

In which quadrant does the angle lie?

Answer

Remember that the coordinate plane is split into four quarters, called quadrants. We label the quadrants in a counterclockwise direction with roman numerals. The upper-right quadrant is called quadrant I, the upper-left quadrant is called quadrant II, the bottom-left quadrant is called quadrant III, and the bottom-right quadrant is called quadrant IV, as shown in the following diagram.

The quadrant an angle lies in is the quadrant in which its terminal side lies when the angle is in standard position. Therefore, to find the quadrant in which the angle lies, we will need to sketch the angle in standard position.

We know that the initial side of any angle in standard position must be coincident with the positive -axis, and its vertex must be at the origin. We also know that the angle is being measured in a clockwise direction, because its measure is a negative number of degrees.

Since there are in a full turn about the origin, we can calculate the number of degrees in a quarter turn to be . This means that after rotating an angle’s terminal side clockwise from the positive -axis, it would lie on the negative -axis, giving us a quadrantal angle measuring .

Likewise, after rotating it clockwise, it would lie on the negative -axis, giving us a quadrantal angle measuring , and after rotating it clockwise, it would lie on the positive -axis, giving us a quadrantal angle measuring . Rotating the terminal side clockwise from the positive -axis would put it back on the positive -axis, giving us a quadrantal angle measuring , as shown in the figure.

This tells us that

any angle measuring between and would have its terminal side in quadrant IV,
any angle measuring between and would have its terminal side in quadrant III,
any angle measuring between and would have its terminal side in quadrant II,
any angle measuring between and would have its terminal side in quadrant I.

An angle measuring would hence appear as follows.

Thus, since is between and , we know that the angle lies in quadrant II or the second quadrant.

Now let’s finish by recapping some key points.

Key Points

An angle is in standard position on the coordinate plane if its vertex is at the origin and one ray is coincident with the positive -axis.
For an angle in standard position on the coordinate plane, the ray that lies on the positive -axis is called the initial side of the angle, and it is where the angle is measured from. The other ray is called the terminal side.
An angle on the coordinate plane that is measured in a counterclockwise direction is positive, while an angle measured in a clockwise direction is negative.
Coterminal angles are angles in standard position that share the same terminal side.
Every angle has an infinite number of positive and negative coterminal angles. These can be found by adding integer multiples of to the angle or subtracting them from the angle.
If the terminal side of an angle in standard position on the coordinate plane lies on one of the axes, the angle is called a quadrantal angle.