Oriented angle

The arms of an angle divide our plane into two non-oriented angles, one larger and one smaller (if they are not equal). To know what angle we are talking about, we introduce the concept of an oriented angle.

Why introduce an oriented angle

Look at the following figure, where the angle is marked AVB.

The question now is which angle we are referring to. Intuitively you probably assumed that I meant the smaller angle, however I could easily be talking about the larger angle. Two different angles formed from two equal semi-parallel lines are shown in the following figure:

Without some additional information, we are unable to specify whether by the term angle AVB we mean the alpha or beta angle. In order to agree on what angle we are actually talking about, the term oriented angle was coined.

Positive and negative direction

The oriented angle helps us to specify which angle we mean. To do this, we need to define two things. We need to specify the arms of the angle in the correct order. Thus we understand the difference between an angle AVB and BVA, even though the arms of the angles are the same - the semi-members VA and VB. If we have an angle AVB, the semi-member VA is called the initial arm of the oriented angle and the semi-member VB is called the terminal arm of the oriented angle. The common point V is called the vertex of the oriented angle.

Okay, we already know that it depends on the order of the arms of the angle. But we still don't know which direction to take to get the angle AVB. In fact, the angle AVB could have been formed in two directions, as the following figure shows:

The starting arm of the angle is the semi-line AB. From here we can take the angle in two directions, shown in the figure by the green arrow and the red arrows. The directions can be named classically clockwise: the green direction is counterclockwise and the red direction is clockwise.

In mathematics, however, we use different labels for these directions, they are positive and negative directions. The positive direction corresponds to the green arrow, the counterclockwise direction. The negative direction corresponds to the red arrow, so it is the clockwise direction.

This is clearly shown in the following figure:

What is an oriented angle

As such, an oriented angle is an ordered pair of semi-parallel lines with a common origin. So we could write an oriented angle as a pair of semi-direct lines, for example as follows:

$$\left<\overrightarrow{VA}, \overrightarrow{VB}\right>,$$

The shorter notation is, of course, this: $\widehat{AVB}$. Furthermore, if the semi-major lines VA and VB are different, then the angles $\widehat{AVB}$ and $\widehat{BVA}$ are different.

We now define the basic size of the oriented angle. The basic magnitude of the oriented $\widehat{AVB}$ is equal to the magnitude of the unoriented angle that results from rotating the initial arm VA to the position of the terminal arm VB in the positive direction, i.e., counterclockwise.

The magnitude of the initial angle is always from the interval $\left<0^\circ, 360^\circ\right)$. Please note that the interval is open from the right. The base angle cannot be equal to 360 degrees. This is because this angle merges with an angle of size zero degrees, so instead of 360 degrees we write zero degrees. If we are moving in arc measure, then the angle is from the interval <0,2π).

In addition to the base size, we also have only the size of the oriented angle. It is defined the same way, only this size can be larger than the base size. Because if we create an angle and rotate a full circle and continue to move the initial arm, we can create an angle of size greater than 360 degrees. If we rotate one and a half circles, we get an angle of size 360 + 180 = 540 degrees. This is a valid size for an oriented angle. However, this angle is identical to an angle of size 180 degrees.

We can easily convert any such oriented angle size to a base size by dividing the integer size by 360 and taking the remainder after division as the base size. So let's try to convert those 540 degrees into a basic size:

$$540 : 360 = 1\quad (\mbox{ Read the rest at } 180)$$

Similarly for the other size. For example, for angles of size 750, 1080 and 2000 degrees we get:

$$\begin{eqnarray} 750 : 360 = 2&\quad& (\mbox{ Read the rest at } 30)\\ 1080 : 360 = 3&\quad& (\mbox{ Read the rest at } 0)\\ 2000 : 360 = 5&\quad& (\mbox{ Read the rest at } 200) \end{eqnarray}$$

Angles have base sizes of 30, 0, and 200 degrees. When calculating in radians, divide by the expression 2π.