In this explainer, we will learn how to identify and measure directed angles and find their equivalent angles.
In order to be able to work with directed angles, we will need to recall some key angle facts.
Definition: Angles at a Point
Angles at a point sum to . In other words, a full rotation makes up . This can be shown on a quadrant diagram, where a rotation is equal to and a rotation is equal to .
We can also measure angles in radians:
Dividing both sides by 2,
We will now explain what we mean by the term “directed angle.”
Definition: Directed Angles
A directed angle is one that has a direction. More specifically, it can be defined as an ordered pair of two rays (called the sides of the angle) with a common starting point (called the vertex).
If an angle is measured in a counterclockwise direction, it is said to be positive, whereas if it is measured in the clockwise direction, it is considered to be negative.
Let us further explore exactly what this definition means. Suppose we had the ordered pair . That is, we have two rays: , the initial side, and , the terminal side, and the two meet at the vertex , as shown.
Then, we can say that is a directed angle. Furthermore, since the angle is measured in a counterclockwise direction, we say that it is positive. It is also possible to measure it in a clockwise direction, which would result in a negative angle, as shown.
On the other hand, if we had the ordered pair , this would correspond to the following directed angle, with two possible measures.
We can see that and are not the same, since the positive and negative measures are swapped around.
In our first example, we will consider how to find these equivalent angles, measured in degrees.
Example 1: Finding the Smallest Positive Equivalent Angle in Degrees
Find the smallest positive equivalent of .
Answer
Let’s imagine that the directed angle is the angle between two rays and , where is the initial side and is the terminal side of the angle.
As is positive, we will measure the angle between these two rays in a counterclockwise direction. We know that a full turn is equal to , so it follows that we will have to do at least one full turn to get to the angle we require.
In fact, , which is still less than , so we will need to complete two full turns.
We need to get to , and . This means that we need to go a further in the counterclockwise direction.
The smallest positive equivalent of is .
We will now consider how to find positive equivalents when given a negative angle measured in degrees.
Example 2: Finding the Measure of the Smallest Positive Angle Equivalent to a Negative One
Find the smallest positive equivalent of .
Answer
Let’s imagine that the directed angle is the angle between two rays and , where is the initial side and is the terminal side of the angle.
As is negative, we will measure the angle between these two rays in a clockwise direction. This means that the angle will be as shown in the diagram.
We need to find the smallest positive equivalent to , so we need to measure the same angle but in the other direction. A positive directed angle indicates we need to measure in the counterclockwise direction.
We recall that angles at a point sum to , so a positive equivalent of is found by subtracting from :
Therefore, the smallest positive equivalent of is .
Before looking at our next example, we will introduce a new term, “coterminal angles.” In the previous question, and were examples of a pair of coterminal angles, as they shared the same initial and terminal sides.
Definition: Coterminal Angles
Coterminal angles share the same initial and terminal sides.
To find a coterminal angle, we can add or subtract (or radians) from the given angle.
Note that since we can add or subtract as many multiples of as we want, there exist an infinite amount of coterminal angles for any given directed angle.
We will now calculate a positive and a negative coterminal angle when the original angle is given in radians.
Example 3: Finding Positive and Negative Measures of Angles Coterminal to a Given Angle
Find one angle with positive measure and one angle with negative measure that are coterminal to an angle with measure .
- ,
- ,
- ,
- ,
- ,
Answer
Let’s consider the angle . We could convert this into degrees by using the fact that radians is equal to :
of = , so, .
However, we will keep our angle in radians as the five answer options are also given in these units.
As our angle is positive, we need to measure the angle in a counterclockwise direction from the initial side, as shown.
We recall that coterminal angles share the same initial and terminal sides. This means that we need to find alternative ways to express the same angle.
We can calculate coterminal angles by adding or subtracting radians from the given angle.
To find another positive angle, we need to keep measuring in the counterclockwise
direction. This means that we need to add to our angle:
An angle with positive measure, which is coterminal to an angle with measure , is radians.
In a similar way, we can find an angle with negative measure by moving in a clockwise direction. This means that we need to subtract from our angle:
An angle with negative measure, which is coterminal to an angle with measure , is radians.
The two angles are and .
In our first three examples, we saw that there are an infinite number of ways to describe a given angle. There will be times where we want to restrict the size of the angle we give. When this is the case, we use the principal angle.
Definition: Principal Angle
The principal angle is the angle between the initial side and the terminal side, measured in a counterclockwise direction, that has an angle whose value in degrees lies in and in radians lies in .
If is our principal angle, then, or .
In the final two examples, we will demonstrate how to calculate the principal angle in radians.
Example 4: Finding Principal Angles
Given the angle , find the principal angle.
Answer
The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value radians, where .
This means that we need to find the coterminal angle to radians that lies between 0 and radians inclusive. Our first step will be to try and simplify the fraction. Since , the angle is equivalent to radians.
A full turn is equal to radians, so we need to calculate how many full rotations we can complete:
, so we can make 45 full turns plus another rotation.
We know that a rotation is equal to radians.
Therefore, the principal angle of is radians.
Example 5: Finding Principal Angles
Given the angle , find the principal angle.
Answer
The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value radians, where .
This means that we need to find the coterminal angle to radians that lies between 0 and radians.
As radians is negative, this angle will be measured in a clockwise direction.
Since is equal to , and a full rotation is equal to radians, we can complete two full rotations and then continue a further radians in the clockwise direction. This is shown in the diagrams below.
Since we are measuring in a clockwise direction, this angle is negative; it is radians. The principal angle must be positive, so we need to find the coterminal angle to this, measured in the counterclockwise direction.
To find the coterminal angle we need, and hence the principal angle, we subtract from :
Therefore, the principal angle of is radians.
We will finish this explainer by recapping some of the key points.
Key Points
- A directed angle is an angle that has a direction; the counterclockwise direction is positive and the clockwise direction is negative.
- Coterminal angles share the same initial and terminal sides. There are an infinite number of equivalent coterminal angles.
- The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value , where or .
