Lesson Video: Directed Angles | Nagwa Lesson Video: Directed Angles | Nagwa

Lesson Video: Directed Angles Mathematics

In this video, we will learn how to identify and measure directed angles and find their equivalent angles.

13:49

Video Transcript

In this video, 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.

We begin by recalling that angles at a point sum to 360 degrees. In other words, a full rotation makes up 360 degrees. This can be shown on a quadrant diagram where a quarter rotation is equal to 90 degrees and a half rotation is equal to 180 degrees. We can also measure angles in radians. Two 𝜋 radians equals 360 degrees. Dividing both sides by two, we see that 𝜋 radians equals 180 degrees. This too can be shown on a quadrant diagram.

We will now explain what we mean by the term directed angle. 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 the 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 𝑂𝐴 and 𝑂𝐵. That is, we have two rays 𝑂𝐴, the initial side, and 𝑂𝐵, the terminal side, and the two meet at the vertex 𝑂 as shown. Then, we can say that 𝑂𝐴, 𝑂𝐵 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 𝑂𝐵, 𝑂𝐴, this would correspond to the following directed angle with two possible measures. We can see that 𝑂𝐴, 𝑂𝐵 and 𝑂𝐵, 𝑂𝐴 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.

Find the smallest positive equivalent of 788 degrees.

Let’s imagine that the directed angle 788 degrees is the angle between two rays 𝑂𝐴 and 𝑂𝐵, where 𝑂𝐴 is the initial side and 𝑂𝐵 is the terminal side of the angle. As 788 degrees is positive, we will measure the angle between these two rays in a counterclockwise direction. We know that a full turn is equal to 360 degrees. So it follows that we will have to do at least one full turn to get the angle we require. In fact, 360 degrees plus 360 degrees equals 720 degrees, which is still less than 788 degrees. So we will need to complete two full turns. We need to get to 788 degrees, and 788 degrees minus 720 degrees equals 68 degrees. This means that we need to go a further 68 degrees in the counterclockwise direction. The smallest positive equivalent to 788 degrees is 68 degrees.

We will now consider how to find positive equivalents when given a negative angle measured in degrees.

Find the smallest positive equivalent of negative 40 degrees.

Let’s imagine that the directed angle negative 40 degrees is the angle between two rays 𝑂𝐴 and 𝑂𝐵, where 𝑂𝐴 is the initial side and 𝑂𝐵 is the terminal side of the angle. As our angle is negative, we will measure the angle between these two rays in a clockwise direction. This means that the 40-degree angle will be as shown in the diagram.

We need to find the smallest positive equivalent to negative 40 degrees. So we need to measure the same angle but in the other direction. A positive directed angle indicates we need to measure in a counterclockwise direction. We recall that angles at a point sum to 360 degrees. So, a positive equivalent of negative 40 degrees is found by subtracting 40 degrees from 360 degrees. Therefore, the smallest positive equivalent of negative 40 degrees is 320 degrees.

Before looking at our next example, we will introduce a new term, coterminal angles. In this question, negative 40 degrees and 320 degrees were examples of a pair of coterminal angles as they share the same initial and terminal sides.

Coterminal angles share the same initial and terminal sides. To find a coterminal angle, we can add or subtract 360 degrees or two 𝜋 radians from the given angle. The diagrams show that 40 degrees, 400 degrees, and negative 320 degrees are coterminal angles since all three angles have the same initial and terminal sides. Note that since we can add or subtract as many multiples of 360 degrees as we want, there exist an infinite amount of coterminal angles for any given directed angle.

We will now calculate a positive and negative coterminal angle when the original angle is given in radians.

Find one angle with positive measure and one angle with negative measure which are coterminal to an angle with measure two 𝜋 over three radians.

Let’s consider the angle two 𝜋 over three radians. We could convert this into degrees by using the fact that 𝜋 radians is equal to 180 degrees. Two-thirds of 180 degrees equals 120 degrees, so two 𝜋 over three radians equals 120 degrees. However, we will keep our angle in radians in this question. 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 two 𝜋 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 two 𝜋 radians to our angle. Two 𝜋 radians is equivalent to six 𝜋 over three radians. This means we need to add two 𝜋 over three and six 𝜋 over three. This is equal to eight 𝜋 over three radians. An angle with positive measure which is coterminal to an angle with measure two 𝜋 over three is eight 𝜋 over three 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 two 𝜋 from our angle. Two 𝜋 over three minus six 𝜋 over three is equal to negative four 𝜋 over three. An angle with negative measure which is coterminal to an angle with measure two 𝜋 over three is negative four 𝜋 over three radians. The two angles are eight 𝜋 over three and negative four 𝜋 over three.

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.

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 the closed interval from zero to 360 and in radians lies in the closed interval from zero to two 𝜋. If 𝜃 is our principal angle, then 𝜃 is greater than or equal to zero degrees and less than or equal to 360 degrees or greater than or equal to zero and less than or equal to two 𝜋 radians.

In our final example, we will demonstrate how to calculate the principal angle in radians.

Given the angle negative 23𝜋 over five, find the principal angle.

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value 𝜃 radians, where 𝜃 is greater than or equal to zero and less than or equal to two 𝜋 radians. This means that we need to find the coterminal angle to negative 23𝜋 over five radians that lies between zero and two 𝜋 radians. As the given angle is negative, it will be measured in a clockwise direction. Since 23𝜋 over five is equal to four and three-fifths 𝜋 and a full rotation is equal to two 𝜋 radians, we can complete two full rotations. We then continue a further three-fifths 𝜋 radians in the clockwise direction as shown in the diagram. Since we are measuring in a clockwise direction, this angle is negative. It is negative three 𝜋 over five 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 three 𝜋 over five from two 𝜋. This is equivalent to 10𝜋 over five minus three 𝜋 over five, which equals seven 𝜋 over five radians. Therefore, the principal angle of negative 23𝜋 over five is seven 𝜋 over five radians.

We will finish this video by recapping some of the 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 𝜃, where 𝜃 is greater than or equal to zero degrees and less than or equal to 360 degrees or greater than or equal to zero and less than or equal to two 𝜋 radians.

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