Lesson Explainer: Directed Angles | Nagwa Lesson Explainer: Directed Angles | Nagwa

Lesson Explainer: Directed Angles Mathematics

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 360. In other words, a full rotation makes up 360. This can be shown on a quadrant diagram, where a 14 rotation is equal to 90 and a 12 rotation is equal to 180.

We can also measure angles in radians: 2𝜋=360.radians

Dividing both sides by 2, 𝜋=180.radians

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 𝑂𝐴,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.

Example 1: Finding the Smallest Positive Equivalent Angle in Degrees

Find the smallest positive equivalent of 788.

Answer

Let’s imagine that the directed angle 788 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 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, so it follows that we will have to do at least one full turn to get to the angle we require.

In fact, 360+360=720, which is still less than 788, so we will need to complete two full turns.

We need to get to 788, and 788720=68. This means that we need to go a further 68 in the counterclockwise direction.

The smallest positive equivalent of 788 is 68.

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 40.

Answer

Let’s imagine that the directed angle 40 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 40 is negative, we will measure the angle between these two rays in a clockwise direction. This means that the 40 angle will be as shown in the diagram.

We need to find the smallest positive equivalent to 40, 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 360, so a positive equivalent of 40 is found by subtracting 40 from 360: 36040=320.

Therefore, the smallest positive equivalent of 40 is 320.

Before looking at our next example, we will introduce a new term, “coterminal angles.” In the previous question, 40 and 320 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 360 (or 2𝜋 radians) from the given angle.

Note that since we can add or subtract as many multiples of 360 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 2𝜋3.

  1. 8𝜋3, 4𝜋3
  2. 5𝜋3, 𝜋3
  3. 8𝜋3, 4𝜋3
  4. 8𝜋3, 4𝜋3
  5. 4𝜋3, 4𝜋3

Answer

Let’s consider the angle 2𝜋3radians. We could convert this into degrees by using the fact that 𝜋 radians is equal to 180:

23 of 180 = 120, so, 2𝜋3=120radians.

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 2𝜋 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 2𝜋 to our angle:

2𝜋3+2𝜋=2𝜋3+6𝜋3=8𝜋3.

An angle with positive measure, which is coterminal to an angle with measure 2𝜋3, is 8𝜋3 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 2𝜋 from our angle: 2𝜋32𝜋=2𝜋36𝜋3=4𝜋3.

An angle with negative measure, which is coterminal to an angle with measure 2𝜋3, is 4𝜋3 radians.

The two angles are 8𝜋3 and 4𝜋3.

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 [0,360] and in radians lies in [0,2𝜋].

If 𝜃 is our principal angle, then, 0𝜃360 or 0𝜃2𝜋.

In the final two examples, we will demonstrate how to calculate the principal angle in radians.

Example 4: Finding Principal Angles

Given the angle 273𝜋3, find the principal angle.

Answer

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value 𝜃 radians, where 0𝜃2𝜋.

This means that we need to find the coterminal angle to 273𝜋3 radians that lies between 0 and 2𝜋 radians inclusive. Our first step will be to try and simplify the fraction. Since 273÷3=91, the angle 273𝜋3 is equivalent to 91𝜋 radians.

A full turn is equal to 2𝜋 radians, so we need to calculate how many full rotations we can complete:

91𝜋2𝜋=912=4512, so we can make 45 full turns plus another 12 rotation.

We know that a 12 rotation is equal to 𝜋 radians.

Therefore, the principal angle of 273𝜋3 is 𝜋 radians.

Example 5: Finding Principal Angles

Given the angle 23𝜋5, find the principal angle.

Answer

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value 𝜃 radians, where 0𝜃2𝜋.

This means that we need to find the coterminal angle to 23𝜋5 radians that lies between 0 and 2𝜋 radians.

As 23𝜋5 radians is negative, this angle will be measured in a clockwise direction.

Since 23𝜋5 is equal to 43𝜋5, and a full rotation is equal to 2𝜋 radians, we can complete two full rotations and then continue a further 3𝜋5 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 3𝜋5 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 3𝜋5 from 2𝜋: 2𝜋3𝜋5=10𝜋53𝜋5=7𝜋5.

Therefore, the principal angle of 23𝜋5 is 7𝜋5 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 𝜃, where 0𝜃360 or 0𝜃2𝜋.

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