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œ‹=360.radians∘

Dividing both sides by 2, πœ‹=180.radians∘

We will now explain what we mean by the term β€œdirected angle.”

Definition: Directed Angles

A directed angle is one that has a direction. 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’s consider rays 𝑂𝐴 and οƒͺ𝑂𝐡 as shown in the diagram. They have the same origin, 𝑂, and the directed angle between 𝑂𝐴 and οƒͺ𝑂𝐡 is being measured in the counterclockwise direction, so we say it is positive.

If we measure the angle in a clockwise direction, then the angle is negative.

In both of these diagrams, 𝑂𝐴 is the initial side of the angle (where we start), and οƒͺ𝑂𝐡 is the terminal side of the angle (where we end).

Since we can continue moving in either direction for as long as we want, it follows that there are infinitely many determinations of the same angle.

Note:

A directed angle is not the same as a standard angle, which must be measured from the positive π‘₯-axis. The initial side of a directed angle can have any orientation.

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 οƒͺ𝑂𝐡, where 𝑂𝐴 is the initial side and οƒͺ𝑂𝐡 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 788–720=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 βˆ’40∘.

Answer

Let’s imagine that the directed angle βˆ’40∘ is the angle between two rays 𝑂𝐴 and οƒͺ𝑂𝐡, where 𝑂𝐴 is the initial side and οƒͺ𝑂𝐡 is the terminal side of the angle.

As βˆ’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 βˆ’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 βˆ’40∘ is found by subtracting 40∘ from 360∘: 360βˆ’40=320.∘∘∘

Therefore, the smallest positive equivalent of βˆ’40∘ is 320∘.

Before looking at our next example, we will introduce a new term, β€œcoterminal angles.” In the previous question, βˆ’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œ‹ 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 2πœ‹3.

  1. 8πœ‹3, βˆ’4πœ‹3
  2. 5πœ‹3, βˆ’πœ‹3
  3. 8πœ‹3, 4πœ‹3
  4. βˆ’8πœ‹3, 4πœ‹3
  5. 4πœ‹3, βˆ’4πœ‹3

Answer

Let’s consider the angle 2πœ‹3radians. We could convert this into degrees by using the fact that πœ‹ radians is equal to 180∘:

23 of 180∘ = 120∘, so, 2πœ‹3=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œ‹ 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œ‹ to our angle:

2πœ‹3+2πœ‹=2πœ‹3+6πœ‹3=8πœ‹3.

An angle with positive measure, which is coterminal to an angle with measure 2πœ‹3, is 8πœ‹3 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œ‹ from our angle: 2πœ‹3βˆ’2πœ‹=2πœ‹3βˆ’6πœ‹3=βˆ’4πœ‹3.

An angle with negative measure, which is coterminal to an angle with measure 2πœ‹3, is βˆ’4πœ‹3 radians.

The two angles are 8πœ‹3 and βˆ’4πœ‹3.

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œ‹].

If πœƒ is our principal angle, then, 0β‰€πœƒβ‰€360∘∘ or 0β‰€πœƒβ‰€2πœ‹.

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

Example 4: Finding Principal Angles

Given the angle 273πœ‹3, find the principal angle.

Answer

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value πœƒ radians, where 0β‰€πœƒβ‰€2πœ‹.

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

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

91πœ‹2πœ‹=912=4512, so we can make 45 full turns plus another 12 rotation.

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

Therefore, the principal angle of 273πœ‹3 is πœ‹ radians.

Example 5: Finding Principal Angles

Given the angle βˆ’23πœ‹5, find the principal angle.

Answer

The principal angle is the counterclockwise angle between the initial side and the terminal side that has a value πœƒ radians, where 0β‰€πœƒβ‰€2πœ‹.

This means that we need to find the coterminal angle to βˆ’23πœ‹5 radians that lies between 0 and 2πœ‹ radians.

As βˆ’23πœ‹5 radians is negative, this angle will be measured in a clockwise direction.

Since 23πœ‹5 is equal to 43πœ‹5, and a full rotation is equal to 2πœ‹ radians, we can complete two full rotations and then continue a further 3πœ‹5 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 βˆ’3πœ‹5 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 3πœ‹5 from 2πœ‹: 2πœ‹βˆ’3πœ‹5=10πœ‹5βˆ’3πœ‹5=7πœ‹5.

Therefore, the principal angle of βˆ’23πœ‹5 is 7πœ‹5 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 πœƒ, where 0β‰€πœƒβ‰€360∘∘ or 0β‰€πœƒβ‰€2πœ‹.

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