# Coterminal Angles – Formula, Definition With Examples

Created on Dec 23, 2023

Updated on January 7, 2024

Welcome to a new journey in your exploration of math with Brighterly! We’re excited to guide you through the intriguing world of coterminal angles. It’s a fascinating topic, shedding light on the unique relationships between angles. Here at Brighterly, we believe in transforming complex math concepts into engaging and understandable ideas, making learning a delightful experience for all our young learners.

## Coterminal Angles – Definition and Explanation

Let’s sail on an exciting mathematical voyage into the world of angles and more specifically, coterminal angles. An essential aspect of geometry and trigonometry, understanding coterminal angles can be quite empowering, making us see math from a fresh perspective. Before we delve into coterminal angles, let’s refresh our understanding of what angles are.

## What Are Angles?

In simple terms, an angle can be defined as the figure formed by two rays, called the sides of the angle, sharing a common endpoint called the vertex. Think of it as the amount of turn between the two lines, measured in degrees or radians.

The world around us is filled with examples of angles. When you open a book, the two sides of the open book form an angle at the spine. The hands of a clock form various angles as time progresses. Even a slice of your favorite pizza forms an angle at its tip!

## Definition of Coterminal Angles

So, what are coterminal angles? Two angles are said to be coterminal if they share the same initial side and the same terminal side. In other words, you can draw two different angles that start and end at the exact same location! It’s like taking two different routes to the same destination.

Coterminal angles might have different angle measurements, but they occupy the same position in space. For instance, 30 degrees and 390 degrees are coterminal, as are -45 degrees and 315 degrees.

## Properties of Angles and Coterminal Angles

### Properties of Angles

Angles, in their simplicity, exhibit numerous fascinating properties. Here are a few:

- Angles can be categorized as acute (less than 90 degrees), right (90 degrees), obtuse (more than 90 degrees but less than 180 degrees), straight (180 degrees), reflex (more than 180 degrees), or full (360 degrees).
- The sum of angles around a point is always 360 degrees.
- The sum of the angles in a triangle always equals 180 degrees.

### Properties of Coterminal Angles

Coterminal angles, on the other hand, exhibit their unique properties:

- Any two angles are coterminal if their difference is a multiple of 360 degrees.
- An angle θ and its coterminal angles can be found using the formula θ + 360n, where n is any integer.
- Positive and negative angles can be coterminal.

## Difference Between Regular Angles and Coterminal Angles

A key difference between regular and coterminal angles lies in their spatial representation. Regular angles are simply a measure of rotation, while coterminal angles share the same position or terminal side despite being different in magnitude. Think of it as having different stories, but ending up in the same place.

## Formula for Coterminal Angles

Navigating to the formula, coterminal angles can be found using a simple, straightforward equation: θ + 360n. Here, θ is the original angle and n can be any integer. By varying the value of n, you can find an infinite number of coterminal angles.

## Understanding the Formula for Coterminal Angles

The formula for coterminal angles is intuitive. Adding or subtracting multiples of a full rotation (360 degrees) doesn’t change the position of the angle. Hence, 360 degrees is used in the formula. The term 360n signifies the addition or subtraction of ‘n’ full rotations to the original angle.

## Writing Formulas for Coterminal Angles

To write a formula for a coterminal angle, identify your original angle θ and decide whether you want to find a positive or negative coterminal angle. For a positive coterminal angle, use a positive integer for n, and for a negative coterminal angle, use a negative integer for n.

For instance, if we have an angle of 45 degrees and we want to find a positive coterminal angle, we might choose n=1, giving us 45 + 360(1) = 405 degrees as a coterminal angle.

## Practice Problems on Coterminal Angles

What’s better to consolidate our understanding than some practice problems? Here are a few:

- Find a positive and negative coterminal angle for 120 degrees.
- If an angle measures 500 degrees, find two coterminal angles.

## Conclusion

Wrapping up our journey into the realm of coterminal angles, we hope you’ve found this exploration as fascinating as we do at Brighterly! Coterminal angles are a brilliant testament to the beautiful complexities and surprising simplicities of mathematics. Whether you’re planning to delve deeper into trigonometry or simply curious about the geometric relationships around you, understanding coterminal angles is an invaluable skill.

Remember, learning is an adventure, not a race. Revisit these ideas, work on the practice problems, and don’t hesitate to dive deep into your questions. As always, Brighterly is here to illuminate your path to knowledge, turning complex concepts into a series of engaging and manageable steps. Keep that bright spark of curiosity alive, and let’s explore the wonders of mathematics together!

## Frequently Asked Questions on Coterminal Angles

### What are coterminal angles?

Coterminal angles are angles that, despite having different measures, share the same initial and terminal sides. They start and end at the exact same location on the plane. For example, angles measuring 30 degrees and 390 degrees are coterminal, because even though they are different in size, they finish at the same position.

### How to find coterminal angles?

Coterminal angles can be found using the formula θ + 360n, where θ is the given angle and n is any integer. If you want to find a positive coterminal angle, use a positive integer for n. To find a negative coterminal angle, use a negative integer for n.

### What is the formula for coterminal angles?

The formula for coterminal angles is θ + 360n. This formula stems from the fact that a full circle is 360 degrees, and adding or subtracting multiples of 360 degrees from an angle gives us angles that are coterminal with the original.

### Can negative angles have coterminal angles?

Absolutely! Negative angles can indeed have coterminal angles. In fact, every angle, whether positive or negative, has an infinite number of coterminal angles. For instance, an angle of -45 degrees is coterminal with an angle of 315 degrees, since -45 + 360(1) = 315.