Exploring Coterminal Angles – A Comprehensive Guide

how to find coterminal angles

Trigonometry, the study of the relationships between angles and sides of triangles, is a fundamental branch of mathematics with numerous practical applications. One essential concept within trigonometry is coterminal angles, which are angles that share the same initial and terminal sides. Understanding coterminal angles is crucial for various applications, such as navigation, engineering, and physics. In this article, we will delve into coterminal angles, their properties, and provide a step-by-step guide on how to find them.

Understanding Coterminal Angles

To grasp the concept of coterminal angles, it’s essential to first understand the basics of angles. An angle is formed when two rays share a common endpoint, known as the vertex. Angles are typically measured in degrees, and a full rotation around a point measures 360 degrees. This 360-degree rotation provides a reference point for coterminal angles.

Coterminal angles are angles that have the same initial and terminal sides as a given angle, but they can differ in their angle measures. In other words, coterminal angles are like different starting points on a continuous circular journey. They are often used to simplify calculations and analyse periodic phenomena, such as periodic functions in trigonometry.

Properties of Coterminal Angles

Before delving into the methods for finding coterminal angles, let’s explore some key properties associated with them

  • Angle Measure Coterminal angles may have different angle measures, but they will always share the same initial and terminal sides.
  • Direction Angles can be positive or negative, depending on the direction of rotation. Positive angles are measured counterclockwise from the initial side, while negative angles are measured clockwise.
  • Infinite Coterminal Angles There are infinitely many coterminal angles for a given angle. This is because you can keep adding or subtracting multiples of 360 degrees to the original angle to find new coterminal angles.
  • Equivalent Angles Coterminal angles are essentially equivalent in terms of their trigonometric properties. This means that they will have the same sine, cosine, and tangent values.

Finding Coterminal Angles

Now, let’s explore how to find coterminal angles systematically

Adding or Subtracting 360 Degrees

The most straightforward way to find coterminal angles is to add or subtract 360 degrees (or 2π radians) to the original angle. This can be done repeatedly to find an infinite number of coterminal angles. For example, if you have an angle of 45 degrees, coterminal angles can be found by adding or subtracting multiples of 360 degrees: 45° + 360° = 405°, 45° – 360° = -315°, and so on.

Using Positive and Negative Angles

To find coterminal angles in a specific direction (counterclockwise or clockwise), you can add or subtract multiples of 360 degrees in the corresponding direction. For instance, to find positive coterminal angles for an angle of -30 degrees, you can add 360 degrees: -30° + 360° = 330°.

Using Radians

If you are working with radians instead of degrees, the same principles apply. To find coterminal angles in radians, add or subtract 2π radians (approximately 6.28) to the original angle.

Reducing to Smallest Positive Coterminal Angle

Sometimes, you may want to find the smallest positive coterminal angle. To do this, repeatedly add or subtract 360 degrees (or 2π radians) until you obtain an angle between 0 and 360 degrees (0 and 2π radians).

Practical Applications

Coterminal angles are not just abstract mathematical concepts; they have numerous practical applications. Here are a few examples

  • Navigation When sailors or pilots use navigation tools, they often work with coterminal angles to determine their position and heading accurately.
  • Engineering In structural engineering, coterminal angles can be used to analyse forces and moments in various directions, ensuring the stability of structures.
  • Physics Coterminal angles are essential for understanding periodic motion and waves, making them invaluable in the field of physics.
  • Trigonometric Identities In trigonometry, coterminal angles play a crucial role in simplifying trigonometric expressions and identities.


What is a Coterminal angle to 45?

In Mathematics, the coterminal angle is defined as an angle, where two angles are drawn in the standard position. Also both have their terminal sides in the same location. For example, the coterminal angle of 45 is 405 and -315.

Is Coterminal 180 or 360?

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa.


Coterminal angles are a fundamental concept in trigonometry that help simplify calculations and analyses of periodic phenomena. Understanding their properties and how to find them is essential for a wide range of applications in mathematics, engineering, physics, and navigation. Whether you are dealing with degrees or radians, the principles for finding coterminal angles remain the same, providing a versatile tool for tackling a variety of problems. So, the next time you encounter angles in your mathematical journey, you’ll be well-prepared to explore their coterminal counterparts.

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