Understanding Coterminal Angles and Converting Radians to Degrees

When dealing with angles in mathematics, particularly in trigonometry, it's important to understand the concept of coterminal angles and how to convert between radians and degrees. This is crucial for solving various trigonometric problems and for proper application in fields such as engineering, physics, and computer science.

Coterminal Angles: Definitions and Properties

Coterminal angles are angles in standard position (angles measured from the positive x-axis) that share the same terminal side but differ in their measures. For example, an angle θ is coterminal with an angle α if α θ ± 2πn, where n is an integer. This property allows us to find equivalent angles within a given interval, typically 0 to 2π radians or 0° to 360°.

Converting Radians to Degrees: A Step-by-Step Guide

To convert an angle from radians to degrees, we use the conversion factor 1 radian 180°/π. Here, we'll work through an example to illustrate the process:

Identify the given angle: Let's consider the angle given as 13π/3 radians. This angle is in radians, and our goal is to find an angle that is coterminal with it and falls within the interval 0 to 2π radians. Calculate the given angle in terms of 2π: 14π/3 radians can be simplified as follows:

14π/3  2π × (14π/3) / (2π)  2π   2π/3

Since 2π radians corresponds to a full circle (360°), we can simplify further to get:

14π/3  2π   2π/3

This shows that 14π/3 radians is coterminal with 2π/3 radians. Therefore, 2π/3 radians is the equivalent angle within the 0 to 2π interval that is coterminal with 14π/3 radians.

Converting Radians to Degrees: Another Example

Let's consider another example to convert the angle 13π/3 radians to degrees and find a coterminal angle within the 0 to 2π interval:

Identify the given angle: 13π/3 radians. Simplify the given angle: 14π/3 radians can be simplified as follows:

14π/3  2π × (14π/3) / (2π)  2π   2π/3

Again, this shows that 14π/3 radians is coterminal with 2π/3 radians. Now, let's convert 2π/3 radians to degrees:

2π/3 radians × (180°/π)  120°

Therefore, 2π/3 radians is equal to 120°.

Conclusion

Understanding coterminal angles and the ability to convert radians to degrees are essential skills in trigonometry. By utilizing the properties of coterminal angles and the conversion factor 1 radian 180°/π, we can easily find equivalent angles and perform necessary calculations. This knowledge is crucial for solving various trigonometric problems and for real-world applications in fields such as physics, engineering, and computer science.