when to use radians or degrees

Guri Bee

2 min read

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21-10-2024

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Radians vs. Degrees: When to Use Which?

The choice between radians and degrees often perplexes students and professionals alike. Both units measure angles, but they differ in their mathematical foundations and practical applications. This article aims to demystify the choice, clarifying when to use radians and when to use degrees.

Understanding the Difference

• Degrees: Degrees are the more familiar unit, dividing a circle into 360 equal parts. This system is widely used in everyday life, such as in navigation, surveying, and clock faces.
• Radians: Radians are a less intuitive but more mathematically powerful unit. They define an angle by the ratio of the arc length to the radius of a circle. This means one radian is the angle subtended by an arc with the same length as the circle's radius.

When to Use Radians

Radians excel in advanced mathematical contexts, particularly in:

• Calculus and Trigonometry: Most trigonometric functions, like sine, cosine, and tangent, are defined in terms of radians. This makes them more elegant and concise in mathematical equations and derivations. For instance, the derivative of sin(x) is cos(x) when x is in radians, but it becomes a more complex formula if x is in degrees.
• Physics and Engineering: Radians are the preferred unit in physics and engineering applications, especially when dealing with oscillatory motions, wave propagation, and rotations. For example, the angular velocity of a rotating object is usually measured in radians per second.
• Computer Science and Programming: Many programming languages and libraries use radians as the default unit for trigonometric calculations. This stems from the mathematical convenience mentioned earlier.

When to Use Degrees

While radians are favored in mathematical and scientific fields, degrees are still prevalent in:

• Everyday Applications: As stated earlier, degrees are used in everyday contexts, including navigation (compass bearings), surveying, and even timekeeping (360 degrees in a clock).
• Visualizations and Illustrations: Degrees are more intuitive to understand and visualize for most people. For instance, a pie chart is easily comprehended when divided into 90-degree sectors.
• Basic Geometry and Trigonometry: For introductory geometry and trigonometry, degrees are often the preferred unit for teaching and learning due to their familiarity.

Example

Imagine a circular object rotating with an angular velocity of 1 radian per second. This means it completes one full rotation every 2π seconds (since 2π radians equal 360 degrees). If the object's velocity were expressed in degrees per second, it would be 360/2π ≈ 57.3 degrees per second.

In Conclusion

While radians and degrees are both valid ways to measure angles, choosing the right unit depends on the context. Radians are preferred in advanced mathematics, physics, and engineering due to their mathematical elegance and consistency with fundamental principles. Degrees, on the other hand, are still essential in everyday applications and provide an intuitive understanding of angles for most individuals.

Note: The information in this article was inspired by discussions and code examples found on GitHub. While I am unable to directly attribute specific users, I acknowledge their contributions to the understanding and application of radians and degrees.