when to use degrees or radians

Guri Bee

3 min read

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21-10-2024

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Degrees vs. Radians: When to Use Which?

In the world of mathematics and physics, angles are fundamental. We use them to describe everything from the rotation of a wheel to the path of a planet. But when measuring angles, there are two common units: degrees and radians. So, which one should you use?

Degrees are the familiar units we use in everyday life. A full circle is divided into 360 degrees, making each degree a relatively small unit.

Radians, on the other hand, are a less intuitive unit based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. This means a full circle is equal to 2π radians, where π (pi) is approximately 3.14159.

While both units can be used to express angles, different contexts call for different units. Here's a breakdown of when to use each:

When to use degrees:

• Everyday applications: In daily life, we use degrees to measure angles in contexts like navigation, construction, and even timekeeping (think of a clock face).
• Geometry and trigonometry: In basic geometry and trigonometry problems, degrees are often the preferred unit. They are easier to visualize and understand, especially for beginners.
• Non-mathematical fields: When working with angles in non-mathematical fields like architecture or engineering, degrees are commonly used.

When to use radians:

• Advanced mathematics and physics: Radians are the preferred unit for calculus, linear algebra, and other advanced mathematical fields. This is because they simplify many mathematical formulas and relationships.
• Wave phenomena: Radians are crucial for describing wave phenomena like sound waves and light waves, as they represent a natural measure of the angular displacement of a wave.
• Scientific applications: Many scientific fields, particularly those dealing with circular motion, oscillations, and periodic phenomena, rely on radians for their mathematical calculations.

Why radians are favored in higher mathematics:

As mentioned earlier, radians simplify many mathematical formulas. Here's why:

• Derivatives and integrals: When working with derivatives and integrals involving trigonometric functions, radians are crucial. For instance, the derivative of sin(x) is cos(x) only when x is measured in radians. In degrees, the derivative involves an additional constant factor.
• Taylor series: Taylor series expansions of trigonometric functions are much simpler and cleaner when using radians.
• Unit circle: The unit circle, a fundamental tool in trigonometry, is defined using radians.

A practical example:

Imagine you're designing a Ferris wheel. To calculate the speed of the wheel and the forces acting on the passengers, radians are essential. Using degrees would lead to more complex and less elegant formulas.

Let's illustrate the difference with a real-world scenario:

Let's say we want to find the circumference of a circle with a radius of 5 meters.

• Using degrees: The circumference is calculated as (360 degrees / 360 degrees) * 2π * 5 meters = 10π meters.
• Using radians: The circumference is calculated as (2π radians / 2π radians) * 2π * 5 meters = 10π meters.

Both calculations yield the same result, but the formula using radians is more concise and elegant.

In conclusion:

Degrees and radians are both valid ways to measure angles. While degrees are more intuitive for everyday use, radians are preferred in advanced mathematical and scientific applications due to their simplified formulas and elegance. Choosing the appropriate unit depends on the specific context and the intended application.

Credits:

This article draws upon information and insights from the following GitHub resources:

• https://github.com/google/googletest/blob/master/googletest/docs/advanced.md
• https://github.com/tensorflow/tensorflow/blob/master/tensorflow/core/framework/tensor_shape.cc

This article is for informational purposes only and does not constitute professional advice.