Degrees and radians. Why do we use radians and not degrees to describe angles in rotational motion?

Understanding how we measure angles is fundamental in rotational motion. Angles can be specified in two main units: degrees and radians.
Degrees are more intuitive for everyday use. A full circle is 360 degrees. For example, turning halfway around is 180 degrees. However, in more technical fields like mathematics, physics, and engineering, radians are frequently favored.
One radian is the angle formed when the length of the arc that subtends the angle is equal to the radius of the circle. This makes 2π radians equal to 360 degrees. Thus, 1 radian equals approximately 57.3 degrees.
This direct relationship with the circle's radius makes radians a natural choice for many analytical purposes.

Radians offer a simplification in trigonometric functions, which are essential in many areas of mathematics. When using radians, the formulas and calculations for these functions become straightforward.
For instance, the derivatives of sine and cosine functions are very simple when angles are in radians. Specifically, \(\frac{d}{dx} \sin(x) = \cos(x)\) and \(\frac{d}{dx} \cos(x) = -\sin(x)\).
This simplicity extends to integration as well. When dealing with trigonometric functions in calculus, the use of radians minimizes the complexity of the integrals.
Therefore, using radians allows for smoother operations on functions, leading to easier and more efficient calculations.

In calculus, radians are much more than a preference—they are almost a necessity. Many integral and derivative formulas are naturally expressed in radians.
For example, the integral of the \(\sin(x)\) function in radians is simply \(-\cos(x)\). If degrees were used, additional conversion factors would complicate the integration process.
In general, when dealing with the calculus of trigonometric functions, radians ensure that the relationships remain direct and comprehensible.
This alignment with natural mathematical expressions is why radians are the standard unit in higher mathematics.

Radians aid in simplifying various mathematical formulas, particularly those involving arc length and sector area.
For example, the formula for the length of an arc is \(s = r\theta\), where \(\theta\) is in radians. This direct proportionality between arc length, radius, and angle is a direct benefit of using radians.
Similarly, the area \(A\) of a sector of a circle can be easily calculated using the formula \(A = \frac{1}{2} r^2\theta\). Again, this simplicity is due to the angle being in radians.
In physics and engineering, where angles frequently relate to periods of motion and rotational dynamics, radians are indispensable. For instance, the formula for angular velocity \(\omega = 2\pi/T\) (where \(T\) is the period) directly ties into radians.
By using radians, formulas stay clean and comprehensive, avoiding unnecessary clutter and complexity.