Why Calculus Only Works in Radians and Not Degrees

Since the inception of calculus, radians have been preferred over degrees for performing mathematical calculations. This article explores the fundamental reasons why radians make trigonometric functions and calculus concepts more intuitive and simplify the process of integration, derivatives, and limits. We will also discuss how using degrees can complicate these calculations and why radians provide a more coherent and simplified framework.

Natural Definition of Angles

Radians are defined based on the unit circle, where an angle is the ratio of the arc length to the radius. This natural relationship makes radians more intuitive for many mathematical concepts. The unit circle, a circle with a radius of one, simplifies the visualization and manipulation of angles and trigonometric functions.

Simpler Derivatives

The derivatives of trigonometric functions are much simpler when expressed using radians. This is a critical factor in calculus, where derivatives form the backbone of the subject.

The derivative of (sin x) with respect to (x) in radians is (cos x). If (x) is given in degrees, the derivative becomes (frac{pi}{180} cosleft(frac{pi}{180} xright)), introducing an unnecessary conversion factor.

Limit Definitions

Many fundamental concepts in calculus, such as limits, are more straightforward when using radians. For example, the following limit holds true:

[lim_{x to 0} frac{sin x}{x} 1]

When (x) is measured in radians. If degrees were used, the limit would not equal 1 without additional conversion, complicating the calculations.

Integration and Series Expansion

The Taylor series expansions of trigonometric functions like (sin x) and (cos x) are derived using radians. Without radians, these expansions would involve additional factors, complicating the process and leading to more convoluted results.

For example, the series expansion for (sin x) in radians is:

[sin x x - frac{x^3}{3!} frac{x^5}{5!} - dots]

Using degrees would necessitate additional factors, increasing the complexity of the series.

Unit Consistency

Radians are dimensionless, which fits well with the requirements of calculus where dimensional consistency is crucial. Using degrees introduces a non-standard unit, leading to confusion and the need for additional conversion factors in calculations. For example, if you want to calculate a trig function or arc length of a sector, you use the radian measure simply and directly. Otherwise, you must first convert the angle measure to radians!

Conclusion

While calculus can technically be performed using degrees, radians provide a more coherent and simplified framework for working with trigonometric functions and calculus concepts overall. This article has demonstrated the advantages of using radians in various aspects of calculus, illustrating why radians are the preferred choice for mathematical simplicity and accuracy.