Radians vs Degrees: Why Calculus Uses Radians

[chandran]

why radians are used in calculus instead of degrees?

 

[HallsofIvy]

Radians have the very nice property that $$lim_{x->0}\frac{sin x}{x}$$= 1[/tex] and $$lim_{x->0}{1- cos x}{x}= 1$$ when x is in radians. As a result the derivative of sin x is cos x and the derivative of cos x is -sin x as long as x is in radians. That's not true if x is measured in degrees. If we insist upon using degrees the corresponding derivatives would be multiplied by $$\frac{180}{\pi}$$.


That's the easy answer. A little deeper- we don't define sine and cosine, in "higher" mathematics in terms of right triangles at all: in a right triangle would have to be between 0 and 90 degrees and we want functions to be defined as many numbers as possible. One definition widely used is this: We are given an xy-coordinate system and the unit circle (the graph of the relation x²+ y²= 1). To find sin t and cos t (for t non-negative), measure around the circumference of the circle, counter clockwise, a distance t (if t< 0, measure clockwise a distance -t). The point at which you end has coordinates, by definition, cos t and sin t. ("by definition"- in other words, whatever the coordinates are, that is how we define cos t, sin t.)

Notice that the variable t in that definition is not measured in degrees OR radians! It is a distance, not an angle. Unfortunately, calculators are designed by engineers, not mathematicians and engineers tend to think of sine and cosine in terms of angles ("phase angles" in electromagnatism have nothing to do with angles!). "Radians" are defined so that the radian measure of an angle is the same as the length of the arc on a unit circle.

 

[M98Ranger]

Radians and degrees are two units of measurement used to measure angles. Degrees are more commonly used in everyday life, while radians are primarily used in mathematics, particularly in calculus.

One of the main reasons why radians are preferred in calculus is because they are a more natural unit of measurement for angles in mathematics. Radians are defined as the ratio of the arc length to the radius of a circle, while degrees are based on dividing a circle into 360 equal parts. This means that radians are a more fundamental and intuitive unit of measurement for angles, as they are directly related to the geometry of a circle.

Furthermore, radians are also more convenient to use in calculus because they simplify trigonometric functions. In calculus, trigonometric functions such as sine and cosine are used extensively to model and solve problems. When angles are measured in radians, these functions can be expressed in terms of simple fractions and their derivatives can be easily calculated. This makes working with trigonometric functions much easier and more efficient in calculus.

Moreover, radians also have a unique property that makes them more suitable for calculus – the arc length of a circle is equal to the angle in radians. This property is known as the "unit circle property" and is crucial for understanding and solving problems in calculus involving circular motion and other related concepts.

In conclusion, radians are used in calculus instead of degrees because they are a more natural and fundamental unit of measurement for angles in mathematics, simplify trigonometric functions, and have a unique property that is essential in calculus. While degrees are more commonly used in everyday life, radians are the preferred unit of measurement in the world of mathematics and specifically in calculus.